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26 April 2001 Physics Letters B 505 (2001) 1–5 www.elsevier.nl/locate/npe Are mirror worlds opaque? R. Foot Research Centre for High Energy Physics, School of Physics, University of Melbourne, Melbourne, Victoria 3010, Australia Received 6 January 2001; received in revised form 8 February 2001; accepted 6 March 2001 Editor: H. Georgi Abstract Over the last few years, many close orbiting (∼ 0.05 A.U.) large mass planets (∼ MJ ) of nearby stars have been discovered. Their existence has been inferred from tiny Doppler shifts in the light from the star and in one case a transit has been observed. Because ordinary planets are not expected to be able to form this close to ordinary stars due to the high temperatures, it has been speculated that the close-in large planets are in fact exotic heavenly bodies made of mirror matter. We show that the accretion of ordinary matter onto the mirror planet (from, e.g., the solar wind from the host star) should make the mirror planet opaque to measured size of the transiting close-in extrasolar ordinary radiation with an effective radius (Rp ) large enough to explain the planet, HD209458b. Furthermore, we obtain the rough prediction that Rp ∝ Ts /Mp (where Ts , is the surface temperature of the ordinary matter in the mirror planet and Mp is the mass of the mirror planet) which will be tested in the near future as more transiting planets are found. We also show that the mirror world interpretation of the close-in extra solar planets explains the low albedo of τ Boo b because the large estimated mass of τ Boo b (∼ 7MJ ) implies a small effective radius of Rp ≈ 0.5RJ for τ Boo. 2001 Elsevier Science B.V. All rights reserved. Over the last few years a number of planets orbiting nearby stars have been discovered (for a review and references see [1]). Their existence has been inferred from tiny Doppler shifts in the light from the star due to its orbit around the center of mass. The magnitude and periodicity of the Doppler shifts can be used to determine the mass × sin I and orbital radius of the planet (where I is the orbital inclination of the planet). In one case, the planet HD209458b transits its star (which means that sin I 1) which allows an accurate determination of the size and mass of the planet [2]. A surprising characteristic of these planets is that some of them have been found which have orbits very close to their star (∼ 0.05 A.U.). The existence of close-in giant planets is surprising because it is thought to be too hot for giant planet formation to ocE-mail address: [email protected] (R. Foot). cur. In Ref. [3] it was suggested that close-in planets might be naturally explained if they are exotic bodies made of mirror matter (rather than ordinary matter as generally assumed). The existence of mirror matter is motivated from particle physics, since mirror particles are predicted to exist if parity and indeed time reversal are unbroken symmetries of nature [4,5]. The idea is that for each ordinary particle, such as the photon, electron, proton and neutron, there is a corresponding mirror particle, of exactly the same mass as the ordinary particle. For example, the mirror proton and the ordinary proton have exactly the same mass. 1 Fur- 1 The mass degeneracy of ordinary and mirror matter is only valid provided that the parity symmetry is unbroken, which is the simplest and theoretically most attractive possibility. For some other possibilities, which invoke a mirror sector where parity is broken spontaneously (rather than being unbroken), see Ref. [6]. 0370-2693/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 3 6 1 - 6 2 R. Foot / Physics Letters B 505 (2001) 1–5 thermore, the mirror proton is stable for the same reason that the ordinary proton is stable, and that is, the interactions of the mirror particles conserve a mirror baryon number. The mirror particles are not produced in Laboratory experiments just because they couple very weakly to the ordinary particles. In the modern language of gauge theories, the mirror particles are all singlets under the standard G ≡ SU(3) ⊗ SU(2)L ⊗ U (1)Y gauge interactions. Instead the mirror particles interact with a set of mirror gauge particles, so that the gauge symmetry of the theory is doubled, i.e., G ⊗ G (the ordinary particles are, of course, singlets under the mirror gauge symmetry) [5]. Parity is conserved because the mirror particles experience right-handed mirror weak interactions and the ordinary particles experience the usual left-handed weak interactions. Ordinary and mirror particles interact with each other predominately by gravity only. 2 At the present time there is a range of experimental evidence supporting the existence of mirror matter. Firstly, it provides a natural candidate for dark matter, which might be mirror stars (and mirror dust, planets, etc.) [10]. There is an interesting possibility that these mirror stars have already been detected experimentally in the MACHO experiments [11]. Secondly, ordinary and mirror neutrinos are maximally mixed with each other if neutrinos have mass [12]. This nicely explains the solar and atmospheric neutrino anomalies. 3 The idea is also compatible with the LSND experiment [12]. Interestingly, maximal ordinary–mirror neutrino oscillations do not pose any problems for big bang nucleosynthesis (BBN) and can even fit the inferred primordial abundances better than the standard model [15]. Finally, there is also tantalizing experimental evidence of the mirror world from the orthopositronium lifetime anomaly which can be explained [16] due to the effects of photon–mirror photon kinetic mixing [7]. Because mirror matter interacts predominately by gravity only, it is not heated up by the ordinary photons emitted by the host star. Thus, any mirror 2 It is possible to have small non-gravitational interactions between ordinary and mirror matter. Assuming gauge invariance and renormalizability the only possibilities are photon–mirror photon kinetic mixing [5,7,8] and Higgs–mirror Higgs mixing [5,9]. 3 For the current experimental status of the mirror world solution to the solar and atmospheric neutrino anomalies, see Ref. [13] and Ref. [14], respectively. matter present in a stellar nebula can form close to the host star without any apparent theoretical problems. In fact such a possibility was effectively predicted by Blinnikov and Khlopov in 1982 [17] where they discussed the possibility of having a close-in mirror planet with an orbit inside the radius of the sun. Interestingly, the “dynamical” mirror image system of a mirror star with an ordinary planet would appear to ordinary observers like us as an isolated ordinary planet. Thus, the recent discovery [18] of isolated planets is not particularly surprising from this perspective [19]. For a close-in ordinary planet the periodic Doppler shift in the frequencies should be of order 10−3 providing a simple test of this idea [19]. At first sight, one might think that a mirror planet would be transparent to ordinary radiation with no scattered or reflected light (i.e., its albedo would be zero). This would be true of a mirror planet composed of 100% mirror matter with zero photon–mirror photon kinetic mixing. In fact even if photon–mirror photon mixing is non-zero, then a pure mirror planet would still be (to an extremely good approximation) transparent (with zero albedo) [20]. However, such an idealized system would not be expected to exist. Even if there was negligible amount of ordinary matter in the mirror planet when it was formed, the mirror planet will accrete ordinary matter from the host star due to that star’s solar wind, and also from comets, asteroids and cosmic rays. Using the sun’s solar wind as a concrete example, the current mass loss of the sun due to the solar wind is estimated to be (see, e.g., Ref. [21]) dM ≈ 3 × 10−14 M /year. (1) dt This implies an accretion rate of ordinary matter onto the mirror planet of roughly, 2 Rp Rp2 dM dM ≈ 2 ∼ 10−2 MJ (2) /Gyr, dt 4rp dt rp2 where Rp is the effective radius of the ordinary matter in the mirror planet, rp is the distance of the planet from the host star and MJ is the mass of Jupiter. Let us now estimate Rp by assuming hydrostatic equilibrium, which should be valid. Denoting the density of ordinary matter in the planet by ρ (o) and the (assumed) much larger density of mirror matter by ρ (m) then the condition for hydrostatic equilibrium is that the pressure (P ) gradient balances the force due R. Foot / Physics Letters B 505 (2001) 1–5 to gravity, i.e., dP = −ρ (o)g, (3) dr where g is the local acceleration due to gravity at a distance r from the center of the planet. Assuming that ρ (m) ρ (o) and taking ρ (m) approximately constant (i.e., independent of r) then g is simply 4πGρ (m) r . (4) 3 Of course this is only valid for r < Rm where Rm is the mirror matter radius. 4 We now need to relate the pressure of the ordinary matter, P , to its density, ρ (o) . First, we assume that the ordinary matter is mainly molecular hydrogen, H2 (which is quite natural if most of it arises due to accretion from the stellar wind from the host star). 5 Second, the ordinary matter inside the mirror planet should be hot because it is heated at its surface by the radiation from the host star. Finally, the ordinary matter doesn’t feel the pressure from the surrounding mirror matter. Because it is hot, low in density and pressure, the ordinary matter should be a gas approximately obeying the ideal gas law: g≈ P= ρ (o) kT , 2mp (5) where k is Boltzmans constant, and 2mp is the molecular hydrogen mass. Substituting Eq. (5) and Eq. (4) into Eq. (3) and solving the resulting differential equation, we obtain the solution: ρ (o) (r) T (0) −r 2 /Rx2 e = ρ (o) (0) T (r) where 3k , Rx ≡ 4πmp Gρ (m) λ for r < Rm , (6) r 1 dr 2 . T (r ) Note that Rx depends on r through the dependence of λ on r. Unfortunately it is not so easy to obtain an accurate estimation for λ because this requires knowledge of the temperature profile of the ordinary matter in the planet. However, a crude lower limit can be obtained by noting that the temperature should increase as r decreases. This means that λ < 1/Ts (where Ts is the “surface temperature”) which allows a lower limit for Rx of Rx 5 × 103 (Ts /103 K)(1 g cm−3 /ρ (m) ) km. (9) In order to estimate the (wavelength dependent) radius of the ordinary matter (Rp ) which we define as the radius within which the radiation from the host star is absorbed or scattered during a transit, we need to know the detailed chemical composition, temperature in addition to the density ρ (o) profile of the ordinary matter. In a recent study, Hubbard et al. [22] have estimated that the pressure where the transiting planet HD209458b becomes opaque to be roughly 10 mbar which corresponds to a density of about ρ (o) ∼ 10−7 g cm−3 [from Eq. (5)]. Since our ordinary matter enriched mirror world should have a similar surface temperature (because for close-in planets the source of the energy emitted is dominated by the irradiation from the host star rather than due to the planets internal energy) to that assumed by Hubbard et al [22], which is ∼ 1500 K, then we may expect that our mirror world should become opaque at about the same density. Thus, assuming a total mass of ordinary matter of about few × 10−4 MJ as suggested by Eq. (2), we then estimate that Rp ≈ 4Rx . (10) ρ (o) (7) and 1 λ≡ 2 r 3 (8) 0 (m) 3 4 For r > R , g = 4π Gρ Rm = GMp . m 3r 2 r2 5 Note that for high temperatures, T 3000 K, H begins to 2 − (o) thereby dissociate into H+ 2 and e , which will increase P /ρ increasing Rx . is a steeply falling distribution Actually, because for r Rx the above estimate of Rp /Rx should be reasonably robust. For example, if we assumed that the pressure or densities, ρ (m) , ρ (o) , were an order of magnitude larger (or smaller), then our estimate of Rp /Rx would change by only about 10% (although Rx itself depends sensitively on ρ (m) ). Of course, if Rp ≈ 4Rx Rm then it means that the distribution of ordinary matter is extended beyond the radius of the mirror matter, Rm , in which case we may expect Rp to be somewhat larger than 4Rx because the ordinary matter density falls off more slowly for r > Rm due to the weaker gravity. 4 R. Foot / Physics Letters B 505 (2001) 1–5 However, as discussed earlier, our largest source of uncertainty in Rp derives from its dependence on λ though Rx . The quantity λ (which we need to evaluate at r ≈ Rp ) should be dominated by the temperature profile in the outer regions (r > 0.6Rp ) where the conditions should not be so different from the temperature profile computed for close-in giant planets made from ordinary matter. This suggests that λ ∼ 1/(5Ts ) which should be accurate to within a factor of two or so. For the transiting planet HD209458b, which is the only planet for which Rp , Mp have been measured, the parameters Rp , Mp are [23] Rp = (1.40 ± 0.17) RJ and Mp = (0.69 ± 0.05) MJ . Since the mass of HD209458b is roughly that of Jupiter, we can use Jupiter as a guide to the most likely size for Rm . 6 This is possible because the surface temperature of Jupiter is dominated by internal energy (rather than by solar irradiation). This leads to an expected radius of Rm ≈ RJ . 7 This implies a ρ̄ (m) ≈ 1 g cm−3 . Thus, we estimate that the effective radius at which the transiting planet HD209458b becomes opaque to be roughly, Rp ≈ 4Rx ∼ RJ , (11) which is consistent with the measured value given our admittedly large theoretical uncertainty. Nevertheless, our simple analysis shows that the transit of HD209458b can be plausibly explained with the mirror planet hypothesis. Furthermore, we can make some rough quantitative predictions. In particular, our simple analysis predicts that Ts Ts ∝ . Rp ≈ 4Rx ∝ (12) (m) Mp ρ Of course, this is only a very rough prediction, especially the dependence on Ts which is just the surface temperature (recall it is really the more complicated function λ that we need in order to determine Rx and hence Rp ). Nevertheless, heuristically it can be understood quite easily. Increasing Mp increases the force of gravity which causes the gas of ordinary matter to become more tightly bound to the mirror planet 6 For ordinary large hydrogen planets, R depends quite weakly p on Mp (e.g., RSaturn /RJ 0.84, while MSaturn /MJ 0.33). 7 If photon–mirror photon kinetic mixing is relatively large, then Rm can be significantly larger because the mirror matter can be heated by transfer of heat from the ordinary to the mirror matter thereby preventing the mirror surface to cool. (thereby decreasing the effective size, Rp ), while increasing the temperature of the gas increases the volume that the gas occupies (thereby increasing Rp ). By contrast, the size of ordinary planets (i.e., planets made mostly of ordinary matter) depends quite weakly on their mass Mp . Thus our hypothesis that the close-in extra-solar planets may in fact be mirror worlds matter model. The rough prediction, Eq. (12) can be tested as soon as another transiting close-in planet is observed (which should occur in the near future given their rate of discovery). This should provide a significant test of the mirror world hypothesis because the radius of ordinary planets depends much more weakly on the mass of the planet. For example, the planet τ Boo b has an estimated mass of about ≈ 7MJ [24] which means that it is about 10 times heavier than HD209458b. √ Thus we predict its effective radius to be roughly 10 times less than the radius of HD209458b, i.e., only about 0.5RJ . On the other hand, for planets made of ordinary matter the radius of τ Boo b is predicted to be [25] Rp 1.2RJ which is only about 15% less than for HD209458b. Our prediction Eq. (12) also has important implications for measurements of reflected light (albedo). In particular a reasonably stringent limit on the albedo exists for the planet τ Boo b. In Ref. [24], they obtain an upper limit on the opposition planet/star flux ratio of < 3.5 × 10−5 (for wavelengths between 387.4 and 586.3 nm) at 99.9% C.L. Given that = p(Rp /rp )2 this translates into a limit on the geometric albedo of the planet, p, of p < 0.22 at 99.9% C.L, assuming a planet radius of Rp 1.2RJ . In comparison, the corresponding mean geometric albedo of Jupiter is about 0.55. However, if τ Boo b is a mirror world then we expect Rp ≈ 0.5RJ , as discussed above. Thus in this case, the “limit” on the albedo is p < 1.3 which is obviously no limit, since p must be less than 1. Thus, the low value of for τ Boo b is explained simply because its effective size is expected to be small in our mirror world interpretation. Of course, for lighter planets, their effective size will be larger, which can make their reflected light easier to detect. In conclusion, we have argued that the hypothesis that the close-in large extra solar planets are in fact mirror worlds can explain the transit of HD209458b. The mirror world is opaque because it would accrete a significant amount of ordinary matter from the so- R. Foot / Physics Letters B 505 (2001) 1–5 lar wind from the host star, which gives the mirror planet an effective radius large enough to explain the transit observations of HD209458b. This explanation can also nicely explain the low effective albedo of τ Boo b. Importantly, the close-in mirror world hypothesis can be tested as more transits of close-in large planets are observed. Thus, we are left with the remarkable prospect that extrasolar planetary astronomy may provide a novel means of testing whether the fundamental interactions of particle physics conserve parity invariance. Acknowledgement The author would like to thank Henry Lew, Sasha Ignatiev and Ray Volkas for discussions. The author is an Australian Research Fellow. References [1] For a review and references on extrasolar planets, see the extrasolar planet encyclopaedia: http://cfa-www.harvard.edu/ planets/encycl.html; See also: M.A.C. Perryman, astro-ph/0005602. [2] D. Charbonneau et al., Astrophys. J. 529 (2000) L15; G.W. Henry et al., Astrophys. J. 529 (2000) L41. [3] R. Foot, Phys. Lett. B 471 (1999) 191. [4] T.D. Lee, C.N. Yang, Phys. Rev. 104 (1956) 256; I. Kobzarev, L. Okun, I. Pomeranchuk, Sov. J. Nucl. Phys. 3 (1966) 837; M. Pavsic, Int. J. Theor. Phys. 9 (1974) 229. [5] R. Foot, H. Lew, R.R. Volkas, Phys. Lett. B 272 (1991) 67. [6] S. Barr, D. Chang, G. Senjanovic, Phys. Rev. Lett. 67 (1991) 2765; R. Foot, H. Lew, hep-ph/9411390; Z.G. Berezhiani, R.N. Mohapatra, Phys. Rev. D 52 (1995) 6607; R. Foot, H. Lew, R.R. Volkas, JHEP 0007 (2000) 032. [7] S.L. Glashow, Phys. Lett. B 167 (1986) 35. [8] B. Holdom, Phys. Lett. B 166 (1985) 196. [9] R. Foot, H. Lew, R.R. Volkas, Mod. Phys. Lett. A 7 (1992) 2567. [10] S.I. Blinnikov, M.Yu. Khlopov, Sov. J. Nucl. Phys. 36 (1982) 472; [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] 5 S.I. Blinnikov, M.Yu. Khlopov, Sov. Astron. 27 (1983) 371; E.W. Kolb, M. Seckel, M.S. Turner, Nature 514 (1985) 415; M.Yu. Khlopov et al., Sov. Astron. 35 (1991) 21; M. Hodges, Phys. Rev. D 47 (1993) 456; Z.G. Berezhiani, A. Dolgov, R.N. Mohapatra, Phys. Lett. B 375 (1996) 26; Z.G. Berezhiani, Acta Phys. Polon. B 27 (1996) 1503; G. Matsas et al., hep-ph/9810456; Z. Silagadze, Mod. Phys. Lett. A 14 (1999) 2321, hepph/0002255; N.F. Bell, R.R. Volkas, Phys. Rev. D 59 (1999) 107301; S.I. Blinnikov, hep-ph/9902305; S.I. Blinnikov, astro-ph/9911138; R.R. Volkas, Y.Y.Y. Wong, Astropart. Phys. 13 (2000) 21; V. Berezinsky, A. Vilenkin, hep-ph/9908257; A.Yu. Ignatiev, R.R. Volkas, Phys. Rev. D 62 (2000) 023508; A.Yu. Ignatiev, R.R. Volkas, Phys. Lett. B 487 (2000) 294; N.F. Bell, Phys. Lett. B 479 (2000) 257; R.M. Crocker, F. Melia, R.R. Volkas, astro-ph/9911292; Z. Berezhiani, D. Comelli, F.L. Villante, hep-ph/0008105. Z. Silagadze, Phys. At. Nucl. 60 (1997) 272; S. Blinnikov, astro-ph/9801015; R. Foot, Phys. Lett. B 452 (1999) 83; R. Mohapatra, V. Teplitz, astro-ph/9902085. R. Foot, H. Lew, R.R. Volkas, Mod. Phys. Lett. A 7 (1992) 2567; R. Foot, Mod. Phys. Lett. A 9 (1994) 169; R. Foot, R.R. Volkas, Phys. Rev. D 52 (1995) 6595. R. Foot, Phys. Lett. B 483 (2000) 151, and references therein. R. Foot, Phys. Lett. B 496 (2000) 169, and references therein. R. Foot, R.R. Volkas, Phys. Rev. D 61 (2000) 043507; R. Foot, R.R. Volkas, Astropart. Phys. 7 (1997) 283. R. Foot, S.N. Gninenko, Phys. Lett. B 480 (2000) 171. S.I. Blinnikov, M.Yu. Khlopov, Sov. J. Nucl. Phys. 36 (1982) 472. M.R. Zapatero Osorio et al., Science 290 (2000) 103; See also: M. Tamura et al., Science 282 (1998) 1095; P.W. Lucas, P.F. Roche, Mon. Not. R. Astron. Soc. 314 (2000) 858. R. Foot, A.Yu. Ignatiev, R.R. Volkas, astro-ph/0010502. R. Foot, A.Yu. Ignatiev, R.R. Volkas, astro-ph/0011156. B.W. Carroll, D.A. Ostlie, Modern Astrophysics, Addison– Wesley, 1996. W.B. Hubbard, astro-ph/0101024. T. Mazeh et al., Astrophys. J. 532 (2000) 55. A. Cameron, astro-ph/0012186; See also: D. Charbonneau, R.W. Noyes, astro-ph/0002489. A. Burrows et al., Astrophys. J. 534 (2000) L97. 26 April 2001 Physics Letters B 505 (2001) 6–14 www.elsevier.nl/locate/npe Magnetic dipole bands in 190Hg: first evidence of excitations across the Z = 82 sub-shell in Hg nuclei A.N. Wilson a,1 , J. Timár b , I. Ahmad c , A. Astier d , F. Azaiez e , M.H. Bergström f,2 , D.J. Blumenthal c , B. Crowell c,3 , M.P. Carpenter c , L. Ducroux d , B.J.P. Gall g , F. Hannachi h , H. Hübel i , T.L. Khoo c , R.V.F. Janssens c , A. Korichi h , T. Lauritsen c , A. Lopez-Martens h , M. Meyer d , D. Nisius c , E.S. Paul f , M.G. Porquet h , N. Redon d , J.F. Sharpey-Schafer f,g,4 , R. Wadsworth a , J.N. Wilson f,1 , I. Ragnarsson j a Department of Physics, University of York, Heslington, York YO10 5DD, UK b Institute of Nuclear Research, H-4001 Debrecen, Hungary c Argonne National Laboratory, Cass Ave, Argonne, IL, USA d IPN Lyon, 69622 Villeurbanne, France e IPN, 91406 Orsay Campus, France f Oliver Lodge Laboratory, University of Liverpool, Liverpool L69 7ZE, UK g IReS, 67037 Strasbourg, France h CSNSM, 91405 Orsay Campus, France i ISKP, Universität Bonn, Germany j Department of Mathematical Physics, Lund Institute of Technology, Box 118, S-22100 Lund, Sweden Received 7 August 2000; received in revised form 9 February 2001; accepted 27 February 2001 Editor: V. Metag Abstract An experiment aimed at studying high-spin states in 190 Hg was performed with the Eurogam II array. The data have revealed the presence of cascades of magnetic dipole transitions with some unexpected properties. Unlike the M1 bands previously observed in the heavier Hg isotopes, these structures have extremely large B(M1)/B(E2) ratios. The observation of a third dipole band with much lower B(M1)/B(E2) values in the same spin/excitation energy regime suggests that the bands may represent configurations occurring in different minima in the potential energy surface. Configuration-dependent Cranked Nilsson–Strutinsky calculations predict the presence of a minimum in the nuclear potential energy surface at a deformation of ε ≈ 0.2, γ ≈ −90◦ , occurring when a proton is excited across the Z = 82 shell-gap into an h9/2 orbital. It is suggested that the bands exhibiting anomalously large B(M1)/B(E2) ratios may be associated with this minimum. 2001 Published by Elsevier Science B.V. PACS: 23.20.-g; 23.20.En; 23.20.Lv; 27.80.+w E-mail address: [email protected] (A.N. Wilson). 1 Present address: Department of Nuclear Physics, Research School of Physical Sciences and Engineering, The Australian National University, Canberra, ACT 0200, Australia. 2 Present address: Niels Bohr Institute, Blegdamsvej 17 DK-2100 Copenhagen, Denmark. 3 Present address: Fullerton Community College, Fullerton, California, USA. 4 Present address: National Accelerator Centre, PO Box 72, Faure, ZA-7131 South Africa. 0370-2693/01/$ – see front matter 2001 Published by Elsevier Science B.V. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 3 3 2 - X A.N. Wilson et al. / Physics Letters B 505 (2001) 6–14 Magnetic dipole bands have now been observed in several isotopes of Hg [1–8]. In each of these cases, the bands display somewhat irregular behaviour with increasing spin and have B(M1)/B(E2) ratios in the range of 1–10 (µN /e b)2 . Such bands are explained as originating from a collective oblate minimum in the potential energy surface of the nucleus at ε ≈ 0.18, γ ≈ −60◦ , being based on configurations involving proton orbitals below the Z = 82 shell-gap. Elsewhere in the same mass region, M1 bands with very large B(M1)/B(E2) ratios and displaying a remarkably regular rotational behaviour have been observed in the Pb isotopes (see, e.g., [9] and references therein). These bands (which are based on the perpendicular coupling of h9/2 and/or i13/2 proton particles to i13/2 neutron holes) have been interpreted within the framework of the Tilted Axis Cranking (TAC) model [10] and are thought to represent a new type of nuclear rotation known as magnetic rotation. In this Letter, we report on the observation of three dipole bands in the nucleus 190Hg. One of these (dipole band A) is similar to the bands that have previously been observed in heavier Hg isotopes and may be directly compared to analogous bands observed in 192 Hg. The other two bands (dipole bands 1 and 2) exhibit remarkably large B(M1)/B(E2) ratios and seem to have a more regular, rotational-like behaviour. All three bands appear to coexist in the same region of the spin/excitation energy plane. Although the properties of bands 1 and 2 appear to bear comparison with the magnetic rotational bands observed in the Pb isotopes, we have chosen to interpret these structures within a Principle Axis Cranking (PAC) approach, where it is more straightforward to fix different configurations within the calculations and follow their behaviour as a function of spin. Some of the limitations of this approach are also discussed. The data presented in this Letter were obtained during an experiment carried out using the Eurogam II array [11]. A beam of 34 S (provided by the Vivitron accelerator at a beam energy of 153 MeV) was incident upon a target consisting of two stacked, thin (500 µg cm−2 ) foils of 160 Gd. Approximately 5 × 108 events (when more than three γ rays were detected within the prompt coincidence time window of ≈ 100 ns) were recorded over a period of 4 days. One of the unique features of the Eurogam II array was the presence of 24 4-element Clover detectors. Along with enhancing 7 the efficiency and granularity of the array, these detectors allow linear polarisation measurements to be extracted from the ratio of “vertical” to “horizontal” scatters (i.e., scatters parallel and perpendicular to the reaction plane) between elements of the same detector [12,13]. Such scattered events were “added-back” and tagged as parallel or perpendicular for the subsequent analysis. Energy-dependent time gates were also applied to the data before the construction of a series of γ 2 -coincidence matrices and a 3-dimensional γ 3 coincidence histogram (a cube) suitable for analysis with the Radware package [14]. A detailed level scheme has previously been established for the normal decay of 190 Hg by Bearden et al. [15]. The high statistics obtained in the present experiment have allowed considerable expansion of the level scheme in both the normal and superdeformed [16] regimes. In this Letter, we focus on the observation of three dipole bands in 190 Hg, all of which were previously unknown. Partial level schemes showing these three bands and the previously known states into which they decay are shown in Fig. 1(a) and (b). The first of these bands, presented in Fig. 1(a) and referred to as dipole band A (DB A), exhibits behaviour common to many of the dipole bands observed in the heavier Hg isotopes, particularly band b of 192 Hg [5,7]. This structure is composed of both M1 and E2 cross-over transitions, and is populated with approximately 2.0% of the total 190Hg intensity. The other two bands, shown in Fig. 1(b), display some extremely unusual features. They appear to consist solely of M1 transitions and display a fairly regular behaviour with increasing spin. These bands are henceforth referred to as dipole bands 1 and 2 (DB 1 and 2). Double-gated spectra showing these bands are presented in Fig. 2(a) and (b). DB 1 is populated with approximately 1% of the reaction channel intensity; DB 2 is somewhat weaker, carrying ≈ 0.5% of the flux into 190Hg. In both cases, the presence of doublets or near doublets within the bands and their subsequent decay paths has been established from the coincidence data; the order in which the γ rays have been placed in the bands derives from intensity considerations alone and is therefore subject to some uncertainty. Both DB 1 and DB A have been linked into the surrounding level scheme and their decay paths to the lower spin states are well-established. Although it is evident from the coincidence data that DB 2 shares 8 A.N. Wilson et al. / Physics Letters B 505 (2001) 6–14 Fig. 1. Partial level schemes showing (a) dipole band A and (b) dipole bands 1 and 2 in 190 Hg observed in the current data. part of the decay path of DB 1, no discrete linking transitions have been identified and thus the absolute excitation energy and spin of the bandhead are not known. Whilst the presence of cross-over transitions provides a strong indication that DB A is comprised of both M1 and E2 transitions, it is necessary to provide some experimental evidence for the multipolarity of the transitions in DB 1 and 2. The dipole nature of the transitions can be shown using the method of Directional Correlations from Oriented Nuclei (DCO) [17]. Linear polarization measurements can be used to show whether a transition is electric or magnetic. Duchêne et al. [13] have performed a detailed study in order to characterise such measurements made with the Clover detectors; the same method is used here, along with the quality factor Q derived in that work. The DCO ratios extracted for DB 1 and 2 are shown in Fig. 3(a) and (b), respectively. The measurements for DB 1 were made in spectra gated on the 216 keV transition; for DB 2, the 354 keV in-band transition was used as a gate. The values shown represent the ratio of the intensities measured in detectors positioned at forward/backward angles (46◦ and 134◦ with respect to the beam direction) to the intensities measured in detectors at angles close to 90◦ . Using this method, the geometry of the Eurogam Phase II array is such that stretched dipole transitions give a ratio of I46+134 /I90 ≈ 1.0 and stretched quadrupole transitions a ratio of I46+134/I90 ≈ 1.8. The DCO A.N. Wilson et al. / Physics Letters B 505 (2001) 6–14 9 Fig. 2. Double-gated spectra showing (a) dipole band 1 and (b) dipole band 2 in 190 Hg. Both spectra were created by summing the cleanest combinations of double gates in the cube (see text). In-band transitions are marked with filled triangles; transitions involved in the decay from the bands to the lower-lying states are marked with open triangles. ratios provide strong evidence that these transitions are stretched dipoles. Fig. 3(c) shows the results of the polarization measurements. Despite the large uncertainties, the ratios clearly point to a magnetic nature for the γ rays forming DB 1. Measurements have also been made for several of the transitions which link DB 1 into the previously known level scheme. On this basis, it can be strongly asserted that DB 1 is comprised of a cascade of M1 transitions, that it is of positive parity (there is some ambiguity due to the presence of a 234 keV transition immediately below the band and a 233 keV transition in the lower part of the decay-out of the band), and that the bandhead spin is 17h̄. Neither DCO nor polarisation measurements have been possible for the second decay branch out of DB 1 (via the 419 and 643 keV transitions) due to the low intensity of this path. It has not been possible to perform decisive polarization measurements for transitions in DB 2, as it is both more contaminated and less strongly populated than DB 1. An additional complication is brought into the analysis by the fact that there are two pairs of self-coincident doublets within the band, making it very hard to establish the exact order of transitions. However, given the strong evidence from the DCO ratios of a stretched dipole nature, it is likely that this band consists of M1 transitions. Two transitions (164 and 202 keV) have been identified as being involved in the decay of this band to the lower part of the level scheme; it is not thought that these transitions are constituents of the band as the DCO ratio measured for the 164 keV transition indicates a stretched quadrupole nature. Some of the flux of DB 2 is observed to pass through the 13+ level de-excited by the 505 keV γ ray in the decay of DB 1. Although no discrete γ rays have been identified which link 10 A.N. Wilson et al. / Physics Letters B 505 (2001) 6–14 Fig. 3. (a) DCO ratios obtained for transitions in dipole band 1 (see text for details). Band members are marked with filled triangles, γ rays involved in the decay-out of the band are marked with open triangles and known stretched quadrupole transitions are marked with open squares. (b) DCO ratios obtained for transitions in dipole band 2. Band member are marked with filled triangles; known stretched quadrupole transitions are marked with open squares. (c) Polarization data measured for transitions in dipole band 1 (indicated by filled triangles) and for transitions decaying out of the bandhead (indicated by open triangles). Known electric (E2 and E1) and magnetic (M1) transitions are shown with open and filled circles, respectively. (d) Branching ratios (B(M1)/B(E2)) for levels in dipole band A (filled/open circles representing the lower- and higher-spin sections) and the lower limits extracted for dipole bands 1 and 2 (filled and open triangles). Typical values for high-K dipole bands fall in the region between the dotted lines and values measured for magnetic rotational bands are in the region between the dashed lines. these two structures, assuming steps of only two or three transitions between them suggests a probable bandhead spin of 20 ± 2 h̄ and an excitation energy similar to that of DB 1. No polarization measurements have been performed for transitions from DB A; however, the complex decay paths to other established levels strongly suggest that the bandhead has I π = 17− . E2 cross-over transi- tions are present in both the lower and upper portions of the band; their disappearance in the middle section and the apparent back-bend around spin I = 23h̄ suggest that a structural change takes place within this sequence. Fig. 3(d) shows the values of B(M1)/B(E2) measured for levels in DB A along with the lower limits extracted for DB 1 and 2. Ratios for the lower and up- A.N. Wilson et al. / Physics Letters B 505 (2001) 6–14 per portions of DB A are indicated by filled and open circles, respectively. Although it is not possible to extract an experimental value for this quantity where no E2 transitions are observed, a lower limit may be obtained. The lower limits extracted for DB 1 and 2 are indicated by filled and open triangles. The extremely high statistics of the data set allows such measurements to be made with a good degree of precision — the lower limits on these ratios for DB 1 and 2 are measured with an uncertainty of less than 10%. The range of typical values for “normal” dipole bands in other Hg isotopes and shears bands in Pb isotopes are indicated by the dotted and dot-dashed lines, respectively. Whilst DB A exhibits branching ratios similar to those measured for the dipole bands observed in the heavier Hg isotopes, the lower limits for DB 1 and 2 indicate that, for these bands, the ratio is an order of magnitude larger and very similar to the ratios obtained for magnetic rotational states. In order to interpret the features of these bands, it is first necessary to consider the deformations and types of excitation available to the nucleus. 190 Hg has 80 protons and thus, in its ground-state, lies just below the Z = 82 shell-closure. The nearest available proton orbitals originate from the (s1/2 , d3/2 ) and h11/2 shells with intruder orbitals coming from the h9/2 and, for higher deformations, i13/2 shells. The neutron Fermi level is below the N = 114 shell gap: this means that the lowest-lying neutron excitations are likely to involve levels from the f7/2 , i13/2 and h9/2 shells, but may also involve levels of (p3/2, f5/2 , p1/2 ) character (i.e., from the shells above N = 114). Although suggestions have been made that proton excitations into the h9/2 orbital might be responsible for some of the M1 bands observed in the Hg isotopes [5], no strong support for these assignments has been found. It has been generally accepted that such excitations are energetically unfavorable at small deformations, and thus that bands of the shears type will not be observed in Hg nuclei. However, the calculations performed for this work suggest that this type of excitation is possible, and may even be favored, at high spins. Potential energy surfaces have been calculated for positive parity states in 190 Hg using the configuration-dependent CNS method, which has been used with great success in describing the properties of smooth-terminating bands in the A ≈ 110 region, as reviewed in Ref. [18]. The method has 11 also been applied to the magnetic bands in Pb isotopes [19]. One of the main advantages of this approach is that large numbers of configurations available to the nucleus in the yrast and near-yrast region can be identified and followed to high spins. Examples of these calculations are shown in Fig. 4, where potential energy surfaces for I = 22+ and I = 26+ are presented. They are constructed from all configurations having positive parity and signature α = 0 (i.e., even spin values) and show the lowest energy among these configurations at each mesh point. At spin I = 22+ (Fig. 4(a)), two minima are clearly visible, one occurring at ε ≈ 0.15, γ ≈ −60◦ and a second at ε ≈ 0.18, γ ≈ −90◦ . The former corresponds to configurations in which no protons have been excited across the Z = 82 shell gap; the latter arise from configurations involving the promotion of one proton into an h9/2 orbital (i.e., excitations across the shell gap). Fig. 4(b), which is calculated at the slightly higher spin of I = 26h̄, shows that the latter becomes more yrast as the spin increases, while the former moves to somewhat lower quadrupole deformation. The very low-lying yrast states in 190Hg are generally built upon configurations with no proton excited across the Z = 82 shell gap, where the nuclear deformation is around ε = 0.12–0.16, with γ in the range [−120◦, −60◦ ]. The magnetic dipole bands in the heavier Hg isotopes can be interpreted as corresponding to these types of configurations. Those bands which display a somewhat smaller B(M1)/B(E2) (≈ 1 (µN /e b)2 ) are thought to arise from the coupling of an h11/2 quasiproton and i13/2 quasineutrons, similar to those configurations giving rise to the structures with bandhead spin I0 = 11h̄ observed in the Au isotopes [20,21]. The results of lifetime measurements in 192 Hg [7] strongly suggest that band b in that nucleus is based on configurations of this type. Fig. 3(d) shows that the B(M1)/B(E2) ratios for levels below the backbend in DB A in 190Hg are clustered around ≈ 1 (µN /e b)2 , while above the backbend the ratio is somewhat higher. These ratios, together with the similar excitation energy and spin ranges over which the structures are observed, point to a similar structure for DB A and band b in 192Hg. It is very difficult to explain the unusual characteristics of DB 1 and 2 if the nucleus remains within the minimum at γ ≈ 60◦ ; the available configurations and 12 A.N. Wilson et al. / Physics Letters B 505 (2001) 6–14 Fig. 4. Total energy surfaces in the (ε, γ )-plane for γ -values in the range (−120◦ , −30◦ ). The surfaces correspond to positive parity at (a) I = 22h̄ and (b) I = 26h̄ in 190 Hg. The contour line separation is 0.1 MeV. The minima at ε ≈ 0.18, γ ≈ −90◦ and ε ≈ 0.15, γ ≈ −60◦ , respectively, define the energies at the respective spin values for the curves drawn by solid lines and filled circles in Fig. 5. collectivity of the nucleus remain such that one would not expect B(M1)/B(E2) ratios of the magnitude observed. We therefore turn to the minimum at γ ≈ 90◦ , predicted by the CNS calculations, for a possible interpretation. The B(E2) strength associated with states in this minimum would be significantly reduced over that arising from a more collective shape with the same value of ε. In addition, these states include a π(h9/2 ) particle excitation, thus creating a situation in which a large B(M1) may be generated (by a perpendicular coupling of the π(h9/2 ) spin vector, as in the Pb isotopes). In order to make a more detailed comparison between the predictions of the configuration-dependant CNS method and the experimental results, calculations have been performed in which several of the near-yrast configurations are followed to high spins. Some of the results are presented in Fig. 5, in which the energies of a few of these low-lying states (relative to a rigid rotor reference) are plotted as a function of spin, together with the data for DB 1. Due to the uncertainty of the results concerning DB 2 (spins, parity and transition order are not fixed), it has not been included in the figure. A.N. Wilson et al. / Physics Letters B 505 (2001) 6–14 13 Fig. 5. Energy relative to a rigid rotation reference as a function of spin showing the result of the configuration-dependent CNS calculations for several low-lying configurations and the experimental data for DB 1 in 190 Hg. The difference in absolute energies arises from the choice of reference energy and other details of the calculations: because of this, it is only meaningful to compare the shapes of the curves. The proton configurations are indicated in the figure; those involving an excitation across the Z = 82 shell-gap are drawn with thick lines. The particles which are almost fully-aligned in these configurations are: triangles, π(h11/2 )−1 , ν(i13/2 )1 (h9/2 , f7/2 )1 and diamonds, π(h11/2 )−1 , ν(i13/2 )2 (h9/2 , f7/2 )2 . Solid and broken lines indicate positive and negative parity, filled and open symbols indicate signature α = 1/2, −1/2, respectively. The calculated bands can be divided into two types: those represented by narrow lines are associated with the minimum at ε ≈ 0.15, γ ≈ −60◦ and define the yrast line for I ≈ 10–20. Relative to the Z = 82, N = 114 core, these have configurations of the type π(d3/2, s1/2 )−2 , with 4 or 5 holes in the ν(i13/2) orbitals, a few particles in the (p3/2 , f5/2 , p1/2 ) levels and the remaining holes in the (h9/2 , f7/2 ) levels. The thick lines indicate structures associated with the minimum at ε ≈ 0.18, γ ≈ −90◦ . They correspond to similar configurations, with the difference of an additional proton excitation from an h11/2 into the h9/2 orbital. They show strong similarities with the results of configuration-dependent CNS calculations performed for the shears bands in the Pb isotopes [19]. These states are perhaps best distinguished by the particles which are almost fully aligned, as indicated in the figure caption. 14 A.N. Wilson et al. / Physics Letters B 505 (2001) 6–14 The calculations clearly predict the presence of low-collectivity bands near yrast at spins above I = 20h̄. The (2) moment of inertia of these bands (as indicated by the strong curvature) is less than half the rigid body value, in good agreement with the behaviour of DB 1. A closer comparison with the data, though, reveals some problems. DB 1 initially appears to be well-reproduced by the state indicated by filled and open triangles (with one aligned i13/2 neutron). However, the calculated band is of negative parity, whereas the experimental data strongly suggests that DB 1 is of positive parity. One drawback of the calculations is that they do not include pairing, which should still be important at the relatively low spins for which the dipole bands are observed. It may be that, with the inclusion of pairing, other states with similar properties (but of positive parity) can be generated at these lower spins. Mixing would also be expected to occur between the different bands. The calculations predict that some of the non-collective structures will be more favored at higher spins (such as that with two aligned i13/2 neutrons, indicated by filled and open diamonds on the figure) where pairing would be less important. No evidence for such states has been found in the data. This may be due to limitations of the statistics; however, it is clear that, with the calculations in their current form, no detailed agreement can be claimed between theory and experiment. In conclusion, the data presented in this paper offer the first evidence in a Hg isotope for the existence of M1 bands involving an excitation across the Z = 82 shell gap. While DB A seems to be analogous to the bands observed in heavier Hg isotopes, DB 1 and 2 show features which suggest that they originate in a minimum at ε ≈ 0.18, γ ≈ −90◦ . This minimum, predicted by configuration-dependent CSN calculations to be near-favored at spins above I ≈ 20h̄, is associated with the excitation of a proton into an h9/2 orbital. This scenario provides a possible explanation for the anomalously large B(M1)/B(E2) ratios associated with these two structures. While the current calculations do not provide detailed agreement with the data, they provide a framework within which an initial understanding of the mechanisms generating the three bands can be achieved. Most importantly, they predict the existence of a previously unobserved minimum in the nuclear potential energy surface of 190 Hg. Additional experimental work, particularly concerning the presence of such bands at higher spins and in other Hg isotopes, as well as more refined calculations including a treatment of the effects of pairing, are required before a thorough understanding of these bands can be achieved. Acknowledgements The authors would like to thank Bob Darlington for the high quality of the targets used in the experiment, and all the staff at CRN associated with the running of the Vivitron and the operation of the Eurogam Phase II array. The Eurogam Project was funded jointly by EPSRC (UK) and IN2P3 (France). One of us (J.T.) acknowledges financial support from the Hungarian Scientific Research Fund, OTKA (contract number T32910). This work was also supported in part by the Swedish Natural Science Research Council and by the U.S Department of Energy, Nuclear Physics Division, under Contract No. W-31-109-ENG-38. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] D. Ye et al., Nucl. Phys. A 537 (1992) 207. J.K. Deng et al., Phys. Lett. B 319 (1993) 63. N. Roy et al., Phys. Rev. C 47 (1993) R903. B. Cederwall et al., Phys. Rev. C 47 (1993) r2443. Y. Le Coz et al., Z. Phys. A 348 (1994) 87. N. Fotiades et al., Z. Phys. 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A 390 (1992) 53. V.P. Janzen et al., Phys. Rev. C 45 (1992) 613. 26 April 2001 Physics Letters B 505 (2001) 15–20 www.elsevier.nl/locate/npe A determination of the 6He + p interaction potential A. de Vismes a , P. Roussel-Chomaz a , W. Mittig a , A. Pakou a,b , N. Alamanos c , F. Auger c , J.-C. Angélique d , J. Barrette e , A.V. Belozyorov f , C. Borcea a,g , W.N. Catford d,h , M.-D. Cortina-Gil i , Z. Dlouhy j , A. Gillibert c , V. Lapoux c , A. Lepine-Szily k , S.M. Lukyanov f , F. Marie c , A. Musumarra c,1 , F. de Oliveira a , N.A. Orr d , S. Ottini-Hustache c , Y.E. Penionzhkevich f , F. Sarazin a,2 , H. Savajols a , N. Skobelev f a GANIL (DSM/CEA, IN2P3/CNRS), BP 5027, 14076 Caen Cedex 5, France b Department of Physics, The University of Ioannina, 45110 Ioannina, Greece c CEA/DSM/DAPNIA/SPhN, Saclay, 91191 Gif-sur-Yvette Cedex, France d LPC, IN2P3/CNRS, ISMRa et Université de Caen, F-14050 Caen Cedex, France e McGill University, 845 Sherbrooke St., Montreal, Quebec H3A 3R1, Canada f FLNR, JINR Dubna, P.O. Box 79, 101 000 Moscow, Russia g Inst. Atomic Physics, P.O. Box MG6, Bucharest, Romania h Department of Physics, University of Surrey, Guildford, Surrey GU2 7XH, United Kingdom i Departamento Fisica de Particulas, Universidad Santiago de Compostela, 15706 Santiago de Compostela, Spain j Nuclear Physics Institute, ASCR, 25068 Rez, Czech Republic k IFUSP-Universidade de São Paulo, C.P. 66318, 05389-970 São Paulo, Brazil Received 4 December 2000; received in revised form 26 February 2001; accepted 2 March 2001 Editor: J.P. Schiffer Abstract The reaction cross section for the halo nucleus 6 He on hydrogen has been measured at 36 MeV/nucleon using the transmission method and a value of σR = 409 ± 22 mb was obtained. A coherent analysis within a microscopic model of this result in conjunction with (p, p) and (p, n) angular distributions has allowed the interaction potential to be uniquely determined. This analysis also allowed the 6 He density distribution to be explored. 2001 Elsevier Science B.V. All rights reserved. PACS: 25.60.Dz; 25.60.Bx; 25.60.Lg Keywords: Reaction cross section; Elastic proton scattering; Charge exchange; Halo nuclei; Microscopic calculations Proton–nucleus elastic scattering has been studied extensively and both phenomenological and micro- 1 Present address: INFN-Laboratori Nazionali del Sud, Via S. Sofia 44, 95123 Catania, Italy. 2 Present address: Department of Physics and Astronomy, Edinburgh EH9 3JZ, Scotland, UK. scopic potential models have been developed [1–6]. In particular, a large amount of experimental data has been successfully interpreted through such models with at most the adjustment of only a few parameters [3–11]. With the advent of radioactive beam facilities, elastic scattering measurements have been extended to 0370-2693/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 3 5 2 - 5 16 A. de Vismes et al. / Physics Letters B 505 (2001) 15–20 neutron-rich nuclei close to the drip lines (see, for example, Refs. [12–15]). However, because of the low beam intensities, such studies often span a limited angular range and cannot probe unambiguously the potential, partly because the effects of the potential cannot be disentangled from the little known density distribution of these nuclei. In this context, reaction crosssection measurements are a valuable tool. Absorbing processes affect the elastic scattering angular distribution and thus reaction cross sections can place restrictions on the amount of absorption, as represented by the imaginary potential [16,17]. In contrast, (p, n) reactions leading to isobaric analog states (IAS) can be used to probe the isovector part of the potential. Consequently, the simultaneous analysis of elastic proton scattering, (p, n) charge exchange and reaction cross section data can provide strong constraints on the interaction potential for neutron-rich nuclei and tell us something about their structure. We have previously reported results on the proton elastic scattering and (p, n) charge exchange reaction on 6 He [12,18]. It was shown in [12] that these first elastic scattering data are not well reproduced by standard proton–nucleus potentials derived from elastic scattering measurements on stable nuclei. A good description of the data was obtained by either a reduction of the amplitude of the real potential or an important increase of the imaginary potential. A description of the charge exchange reaction leading to the IAS of 6 He favored this last option [18]. A more quantitative conclusion was not possible partly because of the limited angular range, (θcm = 15–40 deg), of the elastic scattering data reported in [12]. Very recently a new elastic scattering angular distribution measurement [13], spanning a broader range of angles (θcm = 10–80 deg), was obtained using the MUST detector array [19]. The analysis of the p – 6 He interaction reported here uses these new data. Additionally, as described here, a reaction cross section measurement on a hydrogen target has been undertaken. We report in this Letter on the coherent analysis of three data sets — (p, p) elastic scattering [13], (p, n) charge exchange [18] and the present reaction cross section measurement — within a microscopic framework. The aim of this work was to achieve better description of the interaction potential for 6 He. The experimental details concerning the elastic scattering and charge exchange reactions, have been reported elsewhere [12,13,18]. The p + 6 He reaction cross section forms part of a series of measurements for light stable and neutron-rich nuclei which will be published in the near future [20]. The experimental procedure is thus only briefly summarized here. The 6 He beam was produced by fragmentation of a 60 MeV/nucleon 48 Ca primary beam delivered by the GANIL accelerator complex, and incident on a Be production target, backed by Ta. The secondary ions were subsequently selected using the spectrometer LISE [21]. The reaction cross section, σR , was measured using the transmission method [22]. In such a measurement, the attenuation of the 6 He beam passing through the target is measured and the reaction probability is given by Nf Nreac =1− Ninc Ni Nd e , = 1 − exp −σR (1) A where σR is the reaction cross section, d is the target density, N is the Avogadro number, A is the target mass number and e is the thickness of the target. In the present measurement, liquid hydrogen targets (35 and 70 mg/cm2 thick) were used. By measuring the number of incident and transmitted ions, the reaction cross section may be directly determined. The incident 6 He ions (36.2 MeV/nucleon), were identified by their characteristic energy loss and time-of-flight with respect to the cyclotron RF, using an ionization chamber and a microchannel plate timing detector, placed upstream of the hydrogen target. The transmitted ions were identified using a large area (50 × 50 mm2 ) telescope, set up 6.5 cm downstream of the target. The telescope was composed of a thin (500 µm) position sensitive Si detector, a Si(Li) detector (3500 µm) and a thick stopping CsI scintillator (4.5 cm) [23]. The measured reaction cross section, σR = 409 ± 22 mb, is 10% higher than the empirical prediction of Kox et al. (σR = 365 mb) [24]. The Kox formula reproduces σR well for a wide variety of nuclei, over a broad energy range, and as such the observed enhancement is consistent with the halo structure of 6 He. In order to study the interaction potential and the effect of the density distribution of 6 He in a consisPR = A. de Vismes et al. / Physics Letters B 505 (2001) 15–20 tent manner, calculations using the Jeukenne, Lejeune and Mahaux approach (JLM) [1] of the nuclear interaction potential were undertaken to fit simultaneously the reaction cross section and the angular distributions derived from the elastic scattering [13] and charge exchange reactions [18]. Calculations were performed within a microscopic DWBA approach, in which entrance and exit-channel optical potentials were calculated consistently using the JLM energy and density dependent interaction. The starting point for computing the JLM potentials, is the Brueckner– Hartree–Fock approximation and the Reid hard core nucleon–nucleon interaction which describes, for energies up to 160 MeV, the energy and density dependence of the isoscalar, isovector and Coulomb components of the complex optical potential in infinite matter. The optical potential of a finite nucleus is obtained by using the local density approximation (LDA), that is by substituting the nuclear matter density by the density distribution of the nucleus. The JLM central potential has been extensively studied by Mellema et al. [8] and Petler et al. [9]. It has been particularly successful in describing elastic proton and neutron scattering for light stable nuclei, provided that the imaginary potential is adjusted slightly by a normalization factor (λw ) of around 0.8. In the following, this will be referred to as the standard normalization. As noted earlier, elastic proton scattering brings valuable information on the nuclear interaction potential, while reaction cross sections provide complementary constraints on the imaginary part. As may be seen in Table 1, where calculated reaction cross sections for 6 He + p at 36.2 MeV are displayed for various potentials, variations in the normalization factor for the real potential has little effect, whereas the reaction cross section varies linearly with the amplitude of the imaginary potential. As such, the normalization factor for the imaginary potential derived from elastic scattering can be corroborated using the reaction cross section. Table 1 also demonstrates that the reaction cross section is sensitive to the isovector part of the interaction potential. Previous work has shown that even for such neutron-rich nuclei as 6 He, elastic scattering is almost not sensitive to the isovector part of the potential [25], as opposed to the (p, n) charge exchange reaction to the IAS [26]. 17 Table 1 Reaction cross section as a function of the normalization factors for the real and imaginary JLM potential, λv and λw . For the first set of calculations (first two columns), λw is fixed to the standard value of 0.8, and λisov = 1.4. For the second set of calculations (last three columns), λv is fixed to the standard value of 1. The calculations including a variation in the imaginary potential are performed for two values of the isovector normalization, λisov = 1.4 and 1.0. The 6 He density distribution from the shell model [27] is used λisov = 1.4 λisov = 1.0 λv (λw = 0.8) σR (mb) λw (λv = 1.0) σR (mb) σR (mb) 1 397 1 451 424 0.9 393 0.9 425 399 0.8 388 0.8 397 372 0.7 383 0.7 365 341 0.6 378 0.6 330 307 The analysis subsequently proceeded by iterative fitting of the elastic scattering angular distribution, the reaction cross section and the (p, n) angular distribution. An input to these calculations is the 6 He nucleus density distribution. At first, densities determined via a shell model approach [27] were employed. The best fit of the elastic scattering data obtained with these densities is presented in Fig. 1(a) (solid line), and was obtained with λv = 0.85 and λw = 0.59 (χ 2 = 1.12, see Table 2), irrespective of the isovector component. The latter was adjusted through fits to the (p, n) angular distribution resulting in a normalization of λisov = 1.4 (Fig. 2(a)). Using the potential determined in this manner (λv = 0.85, λw = 0.59 and λisov = 1.4), a reaction cross section of 320 mb was predicted, which is much lower than the measured value. An increase in the imaginary part of the potential is required to reproduce the reaction cross section measurement. From Table 1 it is seen that the imaginary part should be increased to λw = 0.85 to reproduce the experimental reaction cross section. The elastic scattering was then re-examined, by fixing the imaginary normalization factor to λw = 0.85 and by varying the real part of the optical potential. The constrained fit, χ 2 = 3.96, was obtained for λv = 0.88 (Fig. 1(a), dashed line) and represents an adequate compromise for a simultaneous description of the three sets of data. 18 A. de Vismes et al. / Physics Letters B 505 (2001) 15–20 Fig. 1. JLM angular distribution calculations for 6 He(p, p)6 He elastic scattering using the following densities: (a) The shell model density of Karataglidis et al. [27]; (b) The cluster model density of the Surrey group [28]; (c) The cluster model density of Arai, Suzuki and Lovas [29]. In this last case the four different descriptions listed in Table 2 provide angular distributions which are exactly identical within the line width. Solid lines correspond to the best fit. Dashed lines correspond to the constrained fit (see Table 2 and text). The sensitivity to the density distribution was studied by repeating the above iterative procedure for the following theoretical halo density distributions: – A density calculated from a three-body cluster model [28] with a RMS radius similar to the previous shell model one. – A density distribution calculated from an extended three-body cluster model [29] for four different descriptions: (a) a pure three cluster model α + n + n; (b) a three cluster model with the inclusion of a t + t component; (c) an extended three cluster [3N + N] + n + n model; and (d) an extended three cluster model with a t + t component. The RMS radii of all the above densities are listed in Table 2. The details of the calculations are also presented in the same Table. The (p, n) data were Fig. 2. (a) JLM angular distribution calculations for the 6 He(p, n)6 Li∗ reaction. The dashed, solid and (3.56 MeV) dashed-dotted lines correspond to an isovector normalization of λisov = 1.0, 1.4 and 1.8, respectively. (b) JLM angular distribution calculations for the 6 He(p, n)6 Li∗(3.56 MeV) reaction for three different density distributions. The solid line corresponds to the shell model density [27], the dashed line to the three-body cluster model of the Surrey group [28], the dotted line to the Arai–Suzuki–Lovas cluster model [29]. The adopted optical potential normalization factors are those obtained via the constrained fit (Table 2). The isovector normalization factor is 1.4. found to be compatible with an isovector adjustment of λisov = 1.4 for all the densities (Fig. 2(b)). The best fits as well as the constrained fits, obtained with the reaction cross section constraint on the imaginary part of the potential, are shown in Fig. 1(b) and (c), for the densities of Ref. [28] and [29], respectively. The quality of the fits, as noted by the χ 2 values in Table 2, is slightly in favour of the cluster model density distributions of Arai, Suzuki and Lovas. However, neither the elastic scattering nor the reaction cross section can distinguish between the four different descriptions (Table 2) of 6 He in this model. A. de Vismes et al. / Physics Letters B 505 (2001) 15–20 Table 2 Details of the densities used in the JLM calculations and the potential renormalizations (λv , λw ) used to fit the elastic angular distributions. The density indices indicate: (A) Shell model densities from Karataglidis–Amos [27]; (B) Three-body cluster model of Ref. [28]; (C) Three-body cluster model of Arai–Suzuki–Lovas [29] (C-a: a pure three-cluster α + n + n model; C-b: a pure threecluster model with a t + t component; C-c: an extended three-cluster [3N + N] + n + n model; C-d: an extended three cluster model with a t + t component). The RMS radii for the matter, proton and neutron distributions are given. For each density, the first line indicates the best fit results, varying both λv and λw and the second line corresponds to constrained fit, where the imaginary part was set to a value such as to reproduce the reaction cross section results (λisov = 1.4 in both cases) Density rm (fm) rp (fm) rn (fm) A B C-a C-b 2.52 2.53 2.49 2.42 2.03 1.94 1.82 1.81 2.73 2.78 2.77 2.68 C-c 2.34 1.75 2.59 C-d 2.33 1.76 2.57 λv λw χ2 σR (mb) 0.85 0.59 1.12 320 0.88 0.85 3.96 407 0.87 0.66 0.90 337 0.9 0.88 2.65 406 0.88 0.74 0.85 353 0.89 0.92 2.20 407 0.88 0.75 0.80 351 0.89 0.92 2.22 407 0.88 0.85 0.85 365 0.89 1.0 2.10 404 0.88 0.84 0.84 364 0.89 1.0 406 2.13 It should be stressed that the normalization adjustments for both the real and imaginary parts of the optical potential depend on the density distribution of the nucleus. On the other hand, no such dependence within the experimental uncertainties is apparent in Fig. 2(b), where calculations for the (p, n) reaction with various halo densities are compared with the measured angular distribution and show good compatibility, irrespective of the choice of density and of the real and imaginary potentials, with an isovector adjustment of λisov = 1.4. In summary, the reaction cross section p + 6 He has been measured in inverse kinematics at 36 MeV. This result has been incorporated into a consistent microscopic description of earlier (p, p) and (p, n) angular distribution measurements. It was shown that the JLM potential can describe adequately 6 He(p, p) elastic 19 scattering, (p, n) charge exchange and the reaction cross section measurement with an imaginary potential close to the standard one for light stable nuclei. The best description of the ensemble of the data was obtained when the imaginary part of the potential is increased by roughly 10%, and the real part is decreased by roughly 10% in comparison with the standard normalization factors. The need to reduce the real part of the potential was noticed before in the case of nucleus– nucleus elastic scattering involving weakly bound stable nuclei, and was attributed to coupling to the continuum [11,30]. As to the isovector part of the interaction potential, both the reaction cross section and the (p, n) charge exchange angular distribution required an increase of about 40% (λisov = 1.4) with respect to the standard normalization, which is smaller than that for stable nuclei (λisov = 2.5) [31]. It is important to note that such conclusions could be drawn only because a wide range of data was available on the same nucleus at the same beam energy, thus enabling the effects of the different parts of the interaction potential to be disentangled. 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Lapoux et al., in preparation. A. Pakou et al., submitted for publication. 26 April 2001 Physics Letters B 505 (2001) 21–26 www.elsevier.nl/locate/npe Evidence for an l = 0 ground state in 9He L. Chen a,b , B. Blank a,1 , B.A. Brown a,b , M. Chartier a,c,1 , A. Galonsky a,b , P.G. Hansen a,b , M. Thoennessen a,b a National Superconducting Cyclotron Laboratory, East Lansing, MI 48824-1321, USA b Department of Physics & Astronomy, Michigan State University, East Lansing, MI 48824-1321, USA c Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, TN 37831-6368, USA Received 11 January 2001; received in revised form 15 February 2001; accepted 16 February 2001 Editor: J.P. Schiffer Abstract The unbound nuclear systems 10 Li and 9 He were produced in direct reactions of 28 MeV/u 11 Be incident on a 9 Be target. The distributions of the observed velocity differences between the neutron and the charged fragment show a strong influence of final-state interactions. Since the neutron originates in a dominant l = 0 initial state, a selection-rule argument allows a firm l = 0 assignment for the lowest odd-neutron state in 10 Li. We report the results suggesting a very similar unbound state in 9 He, characterized by an s-wave scattering length more negative than −10 fm corresponding to an energy of the virtual state of less than 0.2 MeV. Shell-model calculations cast light on the reasons for the disappearance of the magic shell gap near the drip line. 2001 Published by Elsevier Science B.V. PACS: 21.10.Dr; 25.70.Mn; 27.20.+n Keywords: Stripping reactions with radioactive nuclear beams; 9 He ground state The region of the lightest nuclei offers our only practical possibility for obtaining a glimpse of the structure of nuclear systems that are “beyond the neutron drip line”, i.e., that have no states that are bound with respect to neutron emission. There is a special theoretical interest in the lightest N = 7 isotones where intruders from the 1s0d shell appear. First known in the lightest bound N = 7 nucleus 11 Be [1], this phenomenon provides a sensitive test of theories designed to bridge the 0p–1s0d shells and offers a paradigm for the disappearance of the shell gaps near the neutron drip line. A number of E-mail address: [email protected] (M. Thoennessen). 1 Permanent address: Centre d’Etudes Nucléaires de Bordeaux- Gradignan, BP 120, Le Haut Vigneau, 33175 Gradignan Cedex, France. experiments have searched for the lowest levels in 10 Li, see the recent summary in [2]. The picture emerging is that the p-state, which is the normal ground state for N = 7, and also states with higher angular momentum are detected as relatively narrow resonances in inclusive reactions. On the other hand, these reactions seem to miss the unbound s-states, which do not exhibit a resonance-like structure, but show a rapid rise in cross section at threshold followed by a slow decay towards higher energies. A Breit– Wigner shape is never a good approximation to these. It has taken a different technique based on exclusive studies of the 9 Li + n channel [2–5] at low energy to find a candidate for an l = 0 assignment, supported by indirect arguments. In the case of 9 He [6–8], inclusive measurements identified a resonance at 1.2 MeV assumed to be the ground state. Its relatively narrow 0370-2693/01/$ – see front matter 2001 Published by Elsevier Science B.V. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 3 1 3 - 6 22 L. Chen et al. / Physics Letters B 505 (2001) 21–26 width speaks for an l = 1 assignment and suggested [9] that there is no level inversion in this system. In the following we describe a technique specifically chosen to be sensitive to s strength at low energies, and we report the observation of a low-lying l = 0 level in 9 He and the confirmation of the l = 0 assignment to the lowest neutron state in 10 Li. The experiment is based on a technique that we may refer to as a direct reaction leading to continuum states, and it exploits an approximate selection rule linking the single-particle structure of the initial state to that of the final state. The ideal identification of l = 0 strength would, of course, be elastic neutron scattering on the appropriate radioactive targets, were it not for their short half lives, 0.1–0.8 s. As in the previous work [2–5] we observe instead the reaction channel in which a neutron and the fragment of interest are emitted together. This amounts to observing the final-state interactions and was the original method used for determining the strength of the neutron–neutron interaction [10]. It is applicable if the interaction is of short range, strong and attractive, all of which holds for our cases. However, instead of producing the exit channel in a complex breakup reaction, we have in the new experiment used direct reactions corresponding to single and double proton knockout on the projectile 11 Be. The initial neutron state of this nuclide is dominated [1] by a 1s1/2 single-particle orbital, and it is very difficult to access an odd-parity neutron state without a major rearrangement. On the other hand, reactions on 12 Be with filled and partly filled p3/2 , p1/2 , d5/2 , and s1/2 pairs connect, as we shall show, to both even- and odd-parity final states. We may assume that the two-proton removal to 9 He proceeds via a direct reaction because the large difference in 11 Be neutron separation energy (0.5 MeV) and proton separation energy (20.6 MeV) makes an intermediate step involving proton evaporation very unlikely. The experiment was performed with radioactive beams of the three beryllium isotopes 10,11,12Be, all at 30 MeV/u. They were produced in the National Superconducting Cyclotron Laboratory’s A1200 fragment separator [11] from primary beams of 80 MeV/u 13 C and 18 O interacting in thick production targets, and purified by intermediate aluminum degraders. A thin, fast plastic detector in the flight path upstream of the experimental area provided an event-by-event identi- fication of the incoming particles. The fragments reacted in a 200 mg/cm2 secondary 9 Be target so that the mid-target energy of the projectiles was 28 MeV/u. Charged reaction products and the unreacted secondary beam were deflected by a 1.5 T sweeping magnet, which was part of the fragment detection system [12]. Particle identification of the outgoing fragments was provided by the combination of the energy loss, measured in 58.2 mg/cm2 silicon-strip detectors placed behind the target, and the total energy. The latter was measured in an array of sixteen vertical plastic scintillator bars 1.7 m after the target with photomultipliers placed at either end to determine the vertical position of the charged fragment. The resulting velocity resolution obtained with an incident 11 Be beam expressed in terms of the standard deviation σf becomes 0.08 and 0.13 cm/ns for 9 Li and 8 He, respectively. The neutrons were detected in the two 2 × 2 m2 NSCL Neutron Walls [13] centered around 0◦ . The two walls were 5.0 and 5.5 m behind the target position. The resolution σn on the neutron velocity was 0.24 cm/ns independent of the reaction. The analysis included events inside a maximum angle between projectile residue and neutron of 5◦ for the 10 Li experiment and 10◦ for the 9 He experiment. This selection was taken into account in the theoretical distributions presented below. The acceptance drops rapidly for decay energies greater than 0.5 MeV. Further details of the experimental setup and analysis are published elsewhere [13–15]. It is convenient to present the results in terms of the scalar velocity difference between the almost parallel neutron and fragment [2,3]. As long as the angle between the two is small, this difference is identical to the projection of the velocity difference on the direction of the outgoing fragment. The analysis is based on the sudden approximation. The outcome of the reaction is then obtained by expanding the initial state (a neutron bound to the rest of the projectile) in continuum eigenfunctions representing outgoing waves of the (in principle unknown) final system. Since the initial state does not belong to the same function space as the final states, there is no orthogonality requirement. The single-particle wave functions, both initial and final, were calculated for a Woods–Saxon potential with the radius and diffuseness parameters fixed to 1.25 and 0.7 fm. The potential depths were adjusted to give the (known) eigenenergies for the initial states and specified low-energy scat- L. Chen et al. / Physics Letters B 505 (2001) 21–26 tering parameters (resonance energy or s-wave scattering length as ) for the continuum states [15]. In order to be able to compare with other experiments which cite “energies of resonances” [2], it is convenient to be able to translate a deduced as into an excitation energy scale. To this end we note that for bound states just below the threshold, the eigenenergy is approximately E = −h̄2 /2mas2 . Consequently it seems logical to define the equivalent energy of the virtual state to be the same with opposite sign. As in our previous experiments [2,3] it is necessary to consider the possible presence of a “background” contribution, by which we refer to reaction channels other than the one of interest. In the 11 Be experiments there are two evident candidates for this. The first arises because the spectroscopic factor of the 1s1/2 single-particle configuration is 0.74 [1] with a large part of the remainder representing a much more strongly bound 0d5/2 state coupled to core excitations. Secondly, in the two experiments with 11 Be the low binding of the halo neutron implies that the selection rule used for determining l is relaxed by the recoil momentum of the 9 Li and 8 He residues, of the order of 80 MeV/c in both cases. The transformation to the new center-of-mass system reduces this momentum by an order of magnitude, but it is still comparable to that of the halo. We estimate [15] by extending an analytical expression given by Bertsch et al. [16] that recoil reduces the intensity of the lowest components of the final l = 0 spectrum by approximately 10%, which will appear as a broader component. We have approximated the background by folding the projected neutron and fragment velocity distributions from the coincidence data. These distributions are close to Gaussian shape and quite similar to the thermal distributions used in the previous work. They reflect mainly the limitations in experimental acceptance and differ little between the different projectile–product pairs. We have included such a shape in each fit and allowed the intensity to vary freely. The resulting intensity is 47% for (11 Be, 9 Li + n) and 29% for (11 Be, 8 He + n). A fit to the true coincident events requires a final-state interaction which gives rise to a narrower distribution than the intrinsic momentum distribution of the 11 Be valency neutron, which for comparison is shown in Fig. 2(b). We take this as proof that we are dealing with a real effect. 23 Fig. 1(a) shows the velocity difference spectrum for the 6 He + n system with the 12 Be beam. As mentioned, the selection rule effect is not present here, and the reaction selects the peaks corresponding to forward- and backward-emitted neutrons from the 0p3/2 resonance in 7 He. This was previously observed in the 7 Li(t, 3 He)7 He reaction [17] at an energy of 440 ± 30 keV and with a width of 160 ± 30 keV. The fit to the data from the 12 Be projectile (solid) is a sum of the model calculation with a resonance at 450 keV (short dashes) and the background (dot-dashed). A χ 2 analysis of this and similar results for 11 Be, yielded an energy for the 7 He state of 450 ± 20 keV, in very good agreement with the previous measurement, and Fig. 1. Measured neutron-fragment velocity-difference spectra for 7 He and 10 Li. (a) 7 He was produced in the reaction 9 Be(12 Be, 6 He + n)X and the fit (solid) was adjusted to a p-wave resonance at 450 ± 20 keV (short-dashed), a background contribution (dot-dashed) and a d-component (long-dashed). (b) Three different projectiles 10,11,12 Be producing 10 Li. The intensity (shown in absolute units of 10−7 per bin of 0.2 cm/ns) is normalized to the number of incoming fragments. Note the absence of events from the 10 Be beam, which demonstrates the clean selection of events corresponding to decay of the projectile residue. 24 L. Chen et al. / Physics Letters B 505 (2001) 21–26 Fig. 2. Velocity difference spectra for the reactions of 11 Be leading to 10 Li (a) and 9 He (b). The adjustment assumes s-wave components (short-dashed) characterized by a scattering length as = −25 and −20 fm, respectively, a background contribution (dot-dashed) and, for 10 Li, a p-wave resonance at 0.50 MeV (long-dashed). The curve marked as = 0 in (b) is the distribution calculated assuming no final-state interaction and without background contribution (long-dashed). we present this as an independent determination. The error limits given here and in the following correspond to four units of increase in χ 2 from the best fit, indicating a significant change in the quality of the fit. Fig. 1(b) shows the corresponding spectrum for the 9 Li + n system for the three different projectiles 10,11,12Be. The most striking qualitative result is the almost total absence of 9 Li + n events from 10 Be, which cannot give rise to 9 Li + n in a pure projectile fragmentation process. This proves that our technique, designed to observe projectile fragmentation, discriminates effectively against reaction products, including neutrons, originating in the target. The difference between the 11 Be and the 12 Be spectra also shows the influence of the initial state, the more bound s-state in 12 Be leading to a broader distribution. Fig. 2(a) shows 10 Li data with the potential model fit for the 11 Be projectiles. It confirms the strong finalstate interaction at low energy observed in the previous work [2,3] using 18 O as the projectile. The fact that we see this effect with 11 Be proves that the state must have l = 0, the same as the main single-particle contributor in the projectile. The scattering length is numerically very large, more negative than −20 fm corresponding to an excitation energy of less than 0.05 MeV for the virtual state. The intensity of a p-state assumed to be at 0.5 MeV in the final system is an unconstrained fit. As could be expected from the absence of a suitable initial state, the contribution is much smaller than it was with the 18 O projectile [2,3], which has a full 0p1/2 subshell. For 9 He, only the 11 Be data had enough statistics to be of interest. The velocity distribution shown in Fig. 2(b) requires a final-state interaction characterized by a scattering length of the order of −10 fm (or more negative), corresponding to an energy of the virtual state of 0.0–0.2 MeV. No combination of the background and the intrinsic momentum distribution of the 11 Be valency neutron (corresponding to an s-wave scattering length as of zero) will fit the data. The selection rule again fixes the angular momentum to be zero. This suggests the level scheme shown in Fig. 3, in which the narrow resonances seen in previous work [6–8] are identified as excited levels about 1.2 MeV above threshold. A theoretical spectrum for 9 He calculated in a model space of [(0s)4 (0p)n ] (0h̄ω) for negative parity states and [(0s)3 (0p)(n+1)] plus [(0s)4 (0p)(n−1) (0d1s)(1)] (1h̄ω) for positive parity states is shown for the WBP and WBT interactions of [18] in Fig. 3. The results we discuss for 10 Li and 9 He are predictions (extrapolations) based upon the 0h̄ω and 1h̄ω model and empirical Hamiltonians as described in Ref. [18]. If the configuration space for these nuclei is restricted to [(0p3/2)4 , 0p1/2] and [(0p3/2)4 , 1s1/2 ] for the 1/2− and 1/2+ states, respectively, the 1/2+ comes above the 1/2− state by 2.38 MeV. This restriction is equivalent to that assumed in a spherical Hartree–Fock calculation (with no pairing) and by three-body models for 10 He (11Li) where 8 He (9 Li) are treated as inert (closed-shell) configurations. For the 1/2− state, (0p3/2)4 , 0p1/2 is the only configuration allowed in the p-shell. However for the 1/2+ state the full p-shell space is represented by the three configura- L. Chen et al. / Physics Letters B 505 (2001) 21–26 Fig. 3. Level scheme of 9 He. Theoretical calculations with the WBP and WBT interactions of Warburton and Brown [18] are compared with the result of the present work (marked LC) and previously reported resonances in the 8 He + n exit channel marked KKS [6] and WVO [7,8]. 25 the conditions that these higher excitations are not explicitly present in the low-lying states, and that their effect enters implicitly in terms of the effective oneand two-body matrix elements. Perhaps the close spacing of the p- and s-orbits near the neutron drip line makes the h̄ω restriction more unreliable than it appears to be in nuclei closer to stability. In summary, our results strongly suggest for the first time, that the ground state of 9 He is an unbound s-state with a scattering length of as −10 fm. 9 He represents the lightest possible N = 7 nucleus. Based on this result, the mass assignment for 9 He [21] has to be revised downward by 1 MeV. The appearance of intruder states as the ground states of 9 He continues the trend observed in 11 Be and 10 Li and is in excellent agreement with the model [18] constructed for understanding the 0p−0d1s cross-shell interaction [22,23]. Acknowledgements tions [(0p3/2)4 , 1s1/2 ], [(0p3/2)2 , (0p1/2)2 , 1s1/2], and [(0p3/2)3 , 0p1/2, 1s1/2 ]. The most important mixing is due to the pairing interaction between the first two of these, and this lowers the energy of the 1/2+ state by 3.32 MeV and results in a crossing of the quasiparticle levels. This contribution is present in all N = 7 isotones and reaches a maximum of 4.52 MeV for 11 Be. The comparison presented in Fig. 3 suggests that the theoretical and experimental level spectra are in close agreement. However, the calculated neutron separation energy for the 1/2+ ground state is −4.1 MeV (unbound), in contrast to a value close to zero given by the present experiment. Similarly, the calculated neutron separation energy for the 1/2− state is −4.7 MeV in contrast to the experimental value of −1.2 MeV for the lowest resonance observed in the transfer reactions, which is suggested to be a p-wave resonance due to its narrow shape. Thus the WBT (and WBP) extrapolations appear to be good for the spectrum but poor for the absolute energies with respect to 8 He. 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A 595 (1995) 409. [22] H. Sagawa, B.A. Brown, H. Esbensen, Phys. Lett. B 309 (1993) 1. [23] B.A. Brown, in: R.A. Broglia, P.G. Hansen (Eds.), International School of Heavy-Ion Physics, 4th Course: Exotic Nuclei, World Scientific, Singapore, 1998; B.A. Brown, in: M. de Saint Simon, O. Sorlin (Eds.), Proc. ENAM95, Editions Frontieres, 1995, p. 451. 26 April 2001 Physics Letters B 505 (2001) 27–35 www.elsevier.nl/locate/npe Excited superdeformed K π = 0+ rotational bands in β-vibrational fission resonances of 240Pu M. Hunyadi a , D. Gassmann b , A. Krasznahorkay a , D. Habs b , P.G. Thirolf b , M. Csatlós a , Y. Eisermann b , T. Faestermann c , G. Graw b , J. Gulyás a , R. Hertenberger b , H.J. Maier b , Z. Máté a , A. Metz b , M.J. Chromik b a Inst. of Nucl. Res. of the Hung. Acad. of Sci., H-4001 Debrecen, P.O. Box 51, Hungary b Sektion Physik, Universität München, D-85748 Garching, Germany c Technische Universität München, D-85748 Garching, Germany Received 7 February 2001; accepted 6 March 2001 Editor: V. Metag Abstract The intermediate structure of fission resonances of 240 Pu was observed with an experimental energy resolution of 7 keV in the excitation energy region of E ∗ = 3.8–5.6 MeV using the 239 Pu(d, pf)240 Pu reaction. Two-vibrational resonance groups centered at E ∗ = 4.6 MeV and 5.1 MeV, and attributed to the excitation of three and four β-phonons, were resolved into individual substates, which could be assigned to the low-spin members of K π = 0+ superdeformed (SD) rotational bands. In the region of the lower E ∗ = 4.6 MeV resonance individual moments of inertia of six well separated bands could be extracted for the first time with values of Θ/h̄2 around 157 MeV−1 , close to that of the ground state band in the second well. From the level density of these K π = 0+ band heads the excitation energy of the SD ground state was determined to (2.25 ± 0.20) MeV, in agreement with earlier estimates from excitation functions. 2001 Elsevier Science B.V. All rights reserved. PACS: 21.10.Re; 24.30.Gd; 25.85.Ge; 27.90.+b 1. Introduction After the discovery of fission isomerism in the actinide region, great efforts were directed towards the spectroscopic studies of excited states in superdeformed (SD) nuclei [1–3]. The appearance of such elongated shapes with an axis ratio of 2 : 1 in the actinides is the consequence of a second minimum in the shell-corrected potential energy surface. Lowlying SD excitations were mostly observed in conversion electron experiments, which resulted in the successful identification of rotational bands upon the fission isomeric ground states [4,5]. Only recently, in the second minimum of 240 Pu, which is investigated here at higher excitation energies, a more complete picture of low-lying quadrupole and octupole vibrations was obtained by a detailed γ - and conversion electron spectroscopy [6,7]. In contrast to the first minimum, multiphonon β-vibrations at higher excitation energies can be investigated in the second minimum as transmission resonances in the prompt fission probability, since the β-vibrations manifest themselves as doorway states to fission. A few members of the vibrational series were systematically observed in actinide isotopes just below the fission barrier [3]. Due to the damping of the vibrational motion their large fission width is 0370-2693/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 3 4 1 - 0 28 M. Hunyadi et al. / Physics Letters B 505 (2001) 27–35 distributed over many compound states in their vicinity. While the fine structure due to compound levels in the first minimum (class I states) is not resolved in the present experiment we do resolve, besides the gross structure of the β-vibrational transmission resonances, the intermediate structure due to SD class II compound states due to their lower level density. High energy resolution studies of fission resonances can therefore give spectroscopic information on SD compound states with specific spin J π and spin projection K at excitation energies below the fission barrier. The more one reaches with efficient detector systems into the subbarrier regions of vibrational damping, the better one should be able to observe isolated levels with small widths in the fission channel. The overall structure of the transmission resonances observed in 240 Pu at an excitation energy with respect to the ground state in the first minimum E ∗ = 5.1 MeV [8– 10,13] and at E ∗ = 4.6 MeV [15–18] was repeatedly interpreted as one of the best examples for damped vibrational resonances. In the experiment of Ref. [10] on 240 Pu the E ∗ = 5.1 MeV resonance group was investigated with a high energy resolution of 3 keV, and with the aid of a detailed model description [10] all experimentally resolved class II states being predominantly populated in the (d, p)-reaction could be identified as (K π , J ) = (0+ , 2) states, while the 0+ states were not resolved due to their broad fission width. Until now, the lower E ∗ = 4.6 MeV resonance group was studied only with modest resolution [15,18]. Extending the model description to this group we expect narrow fission widths for all spin values of SD class II states and a reduced level density. In the present work we therefore reinvestigated the E ∗ = 4.6 MeV vibrational resonance with good statistics and high energy resolution and identified for the first time individual K π = 0+ SD rotational band members with spin and parity of J π = 0+ , 2+ , 4+ in this energy region. 2. Experimental method The experiment was carried out at the Munich Accelerator Laboratory employing the 239 Pu(d, pf)240Pu reaction (QGS = 4.309 MeV) with a deuteron beam of Ed = 12.5 MeV, and using an enriched (99.9%) ≈ 30 µg/cm2 thick target of 239Pu2 O3 on a 30 µg/cm2 thick carbon backing. Protons were measured in coin- cidence with the fission fragments. The excitation energy E ∗ of the 240 Pu compound nucleus could directly be deduced from the kinetic energy of the protons, which were analyzed by a Q3D magnetic spectrograph [19] set at ΘLab = 130◦ relative to the incoming beam (Ω = 10 msr). The position in the focal plane was measured by a light-ion focal-plane detector of 1.6 m active length using two single-wire proportional counters surrounded by etched cathode foils [20]. Fission fragments were detected in two position-sensitive avalanche detectors (PSAD) [21] having two wire planes (with delay-line read-out) corresponding to horizontal and vertical directions. Thus the spatial positions of the fragments and their angular correlation with respect to the recoil axis could be determined. The fission detectors covered the range of φ = 0◦ – 100◦ relative to the recoil axis with a solid angle coverage of 24% of 4π (without double-counting of fission fragments). The time resolution between the proton detector and the fission detectors was measured to be 4 ns FWHM after correcting for the time-offlight in the spectrograph. The energy calibration was taken from the 208 Pb(d, p) reaction (QGS = 1.710 ± 0.015 MeV [22]). The experimental energy resolution was measured to be ≈ 7 keV for the calibration lines lying in the focal plane close to the investigated region in 240 Pu. Data were collected during effectively 108 hours with an average beam current of 400 nA, causing a count rate of 20 kHz in the fission detectors, while the focal plane detectors typically operated at 90 Hz. 3. Experimental results and discussion The measured proton energy spectrum from the (d, p) reaction in coincidence with the fission fragments is shown in Fig. 1a in terms of the excitation energy of the compound nucleus 240Pu. Contributions from random coincidences were subtracted. The spectrum is proportional to the product of the fission probability and the known smoothly varying (d, p) cross section, which shows no fine structure [8]. In Fig. 1b we compare our measurement with previous high resolution measurements of the same reaction: The (d, pf)spectrum of Fig. 1b was measured with a resolution of 17 keV by Specht et al. [8]. It is nicely reproduced in all fine structures after folding our spectrum with M. Hunyadi et al. / Physics Letters B 505 (2001) 27–35 29 Fig. 1. (a) Proton coincidence spectrum of the 239 Pu(d, pf) reaction measured in this work. (b) Proton coincidence spectrum at Ed = 12.5 MeV and Θ = 140◦ from Specht et al. [8]. For comparison also spectra folded with a resolution of 17 keV from this work (solid line) and from Glässel et al. [10] (dashed line) are shown. the reduced resolution and applying a global shift of 120 keV to higher energies on the experimental data of Ref. [8] to account partially for the proper Q-value. When folding the more recent measurement by Glässel et al. [10], which was performed at Ed = 12.5 MeV and Θ = 125◦ with an experimental resolution of 3 keV, only a small overall shift of 12 keV was necessary. In this experiment the protons were detected in the Q3D using a multi-wire proportional chamber with digital single wire readout. Severe losses of efficiency of up to 50% become apparent in Fig. 1b for the measurement of Ref. [10] in the lower half of the 5.1 MeV resonance. 3.1. Excited rotational and vibrational bands in the second well Looking at the overall features of Fig. 1a we find for the two transmission resonances at 4.6 MeV and 5.1 MeV the known gross structures with a damping width of about 200 keV. The resolved intermediate structure of the lower resonance shows a regular pattern of well-resolved triplets, with a weaker lower and upper peak which are separated by 19 and 43 keV, respectively, from the stronger central peak. These separations are very close to the 0+ –2+ and 2+ –4+ separation energies (20 keV, 46.6 keV) of the K π = 0+ ground state rotational band in the second minimum [4], strongly suggesting these structures as being due to excited K π = 0+ rotational bands in the SD minimum of 240 Pu. Thus, for the lower resonance at 4.6 MeV, we observe in the intermediate structure for the first time series of “pure” resonance states with K π = 0+ and a spins 0+ , 2+ , 4+ with rotational energy spacings as known from the ground state band in the second minimum [4,9]. This is in contrast to the excitation energy region above ∼ 5 MeV, where we confirm the bunching of peaks ‘without any systematic trend’ [9] observed earlier [8, 10]. The 5.1 MeV resonance region in the spectrum of Glässel et al. [10], where sufficient overlap with our spectrum could be observed, was fitted with K π = 0+ rotational bands assuming a Lorentzian line shape for the band members. The position and the amplitude of each band were treated as individual parameters, 30 M. Hunyadi et al. / Physics Letters B 505 (2001) 27–35 Fig. 2. (a) Proton spectrum of the reaction 239 Pu(d, pf) measured with 7 keV resolution (present work). Also shown is a fit of the data with rotational K π = 0+ bands. The 0+ , 2+ , 4+ picket fences show the rotational bands used in the fit. In the upper part of the 5.1 MeV resonance the fit (thin line), is based on the fitted positions of the rotational bands from Fig. 2d. (b), (c) The full dots represent the a2 (a4 ) – fission angular distribution coefficients determined in this measurement, while the open symbols are the a2 (a4 ) coefficients taken from Glässel et al. [10]. The thick full line results from the fit to spectrum (a). The thin lines correspond to theoretical a2 (a4 ) coefficients for different spin values j of the transferred neutron. (d) Proton spectrum of the reaction 239 Pu(d, pf) from Glässel et al. [10] measured at 3 keV resolution together with a fit of the spectrum using rotational bands as indicated. and a common rotational parameter was used. For the intensity ratio of the band members the value of Ref. [10] was accepted as starting value for the fit procedure. The fit nicely reproduced the experimental data with 13-rotational bands (see Fig. 2d), whose band-head positions, as fixed parameters, were also used to describe the same structure in the present work. Satisfactory description of the experimental data was achieved again by fitting the line width and the relative amplitudes of the bands, as it is shown in Fig. 2a with a grey curve. The 4.6 MeV group and the lower part of the 5.1 MeV group obtained in the present experiment was also fitted by the same procedure, however, the rotational parameters h̄2 /2Θ and the I2+ /I0+ and I4+ /I0+ intensity ratios could be separately determined for the bands with prominent band heads at E ∗ = 4434, 4526, 4625, 4685, 4703 and 4733 keV. In order to prove the spin and K-assignments of the observed compound levels in the second minimum we analyzed the corresponding fission fragment angular correlations with respect to the recoil axis, describing them in the usual way with coefficients a2 and a4 of Legendre polynomials. In Fig. 2 the measured proton coincidence spectrum (Fig. 2a) is shown together with the coefficients a2 (Fig. 2b) and a4 (Fig. 2c). For comparison also the spectrum of Glässel et al. [10] is shown in Fig. 2d together with their a2 and a4 coefficients marked by open circles in Fig. 2b and Fig. 2c. The horizontal lines in Fig. 2b and Fig. 2c show the theoretical a2 and a4 coefficients for different values of the angular momentum j of the neutron transferred M. Hunyadi et al. / Physics Letters B 505 (2001) 27–35 in the (d, p) reaction, leading to 240 Pu states with K = 0 and J = |1/2 ± j |. Due to the low ground state spin (Jiπ = 1/2+ ) of 239Pu, in 240 Pu only low total angular momenta J are populated. 0+ states can easily be identified in the fission fragment angular correlations through their isotropic emission characteristics. The large and positive a2 and a4 coefficients point to a dominant K = 0 character of both resonances. From 240 Pu(γ , f) data [25] a “suppression of the K π = 0− channel by more than two orders of magnitude relative to the K π = 0+ channel” was deduced [10]. For the lower 4.6 MeV resonance a K π = 0− contribution was discussed as well, however, as will be described below, a reasonable description of the fission probability could be achieved assuming a K π = 0+ channel [16– 18]. Thus it seems that a complete spectroscopy of K π = 0+ bands is possible in the regions of vibrational damping. In order to check this completeness and the consistency of the observed level density with the excitation energy in the second minimum, a statistical analysis of the level distances was performed using the band head energies. The statistical distribution of the ratio of experimental and calculated average level distances, using the back-shifted Fermi-gas formula [28], was generated. The shape of the resulting distribution was successfully approximated by a Wigner-type distribution, as it is expected for repelling states with the same angular momentum and parity. A similar analysis was performed in Ref. [10] and Ref. [12]. The χ 2 -value of the level density was minimized by varying the back-shift term of the Fermi-gas formula around the expected ground state energy of the second minimum. The best fit was obtained for an energy of EII = 2.25 ± 0.20 MeV (statistical uncertainty), which is in good agreement with the fission isomer energy obtained from the well-known method of extrapolated excitation functions of various experiments (see Ref. [2]). Next we discuss the spectroscopic results for the vibrational and rotational bands. An important aspect of the collectivity are multi-phonon states in the second minimum (Fig. 3). In the conversion electron measurements of Ref. [6] the phonon energy of the first β-vibrational excitation was determined as h̄ωβ = 769.9 keV. With respect to the excitation energy of the SD ground state (EII = 2.25 ± 0.20 MeV), determined by the statistical analysis of the level distances, the vibrational resonances centered around 4.6 and 31 5.1 MeV can be attributed to three and four β-phonon excitations, respectively. In Fig. 4 all presently known vibrational band heads in the second minimum of 240 Pu are shown. A reduced energy difference between the vibrational states of 0.5–0.6 MeV is expected at higher energies, because the potential well opens up at the top of the barrier. The observation of subsequent β-vibrations in transmission resonances can provide a unique possibility to study slightly anharmonic vibrational series at large nuclear deformations in a more convenient way than in the first minimum, where the high level density cause a complete damping of the β-vibrations. Individual dynamical moments of inertia, which reflect both the nuclear deformation and collective structures of the excitations, could be extracted for the first time in the high excitation energy region of the 4.6 MeV resonance group for the three well separated K = 0+ bands with band head energies of E ∗ = 4434, 4526 and 4625 keV. For the other rotational bands of this resonance group, as for the energy region above E ∗ ∼ 5 MeV only an average moment of inertia could be determined, respectively. The resulting moments of inertia are shown in Fig. 4 together with those for the SD ground state rotational band [9] and the low-lying collective excitations in the second minimum that were observed in recent γ -ray spectroscopy [6] and conversion electron experiments [7], respectively. The moments of inertia of the SD ground state band and the higher phonon bands show a surprising agreement, while larger variations arise for the low-lying excited SD bands. The moments of inertia of the SD ground state band can be reproduced by recent cranking model calculations yielding Θ/h̄2 = 155 MeV−1 [29]. One might have expected that the moments of inertia for the bands built on the third and fourth β-vibrational phonon might be closer to the rigid body moment of inertia because of the reduced pairing at higher excitation energies. However, this is not observed experimentally. Perhaps the selection of bands with strong β-vibrational components results in weaker non-collective contributions. While in the first minimum K π = 0+ vibrations show strong admixtures of pairing vibrations we expect much less admixtures in the second minimum as a significantly larger collectivity of β-vibrations in the second minimum was observed [7]. Therefore pairing vibrations and blocking could be reduced, which might explain 32 M. Hunyadi et al. / Physics Letters B 505 (2001) 27–35 Fig. 3. Vibrational excitations in the second minimum of 240 Pu. The low-lying collective band heads have been observed in Refs. [6] and [7]. Fig. 4. Moments of inertia of rotational bands in the second minimum of 240 Pu as a function of the excitation energy EII with respect to the ground state in the second minimum. Subscripts ‘e’ and ‘o’ denote moments of inertia that have been extracted for the even- and odd-spin members of the respective rotational band. M. Hunyadi et al. / Physics Letters B 505 (2001) 27–35 33 Table 1 Spin dependent properties of the two β-vibrational fission resonances in 240 Pu around E ∗ = 4.6 MeV and 5.1 MeV, respectively. αf (J π ) (αc (J π )) is the normalized relative fission (compound) cross section. Pf (J π ) represents the average spin dependent fission probability, while DI is the (calculated) level spacing for all K-values in the first minimum and DII the (experimentally determined) level spacing in the second minimum (K π = 0+ ). WcII is the width (FWHM) of the fission probability of class II compound levels. In the last two columns the γ -transmission coefficient Tγ I in the first well and the effective transmission coefficient Tf (J π ) are given (see text). Jπ αf (J π ) αc (J π ) 0.19(3) 0.024 Pf (J π ) DI (keV) (all K) DII (keV) (K π = 0+ ) WcII (keV) Tγ I (J π ) Tf (J π ) 15 8.3 × 10−4 3.0 × 10−3 3.3 × 10−3 Resonance: 5.1 MeV 0+ 0.79(12) 0.175 21 2+ 0.66(5) 0.14 0.47(3) 0.039 21 7.5 3.7 × 10−3 4+ 0.15(5) 0.08 0.19(6) 0.027 21 2.6 5.4 × 10−3 1.3 × 10−3 0+ 0.31(6) 0.015 0.042(5) 0.51 2.2 2.3 × 10−4 1.0 × 10−5 1.1 × 10−5 0.4 × 10−5 Resonance: 4.6 MeV 96 2+ 0.59(6) 0.12 0.0098(9) 0.11 96 0.6 1.1 × 10−3 4+ 0.10(5) 0.07 0.0029(13) 0.08 96 0.2 1.5 × 10−3 the observed preservation of the moment of inertia of the SD ground state band. 3.2. Spin dependent fission probabilities Aiming at a deduction and interpretation of the spin-dependent fission probabilities, we discuss in the following relevant properties, which are compiled in Table 1, analog to Ref. [10]. The relative fission cross sections αf (J π ) were obtained from the fit of the spectra in Fig. 2. In the maximum of the 5.1 MeV resonance we obtain (taking an average of the values from our measurement and that of Ref. [10]) a dominance of the 2+ states with αf = 0.66 ± 0.05, which nicely agrees with the value of 0.67 ± 0.06 obtained by Britt et al. [26]. The relative (d, p) compound cross sections αc (J π ) at Ed = 12.5 MeV were obtained in DWBAcalculations for deformed nuclei, where the final Nilsson orbitals were distributed over the compound nuclear levels by strength functions [10]. They agree quite well for the 5.1 MeV resonance with values of αc (J π ) calculated in Ref. [27] for Ed = 13.0 MeV. The excitation energy dependence of αc (J π ) for states in the 4.6 MeV resonance was taken from Ref. [27]. For the energy- and spin-averaged fission probabilities at the maxima of the two resonances values of Pf (5.1 MeV)max = 0.10 and Pf (4.6 MeV)max = 2.0 × 10−3 could be deduced in Ref. [17]. While the upper resonance could be well reproduced in a theoretical description assuming K π = 0+ , for the lower resonance also a contribution from a 0− resonance was discussed. However, a satisfactory description of this resonance could also be achieved with pure K π = 0+ characteristics, assuming a fragmentation of the third β phonon over states with energies of 4.2 MeV (30%), 4.5 MeV (45%) and 4.7 MeV (25%). Using Pf , the average spin dependent fission probability Pf (J π ) can be calculated by Pf (J π ) = Pf · αf (J π )/αc (J π ). The level spacings DI given in the fifth column of Table 1 are calculated with the standard back-shifted Fermi gas formula [28]. However, for the level spacing in the second minimum, DII , the restriction to K π = 0+ bands, in contrast to earlier publications [9,10], causes 0+ , 2+ and 4+ levels to have the same level density. The numbers given in Table 1 are experimentally determined by the fit procedure described earlier. For the FWHM WcII of the fission probability of class II compound levels the relation 2Pf (J π )2 DII · arcos 1 − WcII = π 1 − Pf (J π )2 is deduced in Ref. [10] under the assumption that ΓcII DII . As long as Pf (J π ) is smaller than 34 M. Hunyadi et al. / Physics Letters B 505 (2001) 27–35 √ (1/2) this width WcII can be determined. For larger values of Pf (J π ) the class II levels overlap too strongly. We used this relation to calculate the WcII widths given in Table 1. We cannot calculate a WcII value for the 0+ levels in the centre of the 5.1 MeV resonance, however, within the experimental error of Pf (J π ) a value of 15 keV is estimated. Therefore, a fit with K π = 0+ bands in the upper resonance still appears reasonable. The energy and spin dependence of the averaged fission probability Pf (J π ) can be explained by: (i) a rather complete K-mixing in the first well, (ii) a K-conservation in the second well and (iii) the competition between γ -decay in the first well and fission through the double-humped barrier. The processes can be described by a width Γ or by a transmission coefficient T , which are connected by: T = (2π/D) · Γ , where D is the level spacing of states of the appropriate spin and parity. The γ -“transmission coeffiπ cient” Tγ I (J π ) = (2π/DI (J π )) · ΓγJ increases with π spin and excitation energy, while the decay width ΓγJ is ≈ 23 meV at 5 MeV [10] and changes little with excitation energy (10 meV/MeV [14]), spin J , spin projection K and parity [14]. The double-humped fission barrier predominantly selects K π = 0+ states for transmission resonances out of all possible K-values for a given J -value in the first well. For the lower 4.6 MeV resonance the K-components with K = 0 decay predominantly by γ -decay and the decrease of Pf (J π ) for K π = 0+ states with increasing spin J is approximately proportional to 1/(2J + 1). For the upper 5.1 MeV resonance the decay by fission of K π = 0+ states is faster than the γ -decay (or comparable). Thus the spread of Pf (J π ) with spin is reduced at the upper resonance, explaining also the smooth increase of the I2+ /I0+ ratio with excitation energy. When comparing measured fission probabilities with theory, the averaging over unresolved class I states causes some complications because, e.g., for the averaged fission probability one first has to calculate the probability for individual class I states before averaging: Pf (J π ) = Γf I (J π )/(Γf I (J π ) + Γγ I (J π )) = (F · Γf I )/(Γf I + Γγ I ). Here F is a fluctuation factor, which may be smaller than 1 by up to 30% [14, 24]. In Ref. [11] a formula for the averaged fission probability Pf max at the center of a β-vibrational resonance was deduced under the additionally fulfilled conditions (DI Γγ I and DII > WcII ): h̄ωβ Pf max = π · ΓW PA PB . Tγ I Here PA and PB are the penetrabilities of the inner barrier (EA = 5.8 MeV, h̄ωA = 0.82 MeV) and the outer barrier (EB = 5.45 MeV, h̄ωB = 0.6 MeV) [10]; h̄ωβ = 0.77 MeV is the β-phonon energy in the second minimum; ΓW = γ (1 − Pf ) is the damping width into class II compound states; γ (∼ 200 keV) is the observed resonance width and Tγ I is the γ transmission coefficient. The obtained √ dependence of Pf on the barrier penetrabilities ∝ PA · PB differs from ∝ PA · PB /4(PA + PB )2 for an undamped resonance [23] and from ∝ PA · PB /(PA + PB ), when averaging over the fission width and the γ -width separately [24]. The experimental fission probabilities Pf (J π ) are nicely reproduced by the theoretical value of Pf max for resonance energies of 5.1 MeV and 4.5 MeV. Experimentally it is difficult to determine the exact positions of the β-vibrations, because of centroid shifts introduced by weighting the measured resonance intensities with the steep slope of the barrier penetrabilities. Although the resonance centroid obtained by averaging over the measured intensity results in E ∗ = 4.6 MeV, an improved value of 4.5 MeV can be obtained by taking into account the fragmentation of the resonance strength and the weighting with the absolute fission probability. The fission probabilities Pf (J π ) are strongly spin dependent, but the effective transmission Tf (J π ) = Pf (J π ) · Tγ I (J π )/(1 − Pf (J π )) through the barriers turns out to be rather spin independent. This is expected, because the barrier heights, to first order, vary little for different J members of a rotational band with K π = 0+ . Also the observed reduction of the transmission Tf for spin 4 compared to spin 0 by a factor of ≈ 2.4 for both resonances can be explained, using the Hill–Wheeler formula [2] for the energy dependent penetration probabilities PA and PB . For the 4+ states in the first minimum about 142 keV are bound in rotational motion. Taking into account the change of the moments of inertia for the two barriers and the curthe parabolic barriers one vatures h̄ωA and h̄ωB for √ obtains for the change of (PA · PB ) the same factor of 2.4. This introduction of an effective transmission coefficient Tf is reasonable for Tf Tγ I 1, even when averaging over class I states, because Pf = Tf /(Tf + Tγ I ) Tf /Tγ I Tf /Tγ I . M. Hunyadi et al. / Physics Letters B 505 (2001) 27–35 At the top of the barrier, for overlapping class II states Tf /(Tf + Tγ I ) and Tf /(Tf + Tγ I ) lead to the same results [11]. However, even in the transition region the effective transmission Tf shows a remarkable scaling behaviour in Table 1, where the values for the upper resonance are obtained from the lower resonance by multiplying with a factor of 300. Tf seems to be a very useful quantity and the fluctuation factor F [24] seems to show only small variations. Summarizing, we extended the detailed spectroscopic information obtained in conversion electron and γ spectroscopy for the lowest phonons to the third and fourth β-vibrational phonon by transmission resonance spectroscopy. It will be challenging to reach the intermediate levels between 1.5 and 2.0 MeV in the second minimum at the limit of both methods, because then common levels could be observed and the scaling laws for the fission probability and the barrier penetrabilities could be tested independently. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] Acknowledgement [20] [21] This Letter is dedicated to the 65th birthday of Prof. H.-J. Specht. We acknowledge critical reading of the manuscript by Prof. D. Schwalm and Prof. S.Y. van der Werf. The work has been supported by DFG under HA 1101/6-1 and 436 UNG 113/129/0 the Nederlandse Organisatie voor Wetenschapelijk Onderzoek (NWO), the Hungarian Academy of Sciences under No. 119 and the Hungarian OTKA Foundation No. T23163 and N26675. 35 [22] [23] [24] [25] [26] [27] [28] [29] V.M. Strutinsky, Nucl. Phys. A 95 (1967) 420. C. Wagemans, The Nuclear Fission Process, 1991, CRC Press. S.B. Bjørnholm, J.E. Lynn, Rev. Mod. Phys. 52 (1980) 725. H.J. Specht et al., Phys. Lett. 41B (1972) 43. V. Metag et al., Phys. Rep. 65 (1980) 1. D. Pansegrau et al., Phys. Lett. B 484 (2000) 1. D. Gaßmann et al., Phys. Lett. B 497 (2001) 181. H.J. Specht et al., in: Proc. Symp. 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Ignatyuk et al., Phys. Lett. 29B (1969) 209. J.P. Bondorf, Phys. Lett. 31B (1970) 1. N.S. Rabotnov et al., Sov. J. Nucl. Phys. 11 (1970) 285. H.C. Britt et al., Phys. Rev. 175 (1968) 1525. B.L. Andersen et al., Nucl. Phys. A 147 (1970) 33. T. Rauscher, F.K. Thielemann, K.L. Kratz, Phys. Rev. C 56 (1995) 185. A.V. Afanasjev, P. Ring, in preparation. 26 April 2001 Physics Letters B 505 (2001) 36–42 www.elsevier.nl/locate/npe Production asymmetry of D mesons in γp collisions ✩ G. Herrera a , A. Sánchez-Hernández a , E. Cuautle b , J. Magnin b a Centro de Investigación y de Estudios Avanzados, Apdo. Postal 14 740, México 07000, D.F., Mexico b Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud 150, 22290-180, Rio de Janeiro, Brazil Received 15 November 2000; accepted 26 February 2001 Editor: K. Winter Abstract We study the production asymmetry of charm versus anticharm mesons in photon–proton interactions. We consider photon– gluon fusion plus higher order corrections in which light quarks through vector meson–proton interactions contribute to the cross section. Nonperturbative effects are included in terms of a recombination mechanism which gives rise to a production asymmetry. 2001 Published by Elsevier Science B.V. 1. Introduction In high energy photon–hadron interactions, charm production is expected to be dominantly produced by photon–gluon fusion processes. According with QCD perturbative calculations, this mechanism produces in equal amounts charm and anticharm. However, recent measurements of charm meson production [1] indicate that there are important nonperturbative QCD phenomena in the production process that induce an asymmetry in charm and anticharm production. This phenomena has been observed in hadron– hadron collisions [2] and is well known as “leading effect”. It has been the subject of many models of particle production and several mechanisms have been proposed to explain it [1,3,4]. Here we study the xF distribution of D ± and D 0 mesons produced in photon–proton collisions in the framework of a ✩ This work was supported by CLAF (Brazil) and CONACyT (México), CIEA-JIRA (México). E-mail address: [email protected] (G. Herrera). two-components model that has been used before to successfully describe the asymmetry in pion proton interactions [4]. The production of D mesons in the model is assumed to take place via two different processes, namely QCD parton fusion with the subsequent fragmentation of quarks in the final state and conventional recombination of valence and sea quarks present in a vector meson fluctuation of the photon. The asymmetries obtained with the conventional soft charm component as well as with a hard charm component in the photon, are presented. We compare both with the experimental data available. To quantify the difference in the production of charm and anticharm mesons an asymmetry A is defined as in [1], A(xF , pt ) = Nc − Nc̄ , Nc + Nc̄ (1) where Nc and Nc̄ are the production yields. The asymmetry has been observed to be a function of both xF and the transverse momentum pt . This Letter is organized as follows. The photon gluon mechanism for charm production will be dis- 0370-2693/01/$ – see front matter 2001 Published by Elsevier Science B.V. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 3 3 0 - 6 G. Herrera et al. / Physics Letters B 505 (2001) 36–42 cussed in Section 2. This mechanism is not responsible of a production asymmetry between charm and anticharm. In Section 3 we discuss the contribution of the resolved photon in the frame of a Vector Dominance Model (VDM) component. In Subsection 3.1 we calculate the QCD interaction of the resolved photon. In Subsection 3.2, a recombination mechanism is discussed. While the QCD production mechanisms in photoproduction are the same for charm and anticharm production, recombination favors the formation of D − over D + and D̄ 0 over D 0 . In Section 4 the various components are put together and the asymmetry is estimated. Finally some conclusions are drawn in Section 5. 37 (a) 2. The photon–gluon fusion mechanism In this section we outline the calculation of the photon–gluon fusion contribution at Leading Order (LO) in pQCD to the D-meson inclusive xF distribution. The processes involved in the photoproduction of charm at LO in pQCD are depicted in Fig. 1. The corresponding formula in the parton model is Ec Ec̄ dσ d 3 pc d 3 pc̄ 1 = 0 g(x, Q2 ) where proton and Ec Ec̄ = (b) Fig. 1. According with perturbative QCD, photon–gluon fusion is the main process in charm photo-production. and dx g x, Q2 Ec Ec̄ d σ̂ d 3 pc d 3 pc̄ , (2) is the gluon probability density in the d σ̂ d 3 pc d 3 pc̄ 1 αe αs (Q2 ) (2π)4 δ(pγ + pg − pc − pc̄ )|M|2 . 2ŝ 4(2π)6 (3) The squared invariant matrix |M|2 in terms of the Mandelstam variables is given by (see, e.g., [5,6]) 8 1 m2c − tˆ m2c − û 2 |M| = + 2 9 2 m2c − û mc − tˆ 2 mc m2c +2 + m2c − tˆ m2c − û 2 m2c m2c (4) −2 + m2c − tˆ m2c − û ŝ = xs, √ −yc se , √ yc 2 û = mc − xmT s e , tˆ = m2c − mT (5) where s is the c.m. energy of the γ –p system, m2T = m2c + pT2 , x is the momentum fraction of the gluon inside the proton and yc = (1/2) ln[(Ec − pc )/(Ec + pc )] the rapidity of the c-quark. After integrating out Eq. (2) on the c̄-quark variables and the momentum fraction x, the inclusive cross section for the production of charm (anticharm) is given by √ d σ̂ dσ s xg(x, Q2 ) = (6) dpT2 √ y c dxF 2 E( s − mT e ) d tˆ with παe αs (Q2 ) d σ̂ = |M|2 , ŝ 2 d tˆ mT e−yc . x=√ s − mT e y c (7) 38 G. Herrera et al. / Physics Letters B 505 (2001) 36–42 In Eq. (7), αs is given by αs Q2 = an antiquark differs from the cross section for the production of a quark, but this effect is small [8]. 12π (8) 2 (33 − 2Nf ) log Q Λ2 with Nf = 4 and Λ2 = Λ24 according to the gluon distribution used in Eq. (6). In our calculations we use the GRV-LO gluon distribution [7] with Q2 = 4m2c , mc = 1.5 GeV. The D-meson inclusive xF distribution including fragmentation is given by dz dσc(c̄) xF dσD DD/c (z), z = . = (9) dxF z dx x 3. Light quark corrections to charm photoproduction We will identify the hadron like part of the photon with ρ and ω vector mesons neglecting the contribution of heavier resonances which have smaller couplings to the photon. The ρ and the ω can be regarded as two states systems 1 ρ 0 = √ (uū + d d̄), 2 1 ω = √ (uū − d d̄). 2 We use the Peterson fragmentation function, DD/c (z) = N z[1 − 1 z − ! 2 1−z ] . (10) (11) (12) The photoproduction of D mesons may take place by QCD interaction of the vector meson V with the proton and/or by recombination of its constituent quarks (see Fig. 2). Therefore, in obtaining the differential cross section of the process Vp → DX, we will con- Inclusion of Next to Leading Order (NLO) contributions into the D-meson cross section does not produce appreciable changes neither in the form of the D-meson distribution or in its normalization (see, e.g., [8]). At NLO the cross section for the production of (a) (b) (c) Fig. 2. The resolved photon may interact with the proton via QCD, i.e., quark–antiquark annihilation and gluon–gluon fusion as shown in (a), (b). A nonperturbative interaction (c) may favor charm over anticharm mesons production. G. Herrera et al. / Physics Letters B 505 (2001) 36–42 sider two possible processes, (13) The photon may interact through its constituents with the partons in the proton. In the parton fusion mechanism, D ± D 0 (D̄ 0 ) mesons could be produced via the q q̄(gg) → cc̄ with the subsequent fragmentation of the c(c̄) quark. The contribution of the hadronic (or resolved) component of the photon is given by the usual formula that describes hadron–hadron interactions, qcd dσVDM dxF √ s = dpT2 dy4 2 (14) where x1 fi (x1 , µ2 ) is the parton distribution in the resolved photon, x2 fi (x2 , µ2 ) is the parton distribution in the proton, E is the energy of the fragmenting charm quark and DD/c (z) is the fragmentation function. The partonic cross section in Eq. (14) is given by d σ̂ παs2 (µ) 2 2 (15) | + | = |M |M i,j q q̄ i,j gg ŝ 2 d tˆ with the invariant matrix elements squared and averaged (summed) over initial (final) colours and spins at LO in pQCD given by |Mi,j |2 = 4 (tˆ − m2c )2 + (û − m2c )2 + 2m2c ŝ , 9 ŝ 2 2 m2c (ŝ − 4m2c ) 3 (m2c − tˆ )(m2c − û) (m2c − tˆ )(m2c − û) + m2c (û − tˆ ) ŝ(m2c − tˆ2 ) (m2 − tˆ )(m2c − û) + m2c (tˆ − û) . −6 c ŝ(m2c − û2 ) (16) Writing the four momentum of the incoming and outgoing particles as √ S p1 = (x1 , 0, 0, x1), 2 √ S (x2 , 0, 0, −x2), p2 = 2 pc = mT cosh(yc ), pT , 0, mT sinh(yc ) , pc̄ = mT cosh(yc̄ ), −pT , 0, mT sinh(yc̄ ) , (17) the Mandelstam variables appearing in Eqs. (16) are given by i,j q q̄→cc̄ − −6 3.1. QCD resolved photon contribution x1 fi (x1 , µ)x2 fj (x2 , µ) d σ̂ DD/c (z) , E z d tˆ (m2c − tˆ )(m2c − û) ŝ 2 2 8 (mc − tˆ )(m2c − û) − 2m2c (m2c + tˆ ) + 3 (m2c − tˆ )2 8 (m2c − tˆ )(m2c − û) − 2m2c (m2c + û) + 3 (m2c − û)2 = 12 In order to calculate these two contributions to the total cross section, we assume that the momenta distribution of the quarks of a ρ, ω meson are the same than inside a pion. We will use the GRV parametrization for the parton distribution in the pion. × |Mi,j |2 gg→cc̄ qcd dσ dσ (Vp → DX) = VDM (Vp → DX) dxF dxF dσ rec + VDM (Vp → DX). dxF 39 ŝ = 2m2T (1 + cosh(+y)), tˆ = m2c − m2T (1 + exp(−+y)), û = m2c − m2T (1 + exp(+y)), +y = yc − yc̄ . (18) In our calculation we use the GRV-LO [7] parton distributions in protons and pions, and apply a global factor of 2–3 in order to account for NLO effects. For the fragmentation function we use the Peterson function. 3.2. Charmed meson production by recombination In the scenario described in [9] for π − proton collisions, the annihilation of a u quark from the proton and the ū quark in the pion would liberate the d of the pion which in turn recombines to form a D − 40 G. Herrera et al. / Physics Letters B 505 (2001) 36–42 with Fq (xi ) = xi q(xi ). We use the GRV-LO parametrization for the single quark distributions in Eq. (20). It must be noted that since the GRV-LO [7] distributions are functions of x and Q2 , our F2 (x1 , x2 ) also depends on Q2 . The recombination function is given by R2 (xu,d , xc̄ ) = α Fig. 3. After a fluctuation of the photon to a ρ 0 vector meson, the interaction with the proton may occur in one of the two states. In any case the valence quark in the ρ would be released once the antiquark annihilates with a quark of the proton. and certainly not a D + . On a similar base a ρ 0 proton collision will favor the production of D̄ 0 and D − over D 0 and D + depending on the quantum state of the colliding ρ 0 at the interaction point (see Fig. 3). The production of leading mesons at low pT was described by recombination of quarks long time ago [10]. In recombination models one assumes that the outgoing hadron is produced in the beam fragmentation region through the recombination of the maximum number of valence quarks and the minimum number of sea quarks of the incoming hadron. The invariant inclusive xF distribution for leading mesons is given by 2E dσ rec √ = s σ dxF xF dx1 dx2 F2 (x1 , x2 )R2 (x1 , x2 , xF ), x1 x2 0 (19) where x1 , x2 are the momentum fractions and F2 (x1 , x2 ) is the two-quark distribution function of the incident hadron. R2 (x1 , x2 , xF ) is the two-quark recombination function. The two-quark distribution function is parametrized in terms of the single quark distributions. For recombination of D − , D 0 F2 (x1 , x2 ) = βFd,u;val(x1 )Fc̄;sea (x2 )(1 − x1 − x2 ), (20) xu,d xc̄ xF2 δ(xu,d + xc̄ − xF ), (21) 1 with α fixed by the condition 0 dxF (1/σ ) dσ rec / dxF = 1. Some time ago Barger et al. [11] explained the spectrum enhancement at high xF in Λc production assuming a hard momentum distribution of charm in the proton. Here we will also take a QCD evolved charm distribution, of the form proposed by Barger et al. [11] xc x, Q2 = Nx l (1 − x)k , (22) with a normalization N fixed to dx · xc(x) = 0.005 (23) and l = k = 1. With this values for l and k one tries to resemble the distribution of valence quarks. In contrast with the parton fusion calculation, in which the scale Q2 of the interaction is fixed at the vertices of the appropriated Feynman diagrams, in recombination the value of the parameter Q2 should be used to give adequately the content of the recombining quarks in the initial hadron. We used Q2 = 4m2c . 4. D ± and D 0 (D̄ 0 ) total production The inclusive production cross section of a D meson is then obtained by adding the contribution of recombination, Eq. (19), to the QCD processes from direct photon–gluon interaction, quark–antiquark annihilation and gluon–gluon fusion from the hadronic component of the photon, i.e., dσ tot (D − ) dxF γg qcd dσ dσ rec dσ , = N1 + a b VDM + c VDM dxF dxF dxF (24) G. Herrera et al. / Physics Letters B 505 (2001) 36–42 Fig. 4. Total cross section as a function of xF . Experimental results from [13] and theoretical calculation as in Eqs. (24) and (25) using the GRV distributions. dσ tot (D + ) dxF γg qcd dσ dσ rec dσ = N2 + a b VDM + d VDM dxF dxF dxF Fig. 5. Measured production asymmetry for D − and D + from [1]. The curves show the model result in which a hard charm component (dashed line) and GRV-LO (solid line) in the pion has been considered. The horizontal line at A(xF ) = 0 is for reference only. (25) with qcd dσVDM dσ gg dσ q q̄ = + dxF dxF dxF 41 (26) 1 1 and N1 = 1+ab+ac , N2 = 1+ab+ad the parameters a, b, c and d depend on the contribution of each process to the total cross section. They are fixed in such a way that the differential cross section is well described, before calculating the asymmetry. The resulting inclusive D production cross section dσ tot /dxF (shown in Fig. 4), is used then to construct the asymmetry defined in Eq. (1). The values obtained for the different contributions in Eqs. (24) and (25) are in reasonable agreement with what one would expect. The photon fluctuation to a vector meson is of the order of 1%. Approximately 96% of the total cross section comes from the photon– gluon process. The contribution due to recombination goes from about 1% (for D − ) to about 3% (for D + ). Fig. 5 shows the model prediction for the D − , D + production asymmetry together with the experimental results from the E687 Collaboration [1]. The two curves correspond to the conventional GRV function distribution in the resolved photon and to the distribution proposed by Barger et al. where a hard charm component has been assumed. 5. Conclusions In an earlier work [12] the production asymmetry of Λc was described using the same recombination scheme used here. In hadroproduction the presence of a diquark in the initial state, plays an important role in Λc production. In photo production, however, the production mechanism is somewhat different and the asymmetry is much smaller. The parameters used here are in reasonable agreement with what is physically expected and with the values used in a previous study of production asymmetries in π proton collisions [4]. Changing the values of these parameters may improve the description of the asymmetry but then they may loose meaning in the frame of the asymmetry obtained for hadroproduction. 42 G. Herrera et al. / Physics Letters B 505 (2001) 36–42 Other experimental results for the asymmetry defined as: A(xF ) = σ (D + ) − σ (D − ) σ (D + ) + σ (D − ) (27) are A = −0.0384 ± 0.0096 in xF 0.0 at photon energies of 200 GeV [1] and A = −0.0196 ± 0.0147 in xF 0.2 at photon energies of 80–230 GeV, average energy 145 GeV [13]. The NA14 collaboration studied the asymmetry A(xF ) = σ (D + + D 0 ) − σ (D − + D̄ 0 ) σ (D + + D 0 ) + σ (D − + D̄ 0 ) (28) and obtained A = −0.03 ± 0.05 in xF 0.0 at photon energies of 40–140 GeV and 100 GeV in average [14]. All these results are in good agreement with each other but, the statistical errors are still large. In Ref. [1] experimental data are compared to a model based on string fragmentation. This model gives a larger asymmetry than the one obtained in our approach. The description obtained there is in much better agreement with the experimental results. However, more precise measurements are needed before one can draw a final answer. One would expect that with increasing energy the resolved photon component increases giving rise to a larger contribution of the recombination mechanism which in turn would produce a larger production asymmetry. The HERA experiments should therefore be able to see a production asymmetry. However larger energies of the photon means also smaller values of x for the quarks that participate in the interaction and the hard charm component is expected to play a minor role at small x’s, i.e., the asymmetry should look rather as the one obtained from conventional densities. New experiments will have soon more precise measurements of the asymmetries. This new results will give us a better understanding of the underlying production mechanisms. References [1] E687 Collaboration, P.L. Frabetti et al., Phys. Lett. B 377 (1996) 222. [2] E769 Collaboration, G.A. Alves et al., Phys. Rev. Lett. 77 (1996) 2388; E769 Collaboration, G.A. Alves et al., Phys. Rev. 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C 2 (1998) 477. [13] E691 Collaboration, J.C. Anjos et al., Phys. Rev. Lett. 62 (1989) 513. [14] NA14 Collaboration, M.P. Adamovich et al., Z. Phys. C 60 (1993) 53. 26 April 2001 Physics Letters B 505 (2001) 43–46 www.elsevier.nl/locate/npe Erraticity analysis of pseudorapidity gaps of the NA27 data Wang Shaoshun, Wu Chong Department of Modern Physics, University of Science and Technology of China, Hefei, 230027, PR China Received 20 October 2000; received in revised form 18 December 2000; accepted 18 December 2000 Editor: L. Montanet Abstract The erraticity analysis of pseudorapidity gaps is performed for the data of 400 GeV/c pp collisions. The entropy-like quantities Sq and Σq proposed by R.C. Hwa et al. have been calculated. It is found that Sq and Σq deviate from 1 significantly. The ln Sq versus q has a quite linear behavior, but the ln Σq versus q has only an approximate linear behavior. The same calculations are performed for a Monte-Carlo event sample simulated using the FRITIOF program. It is found that the deviations from the experimental data are rather large. 2001 Elsevier Science B.V. All rights reserved. PACS: 13.85.Hd; 05.45.+b; 12.40.Ee 1. Introduction To study the properties of event-to-event fluctuations of multiparticle production in high energy collisions, one uses the horizontally normalized factorial moments Fq (M) = fq (M)/(f1 (M))q to characterize the spatial pattern of an event [1,2]. The definition of fq (M) is fq (M) = n(n − 1) · · · (n − q + 1) , (1) where n is the number of particles in a bin. M is the number of bins. The average in Eq. (1) is performed over all bins for a fixed event. This method cannot convey all the details of an event, because it is evident from Eq. (1) that only bins with n q can contribute to fq (M), and the positions of the contributing bins have no effect on fq (M). This means that this method is only sensitive to the local fluctuations of the rapidity distribution in an event, but not to the spatial arrangement in rapidity. Especially, E-mail address: [email protected] (W. Shaoshun). when the event multiplicity N is low and the number of bins is high, only a few events have n q, so the statistical fluctuation may be very large and very little information can be obtained [3]. In order to get more information, a new method based on measuring the rapidity gaps has been proposed by R.C. Hwa and Q. Zhang [4]. It seems clear that complementary information accompanying rapidity spikes is provided by rapidity gaps. When N is low, the rapidity gaps can give more information then rapidity spikes. In this paper, an erraticity analysis based on measuring the pseudorapidity gaps has been performed for the data on 400 GeV/c pp collisions in order to check whether the method is useful. 2. The method The details of the method can be found in [4], so we describe it only briefly. The single-particle density distribution in pseudorapidity space is nonflat. In order to reduce the effect of the non-flat 0370-2693/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 0 1 8 - 1 44 W. Shaoshun, W. Chong / Physics Letters B 505 (2001) 43–46 particle density distribution ρ(η), one can use a new cumulative variable X(η) defined by [5,6] η η1 ρ(η ) dη X(η) = ηmax (2) , ηmin ρ(η ) dη where ηmax and ηmin are the extreme points in the distribution ρ(η). In X(η) space, the single-particle density is uniformly distributed from 0 to 1. Consider an event with N particles, labeled by i = 1, 2, . . . , N located in X(η) space at Xi ordered from small to large. The distances between neighboring particles are defined as xi = Xi+1 − Xi , i = 0, . . . , N, (3) with X0 = 0 and XN+1 = 1. Evidently, the xi are the pseudorapidity gaps which satisfy N where the calculation method of Gst q for the MonteCarlo event sample is the same as for Gq . The MonteCarlo event sample has the same number of events and the same multiplicity distribution as the experimental data, but no correlations. The particles are randomly distributed in X(η) space with equal probability. Then we calculate the ratio sq Sq = st , (9) sq and how much degree of Sq deviates from 1 is a measure of erraticity in multiparticle production based on pseudorapidity gaps. Another type of moments which can characterize the pseudorapidity gap distributions is defined as follows: xi = 1. (4) i=0 For each event, we define a new moment 1 q xi , N +1 N Gq = (5) i=0 1 where G0 = 1 and G1 = N+1 . Since Gq fluctuates from event to event, there is a distribution P (Gq ) of Gq for many events. The shape of P (Gq ) characterizes the nature of the event-to-event fluctuations of the gap distribution. In order to quantify the degree of that fluctuation, a new moment was defined in [4] Cp,q = Ne e p 1 p Gq = Gq P (Gq ) dGq , Ne (6) e=1 where e labels an event and Ne is the total number of events. Since the derivative of Cp,q at p = 1 conveys the broadest information on P (Gq ) [4], we shall focus on that derivative. So we have d = −Gq ln Gq , sq = − Cp,q (7) dp p=1 where · · · denotes an average over all events. Since the moment Gq does not filter out statistical fluctuations, we shall estimate the contributions from statistical fluctuations by first calculating st sqst = − Gst (8) q ln Gq , 1 (1 − xi )−q , N +1 N Hq = (10) i=0 where N is the number of particles for an event, xi is as given in Eq. (3). Also these moments receive a dominant contribution from large xi , as do the Gq , but Hq can become 1. Similarly to Gq , the entropy-like measure is defined as σq = −Hq ln Hq , (11) and comparing this with the statistical-only contribution σqst = − Hqst ln Hqst , (12) we obtain the ratio σq Σq = st , σq (13) where Σq is the second erraticity measure based on pseudorapidity gaps. 3. The experimental results In the present investigation, the angular distribution of charged particles produced in pp collisions at 400 GeV/c was measured by using the LEBC films offered by the CERN NA27 collaboration. Details of the measurement are given in [7]. A total of 3950 non-single-diffractive events (N 4) were measured. Among these, there are 3677 events with N 6 and 3050 events with N 8. The accuracy of the W. Shaoshun, W. Chong / Physics Letters B 505 (2001) 43–46 Fig. 1. (a) ln Sq vs. q; (b) ln Sq vs. ln q. Experimental results: ◦ N 4; • N 6; N 8; FRITIOF simulated results: · − · − · − N 4; · · · · · · N 6; − − − − N 8. The solid lines are the linear fits to the experimental data. Table 1 The fit parameters obtained according to (14) and (15) Event sample α χ 2 /NDF α1 χ 2 /NDF N 4 0.237 ± 0.003 0.56 1.2 ± 0.1 10.8 N 6 0.223 ± 0.003 0.66 1.10 ± 0.11 12.4 N 8 0.202 ± 0.002 0.39 1.00 ± 0.09 9.63 pseudorapidity in the region of interest (−2 η 2) is of the order of 0.1 pseudorapidity units. Firstly, we use the cumulative variable X(η) instead of η. The definition of X(η) is given in Eq. (2). According to Eq. (2), the cumulative variations X(η) and X (η) (X (η) for a purely statistical situation, ηmax = 5 and ηmin = −5) have been calculated for N 4, N 6 and N 8 event samples, respectively. The moments Sq are calculated according to Eqs. (3)– (9). The results obtained are shown in Fig. 1(a). It can be seen from Fig. 1(a) that the entropy-like quantities Sq deviate from 1 significantly and that the ln Sq versus q has a quite linear behavior. This means that Sq satisfies the following relationship Sq ∝ eαq . (14) The straight lines are the linear fit to the experimental data. The fit parameters are listed in Table 1. This result is different from the result obtained by Hwa and Zhang. They claim a power law behavior Sq ∝ q α1 . (15) 45 Fig. 2. Normalized multiplicity distributions. • experimental results; solid line for FRITIOF simulated result. In order to see whether the experimental data has power law behavior, the ln Sq versus ln q is plotted in Fig. 1(b). The linear fits to the experimental data are performed. The fit parameters are listed in Table 1. It can be seen from Fig. 1 and Table 1 that Sq has an exponential behavior in q for q 2, the values of Sq and α decrease with increasing cut off at the low part of multiplicity distribution, but the values of α change rather little. In order to compare with the experimental data, we used a Monte-Carlo generator FRITIOF version 7.02 and JETSET 7.3 [8] to simulate the multiparticle production in 400 GeV/c pp collisions. A total of 4500 non-single-diffractive events (N 4) have been created. Among these, there are 3960 events with N 6 and 3296 events with N 8. The normalized multiplicity distribution is plotted in Fig. 2 with solid curve. The experimental data are also shown in Fig. 2 as solid points. It can be seen from Fig. 2 that both are in agreement except for the N = 4 point where the Monte-Carlo event sample is larger than experimental data. The same calculations have been performed for the Monte-Carlo event sample. The results are shown in Fig. 1 as solid curves. We can see that there is no scaling behavior for the Monte-Carlo event sample, although the values of Sq deviate from 1 sufficiently. This means that the FRITIOF version 7.02 cannot reproduce the erraticity behavior of the multiparticle production in hadron–hadron collisions. We calculate the Σq moment for the experimental data and Monte-Carlo event samples. The results are shown in Fig. 3. For the experimental data, the entropy-like quantities Σq deviate from 1 significantly, but ln Σq versus q only has an approximately 46 W. Shaoshun, W. Chong / Physics Letters B 505 (2001) 43–46 Fig. 3. ln Σq vs. q. Experimental results: • N 6; ◦ N 8; FRITIOF simulated results: · − · − · − N 6; · · · · · · N 8. The solid lines are the linear fits to the experimental data. Acknowledgements linear behavior Σq ∝ eβq be seen from our early paper [9]. In that paper we have presented the pseudorapidity distributions for fixed multiplicities. The pseudorapidity distributions are different among different multiplicities which cannot be caused purely by statistical fluctuations. The interesting things are that ln Sq and ln Σq versus q both have linear behavior: ln Sq ∝ αq and ln Σq ∝ βq. It is found that α and β have different behavior when the cut off at the low part of the multiplicity distribution is increased. Whereas α has changed little, β has decreased significantly. The variation of β makes it a more sensitive measure of erraticity, but the stability of α may nevertheless be interesting and useful. The same calculations are performed for a Monte-Carlo event sample simulated using the FRITIOF program. It is found that the deviations from experimental data are large. (16) when q 3. The fit parameters β have significantly decreased with increasing cut off at the low part of the multiplicity distribution. For the N 6 event sample, β = 0.93 ± 0.03; for the N 8 event sample, β = 0.51 ± 0.04. For the Monte-Carlo event sample, ln Σq versus q has no linear behavior. The values of Σq are much smaller than for the experimental data. 4. Conclusion In this paper, the erraticity analysis was performed for the data of 400 GeV/c pp collisions in terms of pseudorapidity gaps. The entropy-like quantities Sq and Σq have been calculated. The fact that Sq and Σq deviate unambiguously from 1 implies that both of them are useful to serve as effective measures of erraticity in multiparticle production. The fact that the fluctuation from event-to-event is large can also We are grateful to the CERN NA27 Collaboration for offering the LEBC films. Project 19975045 supported by the National Natural Science Foundation of China. References [1] Z. Cao, R.C. Hwa, Phys. Rev. D 61 (2000) 074011. [2] W. Shaoshun et al., Phys. Lett. B 458 (1999) 505. [3] J. Fu et al., Phys. Lett. B 472 (2000) 161; L. Lianshou et al., Science in China A 43 (2000) 1215. [4] R.C. Hwa, Q.-H. Zhang, Phys. Rev. D 62 (2000) 0140003. [5] A. Bialas, M. Gazdzicki, Phys. Lett. B 252 (1990) 483. [6] W. Ochs, Z. Phys. C 50 (1991) 339. [7] W. Shaoshun et al., Z. Phys. C 68 (1995) 415. [8] H. Pi, Lund Monte-Carlo for hadron–hadron, hadron–nucleus and nucleus–nucleus collisions, Lund University, Program Draft Code 2931114, 1993; B. Andersson, G. Gustafson, H. Pi, Z. 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Hu ad , I. Iashvili au , B.N. Jin h , L.W. Jones d , P. de Jong c , I. Josa-Mutuberría y , 0370-2693/01/$ – see front matter 2001 Published by Elsevier Science B.V. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 3 0 8 - 2 48 L3 Collaboration / Physics Letters B 505 (2001) 47–58 R.A. Khan s , D. Käfer a,b , M. Kaur s,6 , M.N. Kienzle-Focacci t , D. Kim al , J.K. Kim ap , J. Kirkby r , D. Kiss n , W. Kittel ad , A. Klimentov o,aa , A.C. König ad , M. Kopal as , A. Kopp au , V. Koutsenko o,aa , M. Kräber av , R.W. Kraemer ah , W. Krenz a,b , A. Krüger au , A. Kunin o,aa , P. Ladron de Guevara y , I. Laktineh x , G. Landi q , M. Lebeau r , A. Lebedev o , P. Lebrun x , P. Lecomte av , P. Lecoq r , P. Le Coultre av , H.J. Lee i , J.M. Le Goff r , R. Leiste au , P. Levtchenko ag , C. Li u , S. Likhoded au , C.H. Lin ax , W.T. Lin ax , F.L. Linde c , L. Lista ab , Z.A. Liu h , W. Lohmann au , E. Longo al , Y.S. Lu h , K. Lübelsmeyer a,b , C. Luci r,al , D. Luckey o , L. Lugnier x , L. Luminari al , W. Lustermann av , W.G. Ma u , M. Maity k , L. Malgeri t , A. Malinin r , C. Maña y , D. Mangeol ad , J. Mans aj , G. Marian p , J.P. Martin x , F. Marzano al , K. Mazumdar k , R.R. McNeil g , S. Mele r , L. Merola ab , M. Meschini q , W.J. Metzger ad , M. von der Mey a,b , A. Mihul m , H. Milcent r , G. Mirabelli al , J. Mnich a,b , G.B. Mohanty k , T. Moulik k , G.S. Muanza x , A.J.M. Muijs c , B. Musicar an , M. Musy al , M. Napolitano ab , F. Nessi-Tedaldi av , H. Newman ae , T. Niessen a,b , A. Nisati al , H. Nowak au , R. Ofierzynski av , G. Organtini al , A. Oulianov aa , C. Palomares y , D. Pandoulas a,b , S. Paoletti al,r , P. Paolucci ab , R. Paramatti al , H.K. Park ah , I.H. Park ap , G. Passaleva r , S. Patricelli ab , T. Paul l , M. Pauluzzi af , C. Paus r , F. Pauss av , M. Pedace al , S. Pensotti z , D. Perret-Gallix e , B. Petersen ad , D. Piccolo ab , F. Pierella j , M. Pieri q , P.A. Piroué aj , E. Pistolesi z , V. Plyaskin aa , M. Pohl t , V. Pojidaev aa,q , H. 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Sultanov s , L.Z. Sun u , S. Sushkov i , H. Suter av , J.D. Swain s , Z. Szillasi aq,3 , T. Sztaricskai aq,3 , X.W. Tang h , L. Tauscher f , L. Taylor l , B. Tellili x , D. Teyssier x , C. Timmermans ad , Samuel C.C. Ting o , S.M. Ting o , S.C. Tonwar k , J. Tóth n , C. Tully r , K.L. Tung h , Y. Uchida o , J. Ulbricht av , E. Valente al , G. Vesztergombi n , I. Vetlitsky aa , D. Vicinanza am , G. Viertel av , S. Villa ak , M. Vivargent e , S. Vlachos f , I. Vodopianov ag , H. Vogel ah , H. Vogt au , I. Vorobiev ah , A.A. Vorobyov ag , A. Vorvolakos ac , M. Wadhwa f , W. Wallraff a,b , M. Wang o , X.L. Wang u , Z.M. Wang u , A. Weber a,b , M. Weber a,b , P. Wienemann a,b , H. Wilkens ad , S.X. Wu o , S. Wynhoff r , L. Xia ae , Z.Z. Xu u , L3 Collaboration / Physics Letters B 505 (2001) 47–58 49 J. Yamamoto d , B.Z. Yang u , C.G. Yang h , H.J. Yang h , M. Yang h , J.B. Ye u , S.C. Yeh ay , An. Zalite ag , Yu. Zalite ag , Z.P. Zhang u , G.Y. Zhu h , R.Y. Zhu ae , A. Zichichi j,r,s , G. Zilizi aq,3 , B. Zimmermann av , M. Zöller a,b a I. Physikalisches Institut, RWTH, D-52056 Aachen, Germany 1 b III. Physikalisches Institut, RWTH, D-52056 Aachen, Germany 1 c National Institute for High Energy Physics, NIKHEF, and University of Amsterdam, NL-1009 DB Amsterdam, The Netherlands d University of Michigan, Ann Arbor, MI 48109, USA e Laboratoire d’Annecy-le-Vieux de Physique des Particules, LAPP, IN2P3-CNRS, BP 110, F-74941 Annecy-le-Vieux CEDEX, France f Institute of Physics, University of Basel, CH-4056 Basel, Switzerland g Louisiana State University, Baton Rouge, LA 70803, USA h Institute of High Energy Physics, IHEP, 100039 Beijing, China 7 i Humboldt University, D-10099 Berlin, Germany 1 j University of Bologna and INFN-Sezione di Bologna, I-40126 Bologna, Italy k Tata Institute of Fundamental Research, Bombay 400 005, India l Northeastern University, Boston, MA 02115, USA m Institute of Atomic Physics and University of Bucharest, R-76900 Bucharest, Romania n Central Research Institute for Physics of the Hungarian Academy of Sciences, H-1525 Budapest 114, Hungary 2 o Massachusetts Institute of Technology, Cambridge, MA 02139, USA p KLTE-ATOMKI, H-4010 Debrecen, Hungary 3 q INFN Sezione di Firenze and University of Florence, I-50125 Florence, Italy r European Laboratory for Particle Physics, CERN, CH-1211 Geneva 23, Switzerland s World Laboratory, FBLJA Project, CH-1211 Geneva 23, Switzerland t University of Geneva, CH-1211 Geneva 4, Switzerland u Chinese University of Science and Technology, USTC, Hefei, Anhui 230 029, PR China 7 v University of Lausanne, CH-1015 Lausanne, Switzerland w INFN-Sezione di Lecce and Università Degli Studi di Lecce, I-73100 Lecce, Italy x Institut de Physique Nucléaire de Lyon, IN2P3-CNRS,Université Claude Bernard, F-69622 Villeurbanne, France y Centro de Investigaciones Energéticas, Medioambientales y Tecnologícas, CIEMAT, E-28040 Madrid, Spain 4 z INFN-Sezione di Milano, I-20133 Milan, Italy aa Institute of Theoretical and Experimental Physics, ITEP, Moscow, Russia ab INFN-Sezione di Napoli and University of Naples, I-80125 Naples, Italy ac Department of Natural Sciences, University of Cyprus, Nicosia, Cyprus ad University of Nijmegen and NIKHEF, NL-6525 ED Nijmegen, The Netherlands ae California Institute of Technology, Pasadena, CA 91125, USA af INFN-Sezione di Perugia and Università Degli Studi di Perugia, I-06100 Perugia, Italy ag Nuclear Physics Institute, St. Petersburg, Russia ah Carnegie Mellon University, Pittsburgh, PA 15213, USA ai INFN-Sezione di Napoli and University of Potenza, I-85100 Potenza, Italy aj Princeton University, Princeton, NJ 08544, USA ak University of Californa, Riverside, CA 92521, USA al INFN-Sezione di Roma and University of Rome, La Sapienza, I-00185 Rome, Italy am University and INFN, Salerno, I-84100 Salerno, Italy an University of California, San Diego, CA 92093, USA ao Bulgarian Academy of Sciences, Central Lab. of Mechatronics and Instrumentation, BU-1113 Sofia, Bulgaria ap Laboratory of High Energy Physics, Kyungpook National University, 702-701 Taegu, South Korea aq University of Alabama, Tuscaloosa, AL 35486, USA ar Utrecht University and NIKHEF, NL-3584 CB Utrecht, The Netherlands as Purdue University, West Lafayette, IN 47907, USA at Paul Scherrer Institut, PSI, CH-5232 Villigen, Switzerland au DESY, D-15738 Zeuthen, Germany av Eidgenössische Technische Hochschule, ETH Zürich, CH-8093 Zürich, Switzerland aw University of Hamburg, D-22761 Hamburg, Germany ax National Central University, Chung-Li, Taiwan, ROC ay Department of Physics, National Tsing Hua University, Taiwan, ROC Received 29 January 2001; accepted 15 February 2001 50 L3 Collaboration / Physics Letters B 505 (2001) 47–58 Editor: K. Winter Abstract The process e+ e− → Zγ γ → qq̄γ γ is studied in 0.5 fb−1 of data collected with the L3 detector at centre-of-mass energies between 130.1 GeV and 201.7 GeV. Cross sections are measured and found to be consistent with the Standard Model expectations. The study of the least energetic photon constrains the quartic gauge boson couplings to −0.008 GeV−2 < a0 /Λ2 < 0.005 GeV−2 and −0.007 GeV−2 < ac /Λ2 < 0.011 GeV−2 , at 95% confidence level. 2001 Published by Elsevier Science B.V. 1. Introduction The LEP data offer new insight into the Standard Model of electroweak interactions [1] by investigating the production of three gauge bosons. Results were recently reported on studies of the reactions e+ e− → Zγ γ [2] and e+ e− → W+ W− γ [3,4]. This Letter de+ − scribes the extension of the study of the √ e e → Zγ γ process to centre-of-mass energies, s, between 130 and 202 GeV. Final states with hadrons and isolated photons are considered to select Zγ γ → qq̄γ γ events. In the Standard Model, the e+ e− → Zγ γ process occurs via radiation of photons from the incoming electron and/or positron. One possible diagram is presented in Fig. 1(a). The e+ e− → Zγ γ signal is defined by phase-space requirements on the energies Eγ and angles √ θγ of the two photons, and on the propagator mass s : Eγ > 5 GeV, (1) | cos θγ | < 0.97, √ s − mZ < 2ΓZ, (2) (3) 1 Supported by the German Bundesministerium für Bildung, Wissenschaft, Forschung und Technologie. 2 Supported by the Hungarian OTKA fund under contract numbers T019181, F023259 and T024011. 3 Also supported by the Hungarian OTKA fund under contract numbers T22238 and T026178. 4 Supported also by the Comisión Interministerial de Ciencia y Tecnología. 5 Also supported by CONICET and Universidad Nacional de La Plata, CC 67, 1900 La Plata, Argentina. 6 Also supported by Panjab University, Chandigarh-160014, India. 7 Supported by the National Natural Science Foundation of China. where mZ and ΓZ are the Z boson mass and width. In the following, hadronic decays of the Z boson are considered. Events with hadrons and initial state photons falling outside the signal definition cuts are referred to as “non-resonant” background. A single initial state radiation photon can also lower the effective centre-of-mass energy of the e+ e− collision to mZ , with the subsequent production of a quark–antiquark pair. This photon can be mistaken for the most energetic photon of the e+ e− → Zγ γ → qq̄γ γ process. Two sources can then mimic the least energetic photon: either the direct radiation of photons from the quarks or photons originating from hadronic decays, misidentified electrons or unresolved π 0 s. These background processes are depicted in Figs. 1(b) and 1(c), respectively. In order to compare experimental results with e+ e− → qq̄γ γ matrix element calculations, a further requirement is applied on the angle θγ q between the photons and the nearest quark: cos θγ q < 0.98. (4) This cut avoids collinear divergences. Its inclusion makes the signal definition used here different from the previous one [2]. Signal cross sections calculated with the KK2f √ Monte Carlo program [5] range√from 0.9 pb at s = 130.1 GeV down to 0.3 pb at s = 201.7 GeV. The Zγ γ final state could also originate from the s-channel exchange of a Z boson, as presented in Fig. 1(d). This process is forbidden at tree level in the Standard Model, but it is expected to occur in the presence of Quartic Gauge boson Couplings (QGC) beyond the Standard Model. L3 Collaboration / Physics Letters B 505 (2001) 47–58 (a) (b) (c) (d) 51 Fig. 1. Diagrams of (a) the Standard Model contribution to e+ e− → Zγ γ signal and “non-resonant” background, (b) the background from direct radiation of photon from the quarks, (c) the background from photons, misidentified electrons or unresolved π 0 s originating from hadrons and (d) the anomalous QGC diagram. 2. Data and Monte Carlo samples This measurement uses data collected with the L3 detector [6] at LEP in the years from 1995 through 1999, at centre-of-mass energies between √ √ s = 130.1 GeV and s = 201.7 GeV, for a total integrated luminosity of 0.5 fb−1 . The centre-of-mass energies and the corresponding integrated luminosities are listed in Table 1. Given their relatively low lumi√ √ nosities, the s = 130.1 GeV and s = 136.1 GeV data sample are combined into a single luminosity √ s = 133.1 GeV. Similarly the averaged sample at √ √ s = 161.3 GeV and s = 172.3 √ GeV samples are merged into a single sample at s = 166.8 GeV. The KK2f Monte Carlo program is used to generate e+ e− → qq̄(γ γ ) events, that are assigned to the signal or the background according to the criteria (1)–(4). The hadronisation process is simulated with the JETSET [7] program. Other background processes are generated with the Monte Carlo programs PYTHIA [7] (e+ e− → Ze+ e− and e+ e− → ZZ), KORALZ [8] (e+ e− → τ + τ − (γ )), PHOJET [9] Table 1 Average centre-of-mass energies and corresponding integrated luminosities of the data samples used for this analysis √ s (GeV) Integrated luminosity (pb−1 ) 133.1 12.0 166.8 21.1 182.7 55.3 188.7 176.3 191.6 29.4 195.5 83.7 199.5 82.8 201.7 37.0 (e+ e− → e+ e− hadrons) and KORALW [10] for W+ W− production except for the eνe qq̄ final states, generated with EXCALIBUR [11]. The L3 detector response is simulated using the GEANT [12] and GHEISHA [13] programs, which model the effects of energy loss, multiple scattering 52 L3 Collaboration / Physics Letters B 505 (2001) 47–58 Table 2 Energy dependent criteria for the selection of e+ e− → Zγ γ → qq̄γ γ events √ s (GeV) 133.1 166.8 182.7 188.7 βZ < Eγ1 (GeV) < 0.48 31.9 0.61 55.0 0.64 67.6 and showering in the detector. Time dependent detector inefficiencies, as monitored during data taking periods, are also simulated. 0.66 69.8 191.6 0.66 72.8 195.5 0.67 74.2 199.6 0.69 75.8 201.7 0.70 76.6 criteria on all the other variables are applied. Good agreement between data and Monte Carlo is observed. 4. Results 3. Event selection The e+ e− → Zγ γ → qq̄γ γ selection demands balanced hadronic events with two isolated photons and small energy deposition at low polar angle. Selection criteria on photon energies and angles follow directly from the signal definition as Eγ > 5 GeV and | cos θγ | < 0.97. The invariant mass Mqq̄ of the reconstructed hadronic system, forced into two jets using the DURHAM algorithm [14], is required to be consistent with a Z boson decaying into hadrons, 72 GeV < Mqq̄ < 116 GeV. The main background after these requirements is due to the “non-resonant” production of two photons and a hadronic system. The relativistic velocity βZ = pZ /EZ of the system recoiling against the photons, calculated assuming its mass to be the nominal Z mass, is larger for part of these background events than for the signal and an upper cut is used to reject those events. It is optimised for each centre-of-mass energy, as listed in Table 2. Other classes of background events, shown in Figs. 1(b) and 1(c), are rejected by an upper bound on the energy Eγ 1 of the most energetic photon. This requirement, presented in Table 2, suppresses the resonant return to the Z, whose photons are harder than the signal ones. A lower bound of 17◦ on the angle ω between the least energetic photon and the closest jet is also imposed. This requirement is more restrictive than the similar cut on cos θγ q included in the signal definition. Data and Monte Carlo distributions of the selection variables √ are presented in Fig. 2 for the data collected at s = 192–202 GeV when selection The signal efficiencies and the numbers of events selected in the data and Monte Carlo samples are summarised in Table 3. The dominant background is hadronic events with photons. About half of these are “non-resonant” events. In the remaining cases, they originate either from final state radiation or are fake photons. A clear signal structure is observed in the spectra of the recoil mass √ to the two photons, as presented in Fig. 3 for the s = 192–202 GeV data sample and for the total one. The e+ e− → Zγ γ → qq̄γ γ cross sections, σ , are determined from a fit to the corresponding √ spectra at each s. Background predictions are fixed in the fit. The results are listed in Table 4 with their statistical and systematic uncertainties. The systematic uncertainties on the cross section measurement are of the order of 10% [2]. The main contributions arise from the signal and background Monte Carlo statistics (6%) and a variation of ±2% of the energy scale of the hadronic calorimeter (6%). A variation of ±0.5% of the energy scale of the electromagnetic calorimeter does not yield sizable effects. Other sources of systematic uncertainties are the selection procedure (3%) and the background normalisation (3%). The latter is estimated by varying by 10% the normalisation of the “non-resonant” background, as estimated from a comparison between the KK2f and PYTHIA Monte Carlo predictions for hadronic events with photons, and by 20% that of the other backgrounds. Uncertainties on the determination of the integrated luminosity are negligible. The measurements are in good agreement with the theoretical predictions σ SM , as calculated with L3 Collaboration / Physics Letters B 505 (2001) 47–58 53 Fig. 2. Distributions of (a) the invariant mass Mqq̄ of the hadronic system, (b) the relativistic velocity βZ of the reconstructed Z boson, (c) the energy Eγ 1 of the most energetic photon and (d) the angle ω between the least energetic photon and the nearest jet. Data, signal and background √ Monte Carlo samples are shown for s = 192–202 GeV. The arrows show the position of the final selection requirements. In each plot, the selection criteria on the other variables are applied. 54 L3 Collaboration / Physics Letters B 505 (2001) 47–58 Fig. 3. Recoil mass to the photon pairs in data, Zγ γ and background Monte Carlo for (a) √ s = 192–202 GeV and (b) the total sample. Fig. 4. The cross section of the process e+ e− → Zγ γ → qq̄γ γ as a function of the centre-of-mass energy. The signal is defined by the phase-space cuts of Eqs. (1)–(4). The width of the band corresponds to the Monte Carlo statistics and theory uncertainties. Dashed and dotted lines represent anomalous QGC predictions for a0 /Λ2 = 0.015 GeV−2 and ac /Λ2 = 0.015 GeV−2 , respectively. L3 Collaboration / Physics Letters B 505 (2001) 47–58 55 Table 3 Yields of the e+ e− → Zγ γ → qq̄γ γ selection. The signal efficiencies ε are given, together with the observed and expected numbers of events. qq̄ The right half of the table details the composition of the Monte Carlo samples with Ns denoting the signal, Nb the qq̄ and NbOther the other backgrounds. The uncertainties are statistical only √ qq̄ s (GeV) ε (%) Data Monte Carlo Ns Nb NbOther 133.1 45 4 5.9 ± 0.5 5.0 ± 0.5 0.8 ± 0.2 0.08 ± 0.02 166.8 52 4 6.7 ± 0.3 4.9 ± 0.3 1.4 ± 0.1 0.4 ± 0.1 182.7 51 13 13.6 ± 0.7 10.8 ± 0.6 2.7 ± 0.2 0.06 ± 0.02 188.7 52 38 40.3 ± 2.0 32.5 ± 1.7 7.2 ± 1.1 0.6 ± 0.1 191.6 42 2 5.9 ± 0.4 4.1 ± 0.3 1.8 ± 0.3 0.06 ± 0.02 195.5 46 13 17.5 ± 0.9 12.4 ± 0.7 4.9 ± 0.5 0.2 ± 0.1 199.6 46 14 15.0 ± 0.8 11.5 ± 0.6 3.4 ± 0.5 0.13 ± 0.05 201.7 48 9 6.9 ± 0.5 5.2 ± 0.4 1.7 ± 0.3 0.06 ± 0.02 Table 4 Results of the measurements of the e+ e− → Zγ γ → qq̄γ γ cross section, σ , with statistical and systematic uncertainties. The predicted values of cross sections, σ SM , are also listed √ s (GeV) σ (pb) σ SM (pb) 133.1 0.70 ± 0.40 ± 0.07 0.923 ± 0.012 166.8 0.17 ± 0.13 ± 0.02 0.475 ± 0.006 182.7 0.36 ± 0.13 ± 0.04 0.379 ± 0.004 188.7 0.34 ± 0.06 ± 0.03 0.350 ± 0.004 191.6 0.09 ± 0.09 ± 0.01 0.326 ± 0.004 195.5 0.30 ± 0.11 ± 0.03 0.321 ± 0.004 199.6 0.28 ± 0.11 ± 0.03 0.304 ± 0.004 201.7 0.50 ± 0.18 ± 0.05 0.296 ± 0.003 the KK2f Monte Carlo program, listed in Table 4. The error on the predictions (1.5%) is the quadratic sum of the theory uncertainty [5] and the statistical uncertainty of the Monte Carlo sample generated for the calculation. These results are presented√in Fig. 4 together with the expected evolution with s of the Standard Model cross section. The distribution of the recoil mass to the two photons for the full data sample, presented in Fig. 3(b), is fitted to calculate the ratio RZγ γ between all the observed data and the signal expectation. The background predictions are fixed in the fit, which yields: σ = 0.85 ± 0.11 ± 0.06, σ SM in agreement with the Standard Model. The first uncertainty is statistical while the second is systematic. The correlation of the energy scale and background normalisation uncertainties between data samples is taken into account. RZγ γ = 5. Study of quartic gauge boson couplings The contribution of anomalous QGCs to Zγ γ production is described by two additional dimension-six terms in the electroweak Lagrangian [15,16]: πα ρ, ρ · W a0 Fµν F µν W 4Λ2 πα ρ ·W σ , Lc6 = − 2 ac Fµρ F µσ W 4Λ where α is the fine structure constant, Fµν is the field σ is the weak strength tensor of the photon and W boson field. The parameters a0 and ac describe the strength of the QGCs and Λ represents the unknown scale of the New Physics responsible for the anomalous contributions. In the Standard Model, a0 = ac = 0. A more detailed description of QGCs has recently appeared [17]. Indirect limits on QGCs were derived from precision measurements at the Z pole [18]. L06 = − 56 L3 Collaboration / Physics Letters B 505 (2001) 47–58 Fig. 5. Distributions for the least energetic photon: (a) the energy Eγ 2 , (b) the cosine of its polar angle | cos θγ 2 |, (c) its transverse momentum P tγ 2 with respect to the beam axis. Data, signal and background Monte Carlo are displayed for the full data sample together with QGC predictions. Anomalous values of QGCs are expected to manifest themselves via deviations in the total e+ e− → Zγ γ cross section, as presented in Fig. 4. In the Standard Model, Zγ γ production occurs via bremsstrahlung with the low energy photon preferentially produced close to the beam direction. The QGC s-channel production results instead in a harder energy spectrum and a more central angular distribution of the least energetic photon [16]. Distributions for this photon of the reconstructed energy, the cosine of the polar angle and the transverse momentum for the full data sample are compared in Fig. 5 with the predictions from signal and background Monte Carlo. Predictions in the case of a non zero value of a0 /Λ2 or ac /Λ2 are also shown. They are obtained by reweighting [2] the Stan- dard Model signal Monte Carlo events with an analytical calculation of the QGC matrix element [16]. Monte Carlo studies indicate the transverse momentum as the most sensitive distribution to possible anomalous QGC contributions. A fit to this distribution is performed for each data sample, leaving one of the two QGCs free at a time and fixing the other to zero. It yields the 68% confidence level results: −2 a0 /Λ2 = −0.002+0.003 −0.002 GeV and −2 ac /Λ2 = −0.001+0.006 −0.004 GeV , in agreement with the expected Standard Model values of zero. A simultaneous fit to both the parameters gives L3 Collaboration / Physics Letters B 505 (2001) 47–58 57 Fig. 6. Two-dimensional contours for the QGC parameters a0 /Λ2 and ac /Λ2 . The fit result is shown together with the Standard Model (SM) predictions. the 95% confidence level limits: −0.008 GeV−2 < a0 /Λ2 < 0.005 GeV−2 −0.007 GeV −2 2 −2 < ac /Λ < 0.011 GeV and , as shown in Fig. 6. A correlation coefficient of −57% is observed. The experimental systematic uncertainties and those on the Standard Model e+ e− → Zγ γ → qq̄γ γ cross section predictions are taken into account in the fit. are also measured in the more restrictive phase space obtained by modifying the conditions (2) and (4) into | cos θγ | < 0.95 and cos θγ q < 0.9, respectively. The results are: σ (182.7 GeV) = 0.11 ± 0.11 ± 0.01 pb (SM: 0.233 ± 0.003 pb), σ (188.7 GeV) = 0.28 ± 0.07 ± 0.03 pb (SM: 0.214 ± 0.003 pb), σ (194.5 GeV) = 0.15 ± 0.07 ± 0.02 pb Acknowledgements We wish to express our gratitude to the CERN accelerator divisions for the superb performance and the continuous and successful upgrade of the LEP machine. We acknowledge the contributions of the engineers and technicians who have participated in the construction and maintenance of this experiment. (SM: 0.197 ± 0.003 pb), σ (200.2 GeV) = 0.15 ± 0.07 ± 0.01 pb (SM: 0.185 ± 0.003 pb). The first uncertainty is statistical, the second systematic and the values in parentheses indicate √ the s= Standard Model predictions. The samples at √ 192–196 GeV and s √ = 200–202 GeV are respec√ tively merged into the s = 194.5 GeV and s = 200.2 GeV ones. Appendix A References To allow the combination of our results with those of the other LEP experiments, the cross sections σ [1] S.L. Glashow, Nucl. 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C 13 (2000) 283. [18] A. Brunstein, O.J.P. Éboli, M.C. Gonzales-Garcia, Phys. Lett. B 375 (1996) 233. 26 April 2001 Physics Letters B 505 (2001) 59–63 www.elsevier.nl/locate/npe Experimental limits on the proton life-time from the neutrino experiments with heavy water V.I. Tretyak, Yu.G. Zdesenko Institute for Nuclear Research, Prospekt Nauki 47, MSP 03680 Kiev, Ukraine Received 4 January 2001; accepted 28 February 2001 Editor: K. Winter Abstract Experimental data on the number of neutrons born in the heavy water targets of the large neutrino detectors are used to set the limit on the proton life-time independently on decay mode through the reaction d → n + ?. The best up-to-date limit τp > 4 × 1023 yr with 95% C.L. is derived from the measurements with D2 O target (mass 267 kg) installed near the Bugey reactor. This value can be improved by six orders of magnitude with future data accumulated with the SNO detector containing 1000 t of D2 O. 2001 Elsevier Science B.V. All rights reserved. PACS: 14.20.D; 24.80.+y; 25.40.-h Keywords: Proton life-time; Neutrino detectors; Deuteron 1. Introduction The baryon (B) and lepton (L) numbers are absolutely conserved in the Standard Model (SM). However, many extensions of the SM, in particular, grand unified theories incorporate B and L violating interactions, since in the modern gauge theories conservation of baryon (lepton) charge is considered as approximate law due to absence of any underlying symmetry principle behind it, unlike the gauge invariance in electrodynamics which guarantees the massless of photon and absolute conservation of the electric charge. Therefore, it is quite natural to suppose the decay of protons and neutrons bounded in nuclei. The processes with B = 1, B = 2, (B − L) = 0, (B − L) = 2 have been discussed in literature (see, e.g., [1–3] and references therein), while the disappearance of nucleons (or decay into “nothing”) has been addressed in connection with possible existence of extra dimensions [4–6]. Stimulated by theoretical predictions, nucleon instability has been searched for in many underground experiments with the help of massive detectors such as IMB, Fréjus, Kamiokande, SuperKamiokande and others (for experimental activity see [3,7,8] and references therein). About 90 decay modes have been investigated; however, no evidence for the nucleons decay has been found. A complete summary of the experimental results is given in the Review of Particle Physics [9]. For the modes in which the nucleon decays to particles strongly or electromagnetically interacting in the detector’s sensitive volume, the obtained life-time limits are in the range of 1030 –1033 yr, while for decays to only weakly interacting products (neutrinos) the bounds are up to 10 orders of magnitude lower [9,10]. However, because it is not known a priori which mode of proton decay (from 90 ones listed E-mail address: [email protected] (Y.G. Zdesenko). 0370-2693/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 3 3 4 - 3 60 V.I. Tretyak, Yu.G. Zdesenko / Physics Letters B 505 (2001) 59–63 in [9]) is preferable, the limits on the proton decay independent on the channel are very important. Three approaches were used to establish such limits: (1) In Ref. [11] the bound τ (p →?) > 1.3 × 1023 yr was determined 1 on the basis of the limit for the branching ratio of 232 Th spontaneous fission βSF . It was assumed that the parent 232 Th nucleus is destroyed by the strongly or electromagnetically interacting particles emitted in the proton decay or, in case of proton’s disappearance (or p decay into neutrinos) by the subsequent nuclear deexcitation process. Using the present-day data [12] on the 232 Th half-life T1/2 = 1.405 × 1010 yr and limit βSF < 1.8 × 10−9 %, we can recalculate the value of [11] as τ (p →?) > 1.0 × 1023 yr. (2) In Ref. [13] the limit τ (p →?) > 3 ×1023 yr was established by searching for neutrons born in liquid scintillator, enriched in deuterium, as result of proton decay in deuterium (d → n + ?). (3) In Ref. [14] the limit τ (p → 3ν) > 7.4 × 1024 yr was determined 2 on the basis of geochemical measurements with Te ore by looking for the possible daughter nuclides (130Te → · · · → 129 Xe), while in Refs. [15,16] the bound τ (p → 3ν) > 1.1 × 1026 yr was achieved by the radiochemical measurements with 1710 kg of potassium acetate KC2 H3 O2 placed deep underground (39 K → · · · → 37 Ar). In the experiments (2) and (3) both the baryon number and the electric charge would be not conserved; nevertheless authors suggested that “experimenter would be wise not to exclude such processes from consideration a priori” [14]. The limits [14–16] usually are quoted as “independent on channel” [9], however it is evident that they are valid only for the proton decay into invisible channels (or disappearance), in which the parent nucleus is not fully destroyed (like 232 Th in the experiment [11]). At the same time, bound on the proton decay from the deuterium disintegration requires the less stringent hypothesis on the stability of daughter nuclear system and, hence, it is less model depen1 We recalculated the value quoted in [11] τ (N →?) > 3 × 1023 yr (given for 232 particles: 142 neutrons and 90 protons) for 90 protons which should be taken here into consideration (N is p or n). 2 The value τ (N → 3ν) > 1.6 × 1025 yr quoted in [14] as given for 52 particles (28 neutrons and 24 protons) was recalculated for 24 protons. dent. Such a limit was established in 1970 [13] and is equal τ (p →?) > 3 × 1023 yr at 68% C.L. 3 This value can be improved by using the data from the modern neutrino experiments with heavy water well shielded against cosmic rays and natural radioactivity. With this aim, in present Letter we analyze the measurements of Ref. [17] with the 267 kg D2 O target installed at Reactor 5 of the Centrale Nucleaire de Bugey (France). Further, we show that obtained limit τ (p →?) can be highly improved with the SNO (Sudbury Neutrino Observatory) large volume detector [18] developed mainly for the Solar neutrino investigations and containing 1000 t of D2 O. 2. D2 O experiment at the Bugey reactor The experiment [17] was aimed to measure the cross sections for the disintegration of deuteron by low-energy electron antineutrinos from nuclear reactor through reactions ν e + d → ν e + n + p (neutral currents) and ν e + d → e+ + n + n (charged currents). Events were recognized by the neutrons they produced. The detector was located on the depth of 25 meters of water equivalent (mwe) at 18.5 m distance from the center of the Reactor 5 core at the Bugey site. The cylindrical target tank, containing 267 kg of 99.85% pure D2 O, was surrounded by layers of lead (10 cm) and cadmium (1 mm) to absorb thermal neutrons from external surroundings. The tank with D2 O and Pb–Cd shield was inserted in the center of a large liquid scintillator detector (based on mineral oil) which served as (inner) cosmic ray veto detector. Subsequent layer of lead 10 cm thick was aimed to reduce the flux of external γ quanta with energies Eγ > 2.23 MeV which can photodisintegrate the deuterons creating the background events. However this shielding itself was a significant source of neutrons in the target detector: they were created due to interaction of cosmic rays with Pb. To suppress this background, an additional layer of cosmic ray veto detectors was installed outside the Pb shielding. The outer veto reduced the neutron background in the target by a factor of near 6. 3 Because Ref. [13] is not the source of easy access, and in Ref. [9], where this limit is quoted, there is no indication for confidence level, we suppose that it was established with 68% C.L. V.I. Tretyak, Yu.G. Zdesenko / Physics Letters B 505 (2001) 59–63 Neutrons were detected by 3 He proportional counters installed in the tank with D2 O through reaction 3 He + n → 3 H + p + 764 keV. Further details on the experiment can be found in [17]. The decay or disappearance of proton bounded in deuterium nucleus, which consists only of proton and neutron, will result in the appearance of free neutron: d → n + ?. Thus the proton life-time limit can be estimated on the basis of the neutron rate detected in the D2 O volume when the reactor is switched off. To calculate the lim τ (p →?) value, we use the formula lim τ (p →?) = ε × Nd × t/ lim S, (1) where ε is the efficiency for the neutron’s detection, Nd is number of deuterons, t is the time of measurement, and lim S is the number of proton decays which can be excluded with a given confidence level on the basis of the neutron background (one-neutron events) measured in the experiment. Mean efficiency for single neutrons born isotropically throughout the D2 O volume was determined as ε = 0.29±0.01 [17]. In 267 kg of D2 O there is Nd = 1.605 × 1028 deuterons. Raw one-neutron rate with the reactor down is equal 25.28±0.68 counts per day (cpd) and, corrected for software efficiency (0.444 ± 0.005), this rate is 57.00 ± 1.53 cpd. For very rough estimate of the p life-time (as the first approximation) we can attribute all neutron events to proton decays and obtain the lim S value as 59.5 cpd at 95% C.L. Then, substituting this value in the formula (1) we get limit τ (p →?) > 2.1 × 1023 yr with 95% C.L. 4 However, it is evident that τ limit derived in this way is very conservative because the dominant part of observed neutron rate has other origins rather than proton decay [20–22]. On the other hand, if we suppose that all measured neutron events are belonging to background, then the excluded number of neutrons due to possible proton decay will be restricted only by statistical uncertainties in the measured neutron background. Hence, to estimate value of lim S we can use so-called “one (two, three) σ approach”, in which the excluded number of effect’s events is determined simply as square root of the number of background 4 The similar limit τ (p →?) > 1.9 × 1023 yr with 95% C.L. can be derived from other neutrino deuteron experiment at Krasnoyarsk (Russia) nuclear reactor [19]. 61 counts multiplied by one (two or three) according to the confidence level chosen (68%, 95% or 99%). This method gives us the sensitivity limit of the considered experiment to the proton decay. Applying it we get lim S = 3 cpd (at 95% C.L.), which leads to the bound τ (p →?) > 4 × 1024 yr. Therefore, we can argue that, with the probability close to 100%, estimate of τ limit is within interval 2 × 1023 –4 × 1024 yr. In order to fix life-time limit or at least to narrow this interval, it is necessary to determine the contributions of different sources to the total neutron rate observed. As it was already mentioned, the nature and origins of neutron background in neutrino experiments at nuclear reactors are well known and understood (see, for example, Refs. [20–23]). The main sources are: (i) interaction of cosmic muons (escaped an active veto system) with the detector, passive shield and surrounding materials; (ii) photodisintegration of the deuteron by γ quanta (with Eγ > 2.23 MeV), originated from the radioactive contamination of the detector materials and shield, as well as from environment pollution; (iii) residual (and non-eliminated by the shield) neutron background at the nuclear reactor site. Before coming to details, we would like to remind that crucial characteristics of any neutrino experiment at reactor are the depth of its location and distance from the reactor core [20–23]. Let us prove this statement by Table 1 with parameters of the most advanced experiments and by two short citations, from Ref. [20]: “A striking difference between the two experiments is the amount of overburden, which may be viewed as the main factor responsible for how the experiments compare on detector design, event rate, and signal-to-background”, and from Ref. [21]: “. . . a detector should be located sufficiently deep underground to reduce the flux of cosmic muons — the main source of background in experiments of this type”. It is clear from the Table 1 that neutron background of different experiments is decreased as the depth of their location and distance from the reactor are enlarged. For example, in the Chooz experiment with liquid scintillator the background was reduced roughly by factor 500 as compared with that of Bugey ones 62 V.I. Tretyak, Yu.G. Zdesenko / Physics Letters B 505 (2001) 59–63 Table 1 Main characteristics of the neutrino experiments at nuclear reactor Experiment Bugey-1995 [24] Bugey-1999 [17] Palo Verde [25] Chooz [26] Depth 40 mwe 25 mwe 32 mwe 300 mwe Distance 15 m 18.5 m ≈ 0.9 km ≈ 1 km Detector type Li-loaded scintillator D2 O target + 3 He counters Gd-loaded scintillator Gd-loaded scintillator Detector mass ≈ 0.6 t 267 kg 11 t 5t Neutron background ≈ 100 cpd/t ≈ 100 cpd/t 2.2 cpd/t 0.24 cpd/t [17,24] because the Chooz set up was placed deep underground (300 mwe) and ≈1 km away from nuclear reactor [26]. We recall that Bugey set ups [17,24] were located only 25–40 mwe overburden (which allows to remove the hadronic component of cosmic rays but is not enough to reduce the muon flux significantly) and at 15–18 m distance from the reactor core. Thus, the dominant part of neutron background in [17,24] is associated with the reactor site and muon flux. Indeed, as it was proved by the detail simulation and careful analysis of neutron background in reactor-off periods of the experiment [24], the 67 ± 3% of neutron rate (measured at 15 m distance) are attributed to known origins (see Table 6 in Ref. [24]). On the basis of comparison of different experiments presented in Table 1, and taking into account results of background analysis [24], we can make semi-quantitative and conservative estimation that at least 50% of one-neutron events measured in [17] are caused by the mentioned sources (i)–(iii). Hence, attributing remaining part of one-neutron rate to other unknown background origins, we can accept its value as the excluded number of proton’s decays (lim S = 15 cpd). Finally, substituting this number in the formula (1) we obtain τ (p →?) > 4 × 1023 yr with 95% C.L., which is higher than previous limit [13]. 3. Expected improvements with the SNO solar neutrino detector The Sudbury Neutrino Observatory (SNO) [18] is an unique large Cherenkov detector constructed with an emphasis on the study of Solar neutrinos. The detector, containing 1000 t of 99.917% isotopically pure heavy water, is located in the INCO Creighton nickel mine near Sudbury, Ontario, on the depth of 2039 m (near 6000 mwe); this reduces the muon flux to 70 muons per day in the detector area. Particular attention is payed to minimization of radioactive backgrounds. Near 7000 t of ultra-pure light water shield the central D2 O detector from natural radioactivity from the laboratory walls. All components of the detector are made of selected materials with low radioactivity contamination. Solar neutrinos will be detected through the following reactions with electrons and deuterons: νi + e− → νi + e− (elastic scattering; i = e, µ, τ ), νe + d → e− + p + p (charged current absorption) and νi + d → νi + n + p (neutral current disintegration of deuteron). Near 9600 photomultiplier tubes are used to observe the Cherenkov light produced on the D2 O volume by high energy products. Neutrons released in d disintegration will be detected by neutron capture on deuterons in pure D2 O, or by capture on 35 Cl by dissolving MgCl salt in the heavy water, or by capture on 3 He using proportional counters. Further details can be found in [18]. Extensive Monte Carlo simulations were performed to predict response functions and numbers of expected events due to interaction of the detector with Solar neutrinos, natural radioactivity of various detector components, cosmogenic activities, capture of neutrons, (α, pγ ), (α, nγ ) reactions outside the SNO detector, etc. (f.e., see [27]). Expected number of neutrons from all sources in the D2 O volume is calculated as ≈ 5 × 103 during 1 yr period of exposition, with main contribution from the Solar neutrinos. Efficiency for n detection is equal 83% for n capture on 35 Cl [18,27]. V.I. Tretyak, Yu.G. Zdesenko / Physics Letters B 505 (2001) 59–63 63 Using these unique features of the SNO detector (super-low background, large amount of D2 O and high sensitivity to neutrons), the limit on the proton decay independent on channel can be highly improved. Again for very rough estimate of the p life-time we can conservatively attribute all neutrons in the D2 O volume to proton decays and accept it as the excluded value lim S = 5 × 103 counts. Then, substituting in the formula (1) values of efficiency ε = 0.83, measuring time t = 1 yr, number of deuterons Nd = 6 × 1031 and lim S = 5 × 103 counts, we receive lar neutrino detector (containing 1000 t of heavy water) life-time limit will be improved up to the value τ (p →?) > 4 × 1029 yr, which is, in fact, close to the bounds established for the particular modes of the nucleon decays to charged or strongly interacting particles and would be of a great importance for many extensions of the modern gauge theories. τ (p →?) > 1 × 1028 yr, [1] P. Langacker, Phys. Rep. 71 (1981) 185. [2] M. Goldhaber, P. Langacker, R. Slansky, Science 210 (1980) 851. [3] H.V. Klapdor-Kleingrothaus, A. Staudt, Non-accelerator Particle Physics, IoP, Philadelphia, 1995. [4] F.J. Yndurain, Phys. Lett. B 256 (1991) 15. [5] G. Dvali, G. Gabadadze, G. Senjanovic, hep-ph/9910207. [6] S.L. Dubovsky, V.A. Rubakov, P.G. Tinyakov, hepph/0006046, hep-ph/0007179. [7] D.H. Perkins, Ann. Rev. Nucl. Part. Sci. 34 (1984) 1. [8] R. Barloutaud, Nucl. Phys. B (Proc. Suppl.) A 28 (1992) 437. [9] Particle Data Group, Review of Particle Physics, Eur. Phys. J. C 15 (2000) 1. [10] R. Bernabei et al., Phys. Lett. B 493 (2000) 12. [11] G.N. Flerov et al., Sov. Phys. Dokl. 3 (1958) 79. [12] R.B. Firestone, V.S. Shirley et al. (Eds.), Table of Isotopes, 8th edn., Wiley, New York, 1996. [13] F.E. Dix, Ph.D. thesis, Case Western Reserve University, Cleveland, 1970. [14] J.C. Evans Jr., R.I. Steinberg, Science 197 (1977) 989. [15] E.L. Fireman, in: Proc. Int. Conf. on Neutrino Phys. and Neutrino Astrophys, Neutrino ’77, Baksan Valley, USSR, June 18–24, 1977, Vol. 1, Nauka, Moscow, 1978, p. 53. [16] R.I. Steinberg, J.C. Evans, in: Proc. Int. Conf. on Neutrino Phys. and Neutrino Astrophys, Neutrino ’77, Baksan Valley, USSR, June 18–24, 1977, Vol. 2, Nauka, Moscow, 1978, p. 321. [17] S.P. Riley et al., Phys. Rev. C 59 (1999) 1780. [18] H.H. Chen, Phys. Rev. Lett. 55 (1985) 1534; J. Boger, SNO Collaboration, Nucl. Instrum. Methods A 449 (2000) 172. [19] Yu.V. Kozlov et al., Phys. At. Nucl. 61 (1998) 1268; Yu.V. Kozlov et al., Phys. At. Nucl. 63 (2000) 1016. [20] J. Busenitz, Phys. At. Nucl. 63 (2000) 993. [21] L.A. Mikaelyan, V.V. Sinev, Phys. At. Nucl. 63 (2000) 1002. [22] G. Gratta, Nucl. Phys. B (Proc. Suppl.) 85 (2000) 72. [23] Y. Declais, Nucl. Phys. B (Proc. Suppl.) 70 (1999) 148. [24] B. Achkar et al., Nucl. Phys. B 434 (1995) 503. [25] A. Piepke et al., Nucl. Instrum. Methods A 432 (1999) 392. [26] M. Appolonio et al., Phys. Lett. B 420 (1998) 397. [27] R.G.H. Robertson for the SNO Collaboration, Prog. Part. Nucl. Phys. 40 (1998) 113. which is about five orders of magnitude higher than present-day limit. However, this value can be improved further by accounting the neutron events originating from Solar neutrinos and high energy γ quanta. Number of neutrons born in the D2 O volume due to disintegration νi + d → νi + n + p can be estimated independently using the information on the number of Solar neutrino interaction with the detector volume through neutrino–electron elastic scattering νi + e− → νi + e− . Neutrons created by high energy γ quanta from natural radioactivity in the detector components can be also calculated if the levels of pollution of all materials are measured firmly. In that case the excluded number of neutrons due to possible proton decay will be restricted only by statistical uncertainties of the measured neutron background, i.e., we can √ apply “two σ approach” again. It gives lim S = 2 5000 with 95% C.L., thus the corresponding bound on the proton life-time would be equal to τ (p →?) > 4 × 1029 yr with 95% C.L., which can be considered as the maximal sensitivity of the SNO detector for the proton decay independent on channel. 4. Conclusion The data of the Bugey experiment [17], aimed to measure the cross sections for the deuteron disintegration by antineutrinos from nuclear reactor, were analyzed to set the proton life-time limit. The obtained value τ (p →?) > 4 × 1023 yr at 95% C.L. is higher than the limit established in the previous experiment [13]. With the future data from the SNO So- References 26 April 2001 Physics Letters B 505 (2001) 64–70 www.elsevier.nl/locate/npe Simple solutions of fireball hydrodynamics for self-similar elliptic flows S.V. Akkelin a , T. Csörgő b , B. Lukács b , Yu.M. Sinyukov a , M. Weiner c a Bogolyubov Institute for Theoretical Physics, Kiev 03143, Metrologicheskaya 14b, Ukraine b MTA KFKI RMKI, H-1525 Budapest 114, POB 49, Hungary c Faculty of Science, Eötvös University, Budapest H-1117, Pázmány P.s. 1/A, Hungary Received 19 December 2000; received in revised form 1 February 2001; accepted 19 February 2001 Editor: J.-P. Blaizot Abstract Simple, self-similar, elliptic solutions of non-relativistic fireball hydrodynamics are presented, generalizing earlier results for spherically symmetric fireballs with Hubble flows and homogeneous temperature profiles. The transition from one-dimensional to three-dimensional expansions is investigated in an efficient manner. 2001 Published by Elsevier Science B.V. 1. Introduction Recently, a lot of experimental and theoretical efforts have gone into the exploration of hydrodynamical behavior of strongly interacting hadronic matter in non-relativistic as well as in relativistic heavy ion collisions, see, e.g., [1,2]. Due to the non-linear nature of hydrodynamics, exact hydro solutions are rarely found. Those events, sometimes, even stimulate an essential progress in physics. One of the most impressive historical example is Landau’s one-dimensional analytical solution (1953) for relativistic hydrodynamics [3] that gave rise to a new (hydrodynamical) approach in high energy physics. The boost-invariant Bjorken solution [4], found more than 20 years later, is frequently utilized as the basis for estimations of initial energy densities in ultra-relativistic nucleus–nucleus collisions. E-mail address: [email protected] (T. Csörgő). The obvious success of hydrodynamic approach to high energy nuclear collisions raise interest in an analogous description of non-relativistic collisions, too. The first exact non-relativistic hydrodynamic solution describing expanding fireballs was found in 1979 [5]. It has been generalized for fireballs with Gaussian density and homogeneous temperature profiles [6] as well as for fireballs with arbitrary initial temperature profiles [7] and corresponding, non-Gaussian density profiles. All of these solutions have spherical symmetry and a Hubble-type linear radial flow. However, a non-central collision has none of the mentioned symmetries. The purpose of this Letter is to present and analyze hydro solutions for such cases. The results presented in this Letter may be utilized to access the time-evolution of the hydrodynamically behaving, strongly interacting matter as probed by noncentral non-relativistic heavy ion collisions [8,9]. As the hydro equations have no intrinsic scale, the results are rather general in nature and can be applied to any physical phenomena where the non-relativistic hydrodynamical description is valid. 0370-2693/01/$ – see front matter 2001 Published by Elsevier Science B.V. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 3 5 1 - 3 S.A. Akkelin et al. / Physics Letters B 505 (2001) 64–70 2. Generalization of spherical solutions to elliptic flows Consider an ideal fluid, where viscosity and heat conductivity are negligible, described by the local conservation of matter (continuity equation), the local conservation of energy (energy equation) and the local conservation of momentum (Euler equation): ∂n + ∇(vn) = 0, (1) ∂t ∂ε + ∇(vε) = −P ∇v, (2) ∂t ∂ mn (3) + v∇ v = −∇P , ∂t where m is the mass of a single particle and n = n(t, r), v = v(t, r), P = P (t, r) and ε = ε(t, r) are the local density of the particle number, the velocity, the pressure and the energy density fields, respectively. To complete the set of equations we need to fix the equations of state. For reasons of simplicity we have chosen the equations of state to be those of an ideal, structureless Boltzmann gas: ε(t, r) = 32 P (t, r), (4) P (t, r) = n(t, r)T (t, r). (5) The solutions that are presented in the subsequent parts cannot be trivially generalized to any arbitrary equations of state. Nevertheless, they provide a transparent insight into the collective physical processes in a non-central heavy ion collision. In the following, we utilize the above ideal gas equation of state and rewrite the hydrodynamical equations in terms of the three functions n,v and T . In Ref. [6,7], special classes of exact analytic solutions of fireball hydrodynamics were found assuming spherical symmetry and self-similar Hubble flows. In Ref. [6] a homogeneous temperature profile was assumed, while the general solution for arbitrary, inhomogeneous initial temperature profiles was found in Ref. [7]. In these articles, the concept of self-similarity meant that there is a typical length-scale of the expanding system R = R(t) so that all space–time functions in the hydro equations are of the form F = G(t)H (s) where s = r 2 /R 2 is the so-called (dimensionless) scaling variable. Let us go beyond spherical symmetry and consider three typical lengths of the expanding system: X, Y 65 and Z, all functions of time only. Let us rotate our frame of reference to the major axis of the ellipsoidal expansion, and leave to future applications to relate these major axes to the laboratory frame. Consequently, let us introduce three scaling variables x = rx2 /X2 , y = ry2 /Y 2 and z = rz2 /Z 2 and assume that all space–time functions are of the form of F = G(t)H (x)K(y)L(z). Using this ansatz we find that the continuity equation is satisfied regardless of the density profile if the velocity field is a Hubble-flow field in each principal direction: Ẋ(t) rx , X(t) Ż(t) vz (t, r) = rz . Z(t) vx (t, r) = vy (t, r) = Ẏ (t) ry , Y (t) (6) Although our sole assumption concerning the temperature was the ansatz form already mentioned, we found that the Euler equation requires the temperature to be homogeneous, independent of the coordinate variables: T = T (t). The energy equation is only satisfied if V0 2/3 , T (t) = T0 (7) V (t) where V (t) = X(t)Y (t)Z(t) is the typical volume of the expanding system, while V0 = V (t0 ) and T0 = T (t0 ) are the initial temperature and volume. The homogeneity of the temperature and the Euler equation implied that the density profile is a product of three Gaussians, with different, time dependent radius parameters: ry2 r2 V0 exp − x 2 − n(t, r) = n0 V (t) 2X(t) 2Y (t)2 rz2 − (8) , 2Z(t)2 where n0 = n(0, 0) can be expressed by the total number of particles (N ) as n0 = N 1 . (2π)3/2 V0 (9) The time evolution of the scales are determined (through the Euler equation) by the equations T0 V0 2/3 ẌX = Ÿ Y = Z̈Z = (10) . m V 66 S.A. Akkelin et al. / Physics Letters B 505 (2001) 64–70 This system of non-linear, second-order ordinary differential equations has a unique solution for the scalefunctions if the initial parameters X0 , Y0 , Z0 and Ẋ0 , Ẏ0 , Ż0 are given. Although this solution has not yet been found in an explicite, analytic form, some of its properties are determined in the subsequent parts. 3. Properties of the elliptic solutions Global conservation laws reflect, in general, boundary conditions for solutions, or their behavior at asymptotically large distances. Because of reflection symmetry of densities and velocities, the conservation of momentum, P = 0, is satisfied automatically and gives no non-trivial first integral. On the other hand, the asymptotically fast decreasing of the densities give us the possibility of using total energy conservation as nmv 2 ∂ 3 d r ε+ = 0, (11) ∂t 2 to find the first integral of the system of Eqs. (1)–(3). Substitute (6), (8) into (11), we get: T0 V0 2/3 = A = const. Ẋ2 + Ẏ 2 + Ż 2 + 3 (12) m V Using (10) one can rewrite (12) in the form 1 ∂2 2 X + Y 2 + Z 2 = A, 2 ∂t 2 and find finally (13) R 2 (t) = X2 (t) + Y 2 (t) + Z 2 (t) = A(t − t0 )2 + B(t − t0 ) + C, (14) where m 2 3 m 2 2 = Ẋ + Ẏas2 + Żas Ẋ + Ẏ02 + Ż02 + T0 . 2 as 2 0 2 (17) This relation expresses the equality of the initial flow and internal energy with the asymptotic energy which is present in the form of flow. Although Eqs. (10) are easy to handle with presently available numerical packages, we note that a further simplification of these equations to a non-linear first order differential equation of one variable is possible, if an additional cylindrical symmetry is assumed, corresponding to X(t) = Y (t). One may introduce the angular variable φ = arccos(Z/R) so that 1 X(t) = Y (t) = √ R(t) sin φ(t), 2 Z(t) = R(t) cos φ(t). T0 , m B = 2(X0 Ẋ0 + Y0 Ẏ0 + Z0 Ż0 ), A = Ẋ02 + Ẏ02 + Ż02 + 3 C = X02 + Y02 + Z02 . (Px , Py , Pz ) = m(Ẋ, Ẏ , Ż) and the Hamiltonian H as a rewritten form of Eq. (12): 3 X0 Y0 Z0 2/3 1 2 2 2 . H= P + Py + Pz + T0 2m x 2 XY Z (16) The Hamiltonian equations of motion can be written in terms of Poisson brackets as Ẋ = {X, H }, . . . , P˙x = {Px , H }, . . . . The Lagrangian form of these equations is given by Eqs. (10). Due to the repulsive nature of the potential, the coordinates (X, Y, Z) diverge to infinity for large times. As the potential vanishes for large values of the coordinates, the canonical momenta tend to constant values for asymptotically large times. Eq. (12) expresses the conservation of whole energy (kinetic and potential) of the “particle”, corresponding to H (X, . . . , Px , . . .) = E = const. The resulting Eq. (14) has also great importance for the analysis of approximate analytical solutions. It is worth mentioning another interesting relation that one can get from (12) for asymptotic times tas , when V (tas ) V0 : (15) The simple equation (14) express the general property of elliptic hydrodynamic flows. The value of radius-vector evolves in time similar to a “particle” with coordinates (X, Y, Z) that moves in a noncentral, repulsive potential according to Eqs. (10). In particular, one may introduce the canonical coordinates (X, Y, Z) and the canonical momenta as (18) (19) The time evolution of φ(t) is determined by the following first order equation: 1 3 T0 (X02 Z0 )2/3 E φ̇ 2 = 2 − Ṙ 2 (t) − 2 m R 2 (t) R (t) m 1 (20) × , (sin φ)4/3 (cos φ)2/3 where R(t) is given explicitly by Eq. (14). S.A. Akkelin et al. / Physics Letters B 505 (2001) 64–70 Fig. 1. Top panel shows the time dependence of the major axis of the ellipsoidal (Gaussian) density profile, Eq. (8), while the bottom panel indicates the velocity of expansion at the rms radii, rx = X(t), ry = Y (t) and rz = Z(t), for three-dimensional, self-similar elliptic flows. Solid lines stand for X(t) and vx (t), dashed lines for Y (t) and vy (t), short-dashed lines for Z(t) and vz (t). The initial conditions are: X0 = 4 fm, Y0 = 3 fm, Z0 = 1/4 fm, the initial velocities are all vanishing, T0 /m = 0.1. Due to the Landau-like initial condition (strong initial compression in the z direction), the flow is almost one-dimensional, only small amount of transverse flow is generated in this case. Figs. 1–4 indicate the results of numerical solutions of Eqs. (10). Using Landau-type initial conditions, one confirms that even the full three-dimensional solution results in a small amount of transverse flow generation, while for more general initial conditions, significant amount of transverse flow can be generated. Transverse flow is stronger if the initial conditions are closer to spherical symmetry or, if the fraction of the initial thermal energy is increased as compared to the initial kinetic energy. For more details, see the figure captions. In the last part an approximate, analytic solution is presented that corresponds to Landau-like, onedimensional expansions. 67 Fig. 2. Same as Fig. 1, but for a more spherical initial profile with an initial inwards flow in the z direction. The initial conditions are: X0 = 4 fm, Y0 = 3 fm, Z0 = 3 fm, the initial velocities are Ẋ0 = 0, Ẏ0 = 0, Ż0 = −0.5, while T0 /m = 0.1. Due to the deviation from the Landau-like initial condition, the final flow is almost spherically symmetric, three-dimensional, and a large amount of transverse flow is generated in this case. compressed “ellipsoid”), corresponding to the real situation in non-relativistic heavy ion collisions: Ẋ0 = Ẏ0 = 0, Z0 X0 , Z0 Y0 , (21) and in general case Ż0 = 0. The last reflects the situation when a system is (locally) thermalized before, after or at the moment of full nuclear stopping and transverse expansion starts to develop only after the local thermalization. Then in some time interval t0 t t˜ the hydro evolution is quasi-one-dimensional: X(t) ≈ 1, X0 Y (t) ≈ 1, Y0 (22) and the equation of motion for Z(t) takes the following form T0 Z0 2/3 . Z̈Z = (23) m Z 4. Approximate one-dimensional solutions This equation has an exact analytic solution, 2/3 + (a+ + a− )2 3 , Z 2 (t) = Z 0 Hydrodynamical evolution, which is described by Eqs. (1)–(3), starts from some initial conditions. Consider Landau-type initial conditions (longitudinally where 2 1 2 1/2 1/3 ± Z 0 + Z (t) , a± = 12 Z(t) 4 (24) 68 S.A. Akkelin et al. / Physics Letters B 505 (2001) 64–70 Fig. 3. Same as Fig. 1, but for T0 /m = 0.01. This corresponds to increasing the initial kinetic energy as compared to the internal (thermalized) energy, and in this case the flow becomes approximately one-dimensional again. The initial conditions are: X0 = 4 fm, Y0 = 3 fm, Z0 = 3 fm, the initial velocities are Ẋ0 = 0, Ẏ0 = 0, Ż0 = −0.5, while T0 /m = 0.01. = Z(t) 0 = Z 3u2 t − t˜0 , T0 mu2 3/2 Z0 , 1 T0 2 2 , u = Ż + 3 3 0 m Z0 Ż0 2 T0 t˜0 = t0 − + u . 3u4 m (25) Here t˜0 is the turning point for Z 2 (t) if Ż0 < 0. Using (14) we obtain that the conditions for validity of the solution (22), (24) are satisfied within some time interval t0 t t˜ if A(t − t0 )2 + B(t − t0 ) + Z02 − Z 2 (t) 1. (26) X02 + Y02 The hydrodynamical evolution described by the Eqs. (1)–(3) cannot be continued infinitely in time, because the general criteria of applicability of hydrodynamical description are violated: the mean free path l ∝ 1/(σ n) has to be (much) smaller than the typical length scales of the system, for example the effective geometrical sizes or hydrodynamic lengths. Due to the hydrodynamical expansion the density (8) will decrease with time reaching some critical value that can be estimated utilizing Landau’s criterium, T = mπ , that determines a critical density when hydrodynamic evolution breaks up. Here, we will use a simplified ver- Fig. 4. A more general initial condition may result even in a dominantly two-dimensional expansion. The initial conditions are: X0 = 4 fm, Y0 = 3 fm, Z0 = 3 fm, the initial velocities are Ẋ0 = −0.2, Ẏ0 = 0.1, Ż0 = −0.5, while T0 /m = 0.05. sion of this criterium and suppose the decoupling of hydrodynamical system when density in the center of the system reaches some critical value nf (typically, normal nuclear density). Then the time tf , when the hydrodynamical evolution ends, can be estimated from the condition n(tf , 0) = nf . (27) If the hydrodynamical evolution stops before the condition (26) is violated, tf < t˜, then solutions (22) and (24) describing quasi-one-dimensional expansion give complete hydrodynamical evolution, too. Let us find the conditions for such a situation. Supposing quasi-one-dimensional expansion and using (8) and (22) we get Z(tf ) n0 = . Z0 nf (28) Then using (24) we can find tf . If for t = tf the inequality (26) is satisfied, then we can conclude that tf < t˜ and the one-dimensional expansion is valid approximation until the freeze-out time. It is useful to give simple analytical estimations of the initial hydrodynamic conditions that guarantee quasi-one-dimensional expansion of the nuclear matter. Let us suppose, for simplicity, that Ż0 = 0 and 0 = Z0 , t˜0 = t0 in (24). Supposing hence u2 = Tm0 , Z S.A. Akkelin et al. / Physics Letters B 505 (2001) 64–70 that the upper time limit of quasi-one-dimensional expansion, t˜, is large enough so that 2 ( t˜ ) 4Z02 , Z we can rewrite Eq. (24) in the form: 2 ( t˜ ) 2/3 . 2 ( t˜ ) − 3 Z0 Z Z 2 ( t˜ ) ≈ Z After substitution (29) in (26) we obtain: X02 + Y02 3 Tm0 (t˜ − t˜0 )2 1 . √ Z0 X02 + Y02 3 3 It is easy to see from (30) that for large enough (29) (30) X02 +Y02 Z02 , quasi-one-dimensionality can hold till the longitudinal scale becomes comparable to transversal ones: 2 ( t˜ ) Z 2 ( t˜ ) Z Z 2 ( t˜ ) ≈ ∝ 1. ≈ X2 ( t˜ ) + Y 2 ( t˜ ) X02 + Y02 X02 + Y02 (31) It means that within time t < t˜ solution (22), (24) correctly describes the transformation of longitudinally compressed “ellipsoid” to “spheroid”form. Finally from (28) we get that Z(tf ) < Z( t˜ ) ∝ X02 + Y02 and consequently tf < t˜ if X02 + Y02 n0 (32) < . nf Z0 Under such initial conditions one can expect that whole stage of the hydrodynamical evolution can be correctly described by the approximate quasi-onedimensional solution (22) and (24). 5. Summary and conclusions In this Letter we considered the time evolution of fireball hydrodynamics describing an ideal gas, an elliptic initial density profile, a homogeneous temperature distribution and a Hubble-like flow distribution. For this case, the set of partial differential equations of non-relativistic hydrodynamics have been reduced to a set of ordinary, second order, non-linear differential equations, that can be solved efficiently by presently available numerical packages without the need of sophisticated programming. The initial conditions for these equations are associated with the initial elliptic 69 sizes X0 , Y0 and Z0 , that could be linked with the overlapping geometrical sizes of colliding nuclei in heavy ion collisions and with the dynamics of the compression process during interactions during the pre-thermal time evolution. The general behavior of these hydrodynamical equations is determined analytically and related to the Hamiltonian motion of a particle in a repulsive, noncentral potential. A first integral of motion has been found, corresponding to the conservation of energy in the Hamiltonian problem. It was utilized to obtain an approximate solution for quasi one-dimensional expansions and to determine the domain of applicability of this solution. The importance of the results is given by recent experimental findings in high energy heavy ion reactions, where various elliptic flow patters are observed, see Refs. [10–13] for further details. In future studies, our results could be applied to gain insight into the interpretation of the above mentioned data and to describe nucleus–nucleus collisions with non-relativistic initial energies. Such conditions for a non-relativistic evolution may be reached in the mid-rapidity region near to the softest point of equation of state even in relativistic heavy ion collisions, if the pressure is not strong enough to build up a relativistic transverse flow. Due to the scale invariance of the hydrodynamical equations the solutions described here can also be utilized in other problems related to elliptic flows in nonspherical fireball hydrodynamics. Acknowledgements This research has been supported by a Bolyai Fellowship of the Hungarian Academy of Sciences and by the grants OTKA T024094, T026435, T029158, the US–Hungarian Joint Fund MAKA grant 652/1998, NWO-OTKA N025186, Hungarian–Ukrainian S&T grant 45014 (2M/125-199) and the grants FAPESP 98/2249-4 and 99/09113-3 of Sao Paolo, Brazil. References [1] L.P. Csernai, Introduction to Relativistic Heavy Ion Collisions, Wiley, 1994. [2] D.H. Rischke, Nucl. Phys. A 610 (1996) 88c. 70 S.A. Akkelin et al. / Physics Letters B 505 (2001) 64–70 [3] L.D. Landau, Izv. Akad. Nauk SSSR 17 (1953) 51; L.D. Landau, in: D. Ter-Haar (Ed.), Collected Papers of L.D. Landau, Pergamon, Oxford, 1965, pp. 665–700. [4] J.D. Bjorken, Phys. Rev. D 27 (1983) 140. [5] J. Bondorf, S. Garpman, J. Zimányi, Nucl. Phys. A 296 (1978) 320–332. [6] P. Csizmadia, T. Csörgő, B. Lukács, Phys. Lett. B 443 (1998) 21–25. [7] T. Csörgő, nucl-th/9809011. [8] T. Csörgő, B. Lörstad, J. Zimányi, Phys. Lett. B 338 (1994) 134–140. [9] J. Helgesson, T. Csörgő, M. Asakawa, J. Zimányi, Phys. Rev. C 56 (1997) 2626–2635. [10] A.M. Poskanzer et al., Nucl. Phys. A 661 (1999) 341. [11] K.H. Ackermann et al., nucl-ex/0009011. [12] L.P. Csernai, D. Rohrich, Phys. Lett. B 458 (1999) 454. [13] S.A. Voloshin, A.M. Poskanzer, Phys. Lett. B 474 (2000) 27. 26 April 2001 Physics Letters B 505 (2001) 71–74 www.elsevier.nl/locate/npe 12 Be molecular states in a microscopic cluster model P. Descouvemont 1 , D. Baye Physique Nucléaire Théorique et Physique Mathématique, CP229 Université Libre de Bruxelles, B1050 Brussels, Belgium Received 21 December 2000; accepted 19 February 2001 Editor: J.-P. Blaizot Abstract The 12 Be spectrum is investigated in the Generator Coordinate Method, using microscopic 6 He + 6 He and α + 8 He wave functions. The model is consistent with recent experimental observations of molecular states, but predicts a strong mixing of both configurations, rather than a dominant 6 He + 6 He structure. A negative-parity band is also found. Electromagnetic transition probabilities and partial widths of molecular states are calculated. 2001 Elsevier Science B.V. All rights reserved. Neutron-rich Be isotopes have been extensively studied in recent years [1]. The Borromean nature of 9 Be and 10 Be is responsible for many interesting properties and has been investigated by many authors (see Ref. [2] for recent works). On the other hand, the 11 Be nucleus has attracted much interest because of the well known parity-inversion effect. More recently, experiments aiming at investigate excited states of Be isotopes have been developed with radioactive beam facilities [3–6]. Exotic structures have been found in 9–12 Be where molecular states appear at high energies. New 12 Be states have been observed at Ex = 8.6, 10 and 14 MeV by Korsheninnikov et al. in a p + 12 Be experiment. A 12 Be breakup experiment by Freer et al. [4] indicates the existence of excited states with a significant decay to the α + 8 He and 6 He + 6 He channels. These results lead Freer et al. to the suggestion of 6 He + 6 He molecular states, essentially based on the shape of the rotational band. These new states were subsequently supported by E-mail address: [email protected] (P. Descouvemont). 1 Directeur de Recherches FNRS. Bohlen et al. [5] in a 9 Be(15 N,12 N)12 Be experiment. These authors observe highly excited states in 12 Be and find out that their energies are in good alignment in a J (J + 1) diagram. They could not, however, measure the widths and predict the structure of these states; spin assignments are only tentative. A systematic study of Be isotopes by von Oertzen [7] predicts the existence of rotational bands presenting a strong α clustering. The α clustering is well known in the ground states of 7 Be and 8 Be and may also show up in excited states of heavier isotopes. The antisymmetrized molecular dynamics (AMD) model [8] also suggests molecularlike states in light nuclei, and particularly in Be neutron-rich isotopes. On the other hand, the existence of 6 He + 6 He molecular states has been considered by Ito and Sakuragi [9] in a semi-microscopic coupled-channel calculation. These authors derive an 6 He + 6 He interaction from folding densities and analyze the structure of 12 Be resonances; the α + 8 He configuration, open at the threshold, is however not considered in that work. In the present Letter, we investigate cluster states of 12 Be in the Generator Coordinate Method (GCM) [10]. In this microscopic method, the 12-nucleon hamil- 0370-2693/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 3 4 9 - 5 72 P. Descouvemont, D. Baye / Physics Letters B 505 (2001) 71–74 tonian reads H= 12 i=1 Ti + 12 Vij , (1) i<j =1 where Ti is the kinetic energy of nucleon i, and Vij an effective two-body interaction. Approximate solutions of the Schrödinger equation associated to (1) are obtained in the cluster formalism, which assumes that the 12 nucleons are distributed into two clusters. We consider here two clustering modes: 6 He + 6 He and α + 8 He; both channels are close to each other (the thresholds are located at 8.95 MeV for 6 He + 6 He and 10.11 MeV for α + 8 He). The GCM wave functions in partial wave J π are, therefore, defined as Jπ Ψ J Mπ = Aφ4 φ8 YJM (ρ̂ 4−8 )g4−8 (ρ4−8 ) Jπ (ρ6−6 ), + Aφ6 φ6 YJM (ρ̂ 6−6 )g6−6 (2) n He, where φn corresponds to the wave function of J π to the radial function depending on the relagi−j tive coordinate ρ i−j . Here and in the following, subscripts 4–8 and 6–6 refer to quantities related to the α + 8 He and 6 He + 6 He configurations, respectively. In (2), A is the 12-nucleon antisymmetrizer which ensures the Pauli principle to be exactly taken into account. The internal wave functions φn are built in the one-center harmonic-oscillator model with a full s shell for α and (p 32 )2 and (p 32 )4 configurations for 6 He and 8 He, respectively. It is well known that 6 He presents a halo structure and that accurate wave functions should be more extended than one-center shellmodel wave functions. However, recent microscopic investigations using shell-model [11] or more elaborated 3-cluster functions [12] show that α + 6 He scattering properties at low energies are weakly dependent on the 6 He description. The present model has been already applied to the α + 8 He elastic scattering [11]. In Ref. [11], we analyzed GCM α + 8 He phase shifts and derived equivalent nucleus–nucleus potentials. From that work, we have suggested the existence of 0+ and 0− molecular bands, whose bandheads are located close to the α + 8 He threshold. To investigate the possible existence of 6 He + 6 He molecular states suggested by Freer et al. [4], we extend here this previous calculation by including the 6 He + 6 He channel which, of course, must be taken into account for a reliable investigation of 6 He + 6 He states. Since this channel is symmetric and involves zero-spin nuclei, it only affects positive-parity states. No change is expected for the conclusions on negative-parity resonances. A special attention must be paid to the choice of the nucleon–nucleon force. With standard interactions such as the Volkov [13] or Minnesota [14] potentials, the 6 He + 6 He threshold is overestimated by a few MeV. This problem arises from the shell-model description of 6 He. Since 6 He + 6 He calculations involving 3-cluster wave functions of 6 He are currently not possible, it is necessary to compensate this drawback by adapting the nucleon–nucleon force. As it was done in the past for similar studies (see, for example, Ref. [15]), we use the Volkov V 2 force, with additional Bartlett and Heisenberg terms. This introduces some flexibility and enables us to reproduce the 6 He + 6 He and α + 8 He thresholds simultaneously. The central force is defined by parameters w = 0.55, m = 0.45, b = −h = −0.2374, and is complemented by a zero-range spin–orbit force [16] with amplitude S0 = 30 MeV fm5 . As in Ref. [11], the oscillator parameter is chosen as b = 1.65 fm for α, 6 He and 8 He. The radial functions gi−j (ρ) are expanded over a set of shifted gaussian functions which allows one to write (2) as a linear combination of projected Slater determinants. The calculation of resonance properties (energy, partial widths) is performed in the microscopic R-matrix formalism [17]. In Fig. 1, we present the energy spectrum obtained in 3 different conditions: single-channel α + 8 He or 6 He + 6 He, and the two-channel calculations. Here and in the following, energies are expressed with respect to the α + 8 He threshold. In spite of different clustering assumptions, the three spectra are fairly similar to each other for positive-parity states. This can be understood from the energy curves displayed in Fig. 2. Energy curves give the energy of the system for a fixed distance between the clusters (see Ref. [16] for details). The 0+ and 2+ energy curves present a minimum at small distance, which corresponds to shell-model states. The r.m.s. radii of the low-lying 0+ and 2+ states are about 2.5 fm (see Table 1), characteristic of mass-12 nuclei; beyond J π = 2+ , the energy curves only present a shallow minimum near 3 fm. A striking feature of Fig. 2 is that both configurations give very close energies, the difference being essentially due to the threshold. This means that both bases should describe the same 12 Be states P. Descouvemont, D. Baye / Physics Letters B 505 (2001) 71–74 73 Table 1 Energies (in MeV), r.m.s. radii (in fm), dimensionless reduced widths (in %, at 6 fm) and partial widths (in MeV) of 12 Be states 2 2 Jπ Ecm (r 2 ) θ4−8 θ6−6 Γ4−8 Γ6−6 0+ −8.70 2.55 1.3 0.4 0 0 2+ −5.97 2.52 1.3 0.4 0 0 0+ −1.85 2.97 19.1 9.4 0 0 2+ −0.26 3.03 21.8 9.7 0 0 4+ 3.03 3.50 18.7 12.0 0.26 0.02 17.6 13.4 0.81 0.55 6+ 11.5 1− −0.34 3.06 27.3 – 0 – 3− 1.30 2.92 18.0 – 0.03 – 5− 3.62 2.58 – 0.01 – 3.10 Fig. 1. 12 Be spectra with different conditions of calculation: single-channel α + 8 He or 6 He + 6 He, and two-channel. E2 transition probabilities (in WU) are given for the two-channel calculation. Fig. 2. Energy curves of the α + 8 He (full curves) and 6 He + 6 He (dashed curves) systems. R is the distance between the clusters. and that states with a strong 6 He + 6 He clustering are unlikely. Each spectrum of Fig. 1 shows a 0+ rotational band. Such a band was already predicted in the single-channel calculation of Ref. [11], and no further band is found in the present model. We confirm the existence of a negative-parity band starting near the α + 8 He threshold. The existence of such a band is well established in 10 Be [6], and confirmed by similar calculations [11,12]. In Table 1, we gather some spectroscopic properties of 12 Be: r.m.s. radii, dimensionless reduced widths θ 2 , and partial widths for resonances. The 0+ and 2+ lowlying states present θ 2 values of the order of 1%, characteristic of weakly-deformed states. On the contrary, the states belonging to the molecular bands present radii of about 3 fm, and θ 2 values larger than 10%. In positive-parity, the strong mixing between the α + 8 He and 6 He + 6 He channels is confirmed by the analysis of the partial widths. The reduced widths in both channels are comparable, and it is not possible to assign a definite α + 8 He or 6 He+ 6 He cluster structure to these states. The 0+ and 2+ states are predicted to be bound. For J π = 4+ , the difference of the Q values makes the 6 He+ 6 He partial width significantly lower than the total width (Γ6−6 /Γ = 0.07), in spite of similar reduced widths. The 6+ member is suggested to have similar widths in both channels. In Fig. 1, we also give E2 reduced transition probabilities for the two-channel calculation. The B(E2) + value between the 2+ 1 and 01 states is 6.6 WU, whereas transition probabilities between molecular states are strongly enhanced. This effect is well known, and arises from a similar deformed structure of the states. Strong E2 transitions are a possible way to observe a molecular band; this technique has been used in the past for investigating 12 C + 12 C molecular resonances [18]. Notice that low transition probabilities are expected between states of different bands. This is 74 P. Descouvemont, D. Baye / Physics Letters B 505 (2001) 71–74 band in the 12 Be spectrum. We find that both configurations are nearly equivalent and, consequently, that molecular states can not be considered as 6 He + 6 He states, but as a mixing of 6 He + 6 He and α + 8 He configurations. As for 10 Be [11,12], the GCM suggests a negative-parity band involving narrow resonances with a pure α + 8 He structure, and not observed yet. The head of this band is found near the α + 8 He threshold. Fig. 3. 12 Be states predicted by the GCM (the width is indicated by a vertical bar) and experimental data of Refs. [4,5]. exemplified here with the B(E2) value between the 2+ molecular state and the ground state (1.4 WU). The present results are summed up in the J (J + 1) diagram of Fig. 3. We plot the GCM states with the experimental data of Freer et al. [4] and of Bohlen et al. [5]. As discussed previously, the GCM positive-parity states can not be assigned to a definite 6 He + 6 He structure, but to a mixing of 6 He + 6 He and α + 8 He configurations. In their breakup experiment, Freer et al. [4] observe 4+ , 6+ and 8+ states in the 6 He + 6 He and α + 8 He channels with rather similar energies. The difference being lower than the energy resolution, the configuration mixing predicted by the GCM seems to be supported by the data. The experiment of Bohlen et al. [5] concludes that the observed 12 Be states are deformed, but can not separate the α + 8 He and 6 He + 6 He channels. For J 6, the theoretical energies deviate form a rotational behaviour (for J = 8+ the GCM provides a 10 MeV broad resonance near 30 MeV). This can be explained from two reasons: (i) for large J values, energies are above several open channels which are not included; (ii) the halo structure of 6 He could play a more important role in high-spin states. Further calculations are therefore necessary to improve the theoretical description of these states. In summary, we have performed a microscopic calculation of 12 Be excited states, with α + 8 He and 6 He + 6 He cluster structures. Our results are consistent with recent experiments suggesting a molecular Acknowledgement This Letter presents research results of the Belgian Program P4/18 on interuniversity attraction poles initiated by the Belgian-state Federal Services for Scientific Technical and Cultural Affairs. References [1] H.G. Bohlen et al., Prog. Part. Nucl. Phys. 42 (1999) 17. [2] K. Arai, Y. Ogawa, Y. Suzuki, K. Varga, Phys. Rev. C 54 (1996) 132; Y. Ogawa, K. Arai, Y. Suzuki, K. Varga, Nucl. Phys. A 673 (2000) 122. [3] A.A. Korsheninnikov et al., Phys. Lett. B 343 (1995) 53. [4] M. Freer et al., Phys. Rev. Lett. 82 (1999) 1383. [5] H.G. Bohlen et al., in: B. Rubio et al. (Eds.), Proc. Int. Conf. Experimental Nuclear Physics in Europe, Seville, Spain, AIP Conf. Proc., Vol. 495, 1999, p. 303. [6] M. Freer et al., Phys. Rev. C, in press. [7] W. von Oertzen, Z. Phys. A 357 (1997) 355. [8] H. Horiuchi, Y. Kanada-En’yo, Nucl. Phys. A 616 (1997) 394c. [9] M. Ito, Y. Sakuragi, Phys. Rev. C 62 (2000) 064310. [10] K. Langanke, Adv. Nucl. Phys. 21 (1994) 85. [11] D. Baye, P. Descouvemont, R. Kamouni, Few-Body Systems 29 (2000) 131. [12] K. Fujimura, D. Baye, P. Descouvemont, Y. Suzuki, K. Varga, Phys. Rev. C 59 (1999) 817. [13] A.B. Volkov, Nucl. Phys. 74 (1965) 33. [14] D.R. Thompson, M. LeMere, Y.C. Tang, Nucl. Phys. A 286 (1977) 53. [15] P. Descouvemont, D. Baye, Phys. Rev. C 36 (1987) 1249. [16] D. Baye, N. Pecher, Bull. Cl. Sc. Acad. Roy. Belg. 67 (1981) 835. [17] D. Baye, P.-H. Heenen, M. Libert-Heinemann, Nucl. Phys. A 291 (1977) 230. [18] R.L. Mc Grath, D. Abriola, J. Karp, T. Renner, S.Y. Zhu, Phys. Rev. C 24 (1981) 2374. 26 April 2001 Physics Letters B 505 (2001) 75–81 www.elsevier.nl/locate/npe Shape coexistence and tilted-axis rotation in neutron-rich hafnium isotopes Makito Oi a,b , Philip M. Walker a , Ahmad Ansari c a Department of Physics, University of Surrey, Guildford, Surrey GU2 7XH, United Kingdom b Department of Applied Physics, Fukui University, 3-9-1 Bunkyo, Fukui 910-8507, Japan c Institute of Physics, Bhubaneswar 751 005, India Received 6 October 2000; received in revised form 1 February 2001; accepted 1 March 2001 Editor: J.-P. Blaizot Abstract We have performed tilted-axis-cranked Hartree–Fock–Bogoliubov calculations for a neutron-rich hafnium isotope (182 Hf) whose proton and neutron numbers are both in the upper shell region. We study whether the shell effects play a role in producing high-K isomers or highly gamma-deformed states at high spin. In particular, the possibility of shape coexistence and the effect of wobbling motion are discussed. 2001 Elsevier Science B.V. All rights reserved. PACS: 27.70.+q; 21.10.-k Keywords: High-K states; Gamma deformation; Wobbling motion; Tilted axis cranking model Hafnium isotopes (Z = 72) are best known as nuclei that have high-K isomers (e.g., the K π = 16+ isomer in 178Hf, with a half life t1/2 = 31 yr [1]). From a viewpoint of the Nilsson model, a reason is that proton single-particle levels are filled up to the upper part of the shell where there are many high-Ω states at prolate deformation (Ω is the angular momentum projection on the nuclear symmetry axis). The presence of longlived high-K isomers indicates the existence of axial symmetry to make K = i Ωi , a good quantum number. With recent developments of experimental techniques, such as fragmentation [2] and deep-inelastic reactions [3] in populating high-spin states, the study of nuclei in the well deformed rare-earth region moves away from the β-stable line towards the neutron-rich isotopes. D’Alarcao et al. recently discovered several E-mail address: [email protected] (M. Oi). high-K isomers in the neutron-rich hafnium isotope of 182 Hf [4]. For neutron-rich hafnium isotopes, whose neutron Fermi level is located in a similar position to the proton one (i.e., the upper half of the shell), we can expect an even more important role for high-K isomeric states [5]. At the same time, however, the empty nucleon states near the Fermi surfaces can be considered as hole states, and these may induce substantial gamma deformation which breaks the axial symmetry. A few theoretical investigations of high angular momentum collective states have been reported for such neutron-rich hafnium isotopes: by means of a microscopic method (cranked Hartree–Fock–Bogoliubov method, or C-HFB), the possibility of a collective oblate deformation at high-spin (I = 26h̄) in 180 Hf was predicted [6]; by means of a macroscopic–microscopic method (total routhian surface calculations, or TRS), the existence of non-collective prolate-deformed states becoming yrast, i.e., lowest in energy, for I 0370-2693/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 3 5 0 - 1 76 M. Oi et al. / Physics Letters B 505 (2001) 75–81 10h̄ was shown in 182–186Hf [7]. In the latter study, the collective oblate deformed states become yrast when quite high in spin (I = 36h̄ in 182 Hf). In both of the studies, the rotational states are treated only by the one-dimensional cranking model, but recent developments in the tilted-axis-cranking model [8–10] allow for the possibility of further investigation of these neutron-rich hafnium isotopes. In this Letter, in a microscopic framework of the two-dimensional tilted-axis-cranked HFB method, we analyze the high-spin structures predicted for the neutron-rich hafnium isotope 182 Hf (which has N/Z = 1.53). First, the calculational procedure is briefly described. We solve the HFB equations with a 2dcranking term self-consistently for 182 Hf, following the method of steepest-descent [8]. Nucleon numbers and the total angular momentum are constrained durτ = Nτ (τ = p, n); Jx = Iˆx = ing the iterations; N J cos θ ; Jy = Iˆy = 0; Jz = Iˆz = J sin θ . (Note that the tilt angle θ is measured from the x-axis to see the deviation of the rotation axis from the x-axis. In this study, it corresponds to a principal axis of the quadrupole deformation, perpendicular to the symmetry axis that is chosen to be the z-axis.) We also constrain off-diagonal components of the quadrupole tensor such that the intrinsic coordinate axes coincide with the principal axes of the quadrupole deformation. See Ref. [8] for details. The one-dimensional cranking calculations (or 1dcranking calculations), in which the rotation axis is fixed to be along the x-axis (θ = 0◦ ), are performed as follows. First for J = 0, the Nilsson + BCS state is taken as a trial state. Then cranked HFB states, Ψu (J ), are calculated up to J = 40h̄ with increments J = 0.1h̄. This way of calculation is what we call “upcranking” calculations. Then, using the up-cranking solution at J = 40h̄ as a trial state, “down-cranking” calculations are similarly performed from J = 40h̄ down to J = 0 to obtain Ψd (J ). The calculations of tilted-axis-cranked HFB states (2d-cranked states, or Ψ t (J, θ )) are carried out by starting at a state Ψ t (J, θ = 0◦ ), which are calculated through an up-cranking calculation, and performing a “forward” tilting calculation up to θ = 90◦ with increments θ = 0.5◦ for each (integer) J . Then in a similar manner we make “backward” tilting calculations from θ = 90◦ to θ = 0◦ . At certain tilt angles, the forward and backward tilting results do show interesting differences, particularly, at high spins. Our Hamiltonian consists of a spherical part (H0 ) and a residual part (Vres ) employing the pairingplus-Q · Q interaction. In the hafnium isotopes the hexadecapole interaction can be important, but we have checked that our results below are not affected very much by including the interaction. Thus we omit it in the present study. The Hamiltonian is thus written H = H0 + Vres = N(τ ) τ =p,n i=1 − τ =p,n 2 1 † Qµ Qµ i ci† ci − κ 2 Gτ Pτ† Pτ , µ=−2 (1) in which i means a spherical Nilsson level and i runs all over the model space. Force parameters κ and Gτ are determined in the framework of the Nilsson + BCS model by giving input parameters for the quadrupole deformation (β ini, γ ini) and gap energy (∆ini τ ). (Our definition of γ is taken from p. 8 in Ref. [11], and is opposite to the Lund convention in the sign.) In this paper, we employ a set of the input parameters based on the calculations by Möller et al. [12] because we find it gives a good agreement with experimental ini data. For 182 Hf, we employ ∆ini p = 0.725 MeV; ∆n = ini ◦ ini 0.625 MeV; γ = 0 ; β = 0.270 or 0.268. We will see soon the reason for the two values of β ini . Our single-particle model space is almost the same as the choice of Kumar and Baranger [13] (two major shells in the spherical Nilsson model: N = 4, 5 for proton and N = 5, 6 for neutron), with two extra single-particle orbits, proton i13/2 and neutron j15/2 . The single particle energies are the spherical Nilsson model energies with A-dependent Nilsson parameters [14]. First of all, let us discuss a principal axis rotation (θ = 0◦ ) in 1d- and 2d-cranking calculations. Fig. 1 shows the energy spectra of one-dimensional up- and down-cranking states (Ψu (J ) and Ψd (J ), respectively) and the states of θ = 0◦ in the 2d-cranked calculations, Ψ t (J, θ = 0◦ ), that are obtained through the backward tilting procedure at a given value of J . They are plotted also in the inset of Fig. 1 together with the known experimental values of the g-band. The g-band M. Oi et al. / Physics Letters B 505 (2001) 75–81 77 Fig. 1. Energy spectrum near the yrast line obtained through the tilted-axis-cranking calculations. In the large graph, we show the energy with an arbitrary subtraction of rotational energy, 0.005J (J + 1) MeV for 10h̄ J 40h̄, in order to see excited structures in detail. In the small graph, the energy curve (without subtraction) is shown for the full range of angular momentum 0h̄ J 40h̄. Experimental values for the g-band (open circle) and a K π = (13+ ) isomer (asterisk) are plotted for a comparison. One-dimensional up- and down-cranking are depicted by upward and downward triangles, respectively. When the up- and down-cranking results are the same, the triangles are superposed to give the stars. Two kinds of 1d-cranked calculations are shown as “A” and “B”, which are obtained by β ini = 0.270 (open triangles) and 0.268 (closed triangles), respectively. “A+B” means that the two 1d-cranking solutions “A” and “B” give similar results. Also “u+d” means that the up- and down-cranking calculations show no differences. The thick solid line indicates states of θ = 0◦ obtained in the 2d-up-cranking calculations (Ψ t (J, θ = 0◦ )) while plus signs denote 2d-up-cranking states Ψ t (J, θ = 90◦ ). The tilted rotation minima are represented by diamonds (see Fig. 3(b)). is well reproduced by our one-dimensional cranking calculations. We have two kinds of 1d-cranked states plotted in Fig. 1. They are calculated with the same conditions except for the initial value of β ini . One (denoted as “A” in the graph) is obtained with the initial value of β ini = 0.270 while the other (denoted as “B”) is with β ini = 0.268. The small difference in β ini gives almost no difference in the wave functions at low spin (J 16h̄), but it does lead to a significant difference at high spins. The sensitive dependence of the high-spin HFB solutions to β ini shows that the energy manifold in the variational space near the crossing regions has several local minima that are almost degenerate. The states “A” and “B” imply the possibility of band crossings, 1 which correspond to the regions 17h̄ 1 As Hamamoto et al. pointed out [15], the validity of the cranking model is questionable in the band crossing region because of the semi-classical aspects in the model. However, the model works well outside the crossing region and could explain qualitative features near the crossing region, such as alignments. For a more accurate analysis of the crossing region, we should employ the so-called “variation-after-projection” method [11], the generator coordinate method [18], or the diabatic method [16]. 78 M. Oi et al. / Physics Letters B 505 (2001) 75–81 Fig. 2. (a) Gamma deformation in 1d- and 2d-cranking calculations are shown with respect to J (= Jx ). The symbols are the same as those in Fig. 1. (b) Gamma and beta values at J = 34h̄ in the 2d-cranking calculations are shown with respect to θ . “Fwd” means the forward tilting calculations, while “Bwd” means the backward calculations. J 26h̄ and at 22h̄ J 31h̄, respectively, 2 in the calculations, but a substantial difference between the states is seen in the gamma deformation. Before the crossing regions both of the states have γ 10◦ , while after the crossing regions the states “A” have near-prolate deformation with a negative gamma value (γ −10◦ ) and the states “B” have oblate shape (γ 60◦ ) (see Fig. 2(a) for evolutions in gamma deformation for each solution). From an analysis of our numerical results, the oblate deformation is caused by the gradual alignments of both i13/2 neutrons and h11/2 protons, while, in addition to these alignments, the near-prolate deformation with γ −10◦ is caused by the quick alignment of j15/2 neutrons. (Note that the neutrons in the j15/2 orbits are not part of the usual Kumar–Baranger model space.) The solution “A” is reported in this paper for the first time. Xu et al. found 2 Note that we do not mean here that these values of the crossing angular momentum correspond exactly to the experimental values. In general, the simple self-consistent cranking calculations do not reproduce the value precisely[17]. that there is no stable minimum corresponding to this solution in their TRS calculations [7]. However, they performed their (1d) cranking calculations for given rotational frequencies (ωx ), while we have performed calculations for given (average) angular momentum vectors (Jx , Jy , Jz ). In a spin region 23h̄ J 35h̄, the three kinds of states having different gamma deformations (γ ±10◦ and 60◦ ) are close to each other in energy. This result implies a manifestation of shape coexistence at high spin, or multi-band crossings among bands specified by different gamma deformations (or corresponding rotational alignments). It is interesting to see the energies for the state Ψ t (J, θ = 0◦ ), which are represented by the thick solid line in Fig. 1. The line has two discontinuities, at J = 27h̄ and 34h̄, implying two configuration changes. At lower spin (J 26h̄), Ψ t (J, θ = 0◦ ) follows the 1d-cranking calculations which give rise to nearly prolate shape (γ 10◦ at J 25h̄). Then at J = 27h̄, the gamma deformation changes to γ −10◦ , which is the same as “A”. Finally, at M. Oi et al. / Physics Letters B 505 (2001) 75–81 79 Fig. 3. Energy curves with respect to tilt angle θ . (a) Energy curves at J = 34h̄ and 40h̄. Three states having different gamma deformation are energetically close in the range 0◦ θ 40◦ . For J = 40h̄ there are two minima, at θ = 0◦ and 90◦ , the barrier height between them being only about 50 keV. (b) Energy curve at J = 13h̄. For 12h̄ J 20h̄, there are three minima at θ = 0◦ , 90◦ , and 15◦ . The last minimum implies tilted rotation. (c) Energy curve at J = 4h̄. In the low-spin region (J < 12h̄), there is only one minimum, at θ = 0◦ , i.e., principal axis rotation. J = 34h̄, Ψ t (J, θ = 0◦ ) changes into the same state as “B”, having oblate shape (γ 60◦ ). The result that the states having different gamma deformation are connected by the tilted-axis-cranking solutions indicates the importance of the tilting degree of freedom (θ ) for the search for excited states near band crossings. In Fig. 2(b), we show how the quadrupole deformations (β and γ ) evolve at J = 34h̄ as we vary the tilt angle (θ ). The corresponding energy curves are plotted in Fig. 3(a). There are three types of solutions with different gamma deformation: (i) the solution having no θ -dependence in tilt angles for θ 20◦ is oblate (γ 60◦ ); (ii) the solution which shows a tilted rotation minimum at θ 10◦ has negative gamma deformation (γ −10◦ ); (iii) the solution with a minimum at θ = 90◦ , which may correspond to high-K states, has 0◦ γ 10◦ . The energy difference between the states of type (i) and (ii) is roughly constant and small ( 500 keV), so that these two states can couple to form states with mixed deformation. The energy curves for these states are shallow in the range 0◦ θ 30◦ , so that fluctuations in the rotation axis, or wobbling motion [18], can be expected. However, gamma deformations for each state of type (i) and (ii) are quite constant against variation in the tilt angle up θ 30◦ . Therefore, rather than a picture in which states of type (i) and (ii) are coupled thorough the wobbling motion, we should have a picture where they are connected possibly through γ tunnelling, and where the mixed states wobble around the tilted rotation minimum at θ 10◦ . Nevertheless, a coupling of these mixed states with states of type (iii), is possible through wobbling motion. It is interesting to see in Fig. 3(a) the energy curve at J = 40h̄ corresponding to the states of type (i) above. There are two minima at θ = 0◦ and 90◦ , but the barrier height between them is only 50 keV, not visible in the plot. The corresponding deformation is collective oblate (the y-axis is the symmetry axis) and almost constant over the entire 80 M. Oi et al. / Physics Letters B 505 (2001) 75–81 range of θ . For a strict oblate symmetry, there would be no energy dependence on θ . The projection of the nuclear deformation onto the x–z plane is a circle, so that there is no preference for a direction of (collective) rotation in the x–z plane. For finite triaxiality, it is important to consider wobbling motion in the θ direction. From these discussions, we can deduce that the shape coexistence creates successive backbends (sudden changes in moments of inertia) in the excited rotational bands of θ 0◦ . The first backbend, which is caused by neutron j15/2 alignment, in addition to alignments of neutron i13/2 and proton h11/2 , is expected at lower spins, which corresponds to 17h̄ J 26h̄ in our calculations. The second one, which is caused by de-alignment of the j15/2 neutrons and retaining alignments of the i13/2 neutrons and h11/2 protons, can be seen at higher spins (corresponding to 22h̄ J 31h̄ in the calculations). According to Fig. 1, the second backbend can be more pronounced than the first one. It is also possible that these two backbends are mixed together to create one giant backbend as Hilton and Mang predicted [6], but according to our analysis three types of gamma deformations are involved: γ ±10◦ and 60◦ . Now, let us look at the calculated high-K states. In Fig. 1, states of θ = 90◦ , Ψ t (J, θ = 90◦ ), are shown with “+” symbols. These states correspond to a local minimum at θ = 90◦ in the energy curve (see Fig. 3(b)). In our calculations, this minimum starts to appear at J 8h̄, and becomes the lowest minimum at J = 12h̄ and at higher spins. The θ = 90◦ minimum may be considered approximately to correspond to a high-K state, and the corresponding z-axis cranked state, or Ψ t (J, θ = 90◦ ), is a simulation of nuclear rotation where single-particle angular momenta carry most of the total angular momentum. We have checked that the corresponding gamma deformation is almost zero as in Fig. 2(b) which shows that there is axial symmetry for these states. However, we should note that Jz (= J sin θ ) in the self-consistent cranking calculations is not an eigenvalue but just an expectation value, due to the rotational symmetry breaking by the mean field. Besides, in the tilted-axis-cranked HFB calculations, angular momenta are fully mixed in the sense that even a mixture among even and odd angular momenta happens. This is because the signature symmetry, a discrete sub- group (D2 ) of the rotational group O(3), is broken by tilted rotation [19]. We should therefore keep in mind that the mean-field description of high-K states has a certain limitation. Remembering the above remarks, let us look at our results for Ψ t (J, θ = 90◦ ). The experimental energy of the isomer, which is tentatively assigned to K π = 13+ , is 2.572 MeV relative to the ground state energy, while the numerical values are 2.101 MeV (J = 12h̄), 2.255 MeV (J = 13h̄), and 2.408 MeV (J = 14h̄). Deformations of Ψ t (J = 13h̄, θ = 90◦ ), are calculated to be β = 0.2661 and γ = 0.015◦ . We can consider whether this isomer is yrast or not. (Experimentally, the g-band is identified only up to 8h̄ [1].) Xu et al. calculated that the non-collective prolate state would be yrast at J = 13h̄ [7]. Our self-consistent calculations also show that Ψ t (J, θ = 90◦ ) is lower in energy than principal axis rotation, Ψ t (J, θ = 0◦ ), for 12h̄ J 34h̄, so that in this region the high-K isomeric states can be favoured relative to the collective rotation. Let us use the term “tilted rotation” for the states with local minima, with θ = 0, 90◦ . At J = 13h̄ (see Fig. 3(b)), there is tilted rotation (θ 15◦ ) with an excitation energy of 2.593 MeV, which happens to be quite close to the experimental value of the K π = 13+ isomer. Numerical deformation values for this minimum are β = 0.2704 and γ = 4.62◦ , that is, the shape is nearly prolate. It is possible to consider the tilted rotation to describe a rotational member of a high-K state. However, at this angular momentum, the potential energy curve is shallow in the whole range of θ (see Fig. 3(b)), and the barrier height is the same order of magnitude as uncertainties from the mean-field approximation. The question as to whether the high-K state should be described as either a tilted rotation state or a z-axis cranked state Ψ t (J, θ = 90◦ ) (or coupling of them) should be answered by a quantum mechanical calculation by using angular momentum projection (and/or the generator coordinate method), which we plan to study in the future. At this moment, the description of highK states in the framework of self-consistent tiltedaxis-cranking calculations is reasonably good up to an accuracy of several hundred keV, but more detailed studies are surely necessary. For these shallow energy curves, we can consider the wobbling motion to relate and cause transitions M. Oi et al. / Physics Letters B 505 (2001) 75–81 between high-K isomers and low-K states such as the g-band. For the energy curve at J = 13h̄, there are three minima at θ = 0◦ (principal axis rotation), θ = 15◦ (tilted rotation), and θ = 90◦ (possible high-K states) within 600 keV. If the first minimum represents a rotational member in the g-band and the third (and/or second) minimum represents the one in a high-K band, then the corresponding band crossing is expected to show (experimentally) a strong coupling as evidence for a realization of the wobbling motion. In summary, we have performed tilted-axis-cranked HFB calculations for 182 Hf, and investigated highspin states near the yrast line up to J = 40h̄. For our parameter set, the comparison of the experimental data for the g-band and a high-K isomeric state (K π = 13+ ) with our calculation gives a reasonable agreement within the framework of the mean-field approximation. With our modified single-particle model space based on Kumar–Baranger’s choice, we found a new HFB solution with near-prolate deformation (γ −10◦ ) involving j15/2 neutron alignment at high spin (J 17h̄). We discussed the possibility of shape coexistence among three states in principal axis rotation with different gamma deformation: two with nearprolate shapes (γ ±10◦ ) and the other with oblate shape (γ 60◦ ). An analysis of the possible backbends in yrare states was also given. In addition, we discussed the effect of wobbling motion as a coupling mode between low-K and high-K states. Acknowledgements M.O. would like to acknowledge with thanks support from the Japan Society for the Promotion of Sci- 81 ence (JSPS). He also thanks Drs. N. Onishi, W. Nazarewicz, Y.R. Shimizu, and T. Nakatsukasa for useful discussions. References [1] R.B. Firestone, V.S. Shirley (Eds.), 8th edn., Table of Isotopes, Wiley, New York, 1996. [2] Zs. Podolyák et al., Phys. Lett. B 491 (2000) 225; R. 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Nazarewicz, Phys. Lett. B 252 (1990) 533. [16] I. Hamamoto, Phys. Lett. B 66 (1977) 222. [17] H.J. Mang, Phys. Rep. 18 (1975) 325. [18] M. Oi, A. Ansari, T. Horibata, N. Onishi, Phys. Lett. B 480 (2000) 53. [19] M. Oi, N. Onishi, N. Tajima, T. Horibata, Phys. Lett. B 418 (1998) 1. 26 April 2001 Physics Letters B 505 (2001) 82–88 www.elsevier.nl/locate/npe Form factors for semileptonic B → π and D → π decays from the Omnès representation J.M. Flynn a , J. Nieves b a Department of Physics & Astronomy, University of Southampton, Southampton SO17 1BJ, UK b Departamento de Física Moderna, Universidad de Granada, E-18071 Granada, Spain Received 25 July 2000; received in revised form 23 February 2001; accepted 5 March 2001 Editor: P.V. Landshoff Abstract We use the Omnès representation to obtain the q 2 dependence of the form factors f +,0 (q 2 ) for semileptonic H → π decays from elastic πH → πH scattering amplitudes, where H denotes a B or D meson. The amplitudes used satisfy elastic unitarity and are obtained from two-particle irreducible amplitudes calculated in tree-level heavy meson chiral perturbation theory (HMChPT). The q 2 -dependences for the form factors agree with lattice QCD results when the HMChPT coupling constant, g, takes values smaller than 0.32, and confirm the milder dependence of f 0 on q 2 found in sumrule calculations. 2001 Published by Elsevier Science B.V. 1. Introduction In this Letter we present a description of the form factors f + and f 0 describing semileptonic H → π decays, where H denotes a D or B meson. For the B meson this exclusive semileptonic decay can be used to determine the magnitude of the CKM matrix element Vub , currently the least well-known entry in the CKM matrix. Ultimately, experimental measurements of fB+ (q 2 ) for given momentum-transfer q will be compared directly to theoretical determinations at the same q 2 values to determine |Vub |. In the interim, it may be helpful to consider the decay rate integrated partially or completely over q 2 , but this requires knowledge of the q 2 dependence of the form factors. Lattice calculations and sumrule calculations apply in (different) restricted ranges of q 2 while disE-mail address: [email protected] (J.M. Flynn). persion relations may be used to bound the form factors over the whole q 2 range [1,2], or as a basis for models [3]. A variety of models exists for the whole range of q 2 . One can ensure that general kinematic relations and the demands of heavy quark symmetry (HQS) are satisfied, but an ansatz, such as pole, dipole or other forms, is still required [4,5]. Here we use the Omnès representation to obtain the full q 2 dependence of these form factors from the elastic πH → πH scattering amplitudes. For our application we have an isospin-1/2 channel, with angular momentum J = 1 or 0 for f + and f 0 , respectively. We rely on the following description of the (inverse) amplitude for elastic πH → πH scattering in the isospin I , angular momentum J , channel, with centre-of-mass squared-energy s and masses m and M, respectively [6], TI−1 J (s) = −I 0 (s) − CI J + 0370-2693/01/$ – see front matter 2001 Published by Elsevier Science B.V. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 3 3 3 - 1 1 , VI J (s) (1) J.M. Flynn, J. Nieves / Physics Letters B 505 (2001) 82–88 where VI J is the two-particle irreducible scattering amplitude and CI J is a constant. CI J and VI J are real in the scattering region. This description implements elastic unitarity automatically. Eq. (1) is justified by a dispersion relation for T −1 , where the contributions of the left hand cut and the poles (if any) are contained in −CI J + 1/VI J . I 0 gives the exact contribution from the right hand cut, after any necessary subtractions. 1 The description of Eq. (1) may also be justified by an approach using the Bethe–Salpeter equation. Once TI J is known, we can compute the corresponding phase shift δI J . In turn, δI J can be used in an Omnès representation [8] giving fI J (q 2 )/fI J (0) in terms of an integral involving the phase shift, assuming that at threshold the phase shift should be nπ , where n is the number of bound states in the particular channel considered, and δI J (∞) = kπ , where k is the number of zeros of the scattering amplitude on the physical sheet (this is Levinson’s theorem [9]). We determine VI J from tree level heavy meson chiral perturbation theory (HMChPT) [10], which implements HQS and is a double expansion in powers of 1/M and momenta, where M is the heavy meson mass. The parameter CI J in Eq. (1) partially accounts for higher order contributions in the expansion [6]. We find consistency of our description with lattice results for the D → π [11,12] and B → π [12,13] form factors if we set the HMChPT coupling, g, to values smaller than 0.32. This upper bound is in reasonable agreement with other determinations, but g is not very well known [14,15]. Our model and the Omnès representation are not guaranteed at high energies where inelasticities become important. However, our hypothesis is that only the low-lying states and energies should influence the form factors we consider. A dispersive approach to the f + form factor was taken by Burdman and Kambor [3], who also used HMChPT to calculate the phase shift in πH → πH scattering. Here by working with the inverse amplitude we can ensure that Watson’s theorem and elastic unitarity are satisfied exactly. Moreover, we compute f + and f 0 together to examine whether different 1 I is calculated from a one-loop ‘bubble’ diagram. In the 0 notation of Ref. [7], I 0 (s) = TG ((m + M)2 ) − TG (s), where M and m are the masses of the two propagating particles. 83 behaviours in q 2 are found, consistent with lattice QCD results and allowing extra information from f 0 to be used to constrain f + . 2. Scattering amplitudes and form factors We compute V1/2 , the two-particle irreducible amplitude for πH scattering in the isospin 1/2 channel, π(p1 )H (Mv) → π(p2 )H (Mv + q2 ). Here, v is the four-velocity of the initial heavy meson of mass M. The pion mass is m. We use the direct tree level interaction from the lowest order HMChPT Lagrangian, together with tree diagrams for H ∗ exchange which involve the leading interaction term with coupling g [10,14]. The result is M V1/2 = − 2 3v · p1 + v · p2 f + g 2 (p1 · p2 − v · p1 v · p2 ) 3 1 × + . v · p1 − ∆ v · p2 + ∆ (2) Here, f = 130.7 MeV is the pion decay constant and ∆ = (M∗2 − M 2 )/2M ≈ M∗ − M, where M∗ is the heavy vector meson mass. We subsequently project V1/2 onto the angular momentum 0 and 1 channels. The full scattering amplitude at centre of mass energy-squared s, in the isospin I and angular momentum J channel, is obtained in our approach from Eq. (1). The phase shift δI J is then obtained from TI J (s) = 2iδ (s) 8πis e IJ − 1 , λ1/2 (s, M 2 , m2 ) (3) where λ(x, y, z) = x 2 + y 2 + z2 − 2(xy + yz + zx) is the usual kinematic function. Once the phase shift is known, we use the Omnès representation to obtain the q 2 dependence of the form factors as follows: ∞ q2 f (q 2 ) δI J (s) ds = exp (4) . f (0) π s(s − q 2 ) (m+M)2 In this work we always have I = 1/2. The form factor f + is obtained when J = 1 and depends on J P = 1− resonances, while f 0 is obtained when J = 0 and depends on J P = 0+ resonances. We perform the integral numerically, taking the upper limit as 100 84 J.M. Flynn, J. Nieves / Physics Letters B 505 (2001) 82–88 times the lower limit. 2 The form factors are equal at q 2 = 0: f + (0) = f 0 (0). 3. Semileptonic decays • For J P = 1− we take C = 0 for the D decay because the D ∗ resonance is so close to threshold that we expect it to saturate all the counterterms in HMChPT (compare to vector meson dominance in ππ scattering in ordinary chiral perturbation theory). Calculating C in this case reveals the value C = 8 × 10−6 . We still have the freedom to vary the lowest order coupling constant g in HMChPT. For the B meson decay, we set C = −0.0014 to keep the B ∗ pole at its correct mass. • For J P = 0+ we ignore D ∗ and B ∗ s-channel exchanges, which have the wrong quantum numbers to contribute in this case. These exchanges only contribute because of the heavy meson mass expansion implicit in HMChPT. Instead we keep C non-zero, setting C = −0.0051 for the D-physics case to get a resonance at about 2350 MeV, and C = −0.0016 for B-physics to get a resonance at about 5660 MeV [16]. The process D ∗ → Dπ is kinematically allowed, so the D ∗ is a resonance in Dπ scattering. In HMChPT the decay rates of D ∗ + to D 0 π + and D + π 0 are given to lowest order by The values of C are determined by demanding that T −1 (Re T −1 ) vanishes at the position of a pole (resonance). For the J = 1 channels, V −1 vanishes by construction at the positions of the D ∗ or B ∗ , and so, from Eq. (1), C is independent of g. In the J = 0 channels, g-dependence enters in V −1 , but only through the t-channel tree graphs, and is very weak. C varies by less than 0.5% for 0 < g < 0.45 in the Dmeson case and the dependence is even weaker for the B-meson case. We noted that in using the Omnès representation [8] of Eq. (4), the phase shift at threshold should be nπ , where n is the number of bound states in the channel under consideration. Thus n = 0 in all channels used here except for J P = 1− in the B case where n = 1 to account for the B ∗ . In fact, our model also gives a bound state in the 0+ channel in the B case, which we ignore. One could try to improve the model to avoid this unphysical bound state by replacing C with a function of q 2 (the function should have no right hand cut). 2 f + (q 2 ), where q 2 2 max = (mD − mπ ) , varies by less than Dπ max 1% as the upper limit of integration varies from 50 to 200 times the lower limit, and the variation is smaller at lower q 2 . g 2 p3 Γ D∗ + → D0 π + = , 6πf 2 g 2 p3 Γ D∗ + → D+ π 0 = . 12πf 2 (5) The sum of these rates can also be obtained from the slope of the phase shift at the resonance mass. We find that these two methods agree for a range of g values. The D ∗ exchange is included in our tree level amplitude, and we expect it to saturate the counterterms in HMChPT, so in calculating T1/2,1 we set C = 0 as noted above. Fig. 1 (left) shows the phase shift obtained for J = 1. With input masses, mD = 1864.5 MeV and mD ∗ = 2010 MeV, the D ∗ resonance shows up as the jump of π in the phase at the D ∗ mass. In the J = 0 channel, we tune C to produce a resonance at the expected mass of the D0∗ at 2350 MeV [16]. The J = 0 phase shift is shown on the right in Fig. 1. In the B case, the decay process B ∗ → Bπ is not kinematically allowed and the B ∗ meson is a pole, sitting between the maximum physical q 2 value for 2 = (m − m )2 , and the start of the form factor, qmax B π the physical cut at q 2 = (mB + mπ )2 . Again, we use the physical pseudoscalar and vector meson masses as inputs, mB = 5278.9 MeV, mB ∗ = 5324.8 MeV. The phase shift for the J = 1 case is shown on the left in Fig. 2. The appearance of the B ∗ as a bound state 2 between qmax and q 2 = (mB + mπ )2 is signalled by √ −1 the vanishing of T1/2,1 (s) at s = mB ∗ . The J = 0 phase shift appears on the right of Fig. 2. From the phase shifts we find the form factors f + and f 0 . We perform a simultaneous three-parameter fit to the UKQCD and APE lattice results [11–13] for the form factors f + (q 2 ) and f 0 (q 2 ) which determine the B and D semileptonic decays. The free parameters are the HMChPT coupling constant g and the form factors at q 2 = 0: fB (0) for B → π decays and fD (0) J.M. Flynn, J. Nieves / Physics Letters B 505 (2001) 82–88 Fig. 1. Phase δDπ for the I J = 1/2, 1 (top) and 1/2, 0 (bottom) channels in Dπ scattering. The inset on the left shows the resonance √ at s = mD∗ = 2010 MeV. Phases are calculated with g = 0.21. for D → π decays. 3 The best fit parameters with 39 degrees of freedom are g = 0.21+0.11 −0.21 , fB (0) = 0.39 ± 0.02, fD (0) = 0.60 ± 0.02 with χ 2 /dof = 0.34. (6) Results can be seen in Fig. 3. Errors in the fitted parameters are statistical and have been obtained by increasing the value of the total χ 2 by one unit. A word of caution must be stated about the results for the HMChPT coupling constant g. Scalar channels are almost insensitive to this parameter. For the vector channels, in the case of D meson decay, the resonance is so close to threshold that it completely dominates the process, independent of the value of g, as long as the resonant contribution is more important than 3 To use the results in [11], we take Z eff = 0.88 for the vector V renormalisation constant connecting lattice and continuum results. 85 Fig. 2. Phase δBπ for the I J = 1/2, 1 (top) and I J = 1/2, 0 (bottom) channels in Bπ scattering. Phases are calculated with g = 0.21. the background. This turns out to be true as long as g is greater than 0.001, thus the smallest value g can take is 0.001 and not zero as can be inferred from Eq. (6). To clarify the dependence of our results on g, we show in Fig. 4 both χ 2 and fB (0), fD (0) versus g, for g 0.001. In the first figure the line at χ 2 = 13.28 shows the minimum value of χ 2 , while the line at χ 2 = 14.28 determines the upper error. We also show best fit values, with fixed g, of fB (0) and fD (0) versus g. The points with errors correspond to the results quoted in Eq. (6). We note that fD+ is well-approximated by a simple pole form with the D ∗ giving the pole mass, while fD0 is noticeably ‘flatter’ in q 2 . This is consistent with lattice results. For the B → π case, fB+ is wellapproximated by a pole form with the pole mass of order the B ∗ meson mass. The fB0 form factor has much less q 2 dependence, consistent with the behaviour found in lattice calculations. 86 J.M. Flynn, J. Nieves / Physics Letters B 505 (2001) 82–88 Fig. 3. Form factors in D → π (top) and B → π (bottom) semileptonic decays. The squares (circles) denote f + (f 0 ) from lattice calculations, while the long-dashed (short-dashed) lines denote the fitted curves for f + (f 0 ). Solid symbols are results from UKQCD [11,13], open symbols are results from APE [12]. Fig. 4. Top: chi-squared for the fit described in the text as a function of the HMChPT coupling g, for g 0.001. Bottom: values of +,0 (0) (upper curve) as functions of g. fB+,0 (0) (lower curve) and fD The points with errors on the right are the best fit values of Eq. (6) at g = 0.21. We have also determined the coupling g and form factors at q 2 = 0 separately for D and B decays using independent fits to the UKQCD and APE lattice data for D and B. The best fit values turn out to be the same as in Eq. (6), although gD can be as large as 0.46 while still giving an acceptable chi-squared. To compare with light cone sumrule (LCSR) results, we take the LCSR values fD ∗ gD ∗ Dπ = 2.7 ± 0.8 GeV and fB ∗ gB ∗ Bπ = 4.4 ± 1.3 GeV [17], and combine with lattice calculations of the vector meson decay constants from Becirevic et al. [18] and UKQCD [19], to yield fD ∗ from [18], 0.35+0.12 −0.11, gD = 0.39 ± 0.12, fD ∗ from [19], 0.23 ± 0.08, fB ∗ from [18], gB = (7) fB ∗ from [19]. 0.28+0.10 −0.09 , The values are quite compatible in the B case, less so for D decays, although, as noted above, our fit for gD allowed a large variation above the best-fit value. The value of fD (0) found here agrees well with the LCSR result fD+ (0) = 0.65 ± 0.11 [17], while fB (0) in Eq. (6) is higher than the LCSR value fB+ (0) = 0.28 ± 0.05 [17]. In the D case, the D ∗ resonance is only a few MeV above threshold and the range of q 2 for the semileptonic decay is not large, so one expects a simple pole form for f + to work well. For B physics, the effects of higher resonances and continuum states are evidently more important: such effects are incorporated in LCSR calculations but are not present in the very simple model used here. We address this issue in Section 4 below. Heavy quark symmetry (HQS) is an input in HMChPT. The HQS scaling relations for the B decay 2 are preserved because f + (q 2 )/ form factors at qmax max J.M. Flynn, J. Nieves / Physics Letters B 505 (2001) 82–88 87 4. Extra resonances Fig. 5. Variation of fB+ (0) with g and gres , where the B ∗ coupling is g and a second resonance is added in the J = 1 channel with coupling gres . On the top the resonance mass is mres = 6100 MeV and contours are plotted from fB+ (0) = 0.20 to 0.28 in increments of 0.01; on the bottom mres = 8100 MeV and contours are plotted from fB+ (0) = 0.28 to 0.37 in increments of 0.01. 2 ) is proportional to f 0 (qmax q2 exp max π ∞ (M+m)2 δ+ − δ0 ds . 2 ) s(s − qmax (8) The above result relies on the equality of the form factors at q 2 = 0, f + (0) = f 0 (0). If δ + − δ 0 = π , which√we see is satisfied by our phase shifts at 2 )/f 0 (q 2 ) = 1/(1 − large s, then the ratio f + (qmax max 2 2 qmax /(M + m) ) as demanded by HQS, where M and m are the masses of the heavy meson and the pion, respectively. We have applied the same approach to describe semileptonic D → K decays. Here, it gives form factors flatter than lattice results [11] and the experimental evidence [20]. However, corrections of both types mK /mD and m2K /(4πfπ )2 to the tree level HMChPT results used here are expected to be sizeable in this case. We noted above that our result for fB (0) in Eq. (6) is higher than the LCSR value of around 0.28, while our fit for fB+ (q 2 ) is well-approximated by a simple pole form with pole mass of order mB ∗ . This suggests that deviations from B ∗ pole dominance can become significant at low q 2 . This phenomenon was also noted by Burdman and Kambor [3] who implemented a constrained dispersive model for fB+ . Likewise, lattice results have favoured dipole forms in fits to fB+ [4,12, 13]. To address this issue we have added a second resonance of mass mres by hand in the 2PI J = 1 amplitude V1/2,1 , coupling it like the H ∗ but with its own coupling strength gres . In the D-meson case, we already had a good fit to the lattice results and a consistent value for fD (0). If mres is large enough the extra resonance does not disturb this picture. In the B case, we can easily make fB (0) smaller while still fitting lattice results at large q 2 . In Fig. 5 we show fB (0) as a function of the couplings g and gres for two choices of the extra resonance mass, mres = 6100 MeV, 8100 MeV. The problem in this case is that it is not possible to make a statistically acceptable fit to fB+ and fB0 simultaneously. One could try to add an extra resonance in the J = 0 channel also, but while our choice of mres = 6100 MeV for J = 1 may be motivated by potential models [21] or lattice results [22], we do not know whether or how to set the mass for additional J = 0 resonances, having already set the C values to account for rather poorly known resonances. This emphasises the importance of looking at f + and f 0 together, even though f + is the experimentally accessible form factor. 5. Conclusion Our model is extremely simple, using only tree level HMChPT information for the two particle irreducible amplitude VI J , thereby incorporating only the first excited hadron state. Furthermore we fix to a constant an allowed polynomial in q 2 multiplying the Omnès exponential factor in Eq. (4). Thus, deviations from LCSR results for f + are not unexpected because those calculations incorporate effects of higher resonances and continuum states. Taking our model beyond lead- 88 J.M. Flynn, J. Nieves / Physics Letters B 505 (2001) 82–88 ing order is not possible at present because of the proliferation of undetermined parameters which would appear in the next order of HMChPT and the lack of experimental data to fix them. This is a standard difficulty in using effective theories at higher orders. The simple model presented here gives an excellent description of semileptonic D-decays. For B-decays it gives a good description of the lattice data near 2 and is also compatible within two standard deqmax viations with LCSR predictions at q 2 = 0. Moreover, it provides a framework compatible with heavy quark symmetry, naturally accommodating pole-like behaviour for f + and, simultaneously, non-constant behaviour for f 0 . Previously, as pointed out in [2,4], a difficulty for form factor models with pole-type behaviour for f + was fixing a behaviour for f 0 which satisfied both the relation f 0 (0) = f + (0) and the requirements of heavy quark symmetry. Pole-like behaviour of f + turns out again to be feasible in our model, thanks to the fact that the B ∗ is a bound state rather than a πB resonance. Qualitatively, the results found here are encouraging. However, the larger value found for fB+ (0) compared to that from LCSR calculations would lead to appreciably smaller values for |Vub |. We caution the reader that this should not be taken to indicate a large theoretical spread in the value of |Vub | from exclusive semileptonic B → π decays: one should bear in mind the simplicity of the model used. We indicated how a second resonance in the J = 1 channel can restore compatibility with both LCSR and lattice results for fB+ , although this shifts the problem to making fB0 compatible with the lattice data in a combined fit and emphasises the importance of using information from both form factors. Acknowledgements J.N. acknowledges support under grant DGES PB981367 and by the Junta de Andalucía FQM0225, and thanks the SHEP group for their hospitality during part of this work. J.M.F. acknowledges PPARC for support under grant PPA/G/O/1998/00525. We thank E. Ruiz-Arriola for useful discussions and G. Burdman for communications. References [1] C.G. Boyd, B. Grinstein, R.F. Lebed, Phys. Rev. Lett. 74 (1995) 4603, hep-ph/9412324. [2] L. Lellouch, Nucl. Phys. B 479 (1996) 353, hep-ph/9509358. [3] G. Burdman, J. Kambor, Phys. Rev. D 55 (1997) 2817, hepph/9602353. [4] UKQCD Collaboration, D.R. Burford et al., Nucl. Phys. B 447 (1995) 425, hep-lat/9503002. [5] D. Becirevic, A.B. 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D 62 (2000) 074503, hep-ph/0003130. 26 April 2001 Physics Letters B 505 (2001) 89–93 www.elsevier.nl/locate/npe Fourth generation effects in the Bs → ν ν̄γ decay Yusuf Dinçer Institute of Theoretical Physics, RWTH Aachen, D-52056 Aachen, Germany Received 15 December 2000; received in revised form 12 February 2001; accepted 16 March 2001 Editor: P.V. Landshoff Abstract If the fourth generation fermions exist, the new quarks could influence the branching ratio of the decay Bs → ν ν̄γ . We obtain two solutions of the fourth generation CKM factor Vt∗ s Vt b from the decay of B → Xs γ . With these two solutions we calculate the new contributions of the fourth generation quark to Wilson coefficients of the decay Bs → ν ν̄γ . The branching ratio of the decay Bs → ν ν̄γ in the two cases are calculated. In one case, our results are quite different from that of SM, but almost same in another case. If a fourth generation should exist in nature and nature chooses the former case, this B meson decay could provide a possible test of the fourth generation existence. 2001 Elsevier Science B.V. All rights reserved. 1. Introduction At present the Standard Model (SM) describes very successfully all low energy experimental data. But there is no doubt that from a theoretical point of view SM is an incomplete theory. Among the unsolved problems, such as CP violation, mass spectrum, etc., one of the unsolved mysteries of the SM is the number of generations. In SM there are three generations and, yet, there is no theoretical argument to explain why there are three and only three generations in SM. From the LEP result of the invisible partial decay width of the Z boson it follows that the mass of the extra generation neutrino N should be larger than 45 GeV [1]. There is neither an experimental evidence for a fourth generation nor does any experiment exclude such extra generations. Having this experimental result in mind we can raise the following question: If extra genera- E-mail address: [email protected] (Y. Dinçer). tions really exist, what effect would they have in low energy physics? In [2] effects of the fourth generation quarks on 0 mixing is discussed. In [3] it is MBd,s in B 0 –B argued that the fourth generation quarks and leptons can manifest themselves in the Higgs boson production at the Tevatron and the LHC, before being actually observed. The next generation leads to an increase of the Higgs boson production cross section via gluon fusion at hadron colliders by a factor 6–9. So, the study of this process at the Tevatron and LHC can fix the number of generations in the SM. In [4] the possibility of a fourth generation in the Minimal Supersymmetric Standard Model (MSSM) is explored. It is shown that the new generations must have masses mν , mτ < 86 GeV, mt < 178 GeV and mb < 156 GeV so that the MSSM remains perturbative up to the unification scale MU of the Yukawa couplings. In [5] even the possibility of the fourth generation of quarks without the fourth generation leptons is discussed. In [6] the decay of the top quark into a possible fourth generation b is regarded. The effect 0370-2693/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 3 8 6 - 0 90 Y. Dinçer / Physics Letters B 505 (2001) 89–93 of the fourth generation on the branching ratio of the B → Xs l + l − and the B → Xs γ decays is analysed in [7]. It is also shown that the fourth generation could influence the forward–backward asymmetry of the decay of B → Xs l + l − . The introduction of fourth generation fermions can also affect CP violating parameters / in the Kaon system [8]. In [9] the fourth generation effects in the rare exclusive B → K ∗ l + l − decay are studied. In [10–16], contributions of the new generation to the electroweak radiative corrections were considered. It was shown in [16] that the existing electroweak data on the Z-boson parameters, the W -boson and the top quark masses strongly excluded the existence of the new generations with all fermions heavier than the Z boson mass. However, the same allows few extra generations, if one allows neutral leptons to have masses close to 50 GeV. One promising area in experimental search of the fourth generation, via its indirect loop effects, are the B meson decays. It is hoped that we will find a definite answer on a possible fourth generation at the B factories at SLAC and KEK. In this Letter we study the contribution of the fourth generation in the Bs → ν ν̄γ decay. New physical effects can manifest themselves through the Wilson coefficients, whose values can be different from the ones in the SM [7,17], as well as through new operators [19]. But in this Letter, we will only regard the contribution of the fourth generation in the Wilson coefficients and assume that the operators in SM for three (SM) and four generations (SM4) are the same. For the form factors we use the results from the Light Cone QCD sum rule methods [19]. We use the dipole formula approximation for the form factors which are known with an uncertainty of about 20–30% [19]. However, it will be possible to reduce this uncertainty if the decay Bs → µνγ is detected. Hence, experimental deviations from the SM prediction on the branching ratio for the BS → ν ν̄γ decay of about 20–30% will not necessarily be a signal for new physics. But deviations of more than 30% could be a signal for new physics. And one possibility for new physics could be the extension of the SM to four generations. The Letter is organized as follows. In Section 2 we establish the effective Hamiltonian describing this decay. In Section 3 we present the numerical values. Finally, in Section 4 we summarize the results and give an outlook. 2. The effective Hamiltonian The weak decay of mesons is described by the effective Hamiltonian 10 4GF Ci (µ)Oi (µ), Heff = √ Vt b Vt∗s 2 i=1 (1) where the full set of the operators Oi (µ) and the corresponding expressions for the Wilson coefficients Ci (µ) in the SM are given in [20]. As we mentioned in the introduction, we assume that no new operators appear and clearly the full operator set is exactly the same as in SM. As assumed, the fourth generation changes only the values of the Wilson coefficients C7 (µ), C9 (µ) and C10 (µ), via virtual exchange of the fourth generation up quark t . The above mentioned Wilson coefficients can be written in the following form Vt∗ b Vt s new C (µ), Vt∗b Vt s 7 V ∗ Vt s new C9SM4 (µ) = C9SM (µ) + t ∗b C (µ), Vt b Vt s 9 V ∗ Vt s new SM4 SM C10 (µ) = C10 (µ) + t ∗b C (µ), Vt b Vt s 10 C7SM4 (µ) = C7SM (µ) + (2) (3) (4) where the last terms in these expressions describe the contributions of the t quark to the Wilson coefficients and Vt b and Vt s are the two elements of the (4 × 4)CKM matrix. If the quark b should be very massive then it will also give an additional contribution to the Wilson coefficients on the right hand side of the Eqs. (2)–(4). But we neglect such a contribution in this new Letter. The explicit forms of the C7new , C9new and C10 can easily be obtained from the corresponding Wilson coefficient expressions in SM by simply substituting mt → mt [20,21]. Here, the effective Hamiltonian for the Bs → ν ν̄γ decay in the SM4 is given by Heff = C SM4 s̄γµ (1 − γ5 )b ν̄γ µ (1 − γ5 )ν , (5) where C SM4 = C SM + Vt∗ b Vt s new C Vt∗b Vt s (6) Y. Dinçer / Physics Letters B 505 (2001) 89–93 and where GF α C SM = √ Vt b Vt∗s 2 2π sin2 ΘW x x + 2 3(x − 2) + × ln x , 8 x − 1 (x − 1)2 mt 2 . x= mW 1 I= 2 mB (7) As mentioned, we obtain C new form C SM (7) by substituting mt → mt . The corresponding matrix element for the process Bs → ν ν̄γ is given by M = C SM4 ν̄(p2 )γµ (1 − γ5 )ν(p1 ) × γ (q)|s̄γ µ (1 − γ5 )b|B(q + p), (8) = (p1 + p2 and p1 and p2 are the fiwhere nal neutrinos four momenta. The matrix element γ (q)|s̄γ µ (1 − γ5 )b|B(q + p) can be parametrized in terms of the two gauge invariant and independent form factors f (p2 ) and g(p2 ), namely, q2 )2 γ |s̄γ µ (1 − γ5 )b|B √ g(p2 ) = 4πα µαβσ εα∗ pβ qσ m2B f (p2 ) . + i εµ∗ (pq) − (ε∗ p)qµ m2B (9) Here, εµ and qµ stand for the polarization vector and momentum of the photon, p + q is the momentum of the B meson, g(p2 ) and f (p2 ) correspond to parity conserved and parity violated form factors for the Bs → ν ν̄γ decay. The form factors f (p2 ) and g(p2 ) were calculated in [19] with the light cone QCD sum rules method. As mentioned in [19], the best agreement is achieved with the dipole formulae g p 2 h2 , (1 − p2 /m22 )2 (10) with h1 ≈ 1.0 GeV, m1 ≈ 5.6 GeV, h2 ≈ 0.8 GeV and m2 ≈ 6.5 GeV. For the total decay rate we get Γ (Bs → ν ν̄γ ) = α(C SM4 )2 m5B I, 256π dx (1 − x)3 x f 2 (x) + g 2 (x) . (12) 0 Here x = 1 − 2Eγ /mB is the normalized photon energy. In order to obtain quantitative results we need the value of the fourth generation CKM matrix element |Vt∗ s Vt b |. Following [7], we will use the experimental results of the decays Br(B → Xs γ ) and Br(B → Xc ev̄e ) to determine the fourth generation CKM factor Vt∗ s Vt b . In order to reduce the uncertainties arising from b quark mass, we consider the following ratio R= Br(B → Xs γ ) . Br(B → Xc ev̄e ) (13) In leading logarithmic approximation this ratio can be written as R= |Vt∗s Vt b |2 6α|C7SM4 (mb )|2 , |Vcb |2 πf (m̂c )κ(m̂c ) (14) where the phase factor f (m̂c ) and O(αs ) QCD correction factor κ(m̂c ) [22] of b → cl ν̄ are given by f (m̂c ) = 1 − 8m̂2c + 8m̂6c − m̂8c − 24m̂4c ln m̂4c , (15) 2αs (mb ) 31 3 π2 − (1 − m̂c )2 + . κ(m̂c ) = 1 − 3π 4 2 (16) It is defined m̂c = m2c,pole /m2b,pole . Solving Eq. (14) for Vt∗ s Vt b and taking into account (2) and (15), we get πR|Vcb |2 f (m̂c )κ(m̂c ) ∗ ± SM − C7 (mb ) Vt s Vt b = ± 6α|Vt∗s Vt b |2 × Vt∗s Vt b . C7new (mb ) (17) The values for Vt∗ s Vt± b are listed in Tables 1 and 2. From unitarity condition of the CKM matrix we have h1 ≈ , (11 − p2 /m21 )2 f p2 ≈ 1 91 (11) ∗ Vub + Vcs∗ Vcb + Vt∗s Vt b + Vt∗ s Vt b = 0. Vus (18) If the average values of the CKM matrix elements in the SM are used, the sum of the first three terms in Eq. (18) is about 7.6 × 10−2 . Substituting the value of Vt∗ s Vt(+) b from Tables 1 and 2, we observe that the sum of the four terms on the left-hand side of Eq. (18) is 92 Y. Dinçer / Physics Letters B 505 (2001) 89–93 Table 1 (−) The branching ratios for the solution Vt∗ s Vt b ; mt are pole masses mt [GeV] Vt∗ s Vt b × 10−3 Br(Bs → ν ν̄γ ) [10−8 ] 100 2.39 6.17 150 2.00 6.27 200 1.80 6.39 250 1.69 6.52 300 1.61 6.67 400 1.52 7.03 (−) Fig. 1. The ratio R versus y = (mt /mW )2 ; the upper curve is for Table 2 (+) The branching ratios for the solution Vt∗ s Vt b ; mt are pole masses mt [GeV] Vt∗ s Vt b × 10−2 Br(Bs → ν ν̄γ ) 100 −10.01 2.23 × 10−9 150 −8.37 1.79 × 10−8 200 −7.55 5.22 × 10−8 250 −7.07 1.13 × 10−7 300 −6.75 2.10 × 10−7 400 −6.35 5.58 × 10−7 (+) closer to zero compared to the SM case, since Vt∗ s Vt b is very close to the sum of the first three terms, but with opposite sign. On the other hand if we consider −3 and one order Vt∗ s Vt(−) b , whose value is about 10 of magnitude smaller compared to the previous case. However, it should be noted that the data for the CKM is not determined to a very high accuracy, and the error in sum of first three terms in Eq. (18) is about ±0.6 × 10−2 . It is easy to see then that the value of Vt∗ s Vt(−) b is within this error range. In summary both (+) ∗ Vt s Vt b and Vt∗ s Vt(−) b satisfy the unitarity condition (−) (18) of CKM. Moreover, since |Vt∗ s Vt b | 10−1 × (+) (−) |Vt∗ s Vt b |, Vt∗ s Vt b contribution to the physical quantities should be practically indistinguishable from SM results, and our numerical analysis confirms this expectation. This can also be seen in Fig. 1. There, the quantity R = Γ (Bs → ν ν̄γ )SM4 /Γ (Bs → ν ν̄γ )SM is plotted as a function of y = (mt /mW )2 . For Vt∗ s Vt(−) b this ratio R is approximately one. The greater mt is the more the ratio R differs from unity for the solution (+) Vt∗ s Vt b . the solution Vt∗ s Vt b ; the lower curve is for the solution Vt∗ s Vt b . (−) (+) 3. Numerical analysis In this section we will calculate the branching ratio in SM4 and study the influence of the fourth generation to the branching ratio. In [18] the branching ratio for the decay Bs → ν ν̄γ in SM was found to be of the order of 10−8 . In [19] the branching ratio in SM with three generations was calculated to Br(Bs → ν ν̄γ ) ≈ 7.5 × 10−8 . We use the following numerical values √ sin2 ΘW = 0.2319, α = 1/137, e = 4πα, GF = 1.16639 × 10−5 GeV−2 , mW = 80.22 GeV, mb = 4.8 GeV, mt = 176 GeV, ms = 0.51 GeV, mBs = 5.3 GeV, mc = 1.6 GeV, md = 0.25 GeV, |Vt∗s Vt b | = 0.045. For the values Vt∗ s Vt(±) see Tables 1 and 2. The b results on the branching ratios in SM4 are given in the Tables 1 and 2. For the Bs → ν ν̄γ decay for (±) both solutions Vt∗ s Vt b we see that for the choice Vt∗ s Vt(−) the branching ratios for the decay Bs → b ν ν̄γ calculated in SM4 coincide with the results from SM [18,19]. However, when we choose Vt∗ s Vt(+) b , we observe significant deviations from the SM. We observe that the branching ratio in SM4 is smaller than in SM for values mt < mt . But it increases for enlarging the mass mt . Y. Dinçer / Physics Letters B 505 (2001) 89–93 4. Conclusion In this Letter, we have studied the decay process Bs → ν ν̄γ in the Standard Model with four generations. We obtained two solutions of the fourth generation CKM factor Vt∗ s Vt b from the experimental data of B → Xs γ . We have used the two solutions to calculate the contributions of the fourth generation quark t to the Wilson coefficients. We have calculated the branching ratio for this process in the two cases and compared our results with those from the SM with three generations. We found that the new results are different from that of SM when the value of the fourth generation CKM factor is negative, but almost the same when the value is positive. Therefore, the decay Bs → ν ν̄γ could provide a possible way to probe the existence of the fourth generation if the fourth generation CKM factor Vt∗ s Vt b is negative. Acknowledgement The author wants to thank Prof. L.M. Sehgal for helpful suggestions. References [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [1] Mark II Collaboration, G.S. Abrams et al., Phys. Rev. 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Buras, hep-ph/9806471. 26 April 2001 Physics Letters B 505 (2001) 94–106 www.elsevier.nl/locate/npe Phenomenological issues in the determination of ΓD Eugene Golowich a , Sandip Pakvasa b a Physics Department, University of Massachusetts, Amherst, MA 01003, USA b Department of Physics and Astronomy, University of Hawaii at Manoa, Honolulu, HI 96822, USA Received 14 February 2001; accepted 21 February 2001 Editor: H. Georgi Abstract 0 width difference ΓD experimentally. The current situation is reviewed We consider the issue of determining the D 0 –D and suggestions for further study are given. We propose a number of D 0 decay modes in addition to those studied in the recent E791, FOCUS and BELLE lifetime determination experiments. Then we address prospects for determining CF-CDS strong phase differences, like δKπ which appears in the CLEO study of D 0 → K + π − transitions. We show how to extract δK ∗ π with CDS data and furthermore show when D → KL π data becomes available that δKπ can also be obtained. 2001 Published by Elsevier Science B.V. 1. Introduction 0 mixing are much smaller than those in the kaon, Bd and Bs systems. In the Standard Model, effects of D 0 –D However, charm-related experiments of increasing sensitivity have been carried out, leading to ever-improving bounds on the dimensionless mixing parameters xD ≡ MD /ΓD and yD ≡ ΓD /2ΓD . Most recently, the E791, CLEO, FOCUS and BELLE collaborations have reported on attempts to detect mixing in the D-meson system. This has prompted discussion in the literature as to whether actual D-meson mixing (specifically a nonzero ΓD ) is being seen for the first time [1]. Since a rigorous theoretical prediction for ΓD is unlikely, experimental progress in this area is needed. In this Letter, we discuss specific proposals for further work in lifetime difference measurements and in experimentally determining the strong phase δ (which occurs between Cabibbo-favored and Cabibbo-doubly-suppressed decays). 2. Measurements of lifetime differences The E791 [2], FOCUS [3] and BELLE [4] experiments study the time dependence for D 0 (t) → K + K − (CP = +1 final state) and D 0 (t) → Kπ (CP -mixed final state) under the assumption that CP invariance is assumed. This is reasonable in view of both theoretical expectations based on Standard Model physics and also 0 and introduce the CP recent CLEO results (see Section 3). If we adopt the convention that CP|D 0 = +|D E-mail address: [email protected] (S. Pakvasa). 0370-2693/01/$ – see front matter 2001 Published by Elsevier Science B.V. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 3 2 5 - 2 E. Golowich, S. Pakvasa / Physics Letters B 505 (2001) 94–106 95 eigenstates D 1 = √1 D 0 ± D 0 , 2 2 (1) then |D1 is CP -even and |D2 is CP -odd. It follows from Eq. (1) that 0 D (t) = √1 |D1 (t) + |D2 (t) , 2 (2) where |D1,2 evolve in time with distinct masses and decay widths, |Dk (t) = e−iMk t − 2 Γk t |Dk , 1 k = 1, 2. (3) If the K + K − final state is overlapped with Eq. (2) only the |D1 (t) part contributes, leading to the exponential decay equation ΓK K (t) = AK K e−Γ1 t . (4) For the Kπ final state, we express the time evolution of 0 0 D (t) = f+ (t)D 0 + f− (t)D , D0 as (5) where 1 −iMk t − 1 Γk t 2 f+ (t) = e , 2 2 k=1 1 1 f− (t) = (−)i+1 e−iMi t − 2 Γi t . 2 2 Then the above conditions 1, 2 imply ΓK − π + +K + π − (t) = AKπ e −Γ1 t (6) i=1 +e −Γ2 t = 2AKπ e −(Γ1 +Γ2 )t /2 t cosh (Γ1 − Γ2 ) . 2 (7) The experimental conditions are such that the cosh term in Eq. (7) is nearly unity. Thus the time dependence becomes exponential, allowing determination of (Γ1 +Γ2 )/2. The E791, FOCUS and BELLE experiments measure the quantity yCP , yCP ≡ and find yCP τD 0 →Kπ Γ1 − Γ2 −1= τD 0 →K + K − Γ1 + Γ2 (0.8 ± 2.9 ± 1.0)% (E791), = (3.42 ± 1.39 ± 0.74)% (FOCUS), +1.1 (1.0 +3.8 (BELLE). −3.5 −2.1 )% (8) (9) Due to its superior sensitivity the FOCUS determination dominates, the net result being a positive value for yCP of several per cent at about the two standard deviation level. 2.1. Additional decay modes We urge that additional lifetime studies on CP eigenstates of the neutral D be carried out. It is essential to improve the statistical data base and to acquire a sensitivity beyond the current 2σ level. Beyond that, there is still no experimental input on the pure CP = −1 lifetime. By using lifetimes obtained from pure CP = ±1 modes, one would be determining Γ directly rather than comparing an average of CP = ±1 lifetimes with that of CP = +1. 96 E. Golowich, S. Pakvasa / Physics Letters B 505 (2001) 94–106 There are a number of opportunities for further study, each final state occurring in D 0 decay being a potential candidate. We shall discuss just a limited number of these in the following, citing disadvantages as well as advantages. An important subset of our list has modes which contain a pair of mesons, each of which is selfconjugate under the CP operation. If each member of the pair has spin-zero, the orbital angular momentum is S-wave and the CP value of the two-particle state is simply the product of the individual CP values. If one meson has spin-zero and the other has spin-one, then conservation of angular momentum requires the particles to be in a P-wave. In this case, the CP value of the two-particle state becomes minus the product of the individual CP values. Vertex identification is a key to a successful D 0 lifetime measurement. Starting from its (assumed known) production point, the D 0 will travel unobserved and ultimately decay into some final state particles. In the best case, all these are charged and the decay point becomes well determined. In the worst case, each primary decay product is neutral and, if unstable, decays itself into neutral particles. Then even with calorimetric information, attempting to fix a decay point is problematic. For reference, in the E791, FOCUS and BELLE experiments the had just two particles (both charged) in each final state and the branching fractions detected modes (Kπ and K K) − + were BK π = (3.83 ± 0.09) × 10−2 and BK − K + = (4.25 ± 0.16) × 10−3 [5]. Many D 0 decay modes contain neutral kaons in the final state. The neutral kaons will in turn decay as KS or KL mesons. For a lifetime determination measurement, a KS mode is superior to a KL mode because: (i) the KS detection efficiency is rather larger than the KL detection efficiency, so the statistics will be better for the former, (ii) the KL decay occurs further from the D 0 decay vertex, so its background problem is more severe. Both of these considerations are inherent for any detector. However, since progress in dealing with KL detection is anticipated the KL modes should not be totally disregarded. To summarize, KS detection is easier and can be done now whereas KL is harder and may be done later, although not as well. Finally, we note that PDG listings give branching fractions 0 X (X denotes other final state particles) rather than for D → KS,L X. It will suffice below to use the for D → K approximations ΓD 0 →KS X ΓD 0 →KL X 12 ΓD 0 →K 0 X . (10) These relations are not exact because decay into KS or KL is subject to interference between Cabibbo favored (CF) and Cabibbo doubly suppressed (CDS) modes [6]. We discuss aspects of this interference in the next section. Now we turn to the list of additional possible modes, partitioned according to the CP of the final states and presented as CP = −1, CP = +1 and CP -mixed. Pure CP = −1 modes 1. KS φ: Both the KS and φ decay into charged final states, so this mode is an attractive one as regards particle detection. Since the φ → K + K − transition is a strong decay, it occurs right at the D 0 decay vertex. Also, the φ has a narrow decay width. The branching fraction for this mode is acceptably large (BφKS = (4.3 ± 0.5) × 10−3). 2. KS ω: Although the branching fraction is respectable (BKSω = (1.05 ± 0.2) × 10−2), the ω decays predominantly via the three-body mode π + π − π 0 which renders it more difficult regarding identification of the decay vertex. 3. KS ρ 0 : In this case, the branching fraction is not unattractive (BKSρ 0 = (0.61 ± 0.09) × 10−2 ), and the KS ρ 0 final state would decay into all charged particles. However, the larger width of the ρ 0 (compared to the φ) makes detection relatively more difficult. 4. KS π 0 : This mode has a reasonably large branching fraction (BKS π 0 = (1.06 ± 0.11) × 10−2 ). However, the presence of the π 0 hinders accurate vertex identification. 5. KS η and KS η : Both these modes are potentially interesting since the branching fractions are not highly suppressed (BKS η = (3.5 ± 0.5) × 10−3 and BKS η = (8.5 ± 1.3) × 10−3 ). The problem of vertex ID for a final state η and η would resemble that for a final state ω. E. Golowich, S. Pakvasa / Physics Letters B 505 (2001) 94–106 97 Pure CP = +1 modes 1. π + π − : This mode provides a clean CP = +1 signal but has the disadvantage of a small branching fraction (Bπ − π + = (1.52 ± 0.09) × 10−3 ), about three times less than K + K − . Also backgrounds could be a problem since since there are more π + π − combinations in a typical event (although in the D 0 rest frame, the two pions emerge back-to-back with larger momenta than any other final state). 2. KL η and KL η : Although the branching fractions equal those for KS η and KS η , detection of the KL presents difficulties, as discussed earlier. 3. π 0 φ: Particle ID is more of an issue than for the KS φ mode as the neutral π 0 decays via the chargeless twophoton mode. Even though the branching fraction here is comparatively large (Bπ 0 KS = (1.05 ± 0.11) × 10−2 ), it is not sufficient to compensate for the detection problem. Moreover, other decay modes containing π 0 ’s would be a source of background. 4. KS f0 (980): This mode consists of a scalar-pseudoscalar pair in an S-wave, and has CP = +1 for KS and CP = +1 for f0 (980). Although the KS and f0 (980) each decay into charged particles, the branching ratio is small (BKSf0 (980) = (2.9 ± 0.8) × 10−3 ). Similar comments apply to the KS f0 (1370) final state and to the KS f2 (1270) (except that here the final state is D-wave). 5. φρ 0 : This mode will have positive CP provided the φ and ρ 0 are in an S-wave of D-wave state. Both decay strongly into charged particles, so the decay point will have four emergent tracks. The branching fraction is rather small (Bφρ 0 = (6 ± 3) × 10−4 ). 6. KL π 0 : This final state has the same branching fraction as KS π 0 , but an even greater detection problem due to the KL . In practical terms, vertex identification would be an insurmountable obstacle. Mixed CP = ±1 modes Q=0 . There will For definiteness consider a Dalitz plot analysis for the neutral (Q = 0) three-body state (ππ K) ∗ be resonance bands corresponding to the quasi two-body modes (ρ K)Q=0 and (π K )Q=0 . Although neither the ∗ is a particularly narrow resonance, these decays are CKM dominant so the branching ratios are ρ nor the K relatively large. Specific examples of mixed CP = ±1 modes are: ∗− : This quasi two-particle state has a large branching fraction (B + ∗− = (5.0 ± 0.4) × 10−2 ) and 1. π + K π K 0 . The latter provides a rather ∗ decay modes K ∗− → π 0 K − and K ∗− → π − K there are the two measureable K + − ∗− clean three-body configuration, π (π KS ) where the parentheses stress the K parentage. 2. ρ + K − : The largest branching fraction among all quasi two-body final states for D 0 decay occurs here (Bρ + K − = (10.8 ± 0.9) × 10−2 ). The ρ + decay proceeds through only the mode ρ + → π + π 0 . ∗0 : The branching fraction (B 0 ∗0 = (3.1 ± 0.4) × 10−2 ) is relatively large. The associated three-body 3. π 0 K π K 0 ) and the less useful π 0 (π 0 K − ). configurations will be π 0 (π − K 3. Measurements of wrong-sign D 0 transitions Another study which impacts on determining ΓD is the CLEO experiment [7] which studies the decay rate for D 0 (t) → K + π − . This wrong-sign process can be produced both indirectly, from mixing followed by a CF decay, and directly, from CDS decay. The decay rate is given in the CP -invariant limit by r(t) = e−t RD + RD y · t + RM · t 2 . (11) √ The RD term arises from CDS decay, the RM term from mixing and the RD term from interference between the two. We also have the definitions y ≡ y cos δ − x sin δ, x ≡ x cos δ + y sin δ. (12) 98 E. Golowich, S. Pakvasa / Physics Letters B 505 (2001) 94–106 The parameter δ is the (strong-interaction) phase difference between between the CF and CDS amplitudes, (ch) ≡ δ− + − δ+ − , δ ≡ δKπ (13) where the phases δ− + and δ+ − appear in the amplitudes MD 0 →K − π + = |MD 0 →K − π + | eiδ−+ , MD 0 →K + π − = |MD 0 →K + π − | eiδ+− . (14) (ch) Note that we sharpen the notation for δ (δ → δKπ ) in Eq. (13) because we will encounter several analogous phases in our analysis. The CP -invariant rate formula of Eq. (11) can be generalized to incorporate various sources of CP -violation (CP V ) [8] RD → RD (1 + AD ) (CDS), y → y (1 + AM /2) (mixing), (ch) δKπ (interference), (ch) → δKπ +φ (15) where AD , AM and φ parameterize the extent of CP violation. When the data is fit to include the effects of CP violation none is found, AM = 0.23+0.63 −0.80 ± 0.01, AD = −0.01+0.16 −0.17 ± 0.01, sin φ = 0.00 ± 0.60 ± 0.01. (16) In the same fit one finds at 95% D.L. x = (0 ± 1.5 ± 0.2)% and y = −2.5+1.4 −1.6 ± 0.3 % (17) or equivalently |x | < 2.9% and − 5.8% < y < 1.0%. (18) x y, it is prudent to cite the results as bounds as in Given the present strength of the CLEO signals for and Eq. (18). One expects future experiments to reduce the statistical and systematic uncertainties. Even so, ignorance (ch) of the phase δKπ will hamper efforts to compare the FOCUS/E791/BELLE results with those from CLEO. 3.1. On the determination of y (ch) Can theory alone provide the value of δKπ ? Symmetry considerations are of only limited use. It is known that (ch) δKπ vanishes in the SU(3) invariant world [9,10], and this result has been recognized [11] in discussing aspects (ch) of the wrong-sign D 0 transitions. Thus, calculating the value of δKπ necessarily involves the physics of SU(3) breaking. Unfortunately, our limited understanding of physics in the charm region (especially the complicating effects of QCD) makes it difficult to perform reliable calculations [12]. It is, perhaps, not too surprising to find (ch) rather different statements in the literature about δKπ depending on the underlying approach. In one analysis [13], (ch) the findings of Refs. [14,15] are shown to imply rather small values for δKπ , less than 15◦ . However, the resonance (ch) model of Ref. [16] has considerably greater SU(3) breaking and obtains values as large as δKπ ∼ 30◦ . The largest (ch) value cited for δKπ appears in Ref. [1] which shows that accepting the central values of the FOCUS and CLEO (ch) experiments leads to δKπ in the second quadrant. However, it has been argued [17] that within a reasonable range (ch) of SU(3)-breaking parameters it is not possible to arrive at very large values of δKπ (45◦ or larger) of the type considered in Ref. [1]. In view of this state of affairs, it makes sense to explore what experiment can teach us. E. Golowich, S. Pakvasa / Physics Letters B 505 (2001) 94–106 99 3.2. Doing without KL data Recalling our comments in the previous section on the relative measureability of KS,L modes, we begin by assuming that only a data set not containing final state KL ’s is available. The inclusion of KL data is covered later. Our first conclusion concerns the CDS D → Kπ decays: D 0 → K + π − , D + → K + π 0 , D 0 → K 0 π 0 and + D → K 0 π + . At present, only the first of these has been observed (BK − π + = (1.46 ± 0.30) × 10−4 ). If only KS data is used, then neither D 0 → K 0 π 0 nor D + → K 0 π + modes can be determined experimentally. This can be understood by considering CF and CDS transitions having a neutral kaon in the final state (see Tables 1 and 2). For both CDS transitions D 0 → K 0 π 0 and D + → K 0 π + , the K 0 will decay via the same KS → π + π − mode as 0 π 0 and D + → K 0 π + . Since the CF decays dominate, extracting information about the CF transitions D 0 → K 0 CDS final states containing a K from just KS detection will be impossible. This negates performing a direct (ch) experimental measurement of δKπ . What is the situation for other possible final states like Kρ or K ∗ π ? Clearly, the same no-go result will hold for the D → Kρ decays. This leaves only the case of D → K ∗ π . Since the K ∗ decays strongly into two different charge combinations of Kπ , each D → K ∗ π transition will have two final configurations. Continuing to assume 0 is observed, we obtain the following list (see Tables 3 and 4). that only the KS mode in K 0 and K ∗ Each Kπ arising from K decay is enclosed in parentheses and FS1, FS2 are the two three-body final states per D decay. Each CDS transition with a K ∗0 in the final state has a configuration (FS2) identical to that of a CF ∗0 in the final state. However, the other configurations (FS1) each contain a charged kaon and transition with a K thus distinguish between CF and CDS decays. Thus, all four D → K ∗ π CDS decays can be utilized. In those final states containing a K ∗+ , both configurations FS1 and FS2 will have a unique signature (it is, however, necessary to employ a Dalitz plot analysis to properly identify which ‘Kπ ’ composite is a product of K ∗ decay). For final states with a K ∗0 , there will be a reduction Table 1 0 π CF decays D→K Transition Final state 0π 0 D0 → K KS π 0 0π + D+ → K KS π + Table 2 D → K 0 π CDS decays Transition Final state D0 → K 0 π 0 KS π 0 D+ → K 0 π + KS π + Table 3 ∗ π CF decays D→K Transition FS1 FS2 D 0 → K ∗− π + (K − π 0 )π + (KS π − )π + ∗0 π 0 D0 → K (K − π + )π 0 (KS π 0 )π 0 ∗0 π + D+ → K (K − π + )π + (KS π 0 )π + 100 E. Golowich, S. Pakvasa / Physics Letters B 505 (2001) 94–106 Table 4 D → K ∗ π CDS decays Transition FS1 FS2 D 0 → K ∗+ π − (K + π 0 )π − (KS π + )π − D 0 → K ∗0 π 0 (K + π − )π 0 (KS π 0 )π 0 D + → K ∗+ π 0 (K + π 0 )π 0 (KS π + )π 0 D + → K ∗0 π + (K + π − )π + (KS π 0 )π + factor of 2/3 in the number of events since only the configuration FS1 can be used, i.e., ΓD 0 →(K + π − )π 0 = 23 ΓD 0 →K ∗0 π 0 and ΓD + →(K + π − )π + = 23 ΓD + →K ∗0 π + . (19) Thus we are led to analyze the phenomenology of D → K ∗ π transitions in both the CF and CDS sectors. ∗ π ) decays Cabibbo favored (K ∗ π decays, There are three Cabibbo favored (CF) D → K D 0 → K ∗− π + , ∗0 π 0 , D0 → K ∗0 π + . D+ → K (20) These proceed through the QCD-corrected S = C = ±1 weak hamiltonian, which takes the form [18] (CF) (CF) (CF) HW = c− H− + c+ H+ . (21) (CF) (CF) H− and H+ transform alike under isospin, as the I3 = +1 member of an isotriplet. Under SU(3), however, (CF) (CF) H− belongs to 6 ⊕ 6∗ and H+ to 15 ⊕ 15∗ [19,20]. The coefficients c± encode the short distance, perturbative part of the QCD corrections. At energy scale MW , c± have essentially equal magnitudes. As the energy scale is lowered, the coefficient c− is enhanced whereas c+ is suppressed. (CF) Using just the isospin property of HW , we express the above decay amplitudes as 1 MK ∗− π + = A1 eiδ1 + 12 A3 eiδ3 , MK∗0 π 0 = − √1 A1 eiδ1 + 2 √1 A3 e iδ3 , 2 MK∗0 π + = 32 A3 eiδ3 , (22) ∗ π composites. Observe that these amplitudes where the subscripts represent twice the isospin of the final state K √ obey the sextet-dominance constraints MK∗0 π + = 0 and MK ∗− π + = − 2 MK∗0 π 0 . Upon either expanding the ∗0 π + |[I+ , H(CF) ]|D 0 or utilizing the amplitude relations of Eq. (22), one arrives at the isospin relation 0 = K W sum rule [21–24] √ MK ∗− π + + 2 MK∗0 π 0 − MK∗0 π + = 0. (23) Of interest to us here are the phase difference and amplitude ratio, √ 3ΓK ∗− π + + ΓK∗0 π + − 6ΓK∗0 π 0 2 cos(δ1 − δ3 ) = · , 4 [ΓK∗0 π + (3ΓK ∗− π + + 3ΓK∗0 π 0 − ΓK∗0 π + )]1/2 1/2 2ΓK∗0 π + A3 = . A1 3ΓK ∗− π + + 3ΓK∗0 π 0 − ΓK∗0 π + Kρ see in Table 5. ∗ π, Kπ, The most recent data compilation [5] for the three CF modes D → K 1 Equivalent formulae can be written for Kπ and Kρ. (24) E. Golowich, S. Pakvasa / Physics Letters B 505 (2001) 94–106 101 Table 5 Mode δ1 − δ3 ∗ π K ◦ 103.9◦ +17.2 −17.8◦ ◦ 90.2◦ +7.1 −8.2◦ ◦ 0.0 ± 44.9◦ Kπ Kρ A3 /A1 0.25+0.02 −0.01 0.37 ± 0.03 0.39 ± 0.10 Table 6 ΓD0 →K 0 π 0 /ΓD 0 →K − π + ΓD+ →K 0 π + /ΓD 0 →K − π + 0.5 0.0 0.551 ± 0.006 0.296 ± 0.028 Hypothetical world Real world The preceding equations can of course be used to obtain phase relations in addition to those in the above table, system, e.g., for the Kπ δK − π + − δ3 = 79.5◦, (25) δK 0 π 0 − δ3 = 110.3◦, decays, let us compare physics of the real world with that of and so on. Staying temporarily with the D → Kπ a world which is SU(3) symmetric and in which c+ = 0 (see Table 6). For these rates, at least, the agreement between the real world and the hypothetical world is not unreasonable. That the hypothetical world is, in some sense, nearby the real world will be useful later as a guiding principle in our study of the CDS amplitudes. This comparison between the real world and the hypothetical SU(3) world having c+ = 0 explains the small observed values of A3 /A1 . In an SU(3) symmetric world, the limit c+ = 0 amplitudes. Although the precise values of c± depend on the would correspond to A3 /A1 = 0 for the D → Kπ renormalization scheme (involving both the choice of operator basis and of renormalization scale µ), a typical numerical value is c+ /c− 0.5 for the range 2.0 µ (GeV) 1.5 [18]. The short distance effects embodied in c± account for much of the suppression for A3 /A1 observed in the above table, the rest arising from the operator matrix elements. Operator matrix elements play a much larger role in the kaon system (I = 1/2 rule) where QCD effects are more powerful. Cabibbo doubly suppressed (K ∗ π ) decays ∗ π CF decays of Eq. (20) are the following four D → K ∗ π decays in the Corresponding to the three D → K Cabibbo doubly suppressed (CDS) sector, D 0 → K ∗+ π − , D + → K ∗0 π + , D 0 → K ∗0 π 0 , D + → K ∗+ π 0 . (26) The CDS weak hamiltonian has S = −C = ±1 and is written analogous to Eq. (21), (CDS) (CDS) (CDS) HW = c− H− + c+ H+ , (CDS) but now H− (27) (CDS) and H+ behave differently under isospin, transforming, respectively, as an isosinglet and as the (CDS) (CF) (CDS) I3 = 0 member of an isotriplet. Under SU(3), H− (like H− ) transforms as a member of 6 ⊕ 6∗ and H+ (CF) (like H+ ) transforms as a member of 15 ⊕ 15∗ . Performing isospin decompositions of the CDS decay amplitudes yields √ √ √ √ MK ∗+ π − = 2 Āa ei δ̄1 − 2 Ā3 ei δ̄3 , MK ∗0 π + = 2 Āb ei δ̄1 + 2 Ā3 ei δ̄3 , MK ∗0 π 0 = −Āa ei δ̄1 − 2Ā3 ei δ̄3 , MK ∗+ π 0 = −Āb ei δ̄1 + 2Ā3ei δ̄3 , (28) 102 E. Golowich, S. Pakvasa / Physics Letters B 505 (2001) 94–106 where the CDS isospin moduli and phases are labelled with super-bars. Corresponding to the above four decay CDS decay amplitudes are the four physical observables Āa , Āb , A3 and δ̄1 − δ̄3 . Two distinct I = 1/2 moduli (Āa (CDS) and Āb ) occur because there are two independent sources of the I = 1/2 final state, the isoscalar H− and the (CDS) isovector H+ , √ (−) √ (+) and Āb ≡ 3 A(−) Āa ≡ 3 A1 + A(+) (29) 1 1 − A1 . We will need to determine the phase ∆¯ ∆¯ ≡ δ̄1 − δ̄3 (30) and the moduli ratios r, R, r≡ Ā3 Āa and R ≡ Āb . Āa (31) Since Āa , Āb and Ā3 are moduli, we have r > 0 and R > 0. In addition, we note that Ā3 /Āb = r/R. Taking the absolute square of each relation in Eq. (28) and forming ratios gives 2 + 2r 2 − 4r cos ∆¯ 2 + 2r 2 − 4r cos ∆¯ = , R , 2 1 + 4r 2 + 4r cos ∆¯ 2R 2 + 2r 2 + 4rR cos ∆¯ where the {Rk } are the ratios of CDS decay rates, R1 = R1 ≡ ΓD 0 →K ∗+ π − , ΓD 0 →K ∗0 π 0 R2 ≡ ΓD 0 →K ∗+ π − , ΓD + →K ∗0 π + R3 ≡ R3 = 2 + 2r 2 − 4r cos ∆¯ , R 2 + 4r 2 − 4rR cos ∆¯ ΓD 0 →K ∗+ π − . ΓD + →K ∗+ π 0 (32) (33) By eliminating r and cos ∆¯ from the relations in Eq. (32), one obtains a cubic equation in the variable R. However, there is an unphysical root R = −1, leaving the solution as a root of the quadratic equation R2 R3 (2R1 − 1)R 2 + (2R2 R3 + R1 R3 − 2R1 R2 − R1 R2 R3 )R + R1 R2 − 2R1 R3 = 0. (34) It turns out that to obtain the physical solution it is necessary to choose the square root of the discriminant as positive, √ −b + b2 − 4ac , R= (35) 2a where a = R2 R3 (2R1 − 1), b = 2R2 R3 + R1 R3 − 2R1 R2 − R1 R2 R3 , c = R1 R2 − 2R1 R3 . (36) To see why, let us consider a hypothetical world with c+ = 0. Then since that Ā3 = 0 ⇒ r = 0 and Āb = Āa ⇒ R = 1. (CDS) H− is an isoscalar operator, it follows (37) As a consequence, we have R1 = 2, R2 = 1, R3 = 2, (38) from which the physical solution of Eq. (34) is identified. Returning to the real world, from R one obtains r (2 − R1 )(1 + RR2 ) − 2(1 − R2 R 2 )(1 + R1 ) 1/2 r= (39) , 2(1 − R2 )(1 + R1 ) − 2(1 − 2R1 )(1 + RR2 ) E. Golowich, S. Pakvasa / Physics Letters B 505 (2001) 94–106 103 ¯ and lastly cos ∆, cos ∆¯ = 2 − R1 + 2r 2 (1 − 2R1 ) . 4r(1 + R1 ) (40) (ch) Determining the phase δK ∗ π ∗ ∗ ∗ (ch) (K π) (K π) (K π) In the relation δK − δ+− , the CF phase δ−+ cannot be determined from the relations in Eq. (22) ∗ π ≡ δ−+ because one cannot know the individual values of δ1 and δ3 . However, the first relation in Eq. (22) does allow one (K ∗ π) to solve for δ−+ − δ3 , sin(δ1 − δ3 ) (K ∗ π) . δ−+ (41) − δ3 = tan−1 cos(δ1 − δ3 ) + A3 /2A1 (K ∗ π) Analogously, from the CDS relation in Eq. (28) we can solve for δ+− sin(δ̄1 − δ̄3 ) (K ∗ π) δ+− − δ̄3 = tan−1 , cos(δ̄1 − δ̄3 ) − Ā3 /Āa − δ3 , (42) (K ∗ π) where δ+− is the phase of the CDS D 0 → K ∗+ π − amplitude. (ch) Combining these two relations we find for δK ∗ π ∗ ∗ (ch) (K π) (K π) δK − δ+− ∗ π ≡ δ−+ = δ3 − δ̄3 + tan−1 sin(δ1 − δ3 ) sin(δ̄1 − δ̄3 ) − tan−1 . cos(δ1 − δ3 ) + A3 /2A1 cos(δ̄1 − δ̄3 ) − Ā3 /Āa (43) Under the assumptions that only KS data is used and that isospin is a valid symmetry, we conclude that Eq. (43) will (ch) (ch) be the best one can do in a purely experimental determination of δK ∗ π . An expression for δK ∗ π itself is obtained only via dropping the contribution δ3 − δ̄3 . One might argue that these phases occur in exotic channels and should amplitudes, there is no SU(3) prediction that δ (ch) be individually small. Unlike the case of the D → Kπ K ∗ π = 0. 3.3. Including KL data In the previous subsection, the avoidance of KL data forced us to work with D → K ∗ π decays. The inclusion and D → Kπ decays in the following. Each D → KS,L π mode of KL data allows us to return to the D → Kπ will receive contributions from both CF and CDS sectors. Writing the transition amplitudes for D 0 → KS,L π 0 and D + → KS,L π + in a generic notation, we have MD→KS π = √1 |MCF |eiδCF − |MCDS |eiδCDS , 2 MD→KL π = − √1 |MCF |eiδCF + |MCDS |eiδCDS . (44) 2 The corresponding decay widths will each contain three terms, ΓD→KS π = 12 ΓCF − ΓCF ΓCDS cos(δCF − δCDS ) + 12 ΓCDS , ΓD→KL π = 12 ΓCF + ΓCF ΓCDS cos(δCF − δCDS ) + 12 ΓCDS . (45) 0 π to ignore all but the CF To our knowledge it has been standard in the PDG data compilation for D → K 0 X branching fraction from that contribution by using the “factor of two rule” in Eq. (10) to infer the CF D → K 104 E. Golowich, S. Pakvasa / Physics Letters B 505 (2001) 94–106 of D → KS X. However, some account of sub-dominant terms is made in Ref. [25] by attributing to their neglect a source of 10% systematic error. In terms of Cabibbo counting, the contributions on the right-hand side of Eq. (45) go as 1 : θc2 : θc4 or roughly 1 : 0.05 : 0.002. Taking the sum of decay rates gives ΓD→KS π + ΓD→KL π = ΓCF + ΓCDS . (46) Since no existing facility can deliver 0.2% sensitivity, the ΓCDS contribution to this equation is negligible and one arrives at the kind of relation given earlier in Eq. (10). There is, however, the possibility of observing the O(θc2 ) interference term via the asymmetry measurement [6,26] √ ΓD→KS π − ΓD→KL π ΓCDS A≡ (47) −2 √ cos(δCF − δCDS ), ΓD→KS π + ΓD→KL π ΓCF or more specifically ΓD 0 →K 0 π 0 cos(δK 0 π 0 − δK 0 π 0 ), A00 = −2 ΓD 0 →K 0 π 0 A0+ = −2 ΓD + →K 0 π + cos(δK 0 π + − δK 0 π + ). ΓD + →K 0 π + (48) These asymmetries are O(θc2 ), so signals will occur at about the 5% level. Such measurements are difficult for existing B-factories but hopefully can be performed. The detection of these asymmetries is clearly intriguing because they refer directly to δCF − δCDS . Although the phase differences δK 0 π 0 − δK 0 π 0 and δK 0 π + − δK 0 π + in Eq. (48) are for neutral modes (and not the charged case δK − π + − δK + π − ) it is nonetheless valuable information. At a rigorous level, it follows from the positivity of decay widths that a negative (positive) asymmetry would correspond a phase difference in the first (second) quadrant. Beyond that one is forced into modelling ΓCDS . Since this contributes as a square root, the effect of model dependence is somewhat softened but still may be large. We note that δK 0 π 0 − δK 0 π 0 = 0 in the SU(3) limit. We conclude this section by considering how to implement a complete data set for the D → Kπ decays. It is understood from the preceding discussion that we organize the KS and KL final states into sums and differences. There will be a total of seven D → Kπ decays, of which three provide information on CF physics and four on CDS physics. Defining Γk ≡ pĀ2k /(8πm2D ) (k = 1, 3), we have for CF-related decays ΓK − π + = Γ1 + Γ1 Γ3 cos(δ1 − δ3 ) + 14 Γ3 , 1 1 ΓKL π 0 + ΓKS π 0 = 2 Γ1 − Γ1 Γ3 cos(δ1 − δ3 ) + 2 Γ3 + Γa + 4 Γa Γ3 cos(δ̄1 − δ̄3 ) + 4Γ3 12 Γ1 − Γ1 Γ3 cos(δ1 − δ3 ) + 12 Γ3 , 9 ΓKL π + + ΓKS π + = 4 Γ3 + 2Γb + 4 Γb Γ3 cos(δ̄1 − δ̄3 ) + 2Γ3 94 Γ3 . (49) In the latter two relations, we have made the approximation of discarding O(θc4 ) contributions (in accordance with the discussion around Eq. (46)). The approximate relations are seen to reproduce the content of Eq. (24). For the CDS-related decays we have ΓK + π − = 2Γa − 4 Γa Γ3 cos(δ̄1 − δ̄3 ) + 2Γ3 , ΓK + π 0 = Γb − 4 Γb Γ3 cos(δ̄1 − δ̄3 ) + 4Γ3 , √ √ ΓKL π 0 − ΓKS π 0 = − 2 Γ1 Γa cos(δ1 − δ̄1 ) − 2 2 Γ1 Γ3 cos(δ1 − δ̄3 ) √ √ + 2 Γ3 Γa cos(δ3 − δ̄1 ) + 2 2 Γ3 Γ3 cos(δ3 − δ̄3 ), E. Golowich, S. Pakvasa / Physics Letters B 505 (2001) 94–106 ΓK L π + − ΓK S π + √ √ = 3 2 Γ3 Γb cos(δ3 − δ̄1 ) + 3 2 Γ3 Γ3 cos(δ3 − δ̄3 ), 105 (50) where Γk ≡ pĀ2k /(8πm2D ) (k = a, b, 3). Each term in the first two relations is O(θc4 ) while each term in the latter two are O(θc2 ). As an aid to analyzing these equations, we propose a simplified scenario with δ3 = δ̄3 = 0 and δ1 = π/2. The approximate equations which result are ΓK + π 0 Γb − 4 Γb Γ3 cos δ̄1 + 4Γ3 , ΓK + π − 2Γa − 4 Γa Γ3 cos δ̄1 + 2Γ3 , √ √ √ ΓKL π 0 − ΓKS π 0 − 2 Γ1 Γa sin δ̄1 + 2 Γ3 Γa cos δ̄1 + 2 2 Γ3 Γ3 , √ √ ΓKL π + − ΓKS π + 3 2 Γ3 Γb cos δ̄1 + 3 2 Γ3 Γ3 . (51) In general, one must solve numerically for the unknowns Γa , Γb , Γ3 and sin δ̄1 . Let us point out, however, the qualitative difference between the limiting cases δ̄1 π/2 or δ̄1 0. To study this difference, it is not enough to measure just the K + π − and K + π 0 final states; the KL,S π modes are required as well. It suffices to note here for δ̄1 π/2 that (ΓKL π 0 − ΓKS π 0 )2 /ΓK + π − → Γ1 and (ΓKL π + − ΓKS π + )2 /ΓK + π 0 → Γ3 Γ3 /Γb , whereas for δ̄1 0 both ratios become Γ3 . As δ̄1 proceeds from π/2 to 0, the first ratio decreases but the second increases by almost an order of magnitude. 4. Conclusions The recent FOCUS experiment on ΓD has yielded a signal at the several per cent level. By comparison, this experimental result is over an order-of-magnitude larger than the value yCP 0.8 × 10−3 obtained in a theoretical analysis [15] based on a sum over many D 0 decay modes. In this Letter, we have avoided the temptation to provide a theoretical prediction of our own for ΓD . As stated earlier, we are not aware of any analytic approach in the charm region for which theoretical errors/uncertainties can be controlled. We therefore feel that whether or not the FOCUS result holds up over time is for future experimental work to decide. At the very least, however, the E791, FOCUS, BELLE and CLEO studies serve to stimulate fresh thinking on a subject (D 0 mixing) that has long resisted progress. Our work in this Letter has been to suggest further experimental work which would be of value: 1. We have described in Section 2 both positive and negative aspects of various D 0 decays beyond those used in the E791, FOCUS and BELLE experiments. In particular, we recommend that the KS φ, KS ω and KS ρ 0 modes be given serious attention. Each of these lies within the CP = −1 sector, which heretofore has only been probed indirectly via the mixed-CP case of the D → (K − π + + K + π − ) transition. (ch) 2. In Section 3 we divided our discussion of the strong phase δ ≡ δKπ into two parts: Supposing that accurate data on KL final states is not forthcoming, we concluded that it will not be possible to probe the phase δKπ experimentally, but that the δK ∗ π decays would be accessible. Thus, we propose that branching fractions for the four CDS decays D 0 → K ∗+ π − , K ∗0 π 0 and D + → K ∗+ π 0 , K 0 π ∗+ be studied. At present, there is data only for the D + → K ∗0 π + transition, with a stated uncertainty of about 44%. Although any CDS branching fraction will be very small, the availability of copious charm production at B-factories and hadron colliders allows for the study of this hidden corner of charm physics. 2 We explored the eventuality that accurate data on KL π final states will also be gathered. In principle, the asymmetries of Eq. (48) would provide direct examples of CF-CDS phase differences, but are hindered by the 2 After this Letter was completed, an announcement appeared of a new CLEO measurement, y CP = −(1.1 ± 2.5 ± 1.4)%. [27] This is consistent with previous results (cf. Eq. (9)). They also report a first measurement of the CDS mode D 0 → K + π − π 0 which is a start of the exploration of the K ∗ π CDS modes. 106 E. Golowich, S. Pakvasa / Physics Letters B 505 (2001) 94–106 dependence on CDS branching fractions. A more ambitious program would be to collect the complete set of (Kπ) CDS Kπ data displayed in Eq. (51). In principle, this would allow for a determination of δ+− like that given ∗ in Eq. (43) for K π . Finally, we note that although our approach in this Letter has been limited to what can be learned from just decay rates, the study of Dalitz distributions in multibody final states offers a separate (Kπ) opportunity for attacking the “δ+− problem”. Acknowledgements The research described here was supported in part by the National Science Foundation and by the Department of Energy. We thank Guy Blaylock, Tom Browder, John Donoghue and Harry Nelson for their helpful input and Jonathan Link for a careful reading of the Letter. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] E.g., see: S. Bergmann, Y. Grossman, Z. 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Browder, S. Pakvasa, Phys. Lett. B 383 (1996) 475, hep-ph/9508362. L.-L. Chau, H.-Y. Cheng, Phys. Lett. B 333 (1994) 514, hep-ph/9404207. F. Buccella, M. Lusignoli, A. Pugliese, Phys. Lett. B 379 (1996) 249, hep-ph/9601343. A.F. Falk, Y. Nir, A.A. Petrov, J. High En. Phys. 9912 (1999) 019, hep-ph/9911369. M. Gronau, J. Rosner, U-spin symmetry in doubly cabibbo-suppressed charm meson decays, hep-ph/0010237, submitted to Phys. Lett. B. For reviews see, A.J. Buras, Weak hamiltonian, CP violation and rare decays, hep-ph/9806471; A.J. Buras, Operator product expansion, renormalization group and weak decays, hep-ph/9901409. J. Donoghue, B. Holstein, Phys. Rev. D 12 (1975) 1454. N. Cabibbo, G. Altarelli, L. Maiani, Nucl. Phys. B 88 (1975) 285. T.E. Browder, K. Honscheid, D. Pedrini, Ann. Rev. Nucl. Part. Sci. 46 (1997) 395, hep-ph/9606354. E. Golowich, Proc. of Fourth KEK Topical Conference on Flavor Physics, World Scientific, 1998, pp. 245–259. J. Rosner, Phys. Rev. D 60 (1999) 114026, hep-ph/9905366. M. Gronau, Phys. Rev. Lett. 83 (1999) 4005, hep-ph/9908237. CLEO Collaboration, Phys. Rev. Lett. 78 (1997) 3261, hep-ex/9701008. Z.-Z. Xing, Phys. Rev. D 55 (1997) 196. CLEO Collaboration, Mixing and CP violations in the decay of neutral D mesons at CLEO, Report CLEO CONF 01-1, hep-ex/012006. 26 April 2001 Physics Letters B 505 (2001) 107–112 www.elsevier.nl/locate/npe Effects from the charm scale in K + → π +ν ν̄ Adam F. Falk a , Adam Lewandowski a , Alexey A. Petrov b a Department of Physics and Astronomy, The Johns Hopkins University, 3400 North Charles Street, Baltimore, MD 21218, USA b Newman Laboratory of Nuclear Studies, Cornell University, Ithaca, NY 14853, USA Received 14 December 2000; accepted 28 February 2001 Editor: M. Cvetič Abstract We consider contributions to the rare decay K + → π + ν ν̄ which become nonlocal at the charm scale. Compared to the leading term, such amplitudes are suppressed by powers of m2K /m2c and could potentially give corrections at the level of 15%. We compute the leading coefficients of the subleading dimension eight operators in the effective theory below the charm mass. The matrix elements of these operators cannot all be calculated from first principles and some must be modeled. We find that these contributions are likely to be small, but the estimate is sufficiently uncertain that the result may be as large as the existing theoretical uncertainty from other sources. 2001 Published by Elsevier Science B.V. The search for New Physics relies on experimentally accessible quantities whose Standard Model values can be predicted accurately and reliably. This task is often complicated by nonperturbative hadronic physics, especially when one is interested in the parameters of the Cabibbo–Kobayashi–Maskawa (CKM) matrix. To make progress, it is important to find processes where symmetry can be used to treat low energy QCD effects in a controlled and systematic way. One of these is the rare decay K + → π + ν ν̄. This process is an example of a neutral current S = 1 transition, which in the Standard Model can occur only via one-loop diagrams. The leading contributions to the effective Hamiltonian for this decay are given by α GF Heff = √ 2 2π sin2 ΘW l × (xc ) Vt∗s Vt d X(xt ) + Vcs∗ Vcd XNL l E-mail address: [email protected] (A.F. Falk). × s̄γ ν 1 − γ 5 d ν̄l γν 1 − γ 5 νl , (1) where the index l = e, µ, τ denotes the lepton flavor. The coefficient X(xt ) arises from the top quark loop and is independent of lepton flavor. It is dominated by calculable high energy physics, and has been computed to O(αs ) [1]. Because it grows as m2t , it is large and gives the leading contribution to the decay rate. If this were the sole contribution, the measurement of K + → π + ν ν̄ would yield a direct determination of the combination of CKM parameters |Vt∗s Vt d | [2]. However, due to the smallness of Vt∗s Vt d compared to Vcs∗ Vcd , the charm contribution contained l in the coefficient function XNL (xc ) is significant as well. These terms have been calculated to next-toleading logarithmic order [3]. An important source of error in the calculation comes from the uncertainty in l (x ) depends. the charm quark mass, on which XNL c An important feature of the calculation is the fact that the hadronic matrix element π + |s̄γ ν (1 − γ 5 )d|K + is related via isospin to the matrix element π 0 |s̄γ ν (1 − γ 5 )d|K + responsible for K + → 0370-2693/01/$ – see front matter 2001 Published by Elsevier Science B.V. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 3 4 3 - 4 108 A.F. Falk et al. / Physics Letters B 505 (2001) 107–112 π 0 e+ ν. This largely eliminates the uncertainty due to nonperturbative QCD, up to small isospin breaking effects [4]. However, there remain long distance contributions associated with penguin diagrams containing up quarks which can lead to on-shell intermediate states. Some of these have been estimated in chiral perturbation theory and found to be small [7]. The perturbative contribution from virtual up quarks is tiny, since it is suppressed compared to the charm contribution by m2u /m2c . Summed over neutrino species, the branching fraction for K + → π + ν ν̄ is given by B(K + → π + ν ν̄) 2 Re ξc l Im ξt = κ+ X(x ) + XNL (xc ) t 5 λ 3λ5 l 2 Re ξt (2) + 5 X(xt ) , λ with λ = sin θC ≈ 0.22 and κ+ = rK+ 3α 2 B(K + → π 0 e+ ν) 2π 2 sin4 ΘW λ8 . (3) Here ξi = Vis∗ Vid , and rK + absorbs isospin breaking corrections to the relationship between the decays K + → π 0 e+ ν and K + → π + ν ν̄ calculated in Ref. [4]. In terms of the Wolfenstein parameterization of the CKM matrix [5], the branching ratio may be written as B(K + → π + ν ν̄) = 4.11 × 10−11 · A4 X2 (xt ) 1 × (σ η̄)2 + (ρ0 − ρ̄)2 , σ with −2 σ = 1 − λ2 /2 and ρ0 = 1 + δc , (4) where δc absorbs the charm contribution. A measurement of the branching ratio then constrains the parameters ρ̄ and η̄, which are equal to the Wolfenstein parameters ρ and η up to known corrections of O(λ2 ). The Alternate Gradient Synchrotron (AGS) experiment E949 at Brookhaven and the CKM collaboration at Fermilab propose to obtain measurements of the branching ratio for K + → π + ν ν̄ at the level of 30% and 10%, respectively. The Brookhaven experiment is the successor to AGS-E787, which saw one event in this channel [6]. These experimental prospects then fix the goal for the accuracy of the theoretical prediction at less than 10%. The leading source of theoretical uncertainty is associated with the charm contribution. Calculations at next-to-leading order in QCD yield δc = 0.40 ± 0.07, where the error is due primarily to the uncertainty in the charm mass [3]. The errors from uncomputed terms of order αs2 (mc ) are expected to be small. However, the computation of the charm contribution relies on an operator product expansion which is simultaneously a series in αs and an expansion in higher dimension operators suppressed by powers of mc . The operators which are of higher order in the 1/mc expansion reflect the fact that the penguin loop becomes nonlocal at the relatively low scale mc . One might expect the leading correction from higher order terms to give a contribution to δc of relative size m2K /m2c ∼ 15%, large enough to affect in a noticeable way the extraction of ρ̄ and η̄ from the decay rate. It is important either to verify or to exclude the presence of new terms of such a magnitude. In this Letter we will study the contributions of dimension eight operators to the decay K + → π + ν ν̄. We estimate the correction to δc and comment on the uncertainty induced. After discussing the relevant power counting, we present the calculation of the operator coefficients and an estimation of the correction to the decay rate. We will find a small contribution, but one that need not be negligible. The decay K + → π + ν ν̄ proceeds via the loop processes shown in Fig. 1, which mediate the quark level transition s̄ → d̄ν ν̄. These diagrams contain both short distance and long distance effects, which we separate by computing the effective Hamiltonian density Heff at a low scale µ 1 GeV. The effective Hamiltonian will receive corrections from the charm and top quarks, both of which have been integrated out of the theory, and from highly virtual up quarks. Soft up quarks remain in the theory, and are responsible for long distance corrections. We construct the effective Hamiltonian with an operator product expansion. At leading order, the operator in Heff which contributes to the decay is of dimension six, O (6) = s̄γ ν 1 − γ 5 d ν̄γν 1 − γ 5 ν. (5) The t quark contribution to the coefficient of this operator is obtained by evaluating the diagrams in A.F. Falk et al. / Physics Letters B 505 (2001) 107–112 109 Fig. 1. Penguin and box diagrams responsible for K + → π + ν ν̄. Fig. 1 at the scale µ = MW ≈ mt and matching on to the effective theory below this scale. At the same time, the W and Z are integrated out of the theory, producing four-fermion operators involving up and charm quarks as well. The charm contribution to the operator is then obtained by evaluating the diagrams contributing to the decay at µ = mc . These diagrams look like those in Fig. 1, but with the W and Z propagators replaced by local interactions. Dimensional analysis indicates that the coefficient 2 . of the dimension six operator O (6) scales as 1/MW The diagrams in Fig. 1 are quadratically divergent in 4 , the effective theory below MW , and scale as Λ2 /MW where Λ ∼ MW is an ultraviolet cutoff. The Glashow– Iliopoulos–Maiani (GIM) mechanism ensures that this leading divergence cancels, since it is independent of the mass mq of the virtual quark. The consequence is that the coefficient of O (6) actually scales as 4 . In terms of the Wolfenstein parameter λ, m2q /MW 4 and the the top coefficient has strength λ5 m2t /MW 4 . The top charm coefficient has strength λm2c /MW contribution is significant because of the large top mass, since λ4 m2t /m2c is of order 10. For the purpose of power counting, the operators of dimension eight scale as 2 (6) O , O (8) ∼ MK (6) appearing with generic coefficient C(8) . Dimension4 . The top contribually, C(8) is proportional to 1/MW 2 /m2 relative to (8) tion to C(8) O is suppressed by MK t its contribution to O (6) , leading to an overall strength 2 /M 4 . The corresponding suppression of order λ5 MK W 2 /m2 , so the overall contribution for charm is only MK c (8) 2 /M 4 . Note that of charm to C(8) O scales as λMK W there is now no relative enhancement from the large top mass, so the top contribution to C(8) is suppressed Fig. 2. Diagrams leading to operators of dimension eight. relative to that of charm by λ4 and can be neglected. Furthermore, the contributions in question are independent of mq , so they cancel by the GIM mechanism when the up contribution is included. However, the GIM cancellation is manifest in Feynman diagrams only for contributions which are perturbatively calculable. The long distance contributions involving soft up quarks will differ by factors of order one from their perturbative representations. For 4 , these parts of the diagrams, which scale as 1/MW the GIM cancellation is ineffective. Such long distance contributions have been considered elsewhere [7], and estimated to be small. The GIM cancellation is also spoiled by logarithmic contributions proportional to 2 /M 4 ) ln(m2 /M 2 ). Such terms may be generated (MK c W K by the running of Heff between the scale mc and the low energy scale µ 1 GeV. This is not a large logarithm, numerically, but it allows us nonetheless to identify a GIM violating contribution to Heff . This term, which is generated by intermediate up quarks as shown in Fig. 2, is of the same power-counting size as the long distance contribution. But because the perturbative description of the long-distance part is inaccurate, there is no reason to expect the GIM cancellation to be restored when it is included. 110 A.F. Falk et al. / Physics Letters B 505 (2001) 107–112 The purpose of this Letter is to compute the cor2 /M 4 ) ln(m2 / rections to K + → π + ν ν̄ of order (MK c W 2 MK ). These contributions are well defined, and it is important, in light of the experimental situation discussed above, to determine whether they introduce a theoretical uncertainty at a level competitive with the uncertainty due to mc . Note that pure power counting arguments permit a relative contribution to δc of the order of (m2K /m2c ) ln(m2c /µ2 ), which could be as large as 20%, depending on the value chosen for the hadronic scale µ. We will study the effective Hamiltonian of dimension eight operators, at leading order in αs . This Hamiltonian receives logarithmically enhanced contributions from the up quark loops in Fig. 2. We also must consider the matching corrections at the scale mc ≈ mτ , when the tau lepton is integrated out of the theory. Because the matching function F (mc /mτ ) cannot be approximated by an expansion in mc /mτ , the combination [F (mc /mτ ) − F (mu /mτ )] is a GIM violating finite matching correction which also must be included. The effective Hamiltonian density at the scale µ takes the form Cil (µ)Oil (µ), Heff = (7) l,i where l denotes lepton flavor. As it turns out, there will be two dimension eight operators generated in the theory below mc , O1l = s̄γ ν 1 − γ 5 d(i∂)2 ν¯l γν 1 − γ 5 νl , O2l = s̄γ ν 1 − γ 5 (iD)2 d ν¯l γν 1 − γ 5 νl + 2s̄γ ν 1 − γ 5 iD µ d ν¯l γν 1 − γ 5 (i∂µ )νl + s̄γ ν 1 − γ 5 d ν¯l γν 1 − γ 5 (i∂)2 νl . (8) Fig. 3. Diagrams which could lead to an operator with a gluon field strength. with a gluon field strength, such as O3l = s̄γ ν σ αρ Gαρ 1 − γ 5 d ν¯l γν 1 − γ 5 νl . (9) However, it turns out that contributions to all such operators cancel. l The operators of dimension six that will induce O1,2 in Heff are O4 = s̄γ ν 1 − γ 5 d ūγν 1 − γ 5 u, O5 = s̄γ ν 1 − γ 5 u ūγν 1 − γ 5 d, O6l = ūγν − 43 sin2 θW 1 + γ 5 + 1 − 43 sin2 θW 1 − γ 5 × u ν¯l γ ν 1 − γ 5 νl , O7l = s̄γ ν 1 − γ 5 u ν¯l γν 1 − γ 5 l, ¯ ν 1 − γ 5 νl . O l = ūγ ν 1 − γ 5 d lγ (10) 8 The operator O6l comes from virtual Z exchange, the others from W exchange. The renormalization group equations for O1l and O2l are dC1l = = γ1 C4 C6l + γ1 C5 C6l , dµ dC l γ2j Cjl . µ 2 = = γ2 C7l C8l + dµ µ (11) j The first of these operators does not receive any logarithmic QCD corrections below the scale mc , because it is proportional to a current which is partially conserved. The second does, but we will not include higher order corrections of relative order αs ln(mc /µ). Note that this is not inconsistent with resumming terms of order αsn lnn (MW /mc ). l are generated by the diagrams The operators O1,2 in Fig. 2. In principle, one might have expected the diagrams in Fig. 3 to generate additional operators The anomalous dimensions γ1 , γ1 and γ2 are of order one. The matrix γ2j is of order αs and comes from QCD running below mc ; it will not be included in our analysis. Computing the diagrams in Fig. 2 and solving the renormalization group equations, we find the coefficients at the scale µ, c0 e,µ,τ = 1 − 43 sin2 ΘW G(αs ) log(µ/mc ), C1 2 6MW A.F. Falk et al. / Physics Letters B 505 (2001) 107–112 c0 log(µ/mc ), 2 MW c0 C2τ = − f m2c /m2τ , 2 4MW e,µ C2 =− (12) where α GF c0 = √ Vcs∗ Vcd , 2 2π sin2 θW 6x − 2 4x f (x) = − 2 log x − , 3 (x − 1) (x − 1)2 αs (mc ) −6/25 αs (mb ) −6/23 G(αs ) = 2 αs (mb ) αs (MW ) 12/25 αs (mc ) αs (mb ) 12/23 − , αs (mb ) αs (MW ) e,µ,τ 2 = 0.05 · c0 /MW , e,µ C2 in which case R2 ≈ (pπ + pν̄ )2 = (340 MeV)2 . (13) 2 = 0.69 · c0 /MW , 2 C2τ = 0.28 · c0 /MW . (14) By comparison, the coefficient of the leading charm l (x ) c , which contribution in Eq. (1) is given by XNL c 0 2 2 2 for is 4.0 mc · c0 /MW for l = e, µ and 2.7 m2c · c0 /MW l = τ. To compute the contribution to the decay rate, we l |K + . The also need the matrix elements π + νl ν̄l |O1,2 leading relative corrections come from the interference of O1,2 with O (6) and depend on the ratios Re d[P.S.]| π + νl ν̄l |Oil |K + ∗ π + ν ν̄|O (6)|K + Ri = . d[P.S.]| π + ν ν̄|O (6)|K + |2 (15) The matrix element of the operator O1 is easy to calculate, since it depends only on the lepton momenta. The leptons are treated perturbatively, so the hadronic dependence of the matrix element of O1 is the same as that of O (6) . We then find R1 = (pν + pν̄ )2 = (180 MeV)2 . (16) Unfortunately, the matrix element of O2 cannot be calculated analytically, since it involves the gluon field through the covariant derivative acting on the down quark. We are forced to rely instead on model dependent estimates, which are notoriously unreliable. One ansatz would be to take π + ν ν̄|O2 |K + µ ≈ µ2 π + ν ν̄|O (6)|K + µ , or R2 ≈ µ2 ∼ (650 MeV)2 . Another would be to neglect the gluon field and model the matrix element as π + ν ν̄|O2 |K + = (pπ + pν̄ )2 π + ν ν̄|Q(6)|K + , (18) ∗ V ≈ −V ∗ V . Taking the valand we have used Vus ud cs cd ues mc = 1.3 GeV, mb = 4.5 GeV, ΛMS = 0.35 GeV and µ = mc /2, we find C1 111 (17) (19) Of course, neither of these guesses need be correct within better than an order of magnitude. Fortunately, lattice QCD methods are advancing quickly, to the point that a true unquenched lattice calculation of this matrix element may soon be feasible. For now, we will take these two crude guesses to bracket roughly the actual value of R2 . We now write the branching fraction for K + → + π ν ν̄ as in Eq. (4), with ρ0 = 1 + δc (1 + δ8 ), (20) where δ8 is the new term which we are computing. Summed over lepton species, the contribution of charm at dimension six is given by δc = P0 (xc ) 1 1 l XNL (xc ) · 2 = 4 . 2 A X(xt ) 3λ A X(xt ) (21) l A next to leading order analysis yield P0 = 0.42 ± 0.06, where the error arises in large part from the uncertainty in the charm quark mass [3]. This value of P0 gives δc = 0.40 ± 0.07, where we use X(xt ) = 1.53 ± 0.01 and A = 0.83 ± 0.06. The fractional correction due to dimension eight operators is then δ8 = 1 l (1) (2) C1 R1 + C2l R2 = δ8 + δ8 . 4 3P0 λ (22) l The first term, for which the matrix element is calculable, is negligible in size: with our choice of inputs, δ8(1) = 5.6 × 10−5 . The second, highly uncertain, (2) term is much bigger, with δ8 between 1% and 5% for our adopted range for R2 . On the one hand, even δ8 as large as 5% is somewhat below the existing uncertainty on δc from the value of mc . On the other, if our “upper limit” on the matrix element of O2l were too small by even a factor of two, which need not be unlikely, these contributions would have a significant effect on the extraction of CKM parameters from the branching fraction. 112 A.F. Falk et al. / Physics Letters B 505 (2001) 107–112 We have made a number of approximations in obtaining these results. A potentially important one, within the perturbative calculation, is that we have neglected QCD running below mc . We could include these QCD corrections for O1 simply by incorporating the known running of the coefficients C4 and C5 ; doing so decreases C1 by a factor of two. However, the running of O2 is not equally trivial, since O2 itself is renormalized in QCD. In view of the large uncertainty in the matrix element of O2 , including these QCD corrections would not at this time increase the reliability of our prediction. Of course, the key uncertainty arises not from QCD perturbation theory but from the actual value of π + ν ν̄| O2 |K + . Only a realistic lattice computation will settle the matter. We would argue, in fact, that such a calculation is really required for one to be confident that the effects we have considered do not spoil the extraction of CKM matrix elements from the proposed experiments on K + → π + ν ν̄. This is not the only case where higher dimensional operators can play an interesting role in kaon decays [8,9]. In summary, we have computed the dominant contribution to the coefficients of dimension eight operators contributing to the decay K + → π + ν ν̄. Our best estimate is that this represents a correction of no more than 5% to the leading charm contribution to the decay. However, our ignorance of relevant hadronic matrix elements leaves open the possibility that these contributions could represent an uncertainty as large as or larger than that due to the charm quark mass. A lattice calculation of nonperturbative corrections, and to a lesser extent the inclusion of perturbative QCD cor- rections below the charm scale, will be indispensable to reducing this uncertainty before the planned experiments begin to take their data. Acknowledgements We are grateful to Mark Wise for discussions. A.F. and A.L. are supported in part by the United States National Science Foundation under Grant No. PHY-9404057 and by the United States Department of Energy under Outstanding Junior Investigator Award No. DE-FG02-94ER40869. A.P. is supported in part by the National Science Foundation. A.F. is a Cottrell Scholar of the Research Corporation. References [1] T. Inami, C.S. Lim, Prog. Theor. Phys. 65 (1981) 297; G. Buchalla, A. Buras, Nucl. Phys. B 398 (1993) 285; G. Buchalla, A. Buras, Nucl. Phys. B 400 (1993) 255; M. Misiak, J. Urban, Phys. Lett. B 451 (1999) 161. [2] G. Buchalla, A.J. Buras, Nucl. Phys. B 548 (1999) 309–327. [3] G. Buchalla, A. Buras, Nucl. Phys. B 412 (1994) 106. [4] W.J. Marciano, Z. Parsa, Phys. Rev. D 53 (1996) 1. [5] L. Wolfenstein, Phys. Rev. Lett. 51 (1983) 1945. [6] S. Adler et al., E787 Collaboration, Phys. Rev. Lett. 84 (2000) 3768. [7] M. Lu, M.B. Wise, Phys. Lett. B 324 (1994) 461; C.Q. Geng, I.J. Hsu, Y.C. Lin, Phys. Lett. B 355 (1995) 569; C.Q. Geng, I.J. Hsu, C.W. Wang, Prog. Theor. Phys. 101 (1999) 937. [8] G. Buchalla, G. Isidori, Phys. Lett. B 440 (1998) 170. [9] V. Cirigliano, J.F. Donoghue, E. Golowich, JHEP 0010 (2000) 048. 26 April 2001 Physics Letters B 505 (2001) 113–118 www.elsevier.nl/locate/npe J /ψ suppression: gluonic dissociation vs. colour screening Binoy Krishna Patra, Dinesh Kumar Srivastava Variable Energy Cyclotron Centre, 1/AF Bidhan Nagar, Kolkata 700 064, India Received 9 September 2000; received in revised form 8 January 2001; accepted 28 February 2001 Editor: J.P. Blaizot Abstract We evaluate the suppression of J /ψ production in an equilibrating quark–gluon plasma for two competing mechanisms: Debye screening of colour interaction and dissociation due to energetic gluons. Results are obtained for S + S and Au + Au collisions at RHIC and LHC energies. At RHIC energies the gluonic dissociation of the charmonium is found to be equally important for both the systems while the screening of the interaction plays a significant role only for the larger systems. At LHC energies the Debye mechanism is found to dominate for both the systems. While considering the suppression of directly produced Υ at LHC energies, we find that only the gluonic dissociation mechanism comes into play for the initial conditions taken from the self screened parton cascade model in these studies. Thus we find that a systematic study of quarkonium suppression for systems of varying dimensions can help identify the source and the extent of the suppression. 2001 Published by Elsevier Science B.V. PACS: 12.38M Relativistic heavy ion collision experiments at the CERN SPS are believed [1] to have led to a production of quark–gluon plasma — which existed in the early universe and which may be present in the core of neutron stars. The last two decades have seen a hectic activity towards identifying unique signatures of the quark–hadron phase transition. The suppression of J /ψ production in such collisions has been one of the most hotly debated signals in this connection. The heavy quark pair leading to the J /ψ mesons are produced in such collisions on a very short time-scale ∼1/2mc , where mc is the mass of the charm quark. The pair develops into the physical resonance over a formation time τψ and traverses the plasma and (later) the hadronic matter before leaving E-mail address: [email protected] (D.K. Srivastava). the interacting system to decay (into a dimuon) to be detected. This long ‘trek’ inside the interacting system is fairly ‘hazardous’ for the J /ψ. Even before the resonance is formed it may be absorbed by the nucleons streaming past it [2]. By the time the resonance is formed, the screening of the colour forces in the plasma may be sufficient to inhibit a binding of the cc [3,4]. Or an energetic gluon [5] or a comoving hadron [6] could dissociate the resonance(s). The extent of absorption will be decided by a competition between the momentum of the J /ψ and the rate of expansion and cooling of the plasma, making it sensitive to such details as the speed of sound [7,8]. Thus a study of J /ψ production is poised to provide a wealth of information about the evolution of the plasma and its properties. It has been shown [9] that the nucleonic absorption (the “normal absorption”), operating on the pre- 0370-2693/01/$ – see front matter 2001 Published by Elsevier Science B.V. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 3 4 8 - 3 114 B.K. Patra, D.K. Srivastava / Physics Letters B 505 (2001) 113–118 resonance — which is yet to evolve into a physical particle — is identical for J /ψ, ψ , and χc . This absorption is always present and is brought about by the nucleons (or the Lorentz-contracted partonic clouds) streaming past the pre-resonances, as mentioned earlier. A reliable quantitative estimate within Glauber model is available [9] for this. In the present work we concentrate on the dissociation of the charmonium in quark–gluon plasma due to colour screening and scattering with gluons and ask whether we can distinguish between the two mechanisms. We emphasize that these mechanisms are in addition to nucleonic absorption mentioned earlier. In principle the colour screening is a collective effect, where the presence of a large number of colour quanta modifies the force between c and c so that, above the critical temperature (Tc ∼ 200 MeV), we have: V (r) = −α/r + σ r → V (r) = −α exp(−µD r)/r, (1) where α and σ (the string tension) are phenomenological parameters and µD is the Debye mass. Thus, e.g., the direct production of the J /ψ is inhibited once the Debye mass is more than 0.7 GeV [10]. The gluonic dissociation, on the other hand, is always possible as long as an energetic gluon can be found. They can always be present in the tail of the thermal distributions and thus given sufficient time, a J /ψ can always be dissociated in a plasma of any temperature! Of course in actual practice the QGP will expand and cool and undergo hadronization below the critical temperature Tc , and thus the hot medium will have only a finite life-time. This enriches the competition between the mechanisms of the gluonic dissociation and the Debye screening for the charmonium suppression. In the present work we show that this also provides us with a handle to decipher the extent to which each mechanism contributes to the suppression of J /ψ. Let us assume that a thermally equilibrated plasma is formed in relativistic heavy ion collisions at some time τi and that the elastic scattering among the partons is sufficiently rapid to maintain thermal equilibrium. A large number of studies [11,12] have indicated that the plasma thus produced may not be in a state of chemical equilibrium and that the quark and gluon fugacities are less than unity. We assume that the chem- ical equilibration proceeds dominantly via gg ↔ ggg, gg ↔ qq. (2) Assuming the evolution to proceed according to Bjorken hydrodynamics, the evolution of the parton densities are given by [13]: λ2g λ˙g 1 Ṫ + 3 + = R3 (1 − λg ) − 2R2 1 − 2 , (3) λg T τ λq λ˙q 1 a1 λg λq Ṫ + 3 + = R2 − , (4) λq T τ b1 λq λg 3/4 b2 T 3 τ = const, λg + λq (5) a2 where a1 = 16ζ(3)/π 2 ≈ 1.95, a2 = 8π 2 /15 ≈ 5.26, b1 = 9ζ(3)Nf /π 2 ≈ 2.20, and b2 = 7π 2 Nf /20 ≈ 6.9. The expressions for the density and velocity weighted reaction rates, R3 = 12 σgg→ggg vng , R2 = 12 σgg→qq vng , (6) can be found in Ref. [13]. The results for the time evolution of the fugacities and the temperature for the initial conditions obtained from the self screened parton cascade model [12] for Au + Au collisions at RHIC and LHC energies are given in Ref. [14]. For the S + S collisions we assume that while the initial fugacities are same as those for the Au + Au system, the initial temperatures are estimated by assuming that it scales as Ti ∼ A0.126 . This is motivated by a recent study on the basis of parton saturation [15] which also suggests that the initial number density divided by Ti3 is nearly independent of the mass-number of the nuclei. This, we believe, provides a useful initial guess, even though the conditions envisaged for self screening are not strictly met for S + S at RHIC. For the sake of completeness, we have given the initial conditions in Table 1. It may be noted that these are different from those used in Ref. [5], which were ‘inspired’ by the HIJING model and which had, for example, much smaller fugacities. (We have verified that our computer program fully reproduced the results of Ref. [5], with the initial conditions given there.) We shall also introduce a energy density profile such that, β "(τi , r) = (1 + β) "i 1 − r 2 /R 2 Θ(R − r), (7) B.K. Patra, D.K. Srivastava / Physics Letters B 505 (2001) 113–118 Table 1 Initial values for the time, temperature, fugacities etc. for Au + Au [12] and S + S at RHIC and LHC Au + Au RHIC "i (GeV/fm3 ) 61.40 T (GeV) 0.668 LHC 425 1.02 τ0 (fm) 0.25 0.25 λg 0.34 0.43 λq 0.068 0.086 S+S RHIC "i (GeV/fm3 ) 24.3 LHC 170 T (GeV) 0.531 0.811 τ0 (fm) 0.25 0.25 λg 0.34 0.43 λq 0.068 0.086 where β = 1/2, R is the transverse dimension of the system and r is the transverse distance, and "i is the energy density obtained by taking the initial temperature as Ti and fugacities as λi [12]. The profile plays an important role in defining the boundary of the hot and dense deconfined matter. Having obtained the density of the partons we estimate the Debye mass of the medium as µ2D = κ 2 × 4παs (λg + Nf λq /6)T 2 , (8) where we have arbitrarily taken κ as 1.5 to account for the corrections [16] to the lowest order perturbative QCD which provides the above expression for κ = 1. Results for other values of κ are easily obtained. We shall assume that the J /ψ cannot be formed in the region where µD is more than 0.7 GeV. We can then estimate the survival probability of the directly produced J /ψ as a function of its transverse momentum pT by proceeding along the lines of Ref. [7,8,10]. In order to estimate the gluonic dissociation we recall [17] that the short range properties of the QCD can be used to derive the gluon-J /ψ cross-section as: 2π 32 2 1 (q 0 /"0 − 1)3/2 0 σ (q ) = , 3 3 mC ("0 mC )1/2 (q 0 /"0 )5 (9) 115 where q 0 is the gluon energy in the rest-frame of J /ψ and "0 is the binding energy of the J /ψ. The expression for the thermal average of this crosssection vrel σ is given in Ref. [5]. (See, also Ref. [18] for an interesting alternative approach.) We wish to have a quantitative comparison of these two processes and therefore it is imperative that we compare their results for similar conditions. Thus, exactly as while dealing with Debye screening, we assume that the cc produced initially takes a finite amount of time ∼ 0.89 fm/c in its rest frame to evolve into the physical resonance. This can get large due to time dilation, in the frame of the plasma, leading to the characteristic pT dependence of the survival probability for the J /ψ discussed in the literature. We argue that the gluon-J /ψ cross-section also attains its full value only after the cc pair has evolved into the physical resonance. We assume that this evolution of the cross-section can be parametrized as σ0 (τ/τψ )ν if τ τψ , σ= (10) if τ > τψ , σ0 similarly to the case when the nuclear absorption is considered [19], where σ0 is the cross-section estimated earlier (Eq. (9)). A similar assumption was invoked by Farrar et al. [20] when the QQ-system evolves as it moves away from the point of hard interaction. One may imagine that this amounts to assuming that the effective cross-section scales as the transverse area of the system relative to the size it attains when it is fully formed. In the present work we follow, Blaizot and Ollitrault [19] who have used ν = 2. Farrar et al. [21] have suggested that ν = 1 corresponds to a quantum diffusion of the quarks while ν = 2 would correspond to maximal rapid (classical) expansion. Legrand et al. [22] have used ν = 1 in a recent study. This aspect is in contrast to the work of Xu et al. [5] where a fully formed J /ψ is assumed to exist right at the initial time in the plasma. We shall see that ignoring the formation time leads to an enhanced suppression of the charmonium. We can now easily estimate the time spent by the J /ψ in the deconfined medium for a given pT and get the survival probability following Ref. [5]. In Fig. 1 we show our results for RHIC energies for S + S and Au + Au collisions. We see that the combination of a finite formation time and (reasonably) large 116 B.K. Patra, D.K. Srivastava / Physics Letters B 505 (2001) 113–118 Fig. 1. Survival probability of directly produced J /ψ at RHIC energies due to screening of colour interaction (solid curve) and gluonic dissociation in quark–gluon plasma. The dashed curve gives the latter with inclusion of formation time of the charmonium while the dot-dashed curve gives the same with the assumption that a fully formed J /ψ is available at τ = τi when the plasma is formed. µD required to inhibit the formation of the directly produced J /ψ in the plasma ensures that the mechanism of Debye screening is not effective in suppressing its production. However, the gluonic dissociation leads to a suppression of the J /ψ formation even after the moderating effect of the inclusion of formation time is included. The situation for the larger (and hotter) volume of plasma produced in Au + Au collisions is much richer in detail. We see that while the J /ψs having lower transverse momenta are more strongly suppressed due to the Debye mechanism, those having higher transverse momenta are more suppressed by the mechanism of gluonic dissociation. In fact we see that while the Debye screening has become quite ineffective for pT > 6 GeV, the gluonic dissociation continues to be operative. The different results obtained here compared to authors of Ref. [5] (when the formation time considerations are ignored) are solely due to the SSPC initial conditions (Table 1) used here. The corresponding results at LHC energies are shown in Fig. 2. Now we see that the Debye screening is more effective in suppressing the production of the directly produced J /ψ at all the momenta considered, provided we include the considerations of the forma- Fig. 2. Same as Fig. 1 at LHC energies. Fig. 3. Same as Fig. 1 for Υ at LHC energies. The Debye screening is absent for the initial conditions [12] used here. tion time while evaluating the gluonic dissociation, for both the systems. Of course in a model calculation one can arbitrarily enhance the impact of Debye screening by taking a larger value for the coefficient κ (Eq. (8)). This sensitivity would be useful for determining its precise value [23]. The treatment outlined here can be extended to the case of Υ production studied in great detail by the authors of Ref. [8,23], for example. We give the results only for the LHC energies, for the directly produced Υ (Fig. 3). We find that both for the light as well as the heavy systems the Debye mechanism is not at all able B.K. Patra, D.K. Srivastava / Physics Letters B 505 (2001) 113–118 to inhibit the formation of the directly produced Υ s, though the gluonic dissociation leads to a considerable suppression, with the changes brought about by the inclusion of the formation time seen earlier for the J /ψs. This is easily seen to be the consequence of the initial conditions used here, which have chemically non-equilibrated plasma leading to small Debye mass, even though the temperature is rather high. By the time the Υ is formed the Debye mass drops below the value of ∼1.6 GeV, required to inhibit its formation, causing it to escape unscathed. Before summarizing, let us discuss some of the assumptions made in this work. We have, so far considered only the dissociation of the directly produced J /ψ. Of course, it is well known that up to 30% of the J /ψs seen in these studies may be produced from a decay of χc and up to 10% or so may come from the decay of ψ , which however is also easily dissociated by a moderately hot (confined) hadronic matter and is unable to play a decisive role in distinguishing confined matter from a deconfined matter. In order to include the effect of these resonances, we should also have access to g − χc cross-section. This would involve extending the method of Ref. [24] to the case of charmonium in 1p state. However, this is not quite easy as the assumption "0 ΛQCD used in the above reference are not strictly valid for this case, as the binding energy of χc is only about 240 MeV. Still, from the considerations of binding energy alone, one may expect the g − χc cross-sections to be larger. However, the Debye mass required to inhibit the formation of χc is also smaller and thus this competition between the two mechanisms will continue. The inclusion of the transverse expansion and the possibility of√a different speed of sound than the value of 1/ 3 assumed here will also add to the richness of the information likely to be available from such studies. Of course a full study will additionally include the effect of the nuclear and the co-mover absorption, before these interesting details are investigated. The incorporation of the formation time is interesting for one more reason. The pre-equilibrium stage (before the time τi ) may be marked by presence of gluons of high transverse momenta, as a result of first hard collisions, and one may imagine that they play an important role in suppression of charmonium formation. This is unlikely for two reasons. Firstly, the 117 gluon-charmonium cross-section drops rapidly as the gluon momentum increases [24] after reaching a peak around p ∼ 1 GeV. Secondly we expect these crosssections to be further suppressed during the formation era due to the considerations of the formation time. While considering the suppression of Υ , we found that only the mechanism of gluonic dissociation is playing a role. This happens as the initial conditions used here involve a chemically non-equilibrated plasma. If the initial fugacities were to be larger, the Debye screening would also play a role, which will definitely be a good check on these. In brief, we have seen that while the gluonic dissociation of the J /ψ is always possible, the Debye screening is not effective in the case of small systems at RHIC energies. For the larger systems, the Debye screening is more effective for lower transverse momenta, while the gluonic dissociation dominates for larger transverse momenta. At LHC energies the Debye screening is the dominant mechanism of J /ψ suppression for all the cases and momenta studied. We have also seen that the inclusion of the formation time of the J /ψ plays an interesting role in reducing the role of the gluonic dissociation. As an interesting result, we find the gluonic dissociation to be substantial but the Debye screening to be ineffective for Υ suppression at the LHC energy. This may of course change if different initial conditions and screening criteria [8,23] are employed. Acknowledgements We thank Dr. Dipali Pal for collaboration during the early phases of this work and Prof. Helmut Satz for useful comments. We also thank Prof. Joseph Kapusta for useful correspondence. References [1] See, e.g., M.C. Abreu et al., NA50 Collaboration, Phys. Lett. B 477 (2000) 28; M.C. Abreu et al., Proc. Quark Matter ’99, Nucl. Phys. A 661 (1999); U. Heinz, M. Jacob, nucl-th/0002042; U. Heinz, M. Jacob, in: B.C. Sinha, D.K. Srivastava, Y.P. Viyogi (Eds.), Physics and Astrophysics of Quark–Gluon Plasma, Proc. ICPA’97, Narosa Publishing House, New Delhi, 1998. 118 [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] B.K. Patra, D.K. Srivastava / Physics Letters B 505 (2001) 113–118 C. Gerschel, J. Hüfner, Phys. Lett. B 207 (1988) 253. T. Matsui, H. Satz, Phys. Lett. B 178 (1986) 416. H. Satz, D.K. Srivastava, Phys. Lett. B 475 (2000) 225. X.-M. Xu, D. Kharzeev, H. Satz, X.-N. Wang, Phys. Rev. C 53 (1996) 3051. See, e.g., R. Vogt, Phys. Rep. 310 (1999) 197. M.-C. 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Hwa (Ed.), Quark–Gluon Plasma 2, World Scientific, Singapore, 1995, p. 395. 26 April 2001 Physics Letters B 505 (2001) 119–124 www.elsevier.nl/locate/npe γ 3π and π2γ form factors from dynamical constituent quarks Xiaoyuan Li a , Yi Liao b,c a Institute of Theoretical Physics, The Chinese Academy of Sciences, Beijing 100080, PR China b Institut für Theoretische Physik, Universität Leipzig, Augustusplatz 10/11, D-04109 Leipzig, Germany c Department of Physics, Tsinghua University, Beijing 100084, PR China Received 24 January 2001; received in revised form 14 February 2001; accepted 22 February 2001 Editor: R. Gatto Abstract We study the form factors of the low-energy anomalous π2γ and γ 3π processes in the nonlocal chiral quark model which incorporates the momentum dependence of the dynamical quark mass and realizes correctly the chiral symmetries. The obtained slope parameter for π2γ is in reasonable agreement with the direct experimental results but smaller than the ones invoking vector meson dominance. Our result for the γ 3π form factor interpolates between the two extremes of theoretical approaches, with the largest one provided by the vector meson dominance and the smallest one by the Schwinger–Dyson approach. But all of them are well below the single data point available so far. This situation will hopefully be clarified by the experiments at CEBAF and CERN. 2001 Elsevier Science B.V. All rights reserved. PACS: 11.10.Lm; 12.39.Fe; 13.40.Gp Keywords: Dynamical quark; Anomalous pion–photon interactions; Nonlocal interactions The π 0 γ γ (π2γ ) and γ π + π 0 π − (γ 3π ) processes are the two simplest chiral anomaly-driven processes that involve electromagnetic interactions. A consideration of parity conservation, gauge invariance and Lorentz invariance implies the following structures for their amplitudes: ρ π2γ = µνρσ k1 k2σ Aπ2γ , Aµν ρ γ 3π ν σ γ 3π Aµ = µνρσ p+ p0 p− A . (1) Here k1,2 denote the outgoing momenta of the two photons with Lorentz indices µ and ν, and p+,0,− the incoming momenta of the three pions, for the two processes, respectively. The dynamical information is encoded in the form factors Aπ2γ and Aγ 3π which are Lorentz invariant functions of the relevant momenta. E-mail address: [email protected] (Y. Liao). In the low-energy and chiral limit, they are completely determined by the chiral anomaly as summarized in the Wess–Zumino–Witten action to be [1]: π2γ A0 = e2 Nc , 12π 2 fπ γ 3π A0 = eNc , 12π 2 fπ3 (2) where Nc and fπ are respectively the number of colors and the pion decay constant. Beyond the limit, their dependence on the relevant momenta is a reflection of the detailed strong dynamics. Since these processes involve only one or a few pions, they may provide an ideal testing ground for models of strong interactions. π2γ The excellent agreement of A0 with the experimental value extracted from the on-shell decay of π 0 → γ γ had historically constituted one of the first pieces of firm evidence that quarks carry three colors. When one of the photons is off-shell, the form factor can be parameterized by a slope parameter in the low- 0370-2693/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 3 2 3 - 9 120 X. Li, Y. Liao / Physics Letters B 505 (2001) 119–124 energy region: π2γ Aπ2γ /A0 = 1 + ax, (3) where x = k 2 /m2π describes the virtuality of the offshell photon with momentum k. The slope parameter a has been measured both in the time-like region of k using the Dalitz decay π 0 → e+ e− γ and in the space-like region through the π 0 production in e+ e− collisions. The direct results from TRIUMF and SINDRUM I in the first category are respectively a = 0.026 ± 0.054 [2] and a = 0.025 ± 0.014(stat.) ± 0.026(syst.) [3]. The CELLO group actually measured the form factor in the large space-like region and then extracted the slope parameter by extrapolation using the vector meson dominance to be a = 0.0326 ± 0.0026 [4]. These results are consistent with each other within the quoted errors. Concerning the γ 3π process the experimental situation is less clear. There has been so far one measurement [5] which seems γ 3π than predicted by to favor a larger value of A0 the chiral anomaly. Fortunately this situation will be much improved by the experiments at CEBAF [6] and CERN [7] which will measure the form factor Aγ 3π in a wider range of kinematics. A more precise value of γ 3π A0 can then be extracted and the form factor will be available to distinguish the theoretical results based on hadronic models. The low-energy physics of the lowest-lying pseudoscalars may be described by a chiral Lagrangian which is a tower of terms in increasing order of energy expansion. The structures of terms at each order are completely determined by spontaneously broken chiral symmetries while their coefficients are left free. These parameters may be modelled by properly incorporating the relevant degrees of freedom in the intermediate-energy region. Of special interest in this regard are the quark-based models which may have a close connection to the underlying QCD dynamics. As is well-known, one feature of dynamical quarks is their running mass in the intermediate-energy region, which should have significant effects on low-energy physics when the quarks are integrated out. This point has been nicely taken into account by Holdom and collaborators in their nonlocal constituent quark model [8–10]. Indeed, the coefficients in the O(p4 ) chiral Lagrangian for the lowest-lying pseudoscalars are expressed in terms of convergent integrals of the quark dynamical mass and their phenomenological values are well reproduced. The model has also been successful in modeling the low-energy hadronic contributions to the running QED coupling at the Z boson pole [11], and in understanding the quark–hadron duality [12] and the electroweak couplings of constituent quarks themselves [13]. In this note we shall examine the other aspect of dynamical constituent quarks, namely, their implications on the anomalous sector of the pseudoscalars, especially the form factors of the γ 3π and π2γ processes. Since the Ward–Takahashi identities for flavor symmetries in QCD are built into the model of Holdom et al., we expect that the form factors so obtained should be comparable in quality to the coefficients in the O(p4 ) chiral Lagrangian derived from the model. Our results will be compared with those based on other approaches. The main feature here is that the effects of dynamical quark mass are included in a simplest possible form while at the same time avoiding introducing many free parameters. The nonlocal constituent quark model is an effective theory in the intermediate-energy regime. In this model all physics is assumed to be described by a chiral invariant action quadratic in quark fields. The dynamical quark mass Σ(p) is incorporated into the action and its momentum-dependent nature leads to nonlocal interactions among dynamical quarks, Goldstone bosons and external gauge fields. Let us outline the action involving interactions with the external photon field Aµ (x) relevant to our discussion [10]. The interested reader should consult Refs. [8–10] for a complete account. S = d4 x ψ̄iγµ D µ ψ − d4 x d4 y Σ(x − y)ψ̄(x)ξ(x) × X(x, y)ξ(y)ψ(y), (4) where ψ represents the up and down quark fields with dynamical mass Σ(p) whose Fourier transform is the quantity Σ(x). And Dµ = ∂µ − ieQAµ , Q = diag(2/3, −1/3), ξ = exp(−iπγ5 /fπ ), y X(x, y) = P exp −i π = πaT a, Γµ (z) dzµ , x X. Li, Y. Liao / Physics Letters B 505 (2001) 119–124 121 Γµ = i/2 ξ(∂µ − ieQAµ )ξ † + ξ † (∂µ − ieQAµ )ξ = eQAµ + i/ 2fπ2 (π∂µ π − ∂µ ππ) + · · · , (5) where π a is the pion field, T a is the isospin matrix with Tr[T a T b ] = δab /2, and P stands for pathordering. For convenience, we list in the following the relevant vertices appearing in our calculation of the γ 3π and π2γ amplitudes. The QED vertex between quarks and the photon is modified to be ieQ γµ − (p + p )µ R(p, p ) , (6) where p (p ) denotes the incoming (outgoing) momentum of the incoming (outgoing) quark line (same below), and R(p, k) = Σ(p) − Σ(k) . p2 − k 2 (7) We should mention in passing that the appearance of the R term in the QED vertex just fits the dynamical quark mass Σ appearing in the quark propagator so that the Ward identity still holds. The pion interaction with quarks is of a familiar form generalized from the constant mass case: −fπ−1 γ5 T a Σ(p) + Σ(p ) . (8) The model generally contains nonlinear interactions of pions with quarks and photons due to the nonlinearly realized chiral symmetry and nonlocality. But we found that for the processes considered here only the following interaction involving two pions and two quarks can contribute at one-loop level: i a b T , T Σ(p) + Σ(p + k1 ) 2fπ2 + Σ(p + k2 ) + Σ(p ) + χ T a , T b Σ(p + k2 ) − Σ(p + k1 ) + (k1 − k2 ) · (p + p )R(p, p ) , (9) where the two pions carry the isospin indices a, b and the incoming momenta k1 , k2 respectively. The parameter χ = 0, 1 corresponds to the two versions [9,10] of the model. Since it makes little numerical difference, we shall henceforth take χ = 1, corresponding to Ref. [10]. As one may easily figure out, only the χ term can contribute to Fig. 1(b) for the γ 3π vertex while the first term in Eq. (9) cannot due to symmetry. Let us consider the two processes whose Feynman diagrams are depicted in Fig. 1. Since we are interested (a) (b) (c) Fig. 1. Feynman diagrams for the vertices γ 3π (a and b) and π 2γ (c). The solid, dashed and wavy lines stand for the quark, pion and photon fields, respectively. in the form factors in the low-energy region, we expand the amplitudes in the external momenta. The leading terms must be the same as predicted by the WZW action and thus universal to all models which correctly incorporate the chiral anomaly. In other words, they must be independent of the specific form of Σ(p). This is indeed the case. For example, the leading term in the π2γ amplitude is proportional to the following integral: ∞ dx 0 2 d x √ dx x + Σ 2 ( x ) which is unity independently of Σ as long as Σ is finite in the euclidean space. For the γ 3π process the leading term is contributed only by Fig. 1(a), whose integral can be simplified as ∞ −1 + 0 2 x x d √ √ x + Σ2( x ) x + Σ2( x ) which is always −1/3 for a finite Σ in the euclidean region. The subleading terms depend explicitly on the integrals of Σ which are collected using Mathematica. Aπ2γ has been parameterized in Eq. (3). For the γ 3π process, as will become clear later on, we need to expand up to the O(p4 ) terms to display the kinematic variation of the form factor. Using the Bose symmetry we have Aγ 3π γ 3π A0 = 1 + m−2 π 2 bi Si + m−4 π i=1 6 ci Qi , (10) i=1 where S and Q are symmetrized Lorentz invariants of the momenta p1,2,3 = p+,0,− , S1 = p12 + p22 + p32 , S2 = p1 · p2 + p2 · p3 + p3 · p1 , 2 2 2 Q1 = p12 + p22 + p32 , 122 X. Li, Y. Liao / Physics Letters B 505 (2001) 119–124 Table 1 Results of the coefficients a (in units of 10−2 ), bi (10−2 ) and ci (10−3 ) as a function of the parameter A A m a b1 b2 c1 c2 c3 c4 c5 c6 1 342 1.97 −2.29 −2.32 −1.02 −1.83 −2.23 −1.96 −2.52 −1.58 2 317 1.94 −2.37 −2.32 −0.97 −1.77 −1.95 −1.71 −1.89 −1.22 3 299 1.99 −2.52 −2.44 −1.09 −1.97 −2.13 −1.85 −1.97 −1.28 4 287 2.05 −2.67 −2.59 −1.27 −2.26 −2.45 −2.12 −2.25 −1.47 5 277 2.12 −2.82 −2.73 −1.47 −2.59 −2.83 −2.43 −2.61 −1.71 The mass m (in units of MeV) is determined by the Pagels–Stokar formula. Ignoring the χ term in Eq. (9) would change bi and ci by less than 10%. Q2 = p12 p22 + p22 p32 + p32 p12 , Q3 = p12 p1 · (p2 + p3 ) + p22 p2 · (p3 + p1 ) + p32 p3 · (p1 + p2 ), Q4 = p12 p2 · p3 + p22 p3 · p1 + p32 p1 · p2 , Q5 = p1 · p2 p2 · p3 + p2 · p3 p3 · p1 + p3 · p1 p1 · p2 , Q6 = (p1 · p2 )2 + (p2 · p3 )2 + (p3 · p1 )2 . (11) Note that the explicit factors of mπ are introduced for −4 convenience although m−2 π bi and mπ ci actually do not depend on mπ . The coefficients a, bi and ci are lengthy integrals involving the dynamical quark mass, which is in turn related to fπ by the Pagels–Stokar formula reproduced in the model [8–10]: ∞ x Σ − 12 xΣ Σ Nc 2 dx fπ = (12) 2 , 4π 2 x + Σ2 0 Σ d with = dx Σ. A very simple parameterization for Σ(p) in the euclidean space was suggested by Holdom et al., which incorporates the correct highenergy behavior of the dynamical mass up to logarithms, (A + 1)m3 , (13) p2 + Am2 where m is a typical mass scale of the constituent quark and is related to the parameter A through the Pagels–Stokar formula. Fixing fπ = 84 MeV in the chiral limit we therefore have only one free parameter. Since this simple ansatz is quite successful in reproducing phenomenological values of low-energy quantities as mentioned previously, it will be used in our numerical analysis without further adaptation. Σ(p) = Our results for the coefficients a, bi and ci are presented in Table 1 as a function of the parameter A in the same range of values as used previously, where the mass scale m is of order 300 MeV. Let us first discuss the slope parameter for the π2γ process. We get a stable result of a = 0.02 for the range of A in the table. This is in reasonable consistency with direct results from the Dalitz decays, but smaller than the one extracted from the large space-like region by extrapolation using vector meson dominance. The slope parameter has been studied in other approaches. The free quark loop [14] with 1 2 mπ /m2 , a constant constituent mass m predicts a = 12 which is about 0.014 for m = 330 MeV. In the phenomenological approach of vector meson dominance the momentum dependence of the amplitude derives from the lowest-lying vector resonances and thus a = m2π /m2ρ ∼ 0.03. Chiral perturbation theory is appropriate for dealing with low-energy pion–photon interactions, but it is afflicted in the current case by the unknown counterterm parameters appearing in the O(p6 ) anomalous chiral Lagrangian. Assuming they are again saturated by vector mesons with a mean mass of m2V = (9m2ρ + m2ω + 2m2φ )/12, the sum of loop and counter-term contributions gives a = 0.032 [15]. It is clear that our result is larger than the one in the constant quark mass model but smaller than the ones (both theoretical and experimental) invoking vector meson dominance. For the γ 3π process one has to examine the kinematic variation of the form factor to extract information on the coefficients bi and ci . In all of the three experiments available or approved, the photon and two of the pions, which we assume to be the first and second ones without loss of generality, X. Li, Y. Liao / Physics Letters B 505 (2001) 119–124 are on-shell, (p1 + p2 + p3 )2 = 0, p12 = p22 = m2π . (We take mπ to be the neutral pion mass below and ignore the small isospin breaking in mass.) The experiment at Serpukhov and the one at CERN are of Primakoff type so that the third pion is also onshell, p32 = m2π , while the CEBAF experiment is to be done at a low momentum transfer of p32 ≈ −m2π . Defining the Mandelstam variables s = (p1 + p2 )2 , t = (p2 + p3 )2 and u = (p3 + p1 )2 , the form factor is a function of s and t with other kinematic variables completely fixed. It is then clear that there is no s or t dependence in the O(p2 ) terms of Aγ 3π and this is the reason why we expand up to O(p4 ). We plot in Fig. 2 our numerical results of the form γ 3π factor Aγ 3π /A0 at A = 1 as a function of s with 123 fixed t = −m2π , for the Primakoff case (panel (a)) and the CEBAF case (panel (b)), respectively. Also shown are the results of other approaches, including the free quark loop with a constant constituent mass [16], the Schwinger–Dyson approach in the generalized impulse approximation [17], chiral perturbation theory with vector meson saturation [18], vector meson dominance [19] and its unitarized version [20]. The form factors expanded up to second order in s and t in the free quark loop and the Schwinger–Dyson approaches can be read off in the original papers. The chiral perturbation result augmented with vector meson saturation of counterterms is [18] Aγ 3π γ 3π A0 1 (s + t + u) 2m2ρ m2π 1 1 (s + t + u) ln + − 32π 2 fπ2 3 m2ρ =1+ 5 + (s + t + u) 9 4m2π 2 + f mπ , s + f m2π , t 3 + f m2π , u , (14) where z+1 (1 − x)z ln − 2, z−1 for x < 0, 1 2 2 (1 − x)z 2 arctan − 2, f m ,q = z for 0 < x < 1, 1+z − iπ − 2, (1 − x)z ln 1−z for 1 < x, 1 q2 x= z = 1 − , , x 4m2 γ 3π Fig. 2. The form factor Aγ 3π /A0 at A = 1 (solid curve) as a function of s/m2π (mπ = 135 MeV) for the Primakoff case (panel (a)) and the CEBAF case (panel (b)), respectively. Also shown are the results of the following approaches: the free quark loop with a constant constituent quark mass of 330 MeV [16]; the Schwinger–Dyson approach [17]; chiral perturbation with vector meson saturation [18]; vector meson dominance [19] and its unitarization [20]. (15) which is different from the one quoted for the Primakoff case in Ref. [20]. The phenomenological approach of vector meson dominance gives [19] Aγ 3π γ 3π A0 m2ρ m2ρ m2ρ 1 + + =− 1− , 2 m2ρ − s m2ρ − t m2ρ − u (16) 124 X. Li, Y. Liao / Physics Letters B 505 (2001) 119–124 which is unitarized to be [20] m2ρ m2ρ m2ρ Aγ 3π 1 1 − + + = − γ 3π 2 m2ρ − s m2ρ − t m2ρ − u A0 × (m2ρ − s)(m2ρ − t)(m2ρ − u) m6ρ D1 (s)D1 (t)D1 (u) , m2ρ q2 q2 ln D1 q 2 = 1 − 2 − mρ 96π 2 fπ2 m2π − m2π f m2π , q 2 . 24π 2 fπ2 (17) Note that the results for chiral perturbation and unitarized vector meson dominance are actually shown for γ 3π |Aγ 3π /A0 | since the form factor can become complex in these cases. It is clear from Fig. 2 that the Schwinger–Dyson approach always gives the lowest values of the form factor while the vector meson dominance (especially its unitarized version) predicts the largest values and the steepest change in the kinematic region considered here. It is interesting to notice that in contrast to the case of the vertex π2γ the chiral perturbation theory predicts a much lower value of the γ 3π amplitude than the vector meson dominance does. Our results interestingly interpolate the two extremes and are slightly larger than the one using a constant quark mass of 330 MeV. We have studied the form factors of the lowenergy anomalous π2γ and γ 3π processes in a simple quark-based model which incorporates the momentum dependence of the dynamical quark mass and realizes correctly the chiral symmetries. The obtained slope parameter for π2γ is in reasonable agreement with the direct experimental results from TRIUMF and SINDRUM but smaller than the ones (both theoretical and experimental) invoking vector meson dominance. All theoretical predictions for the γ 3π form factor are well below the single data point available so far. But there are also significant differences among these theoretical results. This situation will hopefully be clarified and distinguished by the experiments at CEBAF and CERN. Acknowledgements The work of X. Li was supported in part by the China National Science Foundation under grant numbers 19835060 and 19875072 and Y. Liao was supported in part by DESY, Germany. X. Li is grateful to K. Sibold and the staff members of ITP at Universität Leipzig for their hospitality during a visit when part of the work was done there. References [1] J. Wess, B. Zumino, Phys. Lett. B 37 (1971) 95; E. Witten, Nucl. Phys. B 223 (1983) 422. [2] F. Farzanpay et al., Phys. Lett. B 278 (1992) 413. [3] R. Meijer Drees et al., Phys. Rev. D 45 (1992) 1439. [4] H.J. Behrend et al., Z. Phys. C 49 (1991) 401. [5] Yu.M. Antipov et al., Z. Phys. C 27 (1985) 21; Yu.M. Antipov et al., Phys. Rev. D 36 (1987) 21. [6] R. Miskimen, K. Wang, A. Yagneswaran (spokesmen), CEBAF Proposal No. PR-94-015, unpublished. [7] M.A. Moinstester, V. Steiner, S. Prakhov, hep-ex/9903017. [8] B. Holdom, J. Terning, K. Verbeek, Phys. Lett. B 245 (1990) 612; Erratum: Phys. Lett. B 273 (1991) 549. [9] J. Terning, Phys. Rev. D 44 (1991) 887. [10] B. Holdom, Phys. Rev. D 45 (1992) 2534. [11] B. Holdom, R. Lewis, Phys. Lett. B 294 (1992) 293. [12] B. Holdom, Phys. Lett. B 292 (1992) 150. [13] X. Li, Y. Liao, Phys. Lett. B 318 (1993) 537. [14] A. Bramon, E. Masso, Phys. Lett. B 104 (1981) 311; Ll. Ametller et al., Nucl. Phys. B 228 (1983) 301; A. Pich, J. Bernabeu, Z. Phys. C 22 (1984) 197. [15] J. Bijnens, A. Bramon, F. Cornet, Phys. Rev. Lett. 61 (1988) 1453; Ll. Amettler et al., Phys. Rev. D 45 (1992) 986. [16] B. Bistrovic, D. Klabucar, Phys. Rev. D 61 (2000) 033006. [17] B. Bistrovic, D. Klabucar, Phys. Lett. B 478 (2000) 127; See also, R. Alkofer, C.D. Roberts, Phys. Lett. B 369 (1996) 101. [18] J. Bijnens, A. Bramon, F. Cornet, Phys. Lett. B 237 (1990) 488. [19] S. Rudaz, Phys. Lett. B 145 (1984) 281; T.D. Cohen, Phys. Lett. B 233 (1989) 467. [20] B.R. Holstein, Phys. Rev. D 53 (1996) 4099. 26 April 2001 Physics Letters B 505 (2001) 125–130 www.elsevier.nl/locate/npe The structure of the Aoki phase at weak coupling R. Kenna, C. Pinto, J.C. Sexton School of Mathematics, Trinity College Dublin, Ireland Received 31 January 2001; received in revised form 22 February 2001; accepted 28 February 2001 Editor: P.V. Landshoff Abstract A new method to determine the phase diagram of certain lattice fermionic field theories in the weakly coupled regime is presented. This method involves a new type of weak coupling expansion which is multiplicative rather than additive in nature and allows perturbative calculation of partition function zeroes. Application of the method to the single flavour Gross–Neveu model gives a phase diagram consistent with the parity symmetry breaking scenario of Aoki and provides new quantitative information on the width of the Aoki phase in the weakly coupled sector. 2001 Published by Elsevier Science B.V. In lattice field theory, there has been considerable discussion on the phase diagrams of theories with Wilson fermions (see, e.g., [1–11]). These can be considered as statistical mechanical systems, and have rich phase structures whose existence is due to lattice artefacts. The Wilson fermion hopping parameter is 0 is the dimensionless bare 0 + d) where M κ = 1/2(M fermion mass and d the lattice dimensionality. It is well known that a system of free Wilson fermions exhibits a phase transition at κ = 1/(2d) and that massless fermions appear at this point in the continuum limit. Discussions concern the extent to which this phase transition persists in the presence of a bosonic field. In QCD, where Wilson fermions couple to gauge fields with a strength given by the dimensionless coupling ĝ, there are two candidates for the phase diagram. In the first, pioneered by Kawamoto, the expectation is that there is a line of phase transitions (the “chiral line”) extending from the strong coupling limit to the weakly coupled one and along which the pion and quark masses vanish [1]. Such vanishing is symp- tomatic of spontaneous chiral symmetry breaking. Approaching the continuum limit, at ĝ = 0, along the chiral line in particular is then expected to recover massless physics. This is still sometimes referred to as the ‘conventional’ picture. The second candidate phase diagram for QCD was determined by Aoki on the basis of comparison with the Gross–Neveu model [2]. The Gross–Neveu model serves as a prototype for QCD [12]. Indeed, except for confinement, it has features similar to QCD. One of these features is asymptotic freedom, so that in the Gross–Neveu model, as in QCD, the continuum limit is taken in the weakly coupled zone. Two features distinguish Aoki’s phase diagram from the earlier ‘conventional’ picture. Firstly, instead of a single critical line, Aoki’s analysis advocates the existence of two lines extending from the strongly to weakly coupled limits, with a number of critical points at ĝ = 0 linked by cusps (see Fig. 1). The region above the cusps and between the two extended lines is often referred to as the Aoki phase. Secondly, the existence of the phase transition in Aoki’s scenario is due to spontaneous parity symmetry breaking within the Aoki phase, as op- E-mail address: [email protected] (R. Kenna). 0370-2693/01/$ – see front matter 2001 Published by Elsevier Science B.V. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 3 2 8 - 8 126 R. Kenna et al. / Physics Letters B 505 (2001) 125–130 posed to chiral symmetry breaking. (Indeed, Wilson fermions explicitly violate chiral symmetry.) This is signaled by a non-zero vacuum expectation value of the pseudoscalar operator π = ψ̄iγS ψ in the thermodynamic limit. The masslessness of the pion is then attributed to the divergence of a correlation length associated with this second order phase transition. In the multiflavour case, flavour symmetry is also broken in the Aoki phase since the pion, whose expectation value is nonvanishing, also carries flavour. The continuum limit has to be approached from outside the Aoki phase since parity and flavour are conserved in the strong interaction. The physical meaning of an approach to the continuum limit from within the Aoki phase is unclear [2]. There exists substantial evidence supporting Aoki’s scenario in the strongly coupled regime [2–7]. In the weakly coupled regime the evidence has, however, been controversial [8] (see [6] for recent discussions on this topic). In asymptotically free theories the weakly coupled region is the appropriate one for the continuum limit. Recently, Creutz [10] questioned whether the Aoki phase, pinched between the arms of cusps, is “squeezed out” at non-zero coupling or whether it only vanishes in the weak coupling limit (see, also, [6,11]). The purpose of this Letter is twofold. We introduce a new type of expansion which is multiplicative rather than additive in nature and from which information on the partition function zeroes of the theory can be extracted in a rather natural way [7]. Secondly, we address the question of the “squeezing out” of the Aoki phase at weak coupling. This multiplicative approach to the single flavour Gross–Neveu model, shows that the width of the central Aoki cusp is O(ĝ 2 ) while the Aoki phase has not yet emerged at this order from the left and right extremes. The Gross–Neveu model is actually a two-dimensional model of fermions only, which interact through a short range quartic interaction [12]. In Euclidean continuum space, the model with a single fermion flavour is given by the four-fermi action (cnm) SGN = d x ψ̄(x)(/ ∂ + M)ψ(x) and where φ(x) and π(x) are auxiliary boson fields. The corresponding Wilson action in terms of dimensionless lattice quantities is SF(W) = SF(0) + S(int) + S(bosons) , where [3] SF(0) = 1 ψ̄(n)ψ(n) 2κ n 1 ψ̄(n)(1 − γµ )ψ(n + µ̂) − 2 n,µ S(int) = ĝ n + ĝ + ψ̄(n + µ̂)(1 + γµ )ψ(n) , (3) φ(n)ψ̄(n)ψ(n) π(n)ψ̄(n)iγS ψ(n), (4) n and S(bosons) = 1 2 φ (n) + π 2 (n) , 2 n (5) and where the auxiliary fields have been rescaled φ → ĝφ, π → ĝπ to explicitly display the order of the interactive part of the action. Here, lattice sites are labeled nµ = −N/2, . . . , N/2 − 1, where N is the number of sites in each of the two directions. We assume N is even. Using the lattice Fourier transform, ψ(n) = (1/Na)2 k ψ(k) exp(ikna), where a is the lattice spacing, the fermionic action can be written 1 ψ̄(q)M (W)(q, p)ψ(p), (6) SF(0) + S(int) = 2 4 N a q,p where the 2N 2 ×2N 2 fermion matrix is M (W) (p, q) = M (0)(p, q) + M (int) (p, q), with free part 2 2 g 2 ψ̄(x)ψ(x) 2 2 , + ψ̄(x)iγS ψ(x) where γS = i −1 γ1 γ2 and the fermion field has 2 spinor components. Bosonizing the action gives for the par (cnm) tition function, ZGN = DφDπDψ̄ Dψ exp(−S), where 1 2 S = d x ψ̄(/ ∂ + M)ψ + 2 φ 2 + π 2 2g + φ ψ̄ψ + π ψ̄iγS ψ , (2) − (1) M (0)(q, p) 1 − = δqp (cos pµ a − iγµ sin pµ a) , 2κ µ (7) R. Kenna et al. / Physics Letters B 505 (2001) 125–130 and interactive part ĝ i(p−q)na φ(n) + π(n)iγS . e M (int) (q, p) = 2 N n (8) It is appropriate to impose antiperiodic boundary conditions in the temporal (1-)direction and periodic boundary conditions in the spatial (2-)direction in coordinate space. With these mixed boundary conditions the momenta for the Fourier transformed fermion fields are pµ = 2π p̂µ /Na, where p̂1 ∈ {−N/2 + 1/2, . . . , N/2 − 1/2} and p̂2 ∈ {−N/2, . . . , N/2 − 1}. Integration over the Grassmann variables gives the full partition function (W) Z = DφDπDψ̄ Dψ exp −SF ∝ det M (W) ∝ (9) λα (p) , α,p with λα (p) the eigenvalues of the fermion matrix and the expectation values being taken over the bosonic fields. In the free field case the eigenvalues of M (0) are easily calculated and found to be λ(0) α (p) = 1 − ηα(0) (p), 2κ where ηα(0) (p) = 2 µ=1 (10) 2 α cos pµ a − i(−) sin2 pµ a, (11) µ=1 are the Lee–Yang zeroes of the free theory [13]. Note that the eigenvalues (10) and the zeroes (11) are degenerate with respect to pµ → −pµ . Furthermore, the lowest zeroes in the free case, and those responsible for the onset of critical behaviour, are two-fold degenerate in two dimensions. These lowest zeroes are ηα (±|p1 |, p2 ) where |p̂1 | = (N − 1)/2 or 1/2 and p̂2 = −N/2 or 0 and impact on the real 1/2κ axis at −2, 0 and 2. Finally note that the zeroes in the upper half plane are given by α = 1, while their complex conjugates correspond to α = 2. The standard additive weak coupling expansion of the full fermion determinant is the Taylor expansion of −1 det M (W) = det M (0) × det M (0) M (W) −1 = det M (0) exp tr ln 1 + M (0) M (int) . (12) 127 This expansion is 2 2 (int) (int) 2N 2N Mii(int) 1 Mij Mj i det M (W) = 1 + − (0) (0) (0) 2 det M (0) i=1 λi i,j =1 λi λj 2 (int) (int) 2N 1 Mii Mjj + + ···, (0) (0) 2 i,j =1 λi λj (13) where the indices i and j stand for the combination of Dirac index and momenta (α, p) which label fermionic matrix elements, so that Mij(int) represents (int) (p, q)|λ(0) (q). Here |λ(0) (q) repreλ(0) α (p)|M β β sents a free fermion eigenvalue. The traces in (13) are just the diagrams which contribute to the vacuum polarization tensor. (int) (int) (int) Setting ti = Mii and tij = tj i = Mij Mj i − (int) (int) Mii Mjj , the ratio of partition functions is, from (13), 2N 2 2N 2 tij det M (W) 1 ti − + ··· . =1+ (0) (0) (0) 2 det M (0) i=1 λi i,j =1 λi λj (14) We note at this point that this expansion is analytic in (0) 1/2κ with poles at 1/2κ = ηi . For the Gross–Neveu model, calculation of the pure bosonic expectation values is particularly simple. Indeed, one has that φ(n) = π(n) = 0 and φ(n)φ(m) = π(n)π(m) = 2δnm . (15) The required bosonic expectation values of the matrix elements are found to be ti ≡ tα,p = 0, (16) tij ≡ t(α,p)(β,q) 2ĝ 2 ρ sin qρ sin pρ α+β = 2 (−1) −1 . N 2 2 µ sin qµ ν sin pν (17) An alternative formulation of the partition function may be obtained by writing the Wilson fermion matrix as M (W) = 1/2κ + H where H is the hopping matrix. The fermion determinant det M (W) = det(1/2κ + H ), is a polynomial in 1/2κ since for finite lattice size these matrices are of finite dimension. Indeed, for an N ×N lattice this polynomial is of degree 2N 2 . Therefore, the bosonic expectation value of the fermion determinant is also a polynomial of the same degree in 128 R. Kenna et al. / Physics Letters B 505 (2001) 125–130 1/2κ and as such may be written in terms of its 2N 2 zeroes, now labeled ηi . We may thus write a ‘multiplicative’ weak coupling expansion as that the O() −1 ) equation to order ĝ is ηn(1) = 0, i (22) while to order ĝ 2 it is ηn(2) = i (23) 2 2 2N 1/2κ − ηi 2N det M (W) ∆i = = 1 − (0) , (0) det M (0) λi λi i=1 i=1 (18) where ∆i = ηi − ηi (0) are the shifts that occur in the zeroes when the bosonic fields are turned on. These are the quantities to be determined. Note that the expression (18) is, like (14), analytic in 1/2κ with (0) poles at ηi . Expanding (18) gives det M (W) det M (0) 2 =1− 2N ∆i (0) i=1 λi 2N 2 2N 2 1 ∆i ∆j + + ···. (0) (0) 2 λ λ i=1 j =i i (19) j In the free fermion theory, the eigenvalues and zeroes of (10) and (11) are two- or four-fold degenerate with respect to momentum inversion. Let {n} denote the nth degeneracy class, so that the Dn eigenvalues (0) (0) (0) λn1 = · · · = λnDn are identical to λn , say. Let 1/2κ = (0) ηn + ) and expand the additive and multiplicative expressions (14) and (19) order by order in ) −1 . Identification of the expansions yields relationships between the known quantities ti , and tij and the shifts in the positions of the zeroes, ∆i , to O() −1 ) and O() −2 ). The O() −1 ) relationship is ∆j ∆ni 1 − ηn(0) − ηj(0) ni ∈{n} j ∈{n} / tni j = (20) , (0) (0) ni ∈{n} j ∈ / {n} ηn − ηj having used (16), while that to O() −2 ) is ∆ni ∆nj = − tni nj . ni ,nj ∈{n},ni =nj (21) ni ,nj ∈{n} These relationships can be considered order by order in the coupling as well. Let ∆i = ηi(1) + ηi(2) + (1) (2) O(ĝ 3 ), where ηi and and ηi are, respectively, the 2 order ĝ and order ĝ shifts in the ith zero. One finds ni ∈{n} ni ∈{n} tni j (0) ηn ni ∈{n},j ∈{n} / (0) − ηj . Also, the O() −2 ) equation, which is entirely O(ĝ 2 ), is 2 ηni (1) = (24) tni nj . ni ∈{n} ni ,nj ∈{n} With relations (22)–(24), the multiplicative expression (19) recovers (14) to O(ĝ 2 ). Now the additive and multiplicative expressions (14) and (18) coincide to O(ĝ 2 ) everywhere in the complex hopping parameter plane and arbitrarily close to any pole. In the free case, the zeroes responsible for criticality are two fold degenerate. For weak enough coupling, one expects these zeroes to govern critical behaviour in the presence of weakly coupled bosonic fields too. From (22) and (24), the first order shifts to two-fold degenerate zeroes are = ± tn1 n2 , ηn(1) (25) i where ni ∈ {n} for i = 1 or 2. The second order equation in the two-fold degenerate case is ηn(2) + ηn(2) = 1 2 (2) tj(2) n1 + tj n2 (0) (0) ηn − ηj j ∈{n} / . (26) Removing the bosonic field expectation values converts the problem into the determination of the eigenvalues of a weakly perturbed matrix whose free eigenvalues are two-fold degenerate. More explicitly, with boson expectation values removed, the eigenvalues are (0) (1) (2) λi = λi − ηi − ηi , which may be determined from conventional time independent perturbation theory. This condition yields enough to fully determine the zeroes to order ĝ 2 . Indeed, the second order shifts are tj ni ηn(2) (27) = . i (0) ηn − ηj(0) j ∈{n} / Now using (17), the O(ĝ) and O(ĝ 2 ) shifts for the erstwhile two-fold degenerate zeroes, ηα (±|p1 |, p2 ) R. Kenna et al. / Physics Letters B 505 (2001) 125–130 129 (for p̂2 = 0 or −N/2), are, respectively, ηα(1) (±|p1 |, p2 ) = ±i ηα(2) (±|p1 |, p2 ) =− 2g 2 N2 2g , N (28) 1 ηα(0) (p) − ηβ(0) (q) (β,q)∈{(α,p)} / . (29) The partition function zeroes are ‘protocritical points’ [14] whose real parts are pseudocritical points. In the thermodynamic limit these become the true critical points of the theory and their determination amounts to determination of the weakly coupled phase diagram, the critical line being traced out by the impact of zeroes on to the real hopping parameter axis. Thus, the phase diagram is given to order ĝ 2 by the limit 1 = lim ηα(0)(p) + ηα(1)(p) + ηα(2) (p) , (30) 2κ N→∞ where p is the momentum corresponding to the lowest zeroes. At order ĝ 0 , the zeroes (11) impact on the real 1/2κ axis at −2, 0 and 2, giving three different continuum limits, corresponding to the nadirs of the three Aoki cusps [2]. The true continuum limit is 1/2κ = 2. From (28) and (29), the O(ĝ 2 )-shift is the shift in the average position of the two zeroes while their relative separation is O(ĝ). In the thermodynamic limit, the O(ĝ) terms in (28) vanish. One finds, numerically, that the imaginary contribution to the O(ĝ 2 ) term (29) also vanishes while the real part becomes an N independent constant. Indeed, the factor 1 1 , (31) 2 (0) N ηα (p) − ηβ(0)(q) (β,q)∈{(α,p)} / approaches approximately 0.77 and −0.77 for (p̂1 , p̂2 ) = (±1/2, 0) and (±(N − 1)/2, −N/2) respectively, and corresponding to the right- and left-most critical lines. Also, (31) is approximately 0.2 and −0.2 for (p̂1 , p̂2 ) = (±(N − 1)/2, 0) and (±1/2, −N/2), respectively, these two lines generating the inner cusp. Therefore, the degeneracy of the free fermion critical point corresponding to the central cusp in Aoki’s phase diagram is indeed lifted and two critical lines emerge in the presence of weak bosonic coupling. These critical lines are 1/2κ ±0.4ĝ 2 . The Aoki phase does not yet emerge to O(ĝ 2 ) from the left- Fig. 1. The phase diagram for the Gross–Neveu model in the weakly coupled region (to O(ĝ 2 )) (dark lines) and a schematic representation of the expected Aoki phase diagram (light curves). and right-most critical points. This is the answer to the question posed by Creutz in [10] at least for the single flavour Gross–Neveu model. Instead the right- and left-most critical lines are 1/2κ ±(2 − 1.54ĝ 2), respectively. This weakly coupled phase diagram is pictured in Fig. 1 (dark lines). From left to right, the critical lines are traced out by the thermodynamic limits of the zeroes indexed by (p̂1 , p̂2 ) = (±(N − 1)/2, −N/2), (±(N − 1)/2, 0), (±1/2, −N/2) and (±1/2, 0), respectively. The lighter curves are a schematic representation of the expected full phase diagram. In conclusion, we have developed a new type of weak coupling expansion which is multiplicative rather than additive in nature and focuses on the Lee–Yang zeroes, or protocritical points, of a lattice field theory with Wilson fermions. This expansion is applied to the Gross–Neveu model, where the existence of an Aoki phase was first suggested. The weakly coupled regime is the one of primary interest as it is there, as with all asymptotically free models, that the continuum limit is taken. The method, applied to the single flavour Gross–Neveu model, yields a phase diagram in this region which is consistent with that of Aoki and the widths of the Aoki cusps are determined to order ĝ 2 . 130 R. Kenna et al. / Physics Letters B 505 (2001) 125–130 Acknowledgements R.K. wishes to thank M. Creutz for a discussion. References [1] N. Kawamoto, Nucl. Phys. B 190 (1981) 617. [2] S. Aoki, Phys. Rev. D 30 (1984) 2653; S. Aoki, Nucl. Phys. B 314 (1989) 79. [3] T. Eguchi, R. Nakayama, Phys. Lett. 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Narayanan, R.L. Singleton Jr., Nucl. Phys. B 518 (1998) 319. [9] H. Gausterer, C.B. Lang, Phys. Lett. B 341 (1994) 46; H. Gausterer, C.B. Lang, Nucl. Phys. B (Proc. Supl.) 34 (1994) 201; V. Azcoiti, G. Di Carlo, A. Galante, A.F. Grillo, V. Laliena, Phys. Rev. D 50 (1994) 6994; V. Azcoiti, G. Di Carlo, A. Galante, A.F. Grillo, V. Laliena, Phys. Rev. D 53 (1996) 5069; I. Hip, C.B. Lang, R. Teppner, Nucl. Phys. B (Proc. Supl.) 63 (1998) 682. [10] M. Creutz, hep-lat/9608024, Talk given at Brookhaven Theory Workshop on Relativistic Heavy Ions, Upton, NY, July 8–19, 1996; M. Creutz, hep-lat/0007032. [11] S. Aoki, Prog. Theor. Phys. Suppl. 122 (1996) 179. [12] D.J. Gross, A. Neveu, Phys. Rev. D 10 (1974) 3235. [13] T.D. Lee, C.N. Yang, Phys. Rev. 87 (1952) 404; T.D. Lee, C.N. Yang, Phys. Rev. 87 (1952) 410. [14] M.E. Fisher, Suppl. Prog. Theor. Phys. 69 (1980) 14. 26 April 2001 Physics Letters B 505 (2001) 131–140 www.elsevier.nl/locate/npe Merons and instantons in laplacian abelian and center gauges in continuum Yang–Mills theory ✩ H. Reinhardt, T. Tok Institut für Theoretische Physik, Universität Tübingen, Auf der Morgenstelle 14, D-72076 Tübingen, Germany Received 22 January 2001; received in revised form 9 February 2001; accepted 22 February 2001 Editor: R. Gatto Abstract Meron, instanton and instanton–antiinstanton field configurations are studied in continuum Yang–Mills theory in laplacian abelian and center gauges in order to detect their monopole and center vortex content. While a single instanton does not give rise to a center vortex, we find center vortices for merons. Furthermore we provide evidence, that merons can be interpreted as intersection points of center vortices. For the instanton–antiinstanton pair, we find a center vortex enclosing their centers, which carries two monopole loops. 2001 Elsevier Science B.V. All rights reserved. PACS: 11.15.-q; 12.38.Aw Keywords: Yang–Mills theory; Center vortices; Maximal center gauge; Laplacian center gauge 1. Introduction At present there are two popular confinement mechanisms: the dual Meissner effect [1–3], which is based on a condensation of magnetic monopoles in the QCD vacuum and the vortex condensation picture [4,5]. Both pictures were proposed long time ago, but only in recent years mounting evidence for the realization of these pictures has been accumulated in lattice calculations. The two pictures of confinement show up in specific partial gauge fixings. Magnetic monopoles arise as gauge artifacts in the so-called abelian gauges proposed by ’t Hooft [6], where the Cartan subgroup H of the gauge group G is left untouched, fixing only the coset G/H . To ✩ Supported by DFG under grant-No. DFG-Re 856/4-1 and DFG-EN 415/1-2. E-mail addresses: [email protected] (H. Reinhardt), [email protected] (T. Tok). be more precise the magnetic monopoles explicitly show up only after the so-called abelian projection, which consists in throwing away the “charged” part of the gauge field after implementing the abelian gauge. Magnetic monopoles appear at those isolated points in space, where the residual gauge freedom is larger than the abelian subgroup. Since the magnetic monopoles arise as gauge artifacts, their occurrence and properties depend on the specific form of the abelian gauge used. For example, monopole dominance in the string tension [7–9] is found in maximal abelian gauge, but not in Polyakov gauge [10] (in Polyakov gauge there is, however, an exact abelian dominance in the temporal string tension). However, in all forms of the abelian gauges considered monopole condensation occurs in the confinement phase and is absent in the deconfinement phase [11]. The vortex picture of confinement, which received rather little attention after some early efforts following 0370-2693/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 3 2 4 - 0 132 H. Reinhardt, T. Tok / Physics Letters B 505 (2001) 131–140 its inception has recently received strong support from lattice calculations performed in the so-called maximal center gauge [12,13], where one fixes only the coset G/Z but leaves the center Z of the gauge group G unfixed. 1 Subsequent center projection, which consists in replacing each link by its closest center element allows the identification of the center vortex content of the gauge fields. Lattice calculations show, that the vortex content detected after center projection produces virtually the full string tension, while the string tension disappears, if the center vortices are removed from the lattice ensemble [12,15]. This fact has been referred to as center dominance. Center dominance persists at finite temperature [16,17] for both the q–q̄ potential (Polyakov loop correlator) as well as for the spatial string tension. The vortices have also been shown to condense in the confinement phase [18]. Furthermore in the gauge field ensemble devoid of center vortices chiral symmetry breaking disappears and all field configurations belong to the topologically trivial sector [15]. Since chiral symmetry breaking and topological properties of gauge fields are usually attributed to instantons and merons [19,20] one may wonder whether and how these field configurations are related to magnetic monopoles and vortices. Unfortunately, both gauge fixing procedures, the maximal abelian gauge and the maximal center gauge, suffer from the Gribov problem [21], both on the lattice as well as in the continuum [22]. To circumvent the Gribov problem, the abelian [23] and center [24, 25] version of the laplacian gauge [26], which are free of Gribov copies, have been introduced. In this Letter we consider the laplacian abelian and center gauges in continuum Yang–Mills theory and study in these gauges field configurations which are considered to be relevant in the infrared sector of QCD like instantons and merons. Center vortices and magnetic monopoles can give an appealing explanation of confinement (see, e.g., Ref. [17]). It is the general consensus that instantons have little to do with confinement but offer an explanation of spontaneous breaking of chiral symmetry [19,20]. Merons can be considered as “half of an instanton with zero radius” and we will 1 The continuum analog of the maximum center gauge has been derived in Ref. [14]. provide evidence that they can be regarded as intersection points of center vortices. The advantage of the abelian and center gauges is that they provide a convenient tool to detect the monopole and vortex content of a field configuration. Previously the monopole content of instantons has been considered in the Polyakov gauge and maximal abelian gauge [27–30]. In maximal abelian gauge a monopole trajectory was found to pass through the center of the instanton in Ref. [27], while an infinitesimal monopole loop around the center of the instanton was found in [29]. These results are consistent with the findings of [31] where an instanton on an S 4 -space–time manifold has been considered and a monopole loop degenerate to a point was found in laplacian abelian gauge. Only for a special choice of the instanton scale one can find a monopole loop of finite size [31]. In Polyakov gauge [32] a static monopole trajectory passes through the center of the instanton [33]. In this gauge the Pontryagin index can be entirely expressed in terms of magnetic monopole charges [34–37]. The vortex content of instanton field configurations has been less understood. The first investigations in this direction have been reported in Refs. [24,25] where a cooled two-instanton configuration and a cooled caloron configuration have been considered in the laplacian center gauge on the lattice. In the former case a vortex sheet was found connecting the positions of the two instantons. In the case of the caloron which can be interpreted as a monopole–antimonopole pair the vortex sheet runs through the positions of monopole and antimonopole, which is expected since in the laplacian center gauge by construction the monopoles are sitting on the vortex sheets. One should, however, keep in mind that the lattice result cannot be straightforwardly transferred to the continuum. Due to the periodic boundary conditions a localized configuration on the lattice corresponds to an array of such configurations in the continuum. In addition the detection of topological charge on the lattice is problematic on its own. In this Letter we study merons, instantons and instanton–antiinstanton pairs in the laplacian abelian and center gauges and extract their monopole and vortex content. We also provide evidence that merons can be interpreted as vortex intersection points. H. Reinhardt, T. Tok / Physics Letters B 505 (2001) 131–140 2. Laplacian abelian and center gauges Since the center Z of a group G belongs to its Cartan subgroup H (center) gauge fixing can be performed in two steps: first one fixes the coset G/H leaving the Cartan subgroup H unfixed, which is referred to as abelian gauge fixing. Secondly one fixes the coset H /Z leaving the center unfixed, which is referred to as center gauge fixing. In this Letter we will consider G = SU(2) and consequently H = U (1). For a recent generalization of the laplacian center gauge fixing to the gauge group SU(N), see Ref. [38]. For SU(2), the laplacian abelian [23] and center [24, 25] gauges are defined as follows. We consider the two lowest-lying eigenvectors ψ1/2 of the covariant Laplace operator D2 in the adjoint representation: D = ∂ + A, ab = f abc Ac . A (2.1) To fix the gauge group up to its Cartan subgroup we D2 into the Cartan gauge rotate the ground state ψ1 of subalgebra, ψ1V (x) = V (x)−1 ψ1 (x)V (x) = h(x)σ3 , h(x) 0, V (x) ∈ SU(2)/U (1). (2.2) This defines the laplacian abelian gauge. Obviously this gauge is ill-defined at points x in space–time where ψ1 (x) = 0 which defines the positions of magnetic monopoles [23]. In a second step the residual U (1) gauge freedom is partially fixed to Z2 by gauge rotating the next-to-lowest eigenvector ψ2 of the covariant Laplacian D2 into the 1–3-plane g(x)−1 V (x)−1 ψ2 (x)V (x) g(x) = l3 (x)σ3 + l1 (x)σ1 , l1 (x) 0, g(x) ∈ U (1)/Z2 . the color vectors ψ1 (x) and ψ2 (x) are trivially colinear, in the laplacian gauge magnetic monopoles lie on the center vortices by construction. 3. Merons and instantons in laplacian center gauge Of specific interest are instanton configurations since they dominate the Yang–Mills functional integral in the semiclassical regime. Moreover these objects carry nontrivial topological charge and are considered to be relevant for the spontaneous breaking of chiral symmetry and for the emergence of the topological susceptibility which by the Witten–Veneziano formula [39,40] provides the anomalous mass of η . The instantons can, however, not account for confinement. Early investigations have introduced merons to explain confinement, which roughly speaking, can be interpreted as half of a zero-size instanton (see below). In view of the recent lattice results supporting the vortex picture of confinement [12,13,15] merons should have some relation to center vortices if they, by any means, give rise to confinement. Furthermore meron pairs behave like instantons concerning the chiral properties (see Ref. [41] and references therein). 3.1. Merons as vortex intersection points In the following we will provide evidence that the merons can be interpreted as vortex intersection points. We will then bring these merons in the laplacian center gauge and in fact detect a center vortex. Merons are topologically nontrivial field configurations defined by (2.3) This so-called laplacian center gauge fixing is equivalent to rotating the component (ψ2V (x))⊥ of ψ2V (x) which is orthogonal to the 3-axis into the 1-direction. Obviously the laplacian center gauge fixing is illdefined when ψ1 (x) and ψ2 (x) are colinear so that (after laplacian abelian gauge fixing) (ψ2V (x))⊥ = 0. The latter condition involves two constraints so that these gauge singularities have codimension 2 and represent vortex singularities [24,25]. In fact one can show that the induced gauge potential g ∂g −1 represents near the vortex singularity a center vortex. Since for ψ1 (x) = 0 133 a (AM )µ = ηµν xν Ta , r2 r 2 = x12 + x22 + x32 + x02 , Ta = 1 σa , 2i (3.1) a dewhich possess Pontryagin index ν = 12 . Here ηµν notes the ’t Hooft symbol. Merons can be considered as half an instanton of vanishing radius. This becomes clear, if one compares the gauge potential of the meron (3.1) with the gauge potential of an instanton a (AI )µ = 2ηµν r2 xν Ta . + ρ2 (3.2) 134 H. Reinhardt, T. Tok / Physics Letters B 505 (2001) 131–140 Furthermore, the vanishing of the radius of the meron implies that the topological density of the meron is localized at a single point (3.6) C 1 Q(x) = δ 4 (x). 2 (3.3) Obviously the meron has the same topological properties as a transversal intersection point of two Z2 center vortex sheets [14]. In the following we will show that the meron, in fact, shows features of an intersection point of vortex sheets. We establish this by considering the Wilson loops around the center of the meron. We will show that for each color component the meron looks near its center like a pair of intersecting Z2 center planes. For this we show that the Wilson loops in the corresponding planes yield center elements. To be more precise we will show that for a color component b the Wilson loops around the center of the meron in the plane (i, j ) and in the plane (b, 0) yield center elements, where the triplet of indices (b, i, j ) is defined by |$bij | = 1. Consider a spherical Wilson loop C in the spatial plane (i, j ). We can use polar coordinates in this plane xi = ρ cos ϕ, xj = ρ sin ϕ, xb = 0 = x0 , |$bij | = 1. (3.4) a a = $akl and η0k = δka of From the properties ηkl a the ’t Hooft symbol ηµν it follows that only the b-component in color space of AM contributes to the Wilson loop, i.e. the calculation of the path-ordered integral simplifies to ordinary integration of AM along the path C: 1 a (AM )µ dxµ = dxµ ηµν x ν Ta 2 x C eventually obtain (AM )µ dxµ = −2π$aij Ta . C = C = dϕ $akl ẋk xl 1 Ta , ρ2 C =e + 2i σa 2π$aij = −1. x0 = ρ cos ϕ, xb = ρ sin ϕ, xi = 0 = xj , |$bij | = 1 C C 2π dϕ ẋ0 (ϕ)xk (ϕ) 1 − ẋk (ϕ)x0 (ϕ) δak 2 Ta ρ 2π (3.5) where the indices k, l run over the values i, j and the integrand is different from zero only for the color component b. Straightforward evaluations yield that the integrand is independent of the angle ϕ, so that we (3.8) a = δ and using the property η0k ak of the ’t Hooft symbol a 1 (AM )µ dxµ = (dx0 xk − dxk x0 )η0k Ta ρ2 0 0 (3.7) The lesson from this calculation is that for the meron only the color component b defined by |$bij | = 1 contributes to the Wilson loop in the (i, j ) plane. Furthermore this Wilson loop equals a center element, which can be interpreted by saying that the b-component of the meron looks like a center vortex piercing the (i, j )-plane with |$bij | = 1. Let us now also show, that a Wilson loop in the plane orthogonal to the (i, j ) plane defined by |$bij | = 1 also receives contribution only from the color component b and yields also a center element. Indeed, for the Wilson loop in the (0, b) plane, which is orthogonal to the (i, j ) plane due to the condition |$bij | = 1, we find, introducing in this plane analogous polar coordinates: =− 1 a dϕ ẋµ (ϕ)ηµν xν (ϕ) 2 Ta ρ 2π Hence, we find for the Wilson loop: W (C) = P exp − (AM )µ dxµ =− dϕ Tb = −2πTb . (3.9) 0 We observe, that for the Wilson loop in the (0, b) plane only the color component b contributes. Thus, indeed the color component b of the meron field looks like the H. Reinhardt, T. Tok / Physics Letters B 505 (2001) 131–140 intersection point of two vortex sheets, one in the (i, j ) and the other in the (0, b) plane, where these indices are related by |$bij | = 1. One also easily shows that in the remaining planes like (i, 0), i = b or (i, k), k = j , |$bij | = 1 the spherical Wilson loops of the b-component of AM around the center of the meron becomes trivial: W (C) = 1. (3.10) The remaining two color components of the meron field also behave like intersection points of two transversal vortex planes (i, j ) and (b, 0) as defined by the condition |$bij | = 1. Thus we have seen, that indeed near its center the meron looks like pairwise intersecting orthogonal center vortex sheets. Now we will analyze the vortex content of the meron in Laplace center gauge. For this purpose we consider the meron on a 4-dimensional sphere S 4 with radius R. On S 4 we use stereographic coordinates xµ , µ = 1, . . . , 4. In these coordinates the metric is conformally flat and reads gµν = 4R 4 δµν . (r 2 + R 2 )2 (3.11) The covariant Laplace operator on S 4 has the form 1 √ Dµ gg µν Dν D2 = √ g = (r 2 + R 2 )4 ∂µ + Aaµ Ta 16R 8 4R 4 b ∂ , + A × 2 T µ b µ (r + R 2 )2 (3.12) where g denotes the determinant of the metric, r 2 = xµ xµ and Ta are the generators of the gauge group in the adjoint representation. Plugging (3.1) into (3.12) results in (r 2 + R 2 )2 2 3 4 2 4 2 ∂r + ∂r − 2 L D = − 2 r∂r r 4R 4 r r + R2 1 2 4 T ·L− 2 T , (3.13) − 2 r r a x µ ∂ and T is a = ad(σ a /2). L where La = −i/2ηµν ν the set of generators of an SU(2) subgroup of the rotation group SO(4) [42]. Introducing the conserved + iT the eigenfunctions angular momentum J = L 135 of the covariant Laplace operator D2 (3.12) can be written in the form ψ(x) = f (r)Y(j,l) (x̂) · σ . (3.14) Here x̂µ = xµ /r and Y(j,l) denote the spherical vector harmonics on S 3 defined by 2 Y(j,l) = l(l + 1)Y(j,l) , L J2 Y(j,l) = j (j + 1)Y(j,l) , 2 T Y(j,l) · σ = t (t + 1)Y(j,l) · σ , (3.15) with t = 1. Substituting f (r) = (r 2 + R 2 )ϕ(r) [31] simplifies the eigenvalue problem problem to 3 (j (j + 1) + l(l + 1) − 1) −∂r2 − ∂r + 2 r r2 4R 4 8R 2 ϕ=λ 2 − 2 ϕ. (3.16) 2 2 (r + R ) (r + R 2 )2 To get the lowest eigenvalue we have to minimize j (j +1)+l(l +1)−1. This quantity becomes minimal for j = l = 1/2 (since the singlet j = l = 0 is excluded by selection rules for t = 1, see (3.15)). Therefore the ground state is 4-fold degenerate and the meron configuration lies on the Gribov horizon for the laplacian center gauge fixing. The four eigenfunctions form the fundamental representation of SO(4). The corresponding spherical harmonics are given by: k Y(1/2,1/2), k = 1, . . . , 4 −x̂4 −x̂3 −x̂2 x̂1 = . (3.17) x̂3 , −x̂4 , x̂1 , x̂2 x̂2 −x̂1 x̂4 x̂3 Taking for instance the 4th eigenvector as the ground state and the 3rd as the first excited state 2 the monopole and vortex content is as follows. We get a static monopole line at x1 = x2 = x3 = 0 and the vortex sheet is the (3, 4)-plane. Another possible choice for the two eigenstates of the covariant Laplacian 1 2 + Y(1/2,1/2) ) and ψ2 = would be ψ1 = ϕ(Y(1/2,1/2) 3 4 ϕ(Y(1/2,1/2) + Y(1/2,1/2)). In this case we identify a magnetic monopole line in the (1, 2)-plane given by x1 = x2 , x3 = x4 = 0 and three center vortex sheets 2 One may think of a suitable infinitesimal perturbation to the gauge potential to select these eigenvectors as the two lowest-lying states. 136 H. Reinhardt, T. Tok / Physics Letters B 505 (2001) 131–140 given by x2 = x1 , x3 = −x4 , x1 = x4 , x2 = −x3 x1 = −x4 , (3.11). With the instanton gauge potential (3.2) xν a Ta (AI )µ = 2ηµν 2 r + ρ2 and x2 = x3 , respectively. The three center vortex sheets intersect at the origin. The meron configuration is SO(4)symmetric and the eigenspace to the lowest eigenvalue of the Laplace operator shares this symmetry. Therefore we can move the vortex planes by arbitrary SO(4) rotations (this corresponds to choosing other linear combinations of the degenerate eigenstates (3.17) of the covariant Laplacian for the gauge fixing). Let us emphasize that the laplacian center gauge fixing of the meron field detects either a single vortex sheet or three center vortex sheets, while the study of the Wilson loop has revealed pairwise intersecting vortex sheets near the meron center. Obviously, highly symmetric configurations like the meron or instanton fields are not faithfully reproduced by the center projection implied by the vortex identification of the laplacian gauge fixing. This is because these configurations are lying on the Gribov horizon. 3.2. Instantons in laplacian center gauge Below we consider a simple instanton and an instanton–antiinstanton pair in the laplacian center gauge in order to reveal its monopole and center vortex content. In the laplacian abelian gauge (which represents a partial gauge fixing of the laplacian center gauge) a simple instanton has been considered recently [31]. We will not stick to the abelian gauge but consider the full laplacian center gauge. In addition, we do not confine ourselves to a single instanton but consider also an instanton–antiinstanton pair. Such a configuration has previously been studied on the Lattice [25]. For a single instanton due to its symmetry the lowest lying eigenvectors of the Laplacian can be found analytically when choosing S 4 as space–time manifold [31]. 3.2.1. The single instanton in laplacian center gauge As for the above discussed meron configuration we use stereographic coordinates xµ on S 4 and the metric the covariant Laplace operator reads (r 2 + R 2 )2 2 3 4 2 8 2 D = ∂r + ∂r − 2 L − 2 T · L 4 r 4R r r + ρ2 2 4 4r 2 − T (3.18) r∂ . − 2 r (r + ρ 2 )2 r 2 + R2 Again the eigenfunctions of D2 have the form (3.14). Depending on the ratio ρ/R between the scale ρ of the instanton and the radius R of the 4-sphere the ground state is 3-fold degenerate for ρ = R and 10-fold degenerate for ρ = R. In the physical case R > ρ (including the infinite volume limit) the ground state is three-fold degenerate and has the form 1 Y(0,1) · σ , (3.19) + r 2) i.e. j = 0 and l = 1, see (3.15). The triplet of functions Y(0,1) is given by 2 x̂1 − x̂22 − x̂32 + x̂42 2(x̂1x̂2 − x̂3 x̂4 ) 2 2 2 2 , −x̂1 + x̂2 − x̂3 + x̂4 , 2(x̂1x̂2 + x̂3 x̂4 ) 2(x̂1x̂3 − x̂2 x̂4 ) 2(x̂2x̂3 + x̂1 x̂4 ) 2(x̂1x̂3 + x̂2 x̂4 ) . (3.20) 2(x̂2x̂3 − x̂1 x̂4 ) −x̂12 − x̂22 + x̂32 + x̂42 To get the monopole and vortex content of the instanton configuration we have to choose one of the three eigenfunctions as the ground state and another as the first excited state. But the only zeros of the eigenfunctions are at the origin. This means that the set of monopoles consists of the origin only, i.e. we have no monopole loop or we can say that the monopole loop is degenerated to a single point. To examine the vortex content of the configuration we have to look for points where two of the three vectors in (3.20) are linearly dependent. But it is easy to see that the three vectors are always perpendicular to each other. Therefore in the laplacian center gauge the instanton with ρ = R does not give rise to center vortex sheets. For the special case R = ρ the ground state is 10fold degenerate. In this case the set of ground states consists of two triplets (j = 0, l = 1 and j = 1, l = 0) and one quadruplet (j = l = 1/2). Choosing eigenfunctions from the quadruplet, see (3.17), as ψ(x) = R(R 2 H. Reinhardt, T. Tok / Physics Letters B 505 (2001) 131–140 ground and first excited state we get the same result as in the meron case, i.e. a monopole line and one or three vortex sheets (see previous subsection). 3.2.2. Instanton–antiinstanton pair in laplacian center gauge We choose here space–time manifold as the direct product of a three-dimensional disc D 3 with radius D and an interval I = [−L0 , L0 ]. We consider a gauge potential describing approximately an instanton–antiinstanton pair: a (AIA )µ = 2 ηµν (xν − zν )B− a + η̄µν (xν + zν )B+ Ta , (3.21) 1 B± = |x ± z|2 + ρ 2 40 . × exp − 1.25|x ± z|/D (3.22) for | x | = D, ψ(−L0 , x) = ψ(L0 , x). the eigenfunctions of D2 in vector spherical harmonics 2 Yj lm on S [43], with j (j + 1), l(l + 1) and m being 2 and J3 : the eigenvalues of J2 , L ψj m = (3.26) Tl (x0 )Rl (r)Yj lm (ϑ, ϕ) · σ . l (3.23) It turns out that the action of D2 on Yj lm · σ does not depend on m. Therefore the eigenvalues of D2 will be (2j + 1)-fold degenerate. The functions Tl (x0 ) have been Fourier expanded in sinus and cosinus functions of the time and in Bessel functions Rl (r) of r. We have solved the eigenvalue problem numerically by calculating the matrix elements of D2 and diagonalizing this matrix. It turned out that the ground state has j = 1 and thus is threefold degenerate. To get rid of the degeneracy we assume that we have an infinitesimal perturbation by εJ32 , such that the ground state has m = 0. We first consider an instanton–antiinstanton configuration where the distance between the centers of instanton and antiinstanton is large compared to the (anti)instanton size ρ (instanton and antiinstanton radii are chosen to be equal ρ). For this case we have chosen the parameters as follows: (3.24) D = L0 = 10, The centers of the two instantons ±zµ are located on the time axis, i.e. zi = 0, i = 1, 2, 3. The eigenfunctions should vanish on the boundary of D 3 and be periodic in the time direction: ψ(x0 , x) = 0 137 The exponential factors in (3.22) are introduced to make the gauge potential nearly vanishing on the boundary of the space–time and to render the Laplace operator selfadjoint. For the considered instanton–antiinstanton configuration the Laplace operator reads T ∂0 D2 = ∂µ ∂µ + 4(B+ − B− ) x · T · ∂ + 4 (x0 − z0 )B− − (x0 + z0 )B+ T · x × ∂ + 4(B+ + B− ) 2 2 + 4 |x − z|2 B− + |x + z|2 B+ T · T + 2 r 2 − x02 + z02 B− B+ − 16B− B+ x · (3.25) T · x · T , where r = | x | and T = ad(T ) is the color spin in the adjoint representation. The Laplace operator + iT is commutes with J2 and J3 , where J = L = −i the total angular momentum and L x × ∂ is the orbital angular momentum. This means we can expand z0 = 1, ρ = 0.1. (3.27) From the zeros of the lowest eigenmode we identified two magnetic charge-1 monopole loops crossing each other near the instanton centers, see Fig. 1. The set of the magnetic monopole loops is symmetric with respect to rotations with angle π around the x1 -, x2 and x3 -axis. To identify √ the center vortices we have chosen ψj =1,y = i/ 2(ψj =1,m=−1 + ψj =1,m=1 ) as the first excited state. The resulting vortex connects at each time x0 all four monopole branches, i.e. the vortex sheet is topologically equivalent to S 2 and encloses the two instanton centers. In Fig. 2 we plotted the vortex in the time-slice x0 = 0. Further we examined the dependence of the monopole and vortex content of the configuration on the distance 2z0 between the instanton centers. Reducing z0 results in smaller monopole loops and at a critical value (z0 = 0.3513) the monopole loops and the vortex sheet disappear. We have also changed the gauge potential (3.21) by a factor of 2, in order to increase the field strength. Accordingly, after laplacian abelian gauge fixing, the 138 H. Reinhardt, T. Tok / Physics Letters B 505 (2001) 131–140 Fig. 3. Plot of the r–x0 -projection (r = x12 + x22 + x32 ) of one of the 4 small magnetic monopole loops and one half of one of the 2 large magnetic monopole loops for the gauge potential 2AIA . The crosses show the positions of the instanton centers. Fig. 1. Plot of the two magnetic monopole loops for the gauge potential (3.21) projected onto the x1 –x2 –x0 space (dropping the x3 -component). The thick dots show the positions of the instantons. Fig. 4. 3-dimensional plot of the monopole loops for the doubled instanton–antiinstanton gauge potential projected onto the x1 –x2 –x0 space. Only one of the 4 small magnetic monopole loops and one half of the 2 large magnetic monopole loops are plotted. The thick dots show the positions of the instanton centers. Fig. 2. Plot of the vortex in the time-slice x0 = 0. The thick dots on the vortex show the positions of the magnetic monopoles. number of magnetic monopole loops increases. We identified 6 magnetic monopole loops — two of them are larger and intersect each other on the x0 axis (similar as in the case with gauge potential AIA , cf. (3.21)), while the other four monopole loops are smaller and separated from each other, cf. Figs. 3 and 4. The set of all magnetic monopole loops is again symmetric with respect to rotations with angle π around the x1 -, x2 - and x3 -axis. The study of configurations with nonzero topological charge (instanton number), like, e.g., instanton– H. Reinhardt, T. Tok / Physics Letters B 505 (2001) 131–140 instanton pairs, is more difficult. This is because in this case one has to implement nontrivial boundary conditions on the fields ψ1 and ψ2 which strongly complicates the numerical solution of the underlying differential equations. 4. Concluding remarks We have studied various field configurations relevant for the infrared sector of QCD in laplacian abelian and center gauges in the continuum. While the gauge does not detect center vortices for single instantons it identifies center vortices for merons and composite instanton–antiinstanton configurations. The absence of center vortices in single instantons is somewhat expected if center vortices are responsible for confinement, which is, however, not explained by instantons. Furthermore we have also shown that for highly symmetric field configurations laplacian center gauge does not necessarily provide a very faithful method for detecting their vortex content, because these configurations lie mostly on the Gribov horizon. A better detector for center vortices is the Wilson loop. From the study of the Wilson loop we have provided evidence that merons can be interpreted as self-intersection points of center vortices. Acknowledgements Discussions, in particular on the numerics, with R. Alkofer, K. Langfeld and A. Schäfke are gratefully acknowledged. 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[37] M. Quandt, H. Reinhardt, A. Schäfke, Phys. Lett. B 446 (1999) 290. [38] Ph. de Forcrand, M. Pepe, Center vortices and monopoles without lattice Gribov copies, hep-lat/0008016. [39] E. Witten, Nucl. Phys. B 156 (1979) 269. [40] G. Veneziano, Nucl. Phys. B 159 (1979) 213. 140 H. Reinhardt, T. Tok / Physics Letters B 505 (2001) 131–140 [41] J.V. Steele, J.W. Negele, Meron pairs and fermionic zero modes, hep-lat/0007006. [42] G. ’t Hooft, Phys. Rev. D 14 (1976) 3432. [43] D.A. Varshalovich, A.N. Moskalev, V.K. Khersonskii, Quantum Theory of Angular Momentum, World Scientific, Singapore, 1988. 26 April 2001 Physics Letters B 505 (2001) 141–148 www.elsevier.nl/locate/npe Testing imaginary vs. real chemical potential in finite-temperature QCD A. Hart a,1 , M. Laine b,c , O. Philipsen b,2 a Department of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3JZ, Scotland, UK b Theory Division, CERN, CH-1211 Geneva 23, Switzerland c Department of Physics, P.O. Box 9, FIN-00014 University of Helsinki, Helsinki, Finland Received 23 January 2001; received in revised form 16 February 2001; accepted 12 March 2001 Editor: P.V. Landshoff Abstract One suggestion for determining the properties of QCD at finite temperatures and densities is to carry out lattice simulations with an imaginary chemical potential whereby no sign problem arises, and to convert the results to real physical observables only afterwards. We test the practical feasibility of such an approach for a particular class of physical observables, spatial correlation lengths in the quark–gluon plasma phase. Simulations with imaginary chemical potential followed by analytic continuation are compared with simulations with real chemical potential, which are possible by using a dimensionally reduced effective action for hot QCD (in practice we consider QCD with two massless quark flavours). We find that for imaginary chemical potential the system undergoes a phase transition at |µ/T | ≈ π/3, and thus observables are analytic only in a limited range. However, utilising this range, relevant information can be obtained for the real chemical potential case. 2001 Published by Elsevier Science B.V. 1. Introduction Given the applications to cosmology and heavy ion collision experiments, it is important to determine the properties of QCD at finite temperatures and baryon densities. For instance, one would like to know the locations of any phase transitions, and the properties of the quark–gluon plasma phase such as its free energy density, or pressure, as well as the spatial and temporal correlation lengths felt by various types of excitations in the system. E-mail address: [email protected] (A. Hart). 1 Current address: DAMTP, University of Cambridge, Cam- bridge CB3 0WA, UK. 2 Current address: Center for Theoretical Physics, MIT, Cambridge, MA 02139, USA. Because the theory is strongly coupled, the only practical first principles method available for addressing these questions is lattice simulations. While there has been steady improvement in the accuracy of results at vanishing baryon density [1], the case of a nonvanishing density is still largely open, despite much work [1–13]. Indeed, introducing a non-vanishing density, or chemical potential, is difficult because it leads to a measure which is not positive definite (this is the so-called sign problem), whereby standard Monte Carlo techniques fail. In this Letter we focus on one of the suggestions for how a finite density system could eventually be addressed with practical lattice simulations. The idea is to first inspect an imaginary chemical potential, whereby the sign problem temporarily disappears, and then relate this to the case of a real chemical 0370-2693/01/$ – see front matter 2001 Published by Elsevier Science B.V. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 3 5 5 - 0 142 A. Hart et al. / Physics Letters B 505 (2001) 141–148 potential. Let us denote by µ the chemical potential for quark number Q, and by µB the chemical potential for baryon number B = Q/3: then µ = µB /3. By µR , µI ∈ R we denote the real and imaginary parts of µ: µ = µR + iµI . (1.1) It is easy to see, by going to momentum space, that physical observables are periodic in µI with the period 2πT . There are then two types of suggestions for how an imaginary µ = iµI could be utilised to obtain information on a system with a real chemical potential, µ = µR . The first idea is related directly to the equation of state, and employs the canonical partition function at fixed quark number ([2–5] and references therein): 1 Z(T , Q) = 2πT +πT dµI Z(T , µ = iµI )e−iµI Q/T , −πT (1.2) where the grand canonical partition function, Z(T , µ) = Tr e In this Letter we study the latter suggestion within the framework of a dimensionally reduced effective theory. As we shall review in the next section, at temperatures sufficiently above the phase transition, the thermodynamics of QCD can be represented, with good practical accuracy, by a simple threedimensional (3d) purely bosonic theory. This can also be done with a chemical potential, both real and imaginary [14]. We then use this theory to measure the longest static correlation lengths in the system for both cases. We find that, for small |µ/T |, the observables are well described by a truncated power series with coefficients determined by fits. We then inspect how well the analytically continued series describes the real data. In principle the free energy density could be addressed with similar effective theory methods [15], but this requires a number of high-order perturbative computations which are not available at the moment for µ = 0. −µQ)/T −(H , 2. Effective theory 2.1. Action (1.3) has been evaluated with an imaginary chemical poten and Q denote the Hamiltonian and quark tial. Here H number operators, respectively, while Q is a number. With an imaginary chemical potential, Z(T , µ = iµI ) or rather the ratio Z(T , µ = iµI )/Z(T , 0), can be determined using standard lattice techniques. What remains is to perform the integral in Eq. (1.2). Of course, this gets more and more difficult in the thermodynamic limit Q → ∞, because oscillations reappear in the Fourier transform. In addition, a Legendre transform would be needed to go from Z(T , Q) to a system in an ensemble with a real chemical potential, Z(T , µ = µR ). The second idea ([11] and references therein) is that, away from possible phase transition lines, the partition function and expectation values for various observables should be analytic in their arguments, in particular in µ/T . Thus, we may attempt a general power series ansatz for the functional behaviour in µ/T , determine a finite number of coefficients with an imaginary chemical potential, and finally analytically continue to real values. The effective theory emerging from hot QCD by dimensional reduction [14,16–22], is the SU(3)+adjoint Higgs model with the action 1 3 Tr Fij2 + Tr[Di , A0 ]2 + m23 Tr A20 S= d x 2 2 (2.1) + iγ3 Tr A30 + λ3 Tr A20 , where Fij = ∂i Aj − ∂j Ai + ig3 [Ai , Aj ], Di = ∂i + ig3 Ai , Fij , Ai , and A0 are all traceless 3 × 3 Hermitian matrices (A0 = Aa0 Ta , etc.), and g32 and λ3 are the gauge and scalar coupling constants with mass dimension one, respectively. The physical properties of the effective theory are determined by the three dimensionless ratios x= λ3 g32 , y= m23 (µ̄3 = g32 ) g34 , z= γ3 g33 , (2.2) where µ̄3 is the MS dimensional regularization scale in 3d. For vanishing chemical potential γ3 = 0 and no term cubic in A0 appears. These ratios are via dimensional reduction functions of the temperature A. Hart et al. / Physics Letters B 505 (2001) 141–148 T /ΛMS and the chemical potential µ/ΛMS , as well as of the number Nf of massless quark flavours; for the case µ = 0, we refer to [22]. The inclusion of quark masses is also possible in principle, but in the numerical part of this work we assume Nf = 2 massless dynamical flavours, the other flavours being approximated as infinitely heavy. The mass parameter m23 , represented by y in Eq. (2.2), turns out to be positive [22]. This guarantees that the 3d theory tends to live in its symmetric phase, A0 ∼ 0, at least on the mean field level. We will return to this issue presently. Compared with the case µ = 0, the dominant changes in the action due to a small chemical potential are now [14] µ Nf ; T 3π 2 z: 0→ y: µ y →y 1+ πT 2 3Nf , 2Nc + Nf (2.3) where Nc = 3. Thus, one new operator is generated in the effective action, and one of the parameters which already existed, gets modified. For real chemical potential, µ = µR , the effective action is thus complex, whereas for imaginary chemical potential, µ = iµI , it is real. 2.2. Ranges of validity There are several requirements for the effective description in Eq. (2.1) to be reliable. They are all related to a sufficiently “weak coupling”, or effective expansion parameter, for a given T /ΛMS , µ/ΛMS , Nf . Let us briefly reiterate them here. First, the perturbative expansions for the effective parameters in Eq. (2.2) have to be well convergent. Inspecting the actual series up to next-to-leading order, it appears that this requirement is surprisingly well met even at temperatures not much above the critical one [22]. Second, the higher-dimensional operators arising in the reduction step which are not included in the effective action in Eq. (2.1), should only give small corrections. This condition is met if the dynamical mass scales described by the effective theory are smaller than the ones ∼ 2πT that have been integrated out. In pure Yang–Mills theory, there is evidence that 143 this can be sufficiently satisfied at temperatures as low as T ∼ 2ΛMS [14,17,22–26]. However, when fermions are included and a real chemical potential is switched on, some of the mass scales increase (see below), and the effective description will become less accurate. Third, the effective 3d theory represents the 4d theory reliably only when it lies in its symmetric phase [22,27] (i.e., A0 ∼ 0). Indeed, for Nf = 0 QCD has a so-called Z(N )-symmetry [28,29], and this symmetry is not fully reproduced by the effective theory. The 4d Z(N ) symmetry is however spontaneously broken for Nf = 0 in the deconfined phase, and even explicitly broken for Nf > 0. In this case broken Z(N ) means that the Polyakov line is approximately unity, corresponding to A0 ∼ 0 and hence to a symmetric phase in terms of the gauge potential. Consequently, the requirement to be in the symmetric phase of the 3d theory is easier to control for Nf > 0. Furthermore, the situation gets even better for real µ = 0. In the effective theory this can be seen, for instance, from the fact that the mass parameter y in Eq. (2.3) grows, which makes it more difficult to depart from A0 ∼ 0. Another stabilising factor is that the unphysical minima correspond to non-zero expectation values for Tr A30 [27], and the imaginary term ∼ i Tr A30 in the action in Eq. (2.1) disfavours such minima according to the standard argument [30]. On the other hand, an imaginary chemical potential µ = iµI , favours those Z(N ) broken minima where the Polyakov line has a non-trivial phase, and correspondingly the gauge potential is non-zero, A0 = 0. Utilising the perturbative effective potential [31], we find that the lowest such minimum becomes degenerate with the symmetric one A0 ∼ 0 already at µI /T = π/3, and increasing µI further it eventually becomes lower than our minimum. Thus there is a (first order) “phase transition” [2]. In the effective theory, this phase transition is triggered by the decrease of the mass parameter y in Eq. (2.3). Moreover, in this case the term ∼ Tr A30 in the action in Eq. (2.1) favours the effective theory remnant of one of the minima with A0 = 0, i.e., a non-trivial phase of the Polyakov line. This phase transition limits the applicability of the effective theory with imaginary chemical potential, since only the symmetric phase is a faithful representation of the 4d theory [22,27]. (For one suggestion on how perhaps to circumvent this problem at least for 144 A. Hart et al. / Physics Letters B 505 (2001) 141–148 Nf = 0, which we shall however not dwell on here, see [32].) In a way, this problem is related to the fact that as one approaches µI /T = π , fermions start to obey Bose–Einstein statistics and become “light” infrared sensitive degrees of freedom (see also [31]), whereby it is no longer legitimate to integrate them out. In summary, the effective theory roughly loses its accuracy with a real chemical potential once even the longest correlation length is shorter than ∼ 1/(2πT ), and with an imaginary chemical potential once |µ/T | exceeds unity. Fortunately, this range of validity contains the parameters that are phenomenologically most relevant. Indeed, heavy ion collision experiments at and above AGS and SPS energies can be estimated to correspond to µB /T 4.0 [33], or a quark chemical potential µ/T 1.3. 2.3. Observables and their parametric behaviour As we have mentioned, the physical observables which we shall study are spatial correlation lengths: we consider operators living in the (x1 , x2 )-plane, and measure the correlation lengths in the x3 -direction. In the presence of µ = 0, there are only two different quantum number channels to be considered, distinguished by the two-dimensional parity P in the transverse plane. The lowest dimensional gauge invariant operators in the scalar (J = 0) channels are: J P = 0+ : 2 2 , Tr A30 , Tr A0 F12 ,... , Tr A20 , Tr F12 J P = 0− : 3 , Tr A20 F12 , Tr A0 F12 , . . . . Tr F12 (2.4) The corresponding 4d operators can be found in [34]. We shall measure whole cross correlation matrices between all (smeared) operators in these channels, but mostly focus on their lowest eigenstates, corresponding to the longest correlation lengths in the 4d finite temperature system. We denote the “energies” of these eigenstates, viz. inverses of correlation lengths, by m. We also examine the overlap of operators of different field contents onto the eigenstates. Since a change µ → −µ can be compensated for by a field redefinition A0 → −A0 in Eq. (2.1), all physical observables must be even under this operation. In the original 4d theory the same statement follows from compensating µ → −µ by a C (or CP) operation. Moreover, since there are no massless modes at µ = 0, we expect the masses to be analytic in µ away from phase transitions. For small values of µ/T , the inverse correlation lengths may thus be written as µ 2 µ 4 m + c2 = c0 + c1 T πT πT 6 µ . +O (2.5) πT We have chosen to include πT in the denominators, because the chemical potential appears with this structure in the effective parameters, cf. Eq. (2.3). Of course, the radii of convergence of such expansions are not known a priori. Here we first check to what extent a truncated series of the type in Eq. (2.5) can accurately describe the data. In the range where this is possible, we determine the {ci } with µ = iµI , and check if the analytically continued result reproduces the independent measurements carried out with µ = µR . 3. Simulations 3.1. Simulation methods We simulate the theory at several µ/T . The values chosen, together with the corresponding continuum parameters, are listed in Table 1. Discretization and lattice–continuum relations [35] are implemented as in [14]. As discussed there, finite volume and lattice spacing effects are expected to be smaller or at most of the same order as the statistical errors for the parameter values we employ. Compared with [14], we have increased the statistics and included many new values of µ/T , in order to carry out more precise fits. For real µ = µR , the action in Eq. (2.1) with parameters as in Eq. (2.3) is complex, which precludes direct Monte Carlo simulations. We must thus carry out simulations using a reweighting technique, which has been explained in detail in [14]. There it was found that physically realistic lattice volumes may be simulated for chemical potentials up to µR /T 4. For imaginary µ = iµI , the action in Eq. (2.1) with A. Hart et al. / Physics Letters B 505 (2001) 141–148 145 Table 1 The parameters used for µ = 0 (cf. Eq. (2.2)). All correspond to T = 2ΛMS , Nf = 2. In addition, x = 0.0919, g32 = 2.92 T, β = 21, volume = 303 , where β determines the lattice spacing (for the detailed relations employed here, see [14]) Real µ Imaginary µ |µ|/T |µ|2 /(π T )2 y z y z 0.50 0.0253 0.49218 0.0338 0.47382 0.0338i 0.75 0.0570 – – 0.46235 0.0507i 1.00 0.1013 0.51970 0.0675 0.44630 0.0675i 1.25 0.1583 0.54035 0.0844 0.42565 0.0844i 1.50 0.2280 0.56558 0.1013 0.40042 0.1013i 1.75 0.3103 0.59540 0.1182 – – 2.00 0.4053 0.62981 0.1351 – – 3.00 0.9119 0.81333 0.2026 – – 3.75 1.4248 0.99914 0.2533 – – 4.00 1.6211 1.07026 0.2702 – – parameters as in Eq. (2.3) is real, and correspondingly we simulate the full action using a Metropolis update. 3.2. Results As a first result, let us note that, as has been the case in several related theories [14,24,36], we again observe a dynamical decoupling of operators, 2 , such that operators involving scalars (Tr A20 , Tr A0 F12 2 etc.) and purely gluonic operators (Tr F12 , etc.) have a mutual overlap consistent with zero. The correlation matrix thus assumes an approximately block diagonal form. We find that the gluonic states remain extremely insensitive to µ/T , and agree well with the masses found in d = 3 pure gauge theory [37]. This situation is illustrated in Fig. 1. The scalar states, on the other hand, show a marked dependence on µ/T , with their masses increasing for real µ and decreasing for imaginary values. For both small real and small imaginary µ/T , the ground state in each channel is scalar in nature, and we plot these states in Fig. 2. Because of the different qualitative behaviours of 3d gluonic and scalar states, we may expect to observe a change in the nature of the ground state excitation at some µR . Indeed, Fig. 1 suggests a level crossing Fig. 1. Inverse correlation lengths in the channel 0+ , for real µ/T . “Scalar” states (Tr A20 , etc.) do depend on µ/T , while 2 , etc.) are practically independent of it. For “gluonic” states (Tr F12 comparison, the horizontal band indicates the 3d pure glue result for 2 [37], converted to our units via g 2 = 2.92 T. Tr F12 3 at µ/T ∼ 4.0. This would mean that the longest correlation length in the thermal system does not get arbitrarily short with increasing density, but rather stays at a constant level. Note that the value of m/T at this crossing is already so large that the effective theory may be inaccurate quantitatively, and in fact in the full 4d theory the flattening off could take place much earlier. However, the qualitative effect should be the same. Next, let us discuss the applicability of the power series ansatz in Eq. (2.5). To this end we perform fits over a range |µ| = 0, . . . , µmax to the inverses of the longest correlation lengths, both for real and imaginary µ. For imaginary µ, we can follow the “analytically continued” metastable branch as long as tunnelling into an unphysical minimum does not become a problem, which in practice means µI /T 1.5. The results are shown for the 0+ channel in Table 2, and for the 0− channel in Table 3. Examining these fits we see that in all cases we have good fits, as demonstrated by the low χ 2 /dof and good Q values. In the case of real µ we find stable and well constrained values for the coefficients as we increase the size of the fitting range. For imaginary µ, due to the breakdown of the effective theory at large values of µI /T , we have fewer significant data points, and 146 A. Hart et al. / Physics Letters B 505 (2001) 141–148 Table 2 Fitting the lowest masses in the channel 0+ from µ = 0 up to µ = µmax . The numbers in parentheses indicate the error of the last digit shown, the coefficients refer to Eq. (2.5), the sub and superscripts R, I denote real or imaginary µ, and Q is the quality of the fit µmax R /T c0R c1R c2R χ 2 /dof Q 1.50 3.952 (37) 3.89 (99) −3.92 (449) 0.175 0.840 2.00 3.956 (35) 3.54 (52) −2.06 (144) 0.145 0.965 3.00 3.965 (32) 3.22 (27) −1.06 (33) 0.216 0.956 4.00 3.983 (30) 2.94 (20) −0.61 (16) 0.607 0.751 χ 2 /dof Q µmax I /T c0I c1I c2I 1.25 3.952 (38) 4.73 (157) −3.07 (933) 0.090 0.914 1.50 3.925 (35) 2.64 (96) −16.89 (443) 1.004 0.390 Table 3 Fitting the lowest masses in the channel 0− . The notation is as in Table 2 µmax R /T Fig. 2. Top: Inverses of longest correlation lengths in the channel 0+ , for real and imaginary µ. Bottom: the same for 0− . consequently the coefficient of the quartic term is much less constrained. As our main result, we can now state that we observe good evidence for analytic continuation in the first non-trivial term, with c1R consistent with c1I in the 0+ channel and similarly for the 0− states. Unfortunately, the data is not accurate enough to make a similar statement for c2R , c2I . Extremely precise measurements would be needed, because the range in |µ/T | available to imaginary chemical potential simulations is very limited. On the other hand, from the phenomenological point of view the first non-trivial coefficient c0R c1R c2R χ 2 /dof Q 1.50 5.839 (69) −0.54 (167) 2.00 5.804 (63) 1.22 (91) 3.00 5.770 (57) 2.18 (47) −0.90 (65) 0.655 0.658 4.00 5.782 (54) 2.01 (35) −0.60 (23) 0.546 0.800 µmax I /T c0I c1I 1.25 5.818 (71) 0.36 (195) −16.36 (1087) 0.298 0.742 1.50 5.857 (65) 2.57 (116) −2.53 (465) 10.33 (722) 0.029 0.971 2.06 (246) 0.429 0.788 c2I χ 2 /dof Q 0.858 0.462 is sufficient, since the series expansion turns out to be in powers of µ2 /(πT )2 , which is small in the most important practical applications. Thus, for phenomenological purposes, it does not seem necessary to invest an extra amount of effort on a more precise determination of the masses in the imaginary µ case. 4. Conclusions In this Letter, we have studied the question as to what extent imaginary chemical potential simulations could be useful for determining the properties of the quark–gluon plasma phase at high temperatures and finite densities. The physical observables we have A. Hart et al. / Physics Letters B 505 (2001) 141–148 measured are static bosonic correlation lengths, but the pattern should be very similar for the free energy density, as long as T > Tc . The method we have used is based on a dimensionally reduced effective field theory. This way we can address both a system with a real and an imaginary chemical potential, as long as their absolute values are relatively small compared with the temperature. For larger absolute values of µ/T = iµI /T , there is a (first order) phase transition, and the effective description breaks down. Despite the fact that we are only working in the quark–gluon plasma phase, we find an interesting structure in the longest correlation length, which decreases first but becomes constant beyond some real value of µ/T , which we estimate to be 4.0. Furthermore, in the region where the effective theory is applicable, we find that direct analytic continuation does seem to provide a working tool for determining correlation lengths. For phenomenological applications, only the first two coefficients in the power series are needed, since we find the expansion parameter to be µ2 /(πT )2 , which is small in heavy ion collision experiments. This is good, since determining more coefficients with imaginary chemical potential would require very precise simulations. 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B 469 (1996) 445, hep-lat/9602006; O. Philipsen, M. Teper, H. Wittig, Nucl. Phys. B 528 (1998) 379, hep-lat/9709145. [37] M.J. Teper, Phys. Rev. D 59 (1999) 014512, hep-lat/9804008. [38] S. Gottlieb, W. Liu, D. Toussaint, R.L. Renken, R.L. Sugar, Phys. Rev. D 38 (1988) 2888; C. Bernard et al., Phys. Rev. D 54 (1996) 4585, heplat/9605028; S. Gottlieb et al., Phys. Rev. D 55 (1997) 6852, heplat/9612020. [39] P. de Forcrand et al., QCD-TARO Collaboration, heplat/0011013. 26 April 2001 Physics Letters B 505 (2001) 149–154 www.elsevier.nl/locate/npe Measuring the spin of the Higgs boson ✩ D.J. Miller a , S.Y. Choi a,b , B. Eberle a , M.M. Mühlleitner a,c , P.M. Zerwas a a Deutsches Elektronen-Synchrotron DESY, D-22603 Hamburg, Germany b Chonbuk National University, Chonju 561-756, South Korea c Université de Montpellier II, F-34095 Montpellier Cedex 5, France Received 9 February 2001; accepted 27 February 2001 Editor: P.V. Landshoff Abstract By studying the threshold dependence of the excitation curve and the angular distribution in Higgs-strahlung at e+ e− colliders, e+ e− → ZH , the spin of the Higgs boson in the Standard Model and related extensions can be determined unambiguously in a model-independent way. 2001 Published by Elsevier Science B.V. 1. Establishing the Higgs mechanism for generating the masses of the fundamental particles, leptons, quarks and gauge bosons, in the Standard Model and related extensions, is one of the principal aims of experiments at prospective e+ e− linear colliders [1]. After the experimental clarification of tantalizing indications of a light Higgs boson at LEP [2] has been stopped, the particle can be discovered at the Tevatron [3] or later at the LHC [4]. Assuming the positive outcome of these experiments, we address in this Letter the question of how the spinless nature and the positive parity of the Higgs boson 1 can be established in a model independent way. Higgs-strahlung, e+ e− → ZH, ✩ (1) Supported in part by the European Union (HPRN-CT-200000149) and by the Korean Research Foundation (KRF-2000-015050009). E-mail address: [email protected] (D.J. Miller). 1 The determination of the parity and the parity mixing of a spinless Higgs boson has been extensively investigated in Refs. [5,6]. provides the mechanism for the solution of this problem. The rise of the excitation curve near the threshold and the angular distributions render the spin-parity analysis of the Higgs boson unambiguous in this channel. Without loss of generality, we can assume the Higgs boson to be emitted from the Z-boson line, Fig. 1(a). Were it emitted from the lepton line, 2 the required H ee coupling would be so large that the state could have been detected as a resonance at LEP, e+ e− → H (γ ), or could be detected at the LHC via resonant H → e+ e− decays, dominating over the H → ZZ (∗) → 4l decay mode which involves two small Z branching ratios. The cross section for Higgs-strahlung in the Standard Model is given by the expression [7] σ [e+ e− → ZH ] = β 2 + 12MZ2 /s G2F MZ4 2 ve + ae2 β , 96πs (1 − MZ2 /s)2 (2) 2 We thank H. Murayama and T. Rizzo for alerting us to this potential loophole. 0370-2693/01/$ – see front matter 2001 Published by Elsevier Science B.V. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 3 1 7 - 3 150 D.J. Miller et al. / Physics Letters B 505 (2001) 149–154 The distribution of the polar angle θ is isotropic near the threshold and it develops into the characteristic sin2 θ law at high energies which corresponds to dominant longitudinal Z production, congruent with the equivalence theorem. Independent information on the helicity of the Z state is encoded in the final-state fermion distributions in the decay Z → f f¯. Denoting the fermion polar angle 4 in the Z rest frame with respect to the Z flight direction in the laboratory frame by θ∗ , the double differential distribution in θ and θ∗ is predicted by the Standard Model to be (a) dσ 1 σ d cos θ d cos θ∗ 9MZ2 γ 2 /4s = sin2 θ sin2 θ∗ β 2 + 12MZ2 /s 1 + 2 1 + cos2 θ 1 + cos2 θ∗ 2γ 2vf af 2ve ae 1 4 cos θ cos θ∗ , (5) + 2 2 2γ (ve + ae2 ) (vf2 + af2 ) (b) Fig. 1. (a) The Higgs-strahlung process, e+ e− → ZH , followed by the subsequent Z boson decay Z → f f¯, and (b) the definition of the polar angles θ and θ∗ , for production and decay, respectively. where ve = −1 + 4 sin2 θW and ae = −1 are the vector and axial-vector√Z charges of the electron; MZ is the Z-boson √ mass, s the centre-of-mass energy, and β = 2p/ s the Z/H three-momentum in the centre-ofmass frame, in units of the beam energy, i.e., β 2 = [1− (MH + MZ )2 /s][1 − (MH − MZ )2 /s]. The excitation curve rises linearly with β and therefore steeply with the energy above the threshold: 3 σ ∼ β ∼ s − (MH + MZ )2 . (3) This rise is characteristic of the production of a scalar particle in conjunction with the Z boson (with only two exceptions, to be discussed later). The second characteristic is the angular distribution of the Higgs and Z bosons in the final state [8], 3 β 2 sin2 θ + 8MZ2 /s 1 dσ = . σ d cos θ 4 β 2 + 12MZ2 /s with γ 2 = EZ2 /MZ2 = 1 + β 2 s/4MZ2 . Again, for high energies, the longitudinal Z polarization is reflected in the asymptotic behaviour ∝ sin2 θ∗ . 2. The helicity formalism is the most convenient theoretical tool for defining observables which uniquely prove the scalar nature of the StandardModel Higgs boson. Denoting the basic helicity amplitude [9] for arbitrary H spin-J , with the azimuthal angle set to zero, by Z(λZ )H (λH )|Z ∗ (m) gW MZ 1 = (6) d (θ )ΓλZ λH , cos θW m,λZ −λH the reduced vertex ΓλZ λH is dependent only on the helicities of the Z and Higgs bosons, λZ and λH respectively, and is independent of the Z ∗ spin component m along the beam-axis by rotational invariance. The standard coupling is split off explicitly. (4) 3 Non-zero width effects can easily be incorporated; see A. Para, note in preparation. 4 Azimuthal distributions provide supplementary information, see Ref. [8]; to match the definitions used in the formulae, the azimuthal angle shown in Fig. 9(a) of Ref. [8] should be denoted (π − φ∗ ). D.J. Miller et al. / Physics Letters B 505 (2001) 149–154 The normality of the Higgs state, nH = (−1)J P, (7) (−1)J which is the product of the spin signature and the parity P, plays an important rôle in classifying these helicity amplitudes. The normality determines the relation between helicity amplitudes under parity transformations. If the interactions which determine the vertex (6) are P invariant, equivalent to CP invariance in this specific case, the reduced vertices are related by ΓλZ λH = nH Γ−λZ −λH . (8) The total cross section for a CP invariant theory is in this formalism then given by, σ= G2F MZ6 (ve2 + ae2 ) 24πs 2 (1 − MZ2 /s)2 × β |Γ00|2 + 2|Γ11|2 + 2|Γ01|2 + 2|Γ10|2 + 2 |Γ12|2 . (9) Correspondingly, the polar angular distributions introduced above can be written, 1 dσ σ d cos θ 3 2 sin θ |Γ00 |2 + 2 |Γ11|2 = 2 4Γ + 1 + cos2 θ |Γ01|2 + |Γ10|2 + |Γ12|2 , (10) and dσ 1 σ d cos θ d cos θ∗ 9 2 = sin θ sin2 θ∗ |Γ00|2 16 Γ 2 + 12 1 + cos2 θ 1 + cos2 θ∗ × |Γ10 |2 + |Γ12 |2 + sin2 θ 1 + cos2 θ∗ |Γ11|2 + 1 + cos2 θ sin2 θ∗ |Γ01 |2 2vf af 2ve ae 2 cos θ cos θ∗ + 2 2 (ve + ae ) (vf2 + af2 ) × |Γ10 |2 − |Γ12 |2 , (11) where Γ 2 corresponds to the square bracket of Eq. (9). 151 The helicity amplitudes of Higgs-strahlung in the Standard Model are given by Γ00 = −EZ /MZ , Γ10 = −1, Γ01 = Γ11 = Γ12 = 0, (12) and the Higgs boson carries even normality: nH = +1. (13) These amplitudes determine uniquely the spinparity quantum numbers of the Higgs boson; this will be demonstrated for a CP invariant theory, for even and odd normality Higgs bosons in Sections 3 and 4, respectively. The analysis will be extended to mixed parity assignments in CP noninvariant theories thereafter. 3. States of even normality J P = 1− , 2+ , 3− , . . . can be excluded by measuring the threshold behaviour of the excitation curve and the angular correlations. 5 The most general current describing the Z ∗ ZH vertex in Fig. 1(a) is given by the expression Jµ = gW MZ Tµαβ1 ...βS ε∗ (Z)α ε∗ (H )β1 ...βJ . cos θW (14) While εα is the usual spin-1 polarization vector, the spin-J polarization tensor εβ1 ...βJ of the state H has the notable properties of being symmetric, traceless and orthogonal to the 4-momentum of the Higgs boson β pHi , and can be constructed from products of suitably chosen polarization vectors. Moreover Tµαβ1 ...βJ , normalized such that Tµα = g⊥µα in the Standard Model, is transverse due to the conservation of the lepton current. These properties strongly constrain the form of the tensor. The most general tensor for spins 2 can be seen in Table 1 (top) together with the resulting helicity amplitudes. (The coefficients ai , bi and ci in Table 1 are independent of the momenta near the threshold.) The leading β dependence of the helicity amplitudes can be predicted from the form of the Z ∗ ZH coupling. Each momentum contracted with the 5 It is well known that the observation of H → γ γ decays or the formation of Higgs bosons, γ γ → H , in photon collisions rules out the spin-1 assignment as a result of the Landau–Yang theorem. 152 D.J. Miller et al. / Physics Letters B 505 (2001) 149–154 Table 1 The most general tensor couplings of the Z ∗ ZH vertex and the corresponding helicity amplitudes for Higgs bosons of spin 2. Here µ... q = pZ + pH , k = pZ − pH and ⊥ indicates orthogonality of a vector or tensor to q µ , t⊥ = t µ... − q µ /sqν t ν... . For spin 3, the helicity J −2 J −1 amplitudes rise ∼ β and ∼ β for even and odd normalities, respectively JP Z ∗ ZH coupling Helicity amplitudes Threshold Even normality nH = + 0+ 1− µα µ a1 g⊥ + a2 k⊥ q α µ µα µβ µα + b3 (q α g⊥ − q β g⊥ ) + b4 (q α g⊥ + q β g⊥ ) 2+ µβ2 c1 (g αβ1 g⊥ 1 Γ10 = −a1 1 2 − M2 ) − 1 b s2β2 + b s Γ00 = β − b1 (s − MZ 3 H 2 2 √ 2 2 − b4 (MZ − MH ) s/(2MZ MH ) µ b1 g αβ k⊥ + b2 q α q β k⊥ µβ Γ00 = (−a1 EZ − 12 a2 s 3/2 β 2 )/MZ µβ + g αβ2 g⊥ 1 ) Γ10 = β(b3 − b4 )s/(2MH ) β Γ01 = β(b3 + b4 )s/(2MZ ) √ Γ11 = β sb1 β Γ00 = µβ + c3 (g⊥ 1 q β2 + g⊥ 2 q β1 )q α µ + c4 (g αβ1 q β2 + g αβ2 q β1 )k⊥ µ + c5 k⊥ q α q β1 q β2 β √ 2/3 2 2 1 7/2 β 4 2 c1 EH (s − MZ − MH ) − 8 c5 s MZ MH 1 2 )) 2/3(−c1 − c2 s 2 β 2 /(4MH 1 2 − M 2 )/√s] − 14 s 2 β 2 [c2 EZ − 2c3 EH + 2c4 (s − MZ H µα + c2 g⊥ q β1 q β2 µβ β Γ10 = √ 2 − M 2 ) + c s 2 β 2 )/(2 2M M ) Γ01 = (2c1 (s − MZ Z H 3 H √ 3/2 2 1 Γ11 = (−c1 EH + 2 c4 s β ) 2/MH Γ12 = −2c1 1 1 1 Odd normality nH = − 0− Γ00 = 0 a1 εµαρσ qρ kσ Γ10 = −iβsa1 1+ µαβρ b1 εµαβρ qρ + b2 ε⊥ kρ µ + b3 εαβρσ qρ kσ k⊥ 2− µαβ1 ρ kρ q β2 µ + c3 εαβ1 ρσ q β2 k⊥ qρ kσ + c4 21 εµαρσ qρ kσ q β1 q β2 + β1 ↔ β2 Γ00 = 0 2 − M 2 ) + 1 s 3/2 β 2 ))/(√sM ) Γ10 = −i(b1 sEH + b2 (EH (MZ H H 2 √ 2 2 1 3/2 2 Γ01 = −i(b1 sEZ + b2 (EZ (MZ − MH ) − 2 s β ))/( sMZ ) 2 − M 2 ) + b s 2 β 2 )/√s Γ11 = −i(b1 s + b2 (MZ 3 H 1 1 1 Γ00 = 0 c1 εµαβ1 ρ q β2 qρ + c2 ε⊥ β 2 − M 2 ) + 1 s 3/2 β 2 ) Γ10 = −iβ c1 sEH + c2 (EH (MZ H 2 √ √ 5/2 2 2 1 + 4 c4 s β 2s/( 3MH ) 2 − M2 ) Γ01 = −iβ c1 sEZ + c2 (EZ (MZ H √ √ 3/2 2 1 − 2 s β ) s/( 2 MZ MH ) √ 2 − M 2 ) + c s 2 β 2 )√s/( 2M ) Γ11 = −iβ(c1 s + c2 (MZ H 3 H Γ12 = 0 β β β D.J. Miller et al. / Physics Letters B 505 (2001) 149–154 Z-boson polarization vector or the H polarization tensor will necessarily give zero or one power of β: βs/2Mj for i = j and λj = 0, for i = j = Z/H pi · εj (λj ) = 0 (15) or λj = ±. Furthermore, any momentum contracted with the lepton current will also give rise to one power of β due to the transversality of the current. Then, one need only count the number of momenta in each term of T µαβ1 ...βJ to understand the threshold behaviour of the corresponding helicity amplitudes. The β dependence of the excitation curve can be derived from the squared β dependence of the helicity amplitude multiplied by a single factor β from the phase space. Spin 0 The spin-0 helicity amplitudes presented in Table 1 (top) have no dependence on β near threshold. Consequently the excitation curve rises linearly in β at threshold, with the single power of β coming from the phase space. This is also the case for the Standard Model, as described in Section 1 and obtained from the spin-0 form factors by setting a1 = 1 and a2 = 0. Spin 1 It is easily seen that all helicity amplitudes vanish near threshold linearly in β, so the excitation curve rises ∼ β 3 , distinct from the Standard Model. Spin 2 The most general spin-2 tensor contains a term with no momentum dependence (∝ c1 ), resulting in helicity amplitudes which do not vanish at threshold if c1 = 0. However, the helicity amplitudes Γ01 and Γ11 contain c1 and are consequently non-zero in this case, leading to non-trivial [1 + cos2 θ ] sin2 θ∗ and sin2 θ [1 + cos2 θ∗ ] correlations which are absent in the Standard Model. Therefore, if the excitation curve rises linearly, not observing these correlations in experiment rules out the spin-2 assignment to the H state. However, if c1 = 0 in the spin-2 case, the excitation curve rises ∼ β 5 near threshold. Spin 3 Above spin-2 the number of independent helicity amplitudes does not increase any more [9]. Consequently, the most general spin-J tensor Tµαβ1 ...βJ is 153 µαβ β a direct product of a tensor T(2) i j isomorphic with the spin-2 tensor and momentum vectors q βk = (pZ + pH )βk as required by the properties of the spin-J wave-function εβ1 ...βJ , µαβ β2 ...βJ T( J ) 1 = µαβi βj β1 T(2) q · · · q βi−1 q βi+1 i<j × · · · q βj−1 q βj+1 · · · q βJ . (16) Contracted with the wave-function, the extra J − 2 momenta give rise to a leading power β J −2 in the helicity amplitudes. The cross section therefore rises near threshold ∼ β 2J −3 , i.e., with a power 3, in contrast to the single power of the Standard Model. 4. It is quite easy to rule out particles of odd normality, J P = 0− , 1+ , 2− , . . . , which may mimic the Standard Model Higgs boson in Higgs-strahlung. Since the helicity amplitude Γ00 must vanish by Eq. (8), the observation of a non-zero sin2 θ sin2 θ∗ correlation in Eq. (11), as predicted by the Standard Model, eliminates all odd normality states. In particular, the assignment of negative parity to the spin-0 state can be ruled out by observing [10] a polar-angle distribution different from the energyindependent 1 + cos2 θ distribution which is characteristic for 0− particle production [8] in contrast to the Standard Model. Nevertheless, in anticipation of the mixed normality scenario we present the helicity amplitudes also for Higgs bosons of odd normality, and spin 2 in Table 1 (bottom). We find a similar picture to the even normality case, where here the excitation curve only presents a linear rise for a particle of spin-1. The generalization to higher spins 3 follows exactly as before, resulting in an excitation curve ∼ β 2J −1 , i.e., with a power 5, at threshold. The above formalism can be generalized easily to rule out a mixed normality state with spin 1. For a Higgs boson of mixed normality one may no longer use Eq. (8) to obtain the simple form of the (differential) cross sections seen in Eqs. (9)–(11). In particular, the polar angle distribution, Eq. (10), is modified to include a linear term proportional to cos θ , indicative of CP violation [6]. The analysis, however, proceeds as in the fixed normality case, since the most 154 D.J. Miller et al. / Physics Letters B 505 (2001) 149–154 general tensor vertex will be the sum of the even and odd normality tensors given in Table 1. A mixed normality Higgs boson of spin 3 may be eliminated by a non-linear rise of the excitation curve at threshold, whereas those of spin-1 and spin-2 may exhibit a linear β dependence, arising from the odd and even tensor contributions, respectively. However, these two exceptions can be ruled out by observing neither [1 + cos2 θ ] sin2 θ∗ nor sin2 θ [1 + cos2 θ∗ ] angular correlations, since a linear excitation curve in both cases requires that both Γ01 and Γ11 be non-zero. lish the J P = 0+ character of the Higgs boson in the Standard Model and related extensions unambiguously. 5. The analyses described above, can be summarized in a few characteristic observations. The key is the threshold behaviour of the excitation curve which is predicted to be linear in β for the J P = 0+ Higgs boson within the Standard Model. The observation of the linear rise, if supplemented by the angular correlations for two exceptional cases, rules out all other J P assignments: σ ∼ s − (MZ + MH )2 References (i) rules out J P = 0− , 1− , 2− , 3± , . . . ; Threshold: (ii) rules out J P = 1+ , 2+ if no [1 + cos2 θ ] sin2 θ∗ sin2 θ [1 + cos2 θ∗ ] correlations. The same rules also eliminate all spin states J 1 for mixed-normality assignments. The rules can be supplemented by other observables which are specific to two interesting cases. By observing a non-zero H γ γ coupling, the spin-1 assignment can be ruled out independently. Moreover, the negative-parity assignment in the spin-0 case would give rise to the energy-independent angular distribution ∼ [1 + cos2 θ ] in contrast to scalar Higgs production, while mixed CP noninvariant 0± assignments can be probed in a linear cos θ dependence of the Higgs-strahlung cross section. As a result, the measurement of the threshold behaviour of the excitation curve for Higgs-strahlung combined with angular correlations can be used to estab- Acknowledgements Thanks go to D.J. Miller for continual encouragement during the project. We are grateful to K. Desch and A. Para for useful experimental advice, and to G. Kramer for discussions and the critical reading of the manuscript. [1] H. Murayama, M.E. Peskin, Ann. Rev. Nucl. Part. Sci. 46 (1996) 533; E. Accomando et al., Phys. 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Khoze, Sov. J. Part. Nucl. 9 (1978) 50; B.W. Lee, C. Quigg, H.B. Thacker, Phys. Rev. D 16 (1977) 1519. [8] V. Barger, K. Cheung, A. Djouadi, B.A. Kniehl, P.M. Zerwas, Phys. Rev. D 49 (1994) 79. [9] G. Kramer, T.F. Walsh, Z. Physik 263 (1973) 361. [10] M. Schumacher, LC-PHSM-2001-003; A. Para, private communication. 26 April 2001 Physics Letters B 505 (2001) 155–160 www.elsevier.nl/locate/npe Lepton flavor violation and radiative neutrino masses Eung Jin Chun Department of Physics, Seoul National University, Seoul 151-747, South Korea Received 20 January 2001; accepted 28 February 2001 Editor: T. Yanagida Abstract Lepton flavor violation in various sectors of the theory can bring important effects on neutrino masses and mixing through wave function renormalization. We examine general conditions for flavor structure of radiative corrections producing the atmospheric and solar neutrino mass splittings from degenerate mass patterns. Also obtained are the mixing angle relations consistent with the experimental results. 2001 Published by Elsevier Science B.V. PACS: 14.60.Pq; 11.30.Hv; 12.60.-i; 12.90.+b Current data coming from the atmospheric [1] and solar neutrino experiments [2] strongly indicate oscillations among three active neutrinos following one of the mass patterns: (i) hierarchical pattern with |m1 |, |m2 | |m3 |, (ii) inversely hierarchical pattern with |m1 | |m2 | |m3 |, (iii) almost degenerate pattern with |m1 | |m2 | |m3 |. In each case, one needs the mass-squared differences m2atm = m232 m231 ∼ 3 × 10−3 eV2 and m2sol = m221 ∼ 10−4 – 10−10 eV2 for the atmospheric and solar neutrino oscillations, respectively [3]. Here we define m2ij ≡ m2i − m2j for the neutrino mass eigenvalues mi . For the mixing angles, we take the standard parameterization of the neutrino mixing matrix U , U = R23 (θ1 )R13 (θ2 )R12 (θ3 ) s3 c2 c2 c3 = −c1 s3 − s1 s2 c3 c1 c3 − s1 s2 s3 s1 s3 − c1 s2 c3 −s1 c3 − c1 s2 s3 s2 s1 c2 c1 c2 E-mail address: [email protected] (E.J. Chun). , (1) where Rij (θk ) is the rotation in the ij plane by the angle 0 θk π and ck = cos θk , sk = sin θk . In our discussion, we neglect CP-violating phases. The atmospheric neutrino data require the νµ − ντ oscillation amplitude Aatm = c22 sin2 2θ1 ≈ 1 implying nearly maximal mixing θ3 ≈ π/4. For the solar neutrino oscillation, the νe − νµ,τ oscillation amplitude Asol = c22 sin2 2θ3 can take either large ∼ 1 or small ∼ 10−3 value (that is, θ1 ∼ π/4 or θ1 1) depending on the solutions to the solar neutrino problem. The mixing element Ue3 is constrained by the reactor experiment on the ν̄e disappearance [4]; |Ue3 | = |s2 | 0.2 for m232 allowed by the atmospheric neutrino data. There is also a limit on the neutrino mass element itself coming from the neutrinoless double beta decay experi ments, |Mee | = | i mi Uei2 | < 0.2 eV [5]. This bound is particularly important for the degenerate pattern (iii) where m2i δm2atm is possible. Note that we restricted ourselves to oscillations among three active neutrinos disregarding the LSND results [6] waiting for a confirmation in the near future. Understanding the origin of neutrino masses and mixing is one of fundamental problems in physics be- 0370-2693/01/$ – see front matter 2001 Published by Elsevier Science B.V. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 3 8 7 - 2 156 E.J. Chun / Physics Letters B 505 (2001) 155–160 yond the standard model. It concerns with the leptonic flavor structure of the theory. As we know, small Majorana neutrino mass textures (at tree level) would be attributed to the effective higher dimensional operator Lα Lβ H2 H2 where Lα denotes the lepton doublet with the flavor index α = e, µ, τ and H2 is the Higgs field coupling to up-type quarks. This operator breaks the total lepton number by 2 units (L = 2). There can be other sectors breaking individual lepton number (with L = 0) which give important contributions to the neutrino mass matrix through radiative corrections. The best-known example is the renormalization group effect of the charged lepton Yukawa couplings [7], which has been studied extensively in recent years [8]. Such an effect provides potentially important origin of tiny neutrino mass splittings, of course, depending on specific tree level neutrino mass textures. In this regard, it is an interesting question whether the desired mass splittings for the degenerate mass patterns (ii) and (iii) can arise from radiative corrections bringing the effect of lepton flavor violation in various sectors of the theory. The purpose of this work is to investigate general conditions for radiative corrections to produce the observed neutrino mass splittings without resorting to any specific models for flavor structure beyond the Yukawa sector. Assuming that various radiative correction contributions are hierarchical, we will identify appropriate flavor structure of the dominant or subdominant contribution which are able to generate the atmospheric or solar neutrino mass splitting. For these loop corrections, we also derive mixing angle relations consistent with the experimental data. Our results would be useful for constructing models of degenerate neutrinos. We also remark that there is another interesting possibility that a mixing angle can be magnified to a large value by a loop correction when it is comparable to a given small mass splitting [9]. The general form of the loop-corrected neutrino mass matrix due to wave function renormalization is given by 1 Mαβ = mαβ + (mαγ Iγ β + Iαγ mγ β ), (2) 2 where mαβ is the tree level neutrino mass matrix and Iαβ is the loop contribution. Note that the above formula is written in the flavor basis where charged lepton masses are diagonal. Often, it is convenient to re-express Eq. (2) in the tree level mass basis of neutrinos; 1 Mij = mi δij + (mi + mj )Iij , (3) 2 where mi is the tree level mass eigenvalue and the loop correction factor Iij is related to Iαβ by the equation, Iij = αβ Iαβ Uαi Uβj , where U is the tree level diagonalization matrix parametrized as in Eq. (1). In this paper, we have nothing to mention about the origin of the flavor structure of mαβ . Given mαβ yielding the degenerate mass pattern of the type (ii) or (iii), we will examine the flavor structure of the loop factor Iαβ which can produce the atmospheric and/or solar neutrino masses and mixing. Before coming to our main point, it is instructive to see where the loop correction Iαβ can come from. To illustrate how the flavor structure of the loop correction Iαβ can arise, we consider one of the most popular model beyond the standard model, namely, the supersymmetric standard model. Like the standard model, it has an inevitable flavor violation in the Yukawa sector with the superpotential, W hα H1 Lα Eαc , (4) where H1 , Lα and Eαc denote the Higgs, lepton doublet and singlet superfields. At the leading log approximation, the Yukawa terms in Eq. (4) give rise to [7,8] h2α MX Iαα ≈ − 2 ln (5) , 8π MZ where MX denote a fundamental scale generating the above-mentioned effective operator and the Z boson mass MZ represents the weak scale. The soft supersymmetry breaking terms included in this model are also potentially important sources of flavor violation. Those terms include sfermion masses and trilinear A-terms which are generically nonuniversal and flavor dependent; Vsoft m2αβ ∗ Lα L†β + Aαβ H1 Lα Eβc + h.c., (6) where we use the same notation for the scalar components of superfields. The effect of slepton masses has been first discussed in Ref. [12]. The off-diagonality and non-degeneracy in diagonal masses of m̃2αβ ∗ give rise to Iαβ = g2 l 2 δ f m̃αα ; m̃2ββ , 8π 2 αβ (7) E.J. Chun / Physics Letters B 505 (2001) 155–160 l = m̃2 /(m̃ m̃ ) for α = β (assuming where δαβ αα ββ αβ l l = 1 and f is an appropriate loop δαβ 1), δαα function of order one. In the similar way, the effect of the A-terms comes from one-loop diagrams with wino/zino and slepton exchange generating ∗ 2 g 2 Aαγ Aβγ H1 , (8) 8π 2 m̃4 where m̃ is a typical mass of the sparticles running inside the loop. In general, the A-terms are not proportional to the (charged lepton) Yukawa couplings, Aαβ ∝ / hαβ = δαβ hα , This can lead to important new contributions to lepton flavor changing loop corrections as above. However, we expect in generic models that Aαβ m̃hτ and thus the sizes of their loop effects are at most comparable to those of the tau Yukawa coupling. Lepton flavor violation can also appear in the superpotential through R-parity and lepton number violating trilinear terms; Iαβ ≈ W λαβγ Lα Lβ Eγc + λαβγ Lα Qβ Dγc , (9) where Q, D c are doublet and down-type singlet quark superfields. The R-parity violating couplings can participate in the renormalization group equation like the charge lepton Yukawa couplings, and thus we get for the couplings λ, λαγ δ λ∗βγ δ MX ln . Iαβ ≈ − (10) 8π 2 MZ For certain combinations of R-parity violating couplings [10], the current experimental bounds are weak enough to give a sizable loop correction Iαβ . In particular, if one takes only one dominant coupling λαγ δ which can be as large as order one, one can have a very large loop correction Iαα . With the generic R-parity violation (9), of course, one may have a finite loop correction to neutrino masses like δmαβ ∝ λαγ δ λβδγ heγ heδ [11] which is not a topic of the present investigation. As we have seen above, there could be rich sources for sizable loop corrections Iαβ in a general class of models. In the below, we discuss the effect of general flavor-dependent loop contributions Iαβ for various degenerate mass patterns at tree level. Let us start with analyzing the conditions to produce the desired mass splittings for the atmospheric as well as solar neutrino oscillations in the case of the fully degenerate 157 mass patterns; |m1 | = |m2 | = |m3 |. For this, we look first for the possible loop corrections giving rise to the atmospheric mass splitting m232/2m20 at the leading order correction. In our analysis, it is assumed that loop corrections Iαβ take hierarchical values and thus the leading order splitting is dominated by one specific Iαβ . For the fully degenerate pattern, there are the following possibilities depending on the CPconserving phases of the tree level mass eigenvalues; I. (m1 , m2 , m3 ) = m0 (−1, −1, 1) or m0 (±1, ∓1, 1) which we call the 1–2 or 1–3 (2–3) degeneracy, respectively. The loop correction can induce nonvanishing off-diagonal components Mij as far as mi + mj = 0 (3). Then, as discussed in Ref. [12], one has a freedom to define the tree level mixing angles in the matrix U in such a way that Iij = Iαβ Uαi Uβj = 0 due to the exact degeneracy mi = mj . It is a simple manner to show this explicitly. In the i–j plane with tree-level degeneracy, the radiatively corrected mass matrix is diagonalized by the rotation Rij (φ) where φ is given by tan 2φ = 2Iij . Ijj − Iii (11) On the other hand, we are free to choose our tree level mixing matrix: U → U = U Rij (φ) where Rij (φ) is given by Rij,kl (φ) = cφ (δik δil + δj k δj l ) + sφ (δik δj l − U written δj k δil ). Then the loop factor Iij = Iαβ Uαi βj in terms of the new mixing matrix U becomes Iij = [2Iij cos 2φ + (Iii − Ijj ) sin 2φ]/2 which vanishes due to the relation (11). Note that this conclusion holds for arbitrary sum of Iαβ . With this properly defined mixing matrix U satisfying the mixing angle relation Iij = 0, the nonvanishing mass-squared differences generated by loop correction can be written as m221 = 2m20 (I22 − I11 ) = 2m20 Iαβ (Uα2 Uβ2 − Uα1 Uβ1 ), m232 = 2m20 (I33 − I22 ) = 2m20 Iαβ (Uα3 Uβ3 − Uα2 Uβ2 ). (12) When a specific component Iαβ gives the dominant contribution in Eq. (12), we need to have (Uα2 Uβ2 − Uα1 Uβ1 )/(Uα3 Uβ3 − Uα2 Uβ2 ) m2sol/m2atm , that 158 E.J. Chun / Physics Letters B 505 (2001) 155–160 is, Uα2 Uβ2 ≈ Uα1 Uβ1 is required to get |m232 | |m221 |. Based on these properties, we are now ready to consider the effect of a loop correction Iαβ . (a) Iαα dominance. For the i–j degeneracy, the relation Iij ∝ Uαi Uαj = 0 implies Uαi = 0 or Uαj = 0. As an immediate consequence, we find that the 1-2 2 degeneracy cannot work for any α since it gives Uα1 2 2 Uα2 0 and thus Uα3 1 contradicting with the 2 1 and U 2 U 2 1/2. empirical results, Ue3 µ3 τ3 In fact, the only possible mixing angle relation is Ue3 = s2 = 0, which can be realized only with the Iee dominance combined with the 1–3 or 2–3 degeneracy. In this case, we get from Eq. (12) m221 m232 cos 2θ3 s32 , (13) which cannot be made small enough for realistic values of the mixing angles satisfying θ1 π/4, θ2 1 and θ3 π/4 or θ3 1. On the other hand, for (αβ) = (µτ ), we get m232 − r m232 /m20 m221 /m20 2 1 + s22 cos 2θ3 2 3c2 + 2s2 sin 2θ3 cot 2θ1 , (14) which becomes vanishingly small for s2 cos 2θ1 1 and cos 2θ3 1. Now we have to check if the mixing angle relation fixed by the condition Iij = 0 for certain combination of (ij ) can be consistent with our consideration. As shown in Ref. [12], the 1–3 or 2–3 degeneracy gives again the desired relation, respectively, s2 = − cot 2θ1 tan θ3 or s2 = cot 2θ1 cot θ3 . (15) s2 Iee /Iµτ Iµτ sin 2θ1 0 Iµµ /Iµτ Iµτ sin 2θ1 Iµτ r 2 sin 2θ1 /2 −rs3 /2c3 Iτ τ /Iµτ Iµτ sin 2θ1 Iµτ r 2 sin 2θ1 /2 rs3 /2c3 Ieµ /Iµτ Iµτ sin 2θ1 Iµτ rc1 c2 sin 2θ3 r/2c1 c2 Ieτ /Iµτ Iµτ sin 2θ1 −Iµτ rs1 c2 sin 2θ3 r/2s1 c2 −Iee −Iee r 2 /2 rc1 s1 s3 /c3 Iτ τ /Iee −Iee −Iee r 2 /2 −rc1 s1 s3 /c3 Ieµ /Iee −Iee Iee rc1 c2 sin 2θ3 −rs1 /c2 Ieτ /Iee −Iee −Iee rs1 c2 sin 2θ3 −rc1 /c2 Iµτ /Iee −Iee Iµµ /Iee which requires the maximal mixing of the solar neutrinos, cos 2θ3 1. (b) Iαβ dominance (α = β). As discussed below Eq. (12), it is useful to notice that we need 0 = Uα2 Uβ2 Uα1 Uβ1 −Uα3 Uβ3 /2 (thelast relation comes from the orthogonality condition i Uαi Uβi = δαβ ) to yield m221 /m232 ≈ 2(I22 − I11 )/3I22 . Then, we can easily rule out the case (αβ) = (eµ) or (eτ ) from the simple observation that m221 2 c1 + sin 2θ cos 2θ 3 3 s1 s2 m232 3 2 s1 or sin 2θ3 , cos 2θ3 − 3 c1 s2 m221 Table 1 Possibilities of radiative generation of m2atm and m2sol in the case of the degeneracy, m1 = −m2 = m3 . The upper and lower box correspond to the cases of the Iµτ and Iee dominance, respectively. The last column shows the correction to Ue3 = s2 from exact bimaximal mixing given subdominant Iαβ Iµτ or Iee 0 0 0 which relates the smallness of the angle θ2 with the large atmospheric neutrino mixing. In sum, the small mass splitting for the atmospheric neutrinos can be obtained only with the dominant loop correction of Iee or Iµτ in the case of the 1–3 or 2–3 degeneracy. Furthermore, this picture can be consistent only with bimaximal mixing θ1 , θ3 π/4 and s2 1 fixed by the mixing angle relation I13 or I23 = 0. Note that all of these are fairly consistent with the neutrinoless double beta decay bound as we have |Mee | = |m0 (c22 cos 2θ3 ± s22 )| |m0 |. Let us turn to the solar neutrino mass splitting. As can be seen from Eqs. (13) and (14), the right values for m221 may arise if the mixing anlges satisfy cos 2θ3 ∼ cos2 2θ1 ∼ m2sol/m2atm , where the second relation is applied only to the Iµτ dominance. However, it appears unnatural to arrange such small values for the tree level mixing angles. It would be more plausible to imagine the situation of exact bimaximal mixing (cos 2θ1 = cos 2θ3 = 0) imposed by certain symmetry in tree level mass matrix. Then, the solar neutrino mass splitting could be generated by a smaller loop correction Iαβ other than Iee or Iµτ . Including now in Eq. (12) this subdominant contribution, one can find the deviations from the leading results, m221 = 0 and s2 = 0. The result of our calculation is summarized in Table 1 in the case of the 1–3 degeneracy. E.J. Chun / Physics Letters B 505 (2001) 155–160 We have shown the explicit angle dependences to notify the sign of mass-squared difference which might be distinguishable by the solar neutrino MSW effect. Similar result can be obtained for the 2–3 degeneracy. A few remarks are in order. The desired size of the loop correction for the atmospheric neutrino masssquared difference Iee,µτ ≈ m2atm /m20 . Thus, the degenerate mass m0 ∼ 1 eV of cosmological interests needs Iee,µτ ∼ 10−3 which is a reasonable value for radiative corrections. As can be seen in Table 1, the ratio m2sol/m2atm is roughly given by r or r 2 depending on the flavor structure of the radiative corrections whereas s2 ∼ r for any cases. If the large angle MSW solution to the solar neutrino problem is realized, one needs r or r 2 of the order 10−2 , that is, a loop correction Iαβ should be smaller than Iee,µτ by factor of 10−2 or 10−1 . For the latter case, we get s2 ∼ 0.1 which is within the reach of future experiments. We note that a supersymmetric model realizing the case with the Iee dominance and r = Iτ τ /Iee has been worked out in Ref. [13]. Another possibility is to have the atmospheric neutrino mass splitting given at tree level and the smaller splitting for the solar neutrino mass is driven by loop corrections. This includes almost full degeneracy |m1 | = |m2 | |m3 | and inverse hierarchy |m1 | = |m2 | |m3 |, both of which can be parametrized as II. (m1 , m2 , m3 ) = m0 (1, ±1, z), where z = ±1 + δa with |δa | = m2atm /2m20 for the almost full degeneracy, or |z| 1 with m20 = m2atm for the inverse hierarchy. We consider the two cases, m1 = ±m2 , separately. (a) m1 = m2 = m0 . As discussed before, the mixing angles satisfy I12 = 0 and the leading contribution to m221 is given by 2m20 (I22 − I11 ). These two quantities are presented in Table 2. One can realize that the Iee dominance does not work at all. For (αβ) = (µµ), (τ τ ), (µτ ), only the small solar mixing angle is consistent since the mixing angle relation s2 ∝ sin 2θ3 has to be put. On the contrary, for (αβ) = (eµ), (eτ ), the large solar mixing can only be allowed since s2 ∝ cos 2θ3 . Imposing these mixing angle relations, one can see that I22 − I11 in Table 2 does not vanish for any (α, β), and thus m2sol ≈ m20 Iαβ . In the case of |z| 1, we thus need Iαβ ≈ m2sol/m2atm 10−2 where the approximate equality is for the large angle 159 Table 2 The mixing angle relation and the loop contribution to m221 for each dominant Iαβ in the case of the degeneracy, m1 = m2 = m3 Iαβ I12 = 0 (I22 − I11 )/Iαβ Iee c22 sin 2θ3 = 0 s sin 2θ Iµµ 2 22 2 = ± sin 2θ3 1 c1 +s1 s2 s2 sin 2θ3 Iτ τ 2 2 2 = ± sin 2θ 1 s1 +c1 s2 −c22 cos 2θ3 Ieµ s2 = cot θ1 cot 2θ3 c12 cos 2θ3 − s2 sin 2θ1 sin 2θ3 s12 cos 2θ3 + s2 sin 2θ1 sin 2θ3 c2 (+c1 sin 2θ3 + s1 s2 cos 2θ3 ) Ieτ s2 = − tan θ1 cot 2θ3 c2 (−s1 sin 2θ3 + c1 s2 cos 2θ3 ) 2s2 Iµτ = tan 2θ1 tan 2θ3 − 21 sin 2θ1 cos 2θ3 − s2 cos 2θ1 sin 2θ3 1+s22 MSW solution and may be a little large value for a radiative correction. Here it is worth noting that, e.g., for the Iτ τ dominance, we have sin2 2θ3 1 and m221 2m20 Iτ τ s12 (c32 − s32 ). (16) For the small solar neutrino mixing to work, we need s32 1 for m221 > 0 (or m221 < 0 for c32 1), which requires from Eq. (16) that Iτ τ > 0. Therefore, Iτ τ given in Eq. (5) for the supersymmetric standard model does not fulfill this condition whereas the usual standard model with Iτ τ ≈ h2τ ln(MX /MZ )/16π 2 can work. (b) m1 = −m2 = m0 . In this case, no mixing angle relation is imposed. Including the effect of the offdiagonal elements M13 and M23 generated from loop correction, we find 1 (z − 1)2 2 I m221 = 2m20 I22 − I11 + 2 (z + 1) 23 1 (z + 1)2 2 I13 , + (17) 2 (z − 1) where I22 − I11 is given in Table 2 for each Iαβ . Depending on the mixing angles fixed at tree level, the leading contribution to m2sol may come from I22 − I11 or the next terms in Eq. (17). That is, we 2 /δ 2 for need to have m2sol/m2atm ∼ Iαβ /δa or Iαβ a 2 for z = ±1 + δa , and m2sol/m2atm ∼ Iαβ or Iαβ |z| 1. Therefore, the values of m2sol for various solutions to solar neutrino problem can be obtained with appropriate values of Iαβ and δa . From Table 2, one can see that the leading term I22 − I11 ∝ Iαβ vanishes for the exact bimaximal mixing with s2 = 0 and cos 2θ3 = 0 in the case of (αβ) = (ee), (µµ), (τ τ ) 160 E.J. Chun / Physics Letters B 505 (2001) 155–160 and (µτ ). Furthermore, for (αβ) = (ee) and (µτ ), I13 and I23 also vanish and thus no splitting can arise at one-loop level. We would like to stress that either the large or small mixing solution to the solar neutrino problem can be realized as we have no mixing angle relation imposed. For the small mixing solution, the solar neutrino mass-squared difference has further suppression by small mixing angles as I22 − I11 ∼ Iαβ sin 2θ3 or Iαβ s2 in the case of (αβ) = (eµ) or (eτ ). In conclusion, we have discussed the general conditions for generating small neutrino mass splittings from the effect of one-loop wave function renormalization of degenerate neutrino masses at tree level. Without assuming any specific model on the structure of lepton flavor violation other than the tree level neutrino mass sector, we identified the flavor dependences of the loop factors which can give rise to the neutrino mass-squared difference and mixings required by the atmospheric and solar neutrino data. For the fully degenerate pattern |m1 | = |m2 | = |m3 |, the atmospheric neutrino mass splitting can be obtained when the dominant loop correction comes from Iee or Iµτ in the cases of m1 = m3 and m2 = m3 . In each case, the smaller loop corrections are examined to generate the solar neutrino mass splitting. For all the cases, it turns out that the so-called bimaximal mixing can only be realized. For the partially degenerate case |m1 | = |m2 | = |m3 |, we identified Iαβ with which the desired mass splitting and the (large or small) mixing angle for the solar neutrino oscillations can be obtained. Our results may be useful for explicit model building along this line. 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Balaji et al., Phys. Rev. Lett. 84 (2000) 5034; K.R.S. Balaji et al., Phys. Lett. B 481 (2000) 33, hepph/0011263. G. Bhattacharyya, hep-ph/9709395; B.C. Allanach, A. Dedes, H.K. Dreiner, Phys. Rev. D 60 (1999) 075014. For recent works, see, e.g., E.J. Chun, S.K. Kang, Phys. Rev. D 61 (2000) 075012; S. Davidson, M. Losada, hep-ph/0010325. E.J. Chun, S. Pokorski, Phys. Rev. D 62 (2000) 053001. P.H. Chankowski, A. Ioannisian, S. Pokorski, J.W.F. Valle, hep-ph/0011150. 26 April 2001 Physics Letters B 505 (2001) 161–168 www.elsevier.nl/locate/npe Prospect for searches for gluinos and squarks at the Tevatron Tripler V. Krutelyov a , R. Arnowitt b , B. Dutta b , T. Kamon a , P. McIntyre a , Y. Santoso b a Department of Physics, Texas A&M University, College Station, TX 77843-4242, USA b Center for Theoretical Physics, Department of Physics, Texas A&M University, College Station, TX 77843-4242, USA Received 27 November 2000; received in revised form 20 February 2001; accepted 26 February 2001 Editor: M. Cvetič Abstract √ We examine the discovery potential for SUSY new physics at a pp̄ collider upgrade of Tevatron with s = 5.4 TeV and luminosity L 4 × 1032 cm−2 s−1 (the Tripler). We consider the reach for gluinos (g̃) and squarks (q̃) using the experimental / T ) of jets + E / T and 1 + jets + E / T (where is an electron or muon) within signatures with large missing transverse energy (E / T channel and the framework of minimal supergravity. The Tripler’s strongest reach for the gluino is 1060 GeV for the jets + E / T channel for 30 fb−1 of integrated luminosity (approximately two years running time). This 1140 GeV for the 1 + jets + E / T channel for 15 (30) fb−1 of integrated is to be compared with the Tevatron where the reach is 440 (460) GeV in the jets + E luminosity. 2001 Published by Elsevier Science B.V. 1. Introduction The Tripler [1] is a proposed energy upgrade of the Tevatron, in which its ring of 4 Tesla NbTi superconducting magnets would be replaced by a ring of 12 Tesla Nb3 Sn magnets. Thanks to improvements in Nb3 Sn technology and in dipole design methodology, it is now possible to extend dipole fields up to and beyond 12 Tesla. Magnets are being developed using several different methodologies at Brookhaven National Lab [2], Fermilab [3], Lawrence Berkeley National Lab [4], and Texas A&M University [5], and a prototype accelerator magnet of 6.5 Tesla has been successfully tested at the Lawrence Berkely National Lab. E-mail address: [email protected] (B. Dutta). The rationale for the Tripler is that the upgrade opens an energy window in which the particles of the Higgs sector and new physics are expected to be produced in a mass range of 1 TeV. The Tripler furthermore accesses this energy window primarily through quark–antiquark annihilation and gluon fusion, whereas the Large Hadron Collider (LHC) will access a similar window primarily through gluon fusion and gluon–quark/quark–quark interaction. The proposed Next Linear Collider (NLC) with its centerof-mass (c.m.) energy of 500 GeV [6] will access a limited energy window through e+ e− annihilation, but with more precise measurements of the parameters of theories. Complementarity has often proved vital in understanding new phenomena at the high-energy frontier. The Tripler would use the existing tunnel, existing p source and injector accelerators, and existing detectors CDF and DØ with minimal changes. With a luminosity of about 4 × 1032 cm−2 s−1 and live 0370-2693/01/$ – see front matter 2001 Published by Elsevier Science B.V. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 3 3 6 - 7 162 V. Krutelyov et al. / Physics Letters B 505 (2001) 161–168 time of 2 × 107 s/year, the Tripler would deliver about 8 fb−1 s/year for each detector [1]. The present Letter is concerned with evaluating the reach of the Tripler for new physics. Ref. [7] analyzed the signals for the Standard Model (SM) Higgs boson at the Tripler and compared them with those at the LHC [8,9]. It is remarkable that the Tripler can discover a Higgs boson up to 680 (600) GeV mass with 40 (10) fb−1 of integrated luminosity, which is close to the triviality upper bound of 710 GeV [10]. A light Higgs boson ( 130 GeV) would be accessible via WWH coupling with 7.5 fb−1 at the Tripler, while its production at the LHC proceeds predominantly via processes involving Yukawa couplings. Another important benchmark for the physics is the potential to discover the particles of supersymmetry (SUSY) [11]. There have been extensive analyses of the discovery potential for SUSY particles, based on minimal supergravity (mSUGRA) model [12] or minimal supersymmetric Standard Model (MSSM), at the Tevatron [13–23] and at the LHC [8,24]. In the Tripler case, SUSY studies on pp → 3 + E / T + X (dominantly from χ̃1± χ̃20 production) and pp → ± ± + jets + E / T (dominantly from g̃ g̃/g̃ q̃ production) have been carried out in Ref. [7]. In this Letter we present a comparative study of the discovery reaches for gluinos and squarks with large missing transverse energy (/ E T ). We examine the signals from jets + E / T and 1 + jets + E / T ( is an electron or muon) at the Tripler and compare these signals at the Tevatron. 2. MSUGRA model To test the reach of the Tripler for gluinos (g̃) and squarks (q̃), we consider SUSY models for which grand unification of the gauge coupling constants occur at a GUT scale MG ≡ 2 × 1016 GeV. These models are consistent with the LEP measurements of αi (i = 1, 2, 3) at the electroweak scale MZ when the renormalization group equations (RGE) are used to run the αi up to MG . We restrict our analysis here to the simplest such model where R-parity is conserved and there are universal soft breaking masses at MG (i.e., mSUGRA). Such models depend on four parameters and one sign: m0 , the universal soft breaking scalar mass at MG ; m1/2 , the universal gaugino mass at MG ; A0 , the universal cubic soft breaking mass at MG ; tan β ≡ H2 /H1 where H1,2 gives rise to (d, u) quark masses, and the sign of µ, the Higgs mixing parameter which appears in the µH1 H2 contribution in the superpotential. (Note that the gluino mass scales approximately with m1/2 , i.e., mg̃ ≈ 2.4m1/2 .) No assumptions are made on the nature of the GUT group which breaks to the SM group at MG . The model used here is the same as that used in LHC analyses by ATLAS and CMS [8,24]. Over almost all of the parameter space, the lightest neutralino (χ̃10 ) is the lightest supersymmetric particle (LSP), and is a natural candidate for cold dark matter [25]. In the present study, we fix A0 = 0 and the sign of µ to be positive (µ > 0) for simplicity, and choose tan β = 3, 10, and 30. Here the ISAJET sign convention for µ [26] is used. The top quark mass is set to 175 GeV. We restrict the parameter space so that the lighter third generation squarks (b̃1 and t˜1 ) remain heavier than the lightest chargino χ̃1± and the next to lightest neutralino χ̃20 , and also develop cuts sensitive to gluinos and squarks. These would decay to the SM particles plus the χ̃10 . For example, g̃ → q q̄ χ̃1± , q̃L → q χ̃1± , followed by χ̃1± → q q̄ χ̃10 or χ̃1± → ± ν χ̃10 . The χ̃10 then would pass through the detector without interaction. Thus, the experimental signatures of pair-produced squarks and gluinos are multi jets and appreciable missing energy associated with either no lepton or some leptons. It should be noted that the event selection with a large jet multiplicity presented later in this Letter is not efficient to detect the production of q̃R q̃¯ R , because each right chiral scalar quark dominantly decays to a quark and a χ̃10 . 3. Monte Carlo simulation We use ISAJET [26] for SUSY and t t¯ events and [27] for all other SM processes (W/Z + jets, dibosons, QCD events) along with TAUOLA [28] and CTEQ3L parton distribution functions [29]. As for SUSY events, we generate all processes for the analyses described in Section 4. For detector simulation we use SHW [30], a simple detector simulation package developed for Run II SUSY/Higgs workshop [23]. The particle identification and misidentification effiPYTHIA V. Krutelyov et al. / Physics Letters B 505 (2001) 161–168 ciencies are parameterized to an expectation for Run II based on the CDF/DØ measurements at Run I in 1992– 1996. The SHW code provides the following objects: electron (e) with isolation, muon (µ) without isolation, hadronically decaying tau lepton (τh ), photon (γ ), jets, / T is the energy imbaland calorimeter-based E /T . E ance in the directions transverse to the beam direction using the calorimeter energies in an event [31]. We modify the SHW code to provide a muon with the isolation and the E / T correction due to muon(s). The pseudorapidity (η) coverage is |η| < 2.0 for e and γ [23]. For µ, τh and tracks |η| is < 1.5, and for jets it is < 4.0 [23]. Jets are formed with a cone size of (R ≡ (η2 + (φ 2 = 0.4. A non-instrumented region of the detector is also simulated as a geometrical acceptance for each object. (For example, SHW will reject a particular object at a rate of 10%, if the fiducial volume in a given pseudorapidity coverage is 90%.) The isolation for an electron is defined to be the calorimeter energy (excluding the electron energy) within (R = 0.4 which is less than 2 GeV. The isolation for a muon is defined as a scalar sum of track momenta (excluding the muon momentum) within (R = 0.4 to be less than 2 GeV. It should be noted that hadronically decaying taus (τh ) are treated as a jet. Throughout the Letter, the leptons and jets are j selected with pT > 15 GeV and ET > 15 GeV, and the reach √ in mass is obtained as 5σ in a significance (≡ NS / NB ) for 15 fb−1 and 30 fb−1 at the Tevatron and 30 fb−1 at the Tripler. Here NS (NB ) is the number of signal (background) events after a set of selection cuts. 4. Results We consider first the jets +E / T channel and proceed to optimize the cuts for SUSY events where mq̃ mg̃ . Our optimized selection is (a) Nj 6; (b) veto on isolated leptons (e or µ); (c) E / T > 200 GeV; (d) minimum azimuthal angle between the E / T direction and j any jet (φ min > 30◦ ; (e) MS ≡ E / T + jet ET > 1000 GeV. Fig. 1 shows the distributions in MS for t t¯, W/Z + jets, dibosons, and QCD events. The SUSY events are also superimposed in the same figure. We require in our analysis NS 30 events. Using these cuts the total SM background is 7.0 fb (Table 1). 163 Fig. 1. Distributions of MS for various SM processes and SUSY / T > 200 GeV in the jets + E /T events with MS > 600 GeV and E analysis at Tripler. Dotted, dashed and solid lines are cumulative contributions from t t¯, W/Z/dibosons, and QCD processes, respectively. Horizontally and vertically hatched histograms are for SUSY events (tan β = 3) with mq̃ mg̃ = 800 GeV and 1000 GeV, respectively. The final cut on MS is set at 1000 GeV. Significances for SUSY events (mq̃ mg̃ ) are plotted as function of mg̃ in Fig. 2. We see that the reach in the gluino mass is ∼ 1000 GeV (corresponding to m1/2 420 GeV). Also there is no significant dependence between the tan β values of 3, 10 and 30, i.e., the Tripler is sensitive to high tan β. In Fig. 3 we plot the significance as a function of m0 at m1/2 = 410 GeV. We see that at the highest gluino mass, the Tripler is sensitive to relatively large m0 , i.e., m0 450 GeV for this channel. In Fig. 3, there are three distinct regions: (i) m0 150 GeV, (ii) 150 m0 300 GeV, (iii) m0 300 GeV. For m0 300 GeV, all sleptons are heavier than χ̃1± and χ̃20 . The jet multiplicity in the SUSY events is determined by the W , Z, and light Higgs boson (h) decays in χ̃1± → W χ̃10 and χ̃20 → hχ̃10 /Z χ̃10 , whose branching ratios are independent of m0 . Thus there is no change in the event topology, but the cross section for q̃ q̃¯ and g̃ q̃ decrease as m0 (i.e., squark masses) increases. For 150 m0 300 GeV, the m0 dependence becomes somewhat gradual. This is be- 164 V. Krutelyov et al. / Physics Letters B 505 (2001) 161–168 Table 1 The background cross sections in fb (after cut) of jets + E / T and 1 + jets + E / T channels. Cuts are specified in the text Tripler Tevatron jets + E /T 1 + jets + E /T t t¯ 2.5 0.06 25 14 W + jets 1.1 0.21 23 48 Z + jets 1.2 0.05 15 6 Process jets + E /T 1 + jets + E /T Diboson 0.0 0.0 1 2 QCD 2.2 0.0 9 0.0 Total 7.0 0.32 73 Fig. 2. Significance as a function of mg̃ (mq̃ mg̃ ) for tan β = 3 (filled circles), 10 (down triangles), and 30 (open circles) in / T channel at the Tripler. jets + E cause, as τ̃1 (and ẽR ) gets lighter than χ̃1± , the decay mode χ1± → τ̃1 ν starts competing with χ̃1± → W ± χ̃10 . Thus the jet multiplicity in the SUSY events involving χ̃1± decay mode is reduced to affect its event acceptance (with Nj 6). In contrast, χ̃20 → τ τ̃1 decay (competing with χ̃20 → hχ̃10 /Z χ̃10 especially for tan β = 10 and 30) does not alter the jet multiplicity. We notice the significance has a tan β dependence 70 Fig. 3. Significance as a function of m0 at m1/2 = 410 GeV (mg̃ 980 GeV) for tan β = 3 (filled circles), 10 (down triangles), and 30 / T channel at the Tripler. m0 150 GeV is (open circles) in jets + E theoretically forbidden for tan β = 30. at a fixed m0 . This can be explained by (a) a change of the third-generation squark masses (especially the t˜1 ), resulting in the change of squark production cross sections and (b) an enhancement of branching ratio for χ̃1± → τ̃1 ν in large tan β region. The decay mode χ̃1± → W ± χ̃10 is dominant in low tan β region, resulting in the change of jet multiplicity. A characteristic V. Krutelyov et al. / Physics Letters B 505 (2001) 161–168 Fig. 4. Significance for 15 fb−1 of luminosity as a function of mg̃ (mq̃ mg̃ ) for tan β = 3 (filled circles), 10 (down triangles), and 30 / T channel at the Tevatron. (open circles) in jets + E change for m0 150 GeV at tan β = 3 and 10, where ẽL and ν̃ become lighter than χ̃1± and χ̃20 , is explained by a monotonic decrease of rates of χ̃1± → W ± χ̃10 and χ̃20 → hχ̃10 decays as m0 decreases. Thus the significance of the SUSY events with Nj 6 is degraded. Fig. 4 gives a comparison of what might be expected at Run II at the Tevatron with 15 fb−1 . Our selection cuts in this case were: (a) Nj 4; (b) veto on isolated leptons; (c) E / T > 100 GeV; (d) (φ min > 30◦ ; j1 j (e) MS2 (≡ E / T +ET +ET2 ) > 350 GeV [14]. The total SM background is 73 fb (Table 1). One sees that the maximum reach for the jets + E /T channel in Fig. 4 is 410 GeV in gluino mass, which rises to 440 (460) GeV for 15 (30) fb−1 of data when mq̃ < mg̃ . (These results are consistent with previous Tevatron studies [13,16].) The latest bound on the χ̃1± mass of 103 GeV from LEPII [32] requires mg̃ 420 GeV, since we have mχ̃ ± mg̃ /3 from gaugino 1 unification. In addition, one may show that Run II will also be able to sample limited range of m0 , i.e., for m0 200 GeV for mg̃ = 420 GeV. Thus there is a 165 Fig. 5. Same as in Fig. 3 (m1/2 = 410 GeV), but in 1 + jets + E /T channel. significant improvement in going from the Tevatron to the Tripler. We consider next the 1 + jets + E / T channel. This channel gives the largest reach for the LHC, and we will see that there are regions of SUSY parameter space where the discovery reach for gluinos is also improved. Here we select events with (a) Nj 4; / T > 200 GeV; (d) (φ min > 30◦ ; (b) N = 1; (c) E (e) MT (≡ 2/ E T pT [1 − cos (φ( ,E / T )]) > 160 GeV; (f) MS > 600 GeV. The MT cut is applied to remove W events. The SM background is 0.32 fb (Table 1). In Fig. 5, we compare the m0 dependence of the significance for tan β = 3, 10 and 30 at m1/2 = 410 GeV (mg̃ 980 GeV). We see here the m0 reach is not as large as in the jets + E / T channel. In this parameter space, the ˜L and the ν̃ are lighter than the χ̃1± and the χ̃20 when m0 150 GeV. Thus, the branching ratios of χ̃1± → ν̃( ˜L ν) and χ̃20 → ˜ increase as m0 decreases, and the significance in the 1 + jets + E / T channel is improved dramatically and is significantly higher than for the jets + E / T channel (see Fig. 3). An interesting feature occurs at m0 140 GeV which distinguishes between the tan β = 3 and 10 scenarios. The SUSY particle masses for these 166 V. Krutelyov et al. / Physics Letters B 505 (2001) 161–168 Fig. 6. Significance for 15 fb−1 of luminosity as a function of m0 for tan β = 3(filled circles), 10 (down triangles), and 30 (open circles) in / T channel at the Tevatron for m1/2 = 160 GeV (mg̃ 1 + jets + E 420 GeV). two tan β values are very close, except for the τ̃1 mass. The τ̃1 mass at tan β = 10 is lighter, so that the branching ratios of χ̃1± → τ̃1 ν and χ̃20 → τ τ̃1 are larger to decrease the 1 + jets + E / T signature. As before, for comparison, we show the significance for m1/2 = 160 GeV in this channel for the Tevatron in Fig. 6. The event selections for this figure was made with the following cuts: (a) Nj 2; (b) N = 1; (c) E / T > 40 GeV; (d) (φ min > 30◦ ; (e) MT < 50 GeV or > 110 GeV; (f) MS2 > 350 GeV. The MT cut is applied to remove W events. The SM background size is 70 fb (Table 1). The significance is found to be always below 5σ for the entire region of parameter space. From Figs. 3 and 5, the significance in jets + E /T and 1 + jets + E / T channels appear to be maximized at m0 = 140–160 GeV and 100 GeV, respectively, for m1/2 = 410 GeV. To obtain the strongest reaches we therefore systematically scan mSUGRA points at tan β = 3 for m1/2 = {360, 400, 440, 470, 500, 540} GeV and m0 = {100, 140, 180, 220, 260} GeV. Fig. 7 shows significance as a function of the gluino mass in both jets +E / T (hatched region bounded by the dashed Fig. 7. Significance as a function of mg̃ for tan β = 3 in jets + E /T (hatched region bounded by the dashed lines) and 1 + jets + E /T (region bounded by the dotted lines) channels at the Tripler. Ranges scanned are 360 m1/2 540 GeV and 100 m0 260 GeV. The / T channel for m0 = 650 GeV and dot-dashed line represents jets + E tan β = 3. lines) and 1 +jets+/ E T (region bounded by the dotted lines) analyses for the above mSUGRA points. We see that the strongest reach in the jets + E / T channel is mg̃ 1060 GeV (m1/2 440 GeV) and mg̃ 1140 GeV (m1/2 480 GeV) in the 1 + jets +E / T channel. The dot-dashed line represents jets + E / T channel for m0 = 650 GeV and tan β = 3. Even for this large m0 , we can see that the 5σ significance can be achieved for mg̃ 900 GeV (mq̃1,2 970 GeV, where mq̃1,2 are the squark masses of the first two generations). 5. Conclusion We have studied the signals for gluinos and squarks within the framework of mSUGRA models in the / T channels for the Tripler jets + E / T and 1 + jets √ +E pp̄ accelerator with s = 5.4 TeV. The Tripler would have a maximum reach of mg̃ 1140 GeV with 30 fb−1 in the 1 + jets + E / T channel (for m0 100 GeV, tan β = 3–10) and mg̃ 1060 GeV in the V. Krutelyov et al. / Physics Letters B 505 (2001) 161–168 jets + E / T channel (for 140 m0 200 GeV, tan β = 3). This gluino mass reach is comparable to the Tripler’s reach of 380 GeV chargino (with 40 fb−1 ) in the trilepton channel [7] via direct chargino–neutralino (χ̃1± –χ̃20 ) production, since gaugino unification implies mχ̃ ± mg̃ /3. The above results can be compared 1 with 440 (460) GeV for the jets + E / T channel for 15 (30) fb−1 of luminosity at the Tevatron. For mg̃ 980 GeV, the Tripler covers relatively large values of / T channel. m0 i.e., to m0 420 GeV in the jets + E Note also, from Figs. 2 and 3, that this gluino and m0 reach of the Tripler is valid for large tan β while the trilepton analysis [7] is sensitive only for small tan β (e.g., tan β = 3). In the above analysis we have set A0 = 0. The results for the maximum Tripler reach are not very sensitive to A0 . Thus there is almost no change for A0 > 0 and for A0 = −1000 GeV, tan β = 3, the gluino reach is increased by about 20 GeV in the 1 + jets + E / T channel. In SUGRA models of this type, the χ̃10 is the LSP and hence is the main candidate for cold dark matter. The astronomical constraints on the amount of relic neutralinos generally implies m0 200 GeV, for m1/2 350–400 GeV. For higher m1/2 , coannihilation effects dominate [33] and for high tan β, m0 can rise to 400–500 GeV [34]. Thus the Tripler would be sensitive to much of the cosmologically interesting part of the parameter space. The LHC gluino reach is mg̃ 2.5 TeV [8,24], which is much higher than the Tripler. The Tripler is however complementary to the LHC in that the production of squarks and gluinos go in part through different channels, as are the detector signals for the charginos and neutralinos. Thus provided SUSY lies sufficiently low to be seen at the Tripler, the two accelerators would be sensitive to different supersymmetric interactions. These results make the proposed energy upgrade of the Tevatron the most appropriate one. Thus a lower energy upgrade would not cover the important parts of the parameter space for Higgs and new physics discovery. A higher energy might increase the Tevatron discovery reach but it would be done at the cost of losing complementarity with the LHC. (Already there is considerable gluon fusion at the Tripler.) 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Santoso, hep-ph/0010244. 26 April 2001 Physics Letters B 505 (2001) 169–176 www.elsevier.nl/locate/npe Neutralino warm dark matter Junji Hisano a,b , Kazunori Kohri c , Mihoko M. Nojiri c a Theory Group, KEK, Oho 1-1, Tsukuba, Ibaraki 305-0801, Japan b TH-devision, CERN, 1211 Geneva 23, Switzerland c YITP, Kyoto University, Kyoto 606-8502, Japan Received 10 January 2001; received in revised form 5 March 2001; accepted 7 March 2001 Editor: T. Yanagida Abstract In the supersymmetric (SUSY) standard model, the lightest neutralino may be the lightest SUSY particle (LSP), and it is is a candidate of the dark matter in the universe. The LSP dark matter might be produced by the non-thermal process such as heavy particle decay after decoupling of the thermal relic LSP. If the produced LSP is relativistic, and does not scatter enough in the thermal bath, the neutralino LSP may contribute as the warm dark matter (WDM) to wash out the small scale structure of O(0.1) Mpc. In this Letter we calculate the energy reduction of the neutralino LSP in the thermal bath and study whether the LSP can be the WDM. If temperature of the production time TI is smaller than 5 MeV, the bino-like LSP can be the WDM and may contribute to the small-scale structure of O(0.1) Mpc. The higgsino-like LSP might also work as the WDM if TI < 2 MeV. The wino-like LSP cannot be the WDM in the favored parameter region. 2001 Published by Elsevier Science B.V. Existence of the dark matter in the universe is one of the important observations for both cosmology and particle physics. The supersymmetric standard model (SUSY SM) provides good candidates for the dark matter in universe, since the R parity stabilizes the lightest SUSY particle (LSP) [1]. In the supergravity scenario the lightest neutralino with mass above O(50) GeV is the LSP. It is produced in the thermal processes of the early universe, and works as the cold dark matter (CDM) which explains the largescale structure in the universe well [2]. On the other hand, it is not yet clear whether the neutralino is consistent with the small-scale structure formation of the universe. It is pointed out that the CDM tends to make cuspy structures in the halo density profiles [3–6]. Such cuspy structures may have E-mail address: [email protected] (M.M. Nojiri). drastic consequences to future observations in the dark matter search [7]. On the other hand, it is argued that the neutralino cuspy profile might be inconsistent with the radio emission from the center of the galaxy [8], although detailed studies are waited for to confirm it. The consistency between the observed structure of sub-galactic scale or cluster of galaxy [9] and the numerical simulation of N body system [10] has been discussed extensively at present. The inflation models with the small-scale perturbation suppressed [11] and some new candidates of the dark matter [12,13] have been proposed in order to explain it well. Here we consider the neutralino dark matter produced by non-thermal processes. If the neutralino dark matter is produced after decoupling of the thermal relic neutralino, such neutralinos might remain without annihilating and contribute as the dark matter. The thermal relic of the LSP on the other hand may be washed away by the entropy production associ- 0370-2693/01/$ – see front matter 2001 Published by Elsevier Science B.V. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 3 9 5 - 1 170 J. Hisano et al. / Physics Letters B 505 (2001) 169–176 ated with the non-thermal production. Such a situation is realized in some models, those are, the heavier moduli decay with mass of the order of 10–100 TeV at the late time [14,15], evaporation of cosmological defects [13], and so on. In those cases the produced LSP can be highly relativistic compared with the thermal background. If the LSP keeps most of its energy from the scattering processes in the thermal bath till the matter-radiation equality, the LSP behaves as the warm dark matter (WDM). The smallscale structure in the universe within the comoving free-streaming scale at the matter-radiation equality is washed out, and the cuspy profiles in the halo would not be formed [13]. If the reduction of the LSP energy by the scattering can be neglected, the comoving free-streaming scale at the matter-radiation equality Rf is given as tEQ Rf = v(t ) dt a(t ) tI 2v0 tEQ (1 + zEQ )2 × log 1 1 , 1+ 2 + 2 v (1 + zEQ ) v0 (1 + zEQ ) 0 (1) where zEQ and tEQ are the red shift and cosmic time for the matter-radiation equality [16]. The v0 is the current velocity of the LSP, v0 = T0 E I , TI mχ̃ 0 (2) 1 where T0 and TI are temperatures for the current cosmic microwave background radiation and the production time of the LSP, EI and mχ̃ 0 are the energy at the 1 production time and the mass for the LSP. In order to explain the small-scale structure of O(0.1) Mpc well, v0 is preferred to be 10−(7−8) from Eq. (1), and this means [13] m 0 EI v0 χ̃1 7 = 2.1 × 10 × (3) . TI 50 GeV 10−7 Other energetic particles are likely associated with the LSP production. Therefore we mainly consider a case where TI at the production time is larger than about a few MeV so that the standard nucleosynthesis works. We will come back to this point later. 1 So far we assumed that the LSP does not lose its relativistic energy significantly in the scattering processes in the thermal bath. If the LSP is gravitino or axino, it does not lose its energy by the scattering because it couples with particles of the SUSY SM very weakly [16]. However, the neutralino LSP is weak interacting, and it may lose most of its energy by the scattering processes in thermal bath. In this Letter we calculate the energy reduction in the successive scattering of the relativistic LSP in the thermal bath. We find the energy reduction is suppressed and the neutralino could be the WDM if some conditions are satisfied. For v0 is 10−(7−8) , TI cannot exceed over ∼ 5 MeV for the bino-like LSP, ∼ 2 MeV for the higgsino-like LSP. The wino-like LSP cannot be the WDM. First, we review nature of the neutralino LSP. The neutralinos are composed of bino, wino, and two higgsinos. The mass matrix is MN MB 0 = −m s c Z W β mZ s W s β 0 MW mZ cW cβ −mZ cW sβ −mZ sW cβ mZ cW cβ 0 −µ mZ sW sβ −mZ cW sβ . −µ 0 (4) Here MB, MW , and µ are the bino, wino, and supersymmetric higgsino masses, respectively. cβ (≡ cos β) and sβ (≡ sin β) are for a mixing angle of the vacuum expectation values of the Higgs bosons, and cW (≡ cos θW ) and sW (≡ sin θW ) for the Weinberg angle. If MB µ, MW , the LSP is bino-like. On the other hand, if MW or µ is smaller than the others, the LSP is wino- or higgsino-like. In the wino- and higgsinolike cases, the LSP and the lighter chargino are degenerated in masses. The next lightest SUSY particle would be important for our discussion since the inelastic scattering of the LSP contributes to the energy reduction. 1 One might worry that the LSP is relativistic at the nucleosynthesis era and it may change the expansion rate significantly. Assuming that the LSP is the dark matter of the universe and Eq. (3), the energy density of the LSP at the nucleosynthesis era is ∼ 0.2% (v0 /10−7 ) of that of three neutrinos, and it does not give any significant effect on the nucleosynthesis. J. Hisano et al. / Physics Letters B 505 (2001) 169–176 In the minimal supergravity model, the binocomponent is dominant in the LSP. This is because M1 ∼ 0.5M2 and µ tends to be larger than the gaugino masses due to the radiative breaking condition. However, if the universal gaugino mass condition at the gravitational scale is broken, LSP can be wino-like. Especially, in the anomaly mediation SUSY breaking model, the wino-component dominates over the others since the gaugino masses are proportional to the oneloop beta function of the gauge coupling constants in the SUSY SM [17]. For a very large universal scalar mass compared to the gaugino masses in the minimal supergravity model [18] or breakdown of universality of the scalar masses may lead to the higgsino-like LSP. In this Letter we do not assume any specific SUSY breaking models and discuss each the neutralino LSPs. The energy loss of the relativistic LSP depends on the temperature at the LSP production time, TI . If the LSP is produced below TC , m 0 1/2 v0 −1/2 χ̃1 TC = 6.3 MeV (5) , 50 GeV 10−7 it is typically non-relativistic in the CM frame of the scattering processes with particles in the thermal bath. In this case the energy reduction par one scattering r(≡ E/E) is qE sin2 (θ/2) sin2 (η/2). m2 0 two-body elastic scattering in the thermal bath. The evolution of the LSP energy is given as dE = −H E − gi dt gi i = d 3 q −q/T dσi dr e (rE)vrel 3 (2π) dr 16 (i) 2 (i) 2 E 4 T 6 AL + AR . π3 m4 0 Here, g22 (e) Here, q is the energy of a particle in the thermal bath, which is ∼ 3T , and θ is the relative angle between the LSP and the particle in the thermal bath, and η the scattering angle of the LSP in the CM frame. Here we take a leading term of O(q/E). Then, the energy reduction is suppressed by O(T E/m2 0 ) when AR = g22 2 2 mZ cW (ν) g22 r = sin2 (η/2). (7) Provided that the event rate is faster than the Hubble expansion, the LSP loses the energy quickly so that LSP scattering becomes non-relativistic in the CM frame. First, we consider the case where TI TC and calculate the energy reduction of the LSP due to the (9) χ̃1 χ̃1 χ̃1 dσi d 3 q −q/T dr. e (rE)vrel 3 dr (2π) (8) Here H is the Hubble parameter. The index i is for spices of particle in the thermal bath with the degrees of freedom gi , vrel and σi are the relative velocity and the cross section of the elastic scattering between the LSP and the particle in the thermal bath. Since the LSP is neutral and TI is smaller than TC and larger than 1 MeV, i = e− , νe , νµ , ντ , and the anti-particles. The contributing diagrams to the energy reduction come from the Z boson and slepton exchanges. We will take a massless limit for the particles in the thermal bath for simplicity. The explicit calculation gives AL = T TC . On the other hand, if the LSPs are produced above TC , they are relativistic in the CM frame of the scattering processes at the production time, and the energy reduction is unsuppressed as i (6) r =4 171 (e) AL = 2 m2Z cW 2 m2Z cW C11 Le − C11 Re + C11 Lν − g22 2mẽ2L 2g22 m2ẽL ([ON ]12 + [ON ]11 tW )2 , ([ON ]11 tW )2 , g22 2m2ν̃L ([ON ]12 − [ON ]11 tW )2 , A(ν) R = 0, where C11 = ([ON ]213 − [ON ]214 ) with [ON ] the diago2 , R = Qs 2 , nalization matrix of MN , Li = T3 + QsW i W and tW ≡ tan θW . The momentum transfers on the propagators of the exchanged particles are negligible compared with the masses, thus we replace the propagators of Z boson and sleptons to their mass squares m2Z and m2˜ , respectively. By solving Eq. (8), the LSP l energy at the radiation-matter equality is given as TEQ E 1− , EEQ = EI (10) TI E eff 172 J. Hisano et al. / Physics Letters B 505 (2001) 169–176 where √ E 24 5 −1/2 = g∗ E eff 7π 9/2 (i) 2 (i) 2 Mpl E 3 T 4 I I A + A × . L R 4 m 0 i bino limit, (11) Here, g∗ is total number of the effective degrees of freedom for at the temperature TI . We assume that the universe is radiation dominant and use H = (4π 3 /45)1/2g∗ 1/2 T 2 /Mpl for the Hubble parameter. The first bracket in the right-handed side in Eq. (10) comes from the red-shift due to the expansion of the universe, and the second one is the effect from the scattering of the LSP in the thermal bath. Here we expand EEQ by EI TI /m2 0 and keep the leading Here mχ̃ 0 MB, and we take mẽR = mẽL = mν̃L 1 (≡ ml˜). In order to suppress the energy reduction below the 10% so that the LSP can behave as the WDM, TI should be smaller than 1.1 (3.1) MeV for v0 = 10−7 (10−8 ), mχ̃ 0 = 50 GeV and ml˜ < 1 TeV. 1 This value corresponds to EI = 24 (6.5) TeV from Eq. (3). If the LSP is heavier, the energy reduction is suppressed more, and a slightly larger TI is possible. For mχ̃ 0 = 200 GeV, TI should be smaller than 1.4 term in Eq. (11), assuming the energy reduction from the scattering is small. When ( E/E)eff is larger than one, TI is replaced to the temperature at which the elastic scattering becomes ineffective to the LSP energy reduction, and EI is given by the LSP energy at the TI . This means that our result is conservative. In Eq. (11), ( E/E)eff is suppressed by TI4 . This comes from the suppression in the amplitude and the phase space, in addition to the energy reduction in the non-relativistic limit of the LSP (Eq. (6)). The momentum transfers in the scattering processes (∼ ET ) are smaller than the exchanged particle masses in the amplitude, and the phase space of the elastic scattering is also suppressed by ET /m2 0 . Thus, the event rate par 5 ml˜ −4 v0 2 TI Γ = 3.8 . H 1 TeV 10−7 1 MeV E E eff χ̃1 χ̃1 χ̃1 a Hubble time is smaller in lower temperature by ∝ T 3 as 1 d 3 q −q/T Γ ≡ gi e vrel σi H H (2π)3 i √ 45 5 −1/2 (i) 2 (i) 2 Mpl E 2 T 3 AL + AR g∗ . = 16π 9/2 m2 0 i χ̃1 (12) Note that the energy reduction ( E/E)eff is dominated by the contribution at T = TI and is not sensitive to TEQ . If the LSP is bino-like, the Z boson exchange contribution is suppressed by m2Z /µ2 in the amplitude. Then, the slepton exchange contribution dominates if µ is larger than the slepton masses. Taking the pure m 0 −1 ml˜ −4 χ̃1 50 GeV 1 TeV 7 3 TI v0 × . (13) −7 10 1 MeV = 3.9 × 10−2 1 (3.7) GeV for v0 = 10−7 (10−8 ). This means that EI < 118 (32) TeV. Note that calculation of the energy reduction rate is valid only when Γ /H > 1. In the bino dominant limit, the event rate of the elastic scattering process by the slepton exchange par a Hubble time is (14) The event rate is not still sufficiently suppressed compared with the Hubble expansion. In Eq. (13) we took the slepton masses 1 TeV. However, some SUSY breaking models predict much heavier sleptons, which is not necessarily in conflict with the naturalness argument [19]. When sfermions are heavy, the thermal component of the bino-like LSP cannot annihilate sufficiently in the thermal processes so that the energy density might be too large beyond the critical density. However, if the huge entropy is supplied in the non-thermal process as mentioned before, it can be diluted and be harmless. When the slepton exchange is sufficiently suppressed, the Z boson exchange becomes dominant in the energy reduction of the LSP. The energy reduction by the Z boson exchange is given as E E eff = 6.9 × 10−3 m 0 −1 χ̃ 1 µ 1 TeV 50 GeV 7 3 TI v0 × cos2 2β. 10−7 1 MeV −4 (15) J. Hisano et al. / Physics Letters B 505 (2001) 169–176 Here we used the approximated solution C11 = 2 cos 2β/µ2 for m , M µ. Recent LEP II −m2Z sW Z B searches of the light Higgs boson prefer | cos 2β| > 0.53 [20]. Since the higgsino mass µ is related with the Higgs boson mass, we cannot take too large a value for µ compared to the weak scale from the naturalness argument. From Eq. (15), TI should be smaller than 5 MeV assuming µ is smaller than 1 TeV, mχ̃ 0 = 1 200 GeV, and v0 = 10−8 . Next, let us consider the higgsino-like LSP. In this case the slepton exchange contribution is suppressed by the small gaugino components and the Yukawa coupling constants, and the Z boson exchange contribution dominates in the elastic scattering processes. Then, m 0 −3 −2 mB E χ̃1 = 2.4 E eff 100 GeV 500 GeV 3 7 v0 TI × cos2 2β. (16) 10−7 1 MeV Here, mχ̃ 0 µ and it should be larger than about 1 100 GeV from negative search for the higgsino-like chargino. When the gaugino masses are heavier than µ and mZ , C11 is given as c2 m2 s 2 C11 = ∓ Z W + W cos 2β, (17) 2µ mB mW for µ positive (negative). It further reduces to C11 = ∓ 2 4 m2Z sW cos 2β 3 µmB (18) 2 by using the GUT relation MB/MW = 5/3tW 1/2. In Eq. (16) we used this formula for simplicity. In order to suppress the energy reduction by the elastic scattering, TI should be smaller than 0.85 (2.3) MeV for v0 = 10−7 (10−8) and a relatively heavy LSP mass mχ̃ 0 = 200 GeV as far as the gaugino masses are 1 smaller than 1 TeV. We saw that the elastic scattering of the higgsinolike LSP is suppressed for heavier gaugino masses. However, we have to check if the inelastic scattering of the LSP by the W boson exchange does not contribute to the energy reduction. As we noted before, the chargino is degenerate with LSP in masses. The Boltzmann suppression, exp(−(mχ̃ 0 mχ̃ )/2ET ), may not 1 be too small. Furthermore the coupling with W boson 173 is not suppressed at all. Therefore the processes, such as χ̃10 e− → χ̃1− ν, may be important over the Boltzmann suppression factor. The mass difference between the chargino and the LSP is mχ̃ ≡ mχ̃ + − mχ̃ 0 1 1 2 2 1 ± sin 2β m2Z sW 1 ∓ sin 2β m2Z cW = + 2 MB 2 MW (19) 2 m2Z sW 4 1 = (20) ∓ sin 2β , MB 3 3 and this is about 5 GeV for MB = 500 GeV. The energy reduction by one scattering of χ̃10 e− → χ̃1− ν is r =4 mχ̃ qE sin2 (θ/2) sin2 (η/2) − 2 . 2 mχ̃ 0 m 0 χ̃1 (21) 1 From the kinematics, r is positive definite. Each chargino decay also reduces the energy of the order of mχ̃ /mχ̃ 0 . The event rate of the inverse inelastic 1 scattering processes of chargino, such as χ̃1− νe → χ̃10 e− , is suppressed by 120π(T / mχ̃ )3 compared with the decay rate, thus contribution to the energy reduction is negligible. The event rate of the inelastic scattering of the higgsino-like LSP par a Hubble time is √ 3 5 −1/2 4 Mpl ET 2 −mχ̃ 0 mχ̃ /(2ET ) Γ 1 = g∗ g2 e H 4π 3/2 m4W mχ̃ ET × + 6 2 NF , (22) mχ̃ 0 m 0 1 χ̃1 where NF is the number of the inelastic processes. When the mass difference is larger than 2 2TI EI TI v0 (23) = 850 MeV , mχ̃ 0 10−7 1 MeV 1 the inelastic processes are suppressed by the Boltzmann factor. Then, Γ /H is sensitive to T , v0 , and mχ̃ . If v0 = 10−7 and mχ̃ = 5 GeV, Γ /H is of the order of 105 even for T = 1 MeV, and the energy reduction by the inelastic scattering cannot be suppressed. On the other hand, if v0 = 10−8 and mχ̃ = 5 GeV, it is 10−20 (33) for T = 1(2) MeV, 174 J. Hisano et al. / Physics Letters B 505 (2001) 169–176 and the energy reduction may be suppressed. The higgsino-like WDM is marginally viable. Note that the energy of the chargino produced by the inelastic scattering is also reduced by the electromagnetic interaction. The life time of the higgsino-like chargino is τχ̃−1 − = ND 1 g24 mχ̃ 5 960π 3 m4W (24) with ND the number of the decay modes. The energy reduction by the electromagnetic interaction is given as π 3 α2 dE =− ΛT 2 , dt 3 i (25) = where Λ is of the order of 1 [21]. Then, the energy reduction rate of the higgsino-like chargino in one life is 2 E T −3 ∼ 1.3 × 10 ND Λ E 1-life 1 MeV m 0 −1 mχ̃ −5 χ̃1 × . (26) 100 GeV 5 GeV This effect may be harmless if mχ̃ = 5 GeV. When the LSP is wino-like, the elastic scattering can be suppressed if the slepton and the higgsino masses are heavy, similar to the bino-like LSP. However, when the Z boson contribution is suppressed by raising the higgsino mass, the chargino and the LSP become more degenerate in masses than in the higgsino-like case as mχ̃ = m4Z 2 2 s c sin2 2β MBµ2 W W the energy reduction becomes maximum at T TC and the LSP loses the relativistic energy till the temperature goes down to TC . As an example, we present the energy reduction of the bino-like LSP by the elastic scattering since the constraint on the TI is the weakest among the neutralino LSPs. Assuming the slepton exchange is suppressed by the heavy masses, the Z boson contribution to the energy reduction when the typical momentum transfer is much larger than m2Z (ET m2Z ) is expressed by d 3q dσi dr gi e−q/T (rE)vrel dr (2π)3 (27) for MW , mZ MB , µ. If µ is 1 TeV and MB is 100 GeV, mχ̃ is about 100 MeV. The Γ /H for the inelastic scattering by the W boson exchange is of the order of 105 for mχ̃ = 100 MeV even if T = 1 MeV and v0 = 10−8 . Since either the Z or W boson exchange contributions cannot be suppressed, the wino-like LSP cannot be the WDM. Finally, we discuss the case for TI TC . In this case, the momentum on the exchanged particle is not negligible, and the event rate becomes larger than in the case of the lower temperature. Therefore g4 t 4 m4 ζ 2 W3 L2i + Ri2 Z4 cos2 2βT 2 , 64π µ (28) i where ζ = (2 log(4ET /m2Z ) − 5 − 2γ ). The energy reduction rate is given as −4 m 0 −1 E µ χ̃1 = 1.0 × 103 ζ E eff 50 GeV 1 TeV −1 TC v0 cos2 2β × −7 10 100 MeV (29) and it is difficult for the LSP to keep the relativistic energy. In this Letter we calculate the energy reduction of the LSP which is produced by the non-thermal process and study whether the LSP can be the warm dark matter or not. If the temperature of the production time TI is smaller than 5 MeV, the bino-like LSP can be the WDM and may contribute to the small-scale structure of O(0.1) Mpc. The higgsino-like LSP might also work as the WDM if TI < 2 MeV. The wino-like LSP cannot be the WDM. We now discuss the some of the aspects on the mechanism to produce relativistic neutralino. Here we discuss the LSP produced from heavy moduli decay. Such a moduli might dominate the energy density of universe before its decay. Therefore, the moduli decay induces the large entropy production and reheating. In this case, the LSP energy density over the entropy density at present is estimated by m 0 χ̃1 0 mχ̃ 0 Yχ̃ 0 0.75 × 10−6 GeV N χ̃ 1 1 1 100 GeV J. Hisano et al. / Physics Letters B 505 (2001) 169–176 × TR 1 MeV mφ 100 TeV −1 , (30) where TR is the reheating temperature and we iden 0 is average number of the LSP tify TR ≡ TI . N χ̃1 from a moduli decay. On the other hand, mχ̃ 0 Yχ̃ 0 ∼ 1 1 10−9 GeV is preferred as the dark matter density. This 0 must be a order of 10−3 . Such a small leads N χ̃1 branching ratio is expected for the case where the moduli decay into gravitino is suppressed [15]. Because the small branching ratio also means that a lot of energetic particles are produced associated with the moduli decay, we must consider the effect on the initial condition of big-bang nucleosynthesis (BBN). If we simply assume that the decay products are only photons, we get the relatively mild constraint TR 0.7 MeV. This lower bound comes from the condition to thermalize the neutrinos and its effects on the neutron to proton ratio in BBN epoch. On the other hand, if the massive particle decays into quarks and gluons, a lot of hadrons would be emitted into the thermal plasma, and they might change the neutron to proton ratio through the strong interaction before BBN starts. In this case, TR 2.5–4 MeV is allowed to agree with the observational light element abundances [22]. For the neutralino LSP to stay warm and produced above 1 MeV, the LSP must be either nearly pure bino or higgsino, in order to suppress the scattering in the thermal bath. This means counting rate at the conventional dark matter detectors would be very small. Discovery of the dark matter signal in any forthcoming experiments [23] will suggest the LSP is not the warm dark matter. For the bino-like LSP the slepton masses also need to be very heavy. If deviation of the muon anomalous magnetic moment from the standard model prediction is observed, the warm binolike LSP is disfavored [24]. Note added After completion of this work, there appears a paper where the energy reduction of the WIMP by scattering in the thermal bath is also discussed [25]. They assume the WIMP is produced by the non-thermal process, but the WIMP is not the LSP. Also, they impose Γ /H < 1 175 for the scattering processes, and do not calculate the energy reduction rate of the WIMP by scattering. Acknowledgements We would like to thank R. Brandenberger, L. Roszkowski, and Y. Suto for useful discussions. This work was supported in part by the Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Sports and Culture of Japan, on Priority Area 707 “Supersymmetry and Unified Theory of Elementary Particles” (J.H.) and Grant-in-Aid for Scientific Research from the Ministry of Education (12047217, M.M.N.). References [1] For reviews, K. 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Santoso Center for Theoretical Physics, Department of Physics, Texas A&M University, College Station, TX 77843-4242, USA Received 1 March 2001; accepted 8 March 2001 Editor: M. Cvetič Abstract We examine the recently observed deviation of the muon g − 2 from the Standard Model prediction within the framework of gravity mediated SUGRA models with R-parity invariance. Universal soft breaking (mSUGRA) models, and models with nonuniversal Higgs and third generation squark/slepton masses at MG are considered. All relic density constraints from stau– neutralino co-annihilation and large tan β NLO corrections for b → sγ decay are included, and we consider two possibilities for the light Higgs: mh > 114 GeV and mh > 120 GeV. The combined mh , b → sγ and aµ bounds give rise to lower bounds on tan β and m1/2 , while the lower bound on aµ gives rise to an upper bounds on m1/2 . These bounds are sensitive to A0 , e.g., for mh > 114 GeV, the 95% C.L. is tan β > 7(5) for A0 = 0(−4m1/2 ), and for mh > 120 GeV, tan β > 15(10). The positive sign of the aµ deviation implies µ > 0, eliminating the extreme cancellations in the dark matter neutralino–proton detection cross section so that almost all the SUSY parameter space should be accessible to future planned detectors. Most of the allowed parts of parameter space occur in the co-annihilation region where m0 is strongly correlated with m1/2 . The lower bound on aµ then greatly reduces the allowed parameter space. Thus using 90% C.L. bounds on aµ we find for A0 = 0 that tan β 10 and for tan β 40 that m1/2 = (290–550) GeV and m0 = (70–300) GeV. Then the tri-lepton signal and other SUSY signals would be beyond the Tevatron Run II (except for the light Higgs), only the τ̃1 and h and (and for part of the parameter space) the ẽ1 will be accessible to a 500 GeV NLC, while the LHC would be able to see the full SUSY mass spectrum. 2001 Published by Elsevier Science B.V. The remarkable accuracy with which the muon gyromagnetic ratio can be measured makes it an excellent probe for new physics beyond the Standard Model. The recently reported result of the Brookhaven E821 experiment now gives a 2.6σ deviation from the predicted value of the Standard Model [1]: aµ − aµSM = 43(16) × 10−10, exp (1) where aµ = (gµ − 2)/2. Efforts were made initially to calculate a possible deviation from the Standard Model within the framework of global supersymE-mail address: [email protected] (B. Dutta). metry (SUSY) [2]. However, one may show that in the limit of exact global supersymmetry, aµSUSY will vanish [3], and thus one needs broken supersymmetry to obtain a nonzero result. The absence of a phenomenologically viable way of spontaneously breaking global supersymmetry made realistic predictions for these models difficult. In contrast, spontaneous breaking of supersymmetry in supergravity (SUGRA) is easy to achieve, and the advent of supergravity grand unified models [4] led to the first calculations of aµSUGRA [5,6], of which [6] was the first complete analysis. Since that time there have been a number of papers updating that result. (See, e.g., [7].) 0370-2693/01/$ – see front matter 2001 Published by Elsevier Science B.V. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 3 7 0 - 7 178 R. Arnowitt et al. / Physics Letters B 505 (2001) 177–183 In SUGRA models, the spontaneous breaking of supersymmetry triggers the Higgs VEV and hence the breaking of SU(2) × U (1), relating then these two mass scales. Thus the scale of the new SUSY masses is predicted to be ∼ 100 GeV – 1 TeV. It was then possible to predict in [6] that the SUGRA contributions would be comparable or larger than the electroweak contribution, 15.2(4) × 10−10 [8], in accord with the now observed deviation of Eq. (1). This scale for the SUSY masses was further confirmed by the LEP data showing that consistency with grand unification could be obtained if the SUSY masses also lie in the above range [9]. Finally, we note that SUGRA models with R-parity invariance predict a dark matter candidate (the lightest neutralino) with the astronomically observed amount of relic density if the SUSY masses again lie in this range. It is thus reasonable to investigate whether the obexp served deviation from aµ can be understood within the framework of SUGRA models, and in this Letter we consider gravity mediated SUSY breaking with R-parity invariance for models with universal soft breaking masses (mSUGRA) and also models with nonuniversal masses in the Higgs and third generation sector. SUGRA models have a wide range of applicability including cosmological phenomena and accelerator physics, and constraints in one area affect predictions in other areas. In particular, as first observed in [5] and emphasized in [10], that aµ increases with tan β, as do dark matter detection rates. Thus as we will see, the deviation of Eq. (1) will significantly effect the minimum neutralino–proton cross section, σχ̃ 0 −p , for terrestrial detectors. Even more significant 1 is the fact that the astronomical bounds on the χ̃10 relic density restrict the SUSY parameter space and hence the SUGRA predictions for aµ as well as what may be expected to be seen at the Tevatron RUN II and the LHC. In order to carry out this analysis, however, it is necessary to include all the co-annihilation effects for large tan β, as well as the large tan β corrections to mb and mτ (which are needed to correctly determine the corresponding Yukawa coupling constants) and the large tan β NLO corrections to the b → sγ decay [11]. In addition, the light Higgs (h) mass bounds play an important role in limiting the SUSY parameter space and it is necessary to include the one and two loop corrections, and the pole mass corrections. The above corrections for dark matter (DM) calculations were carried out in [12], and we will use the same corrections here. Recently several papers have appeared analysing the SUGRA contribution to aµ in light of the final LEP bounds on mh and the deviation of Eq. (1) [13–16]. Relic density constraints were not considered in Refs. [15,16] and coannihilation effects apparently not included in Refs. [13,14]. Also Refs. [13,15,16] do not seem to have included the constraints from the b → sγ decay. As will be seen below, these effects are of major importance in determining the SUGRA predictions. Before proceeding on, we state the range of parameters we assume. We take a 2σ bound of Eq. (1), 11 × 10−10 < aµSUGRA < 75 × 10−10, (2) a 2σ bound on the b → sγ branching ratio, 1.8 × 10−4 < BR(b → sγ ) < 4.5 × 10−4 , and a neutralino relic density range of 0.02 < Ωχ̃ 0 h2 < 0.25. (Assum1 ing a lower bound of 0.1 does not affect results significantly.) The b-quark mass is assumed to have the range 4.0 GeV < mb (mb ) < 4.4 GeV. We consider two bounds on the Higgs mass: mh > 114 GeV and mh > 120 GeV. The first is the current LEP bound and the second is likely within reach of the Tevatron Run II. However, the theoretical calculations of mh have still some uncertainty as well as uncertainty in the t-quark mass, and so we will conservatively interpret these bounds to mean that our theoretical values obey mh > 111 GeV and 117 GeV, respectively. (Our calculations of mh are consistent with [17].) The scalar and gaugino masses at the GUT scale obey (m0 , m1/2 ) < 1 TeV. We examine the range 2 < tan β < 40, and the cubic soft breaking mass is parameterized at the GUT scale by |A0 | < 4m1/2 . Nonuniversal masses deviate from universality according to m20 (1 + δ) where −1 < δ < +1. Other parameters are as in [12]. We consider first the mSUGRA model, which depends on the four parameters m0 , m1/2 , A0 , tan β = H2 /H1 (where H(1,2) give rise to (d, u) quark masses) and the sign of the µ parameter of the Higgs mixing part of the superpotential (W = µH1 H2 ). The SUSY contribution to aµ arises from two types of loop diagrams, i.e., those with chargino–sneutrino intermediate states, and those with neutralino–smuon intermediate states. The dominant contribution arises from the former term with the light chargino (χ̃1± ). For R. Arnowitt et al. / Physics Letters B 505 (2001) 177–183 2 , moderate or large tan β, and when (µ ± m̃2 )2 MW one finds m2µ 1 tan β α aµSUGRA ∼ = 4π sin2 θW mχ̃ ± µ 1 − m̃22 1 µ2 m̃2 2 1 + 3 22 MW µ F (x), × 1− 2 µ (1 − m̃22 )2 2 µ (3) where m̃i = (αi /αG )m1/2 , i = 1, 2, 3, are the gaugino masses at the electroweak scale and αG ∼ = 1/24 is the GUT scale gauge coupling constant. (One has mχ̃ ± ∼ = m̃2 ∼ = 0.8m1/2, and the gluino (g̃) mass is 1 mg̃ ∼ = m̃3 .) In Eq. (3), the form factor is F (x) = (1 − 3x)(1 − x)−2 − 2x 2(1 − x)−3 ln x, where x = (mν̃ /mχ̃ ± )2 . The sneutrino and chargino masses being related to m0 and m1/2 by the renormalization group equations (RGE) [18]. (The contribution from the heavy chargino, χ̃2± reduces this result by about a third.) One finds for large m1/2 that F (x) ∼ = 0.6 so that aµ decreases as 1/m1/2, while for large m0 , F decreases as ln(m20 )/m20 (exhibiting the SUSY decoupling phenomena). Eq. (3) exhibits also the fact discussed in [10,19] that the sign of aµSUGRA is given by the sign of µ. Eq. (1) thus implies that µ is positive (as pointed out in [14–16]). This then has immediate consequences for dark matter detection. Thus as discussed in [12,20,21], for µ < 0, accidental cancellations can occur reducing the neutralino–proton cross section to below 10−10 pb over a wide range of SUSY parameters, and making halo neutralino dark matter unobservable for present or future planned terrestrial detectors. Thus this possibility has now been eliminated, and future detectors (e.g., GENIUS) should be able to scan almost the full SUSY parameter space for m1/2 < 1 TeV. The lower bound of Eq. (1) plays a central role in limiting the µ > 0 SUSY parameter space, particularly when combined with the bounds on the Higgs mass and the b → sγ constraints. As seen above, lowering tan β can be compensated in aµ by also lowering m1/2 . However, mh decreases with both decreasing tan β and decreasing m1/2 . Thus the combined Higgs and aµ bounds put a lower bound on tan β. This bound is sensitive to A0 since A0 enters in the L–R mixing in the stop (mass)2 matrix and affects the values of the stop masses. We find for mh > 111 GeV (i.e., the 179 114 GeV experimental bound), that tan β > 7 for A0 = 0, and tan β > 5 for A0 = −4m1/2 . At higher mh the bound on tan β is more restrictive. Thus for mh > 117 GeV (corresponding to an experimental 120 GeV bound), one has tan β > 15 for A0 = 0, and tan β > 10 for A0 = −4m1/2. As the Higgs mass increases, the bound on tan β increases. As discussed in [12,21–23], for large tan β, the relic density constraints leave only co-annihilation regions possible, and these are very sensitive to the value of A0 . Fig. 1 exhibits the allowed regions in the m0 –m1/2 plane for tan β = 40, mh > 111 GeV for A0 = 0, −2m1/2 , and 4m1/2 (from bottom to top). The corridors terminate at low m1/2 due to the b → sγ and mh constraints. Without the aµ constraint, the corridors would extend up to the end of the parameter space (m1/2 = 1 TeV). We see also that the relic density constraint effectively determines m0 in terms of m1/2 in this region. The lower bound of Eq. (1), however, cuts off these curves (at the vertical lines) preventing m0 and m1/2 from getting too large. Thus for large tan β, the gµ − 2 experiment puts a strong constraint on the SUSY parameter space. The restriction of the SUSY parameter space by the aµ constraint affects the predicted dark matter detection rates. Thus the exclusion of the large m0 and large m1/2 domain of Fig. 1 generally raises the Fig. 1. Corridors in the m0 –m1/2 plane allowed by the relic density constraint for tan β = 40, mh > 111 GeV, µ > 0 for A0 = 0, −2m1/2 , 4m1/2 from bottom to top. The curves terminate at low m1/2 due to the b → sγ constraint except for the A0 = 4m1/2 which terminates due to the mh constraint. The short lines through the allowed corridors represent the high m1/2 termination due to the lower bound on aµ of Eq. (1). 180 R. Arnowitt et al. / Physics Letters B 505 (2001) 177–183 Fig. 2. σχ̃ 0 −p as a function of the neutralino mass mχ̃ 0 for 1 1 tan β = 40, µ > 0 for A0 = −2m1/2 , 4m1/2 , 0 from bottom to top. The curves terminate at small mχ̃ 0 due to the b → sγ 1 constraint for A0 = 0 and −2m1/2 and due to the Higgs mass bound (mh > 111 GeV) for A0 = 4m1/2 . The curves terminate at large mχ̃ 0 due to the lower bound on aµ of Eq. (1). Fig. 3. σχ̃ 0 −p as a function of mχ̃ 0 for tan β = 10, µ > 0, 1 1 mh > 111 GeV for A0 = 0 (upper curve), A0 = −4m1/2 (lower curve). The termination at low mχ̃ 0 is due to the mh bound for 1 A0 = 0, and the b → sγ bound for A0 = −4m1/2 . The termination at high mχ̃ 0 is due to the lower bound on aµ of Eq. (1). 1 1 lower bounds on the neutralino–proton cross section. In Fig. 2 we have plotted σχ̃ 0 −p as a function of mχ̃ 0 1 1 for tan β = 40 for the allowed corridors for A0 = −2m1/2, 4m1/2 and 0 (bottom to top). The curves terminate at high mχ̃ 0 due to the lower bound on 1 aµ of Eq. (1). (Note that mχ̃ 0 ∼ = 0.4m1/2.) Again 1 one sees the sensitivity of results to the value of A0 , both for the high mχ̃ 0 termination point and 1 for the magnitude of the cross section. Over the full range one has that σχ̃ 0 −p > 6 × 10−10 pb, and 1 hence should generally be accessible to future planned detectors. If we reduce tan β, one might expect the minimum value of σχ̃ 0 −p to significantly decrease. However, the 1 aµ bound then becomes more constraining, eliminating more and more of the high m1/2 , high m0 region. This is shown in Fig. 3 where the minimum value of σχ̃ 0 −p is plotted as a function of mχ̃ 0 , for 1 1 tan β = 10, µ > 0, mh > 111 GeV, for A0 = −4m1/2 (lower curve), A0 = 0 (upper curve). The A0 = 0 curve terminates at low mχ̃ 0 due to the Higgs mass 1 bound, while the A = −4m1/2 terminates due to the b → sγ constraint. The termination at high mχ̃ 0 is due 1 to the aµ lower bound of Eq. (1). We see that the parameter space is now quite restricted, and so even though tan β is quite reduced, we find σχ̃ 0 −p > 4 × 10−10 pb. 1 The co-annihilation region begins at mχ̃ 0 140 GeV, 1 and so the earlier part of these curves lie in the nonco-annihilation domain. If we raise mh and require mh > 117 GeV (corresponding to an experimental bound of 120 GeV), then mh controls the termination of the curves at low mχ̃ 0 . 1 Thus for tan β = 40, the curves of Fig. 2 start at mχ̃ 0 = 1 200 GeV for A0 = −2m1/2 , at mχ̃ 0 = 215 GeV for 1 A0 = 0, and at mχ̃ 0 = 246 GeV for A0 = 4m1/2 (i.e., 1 the A0 = 4m1/2 curve is almost completely eliminated by the mh constraint). One has thus only a narrow range of allowed mχ̃ 0 . The allowed range becomes 1 even narrower with decreasing tan β, and the entire parameter space is eliminated when tan β = 10. We turn next to consider nonuniversal soft breaking models with nonuniversal masses at MG in the third generation squarks and sleptons and in the Higgs masses: m2H1 = m20 (1 + δ1 ), m2H2 = m20 (1 + δ2 ), m2qL = m20 (1 + δ3 ), m2tR = m20 (1 + δ4 ), m2τR = m20 (1 + δ5 ), m2bR = m20 (1 + δ6 ), m2lL = m20 (1 + δ7 ). (4) Here q̃L = (t˜L , b̃L ) squarks, l˜L = (ν̃τ , τ̃L ) sleptons, etc. and we assume −1 < δi < +1. As discussed in [12], the value of µ significantly controls both the relic density and σχ̃ 0 −p , and one may understand 1 qualitatively how µ varies from its analytic expression R. Arnowitt et al. / Physics Letters B 505 (2001) 177–183 which is valid for low and intermediate tan β: t2 1 1 − D0 1 − 3D0 + 2 + (δ3 + δ4 ) µ2 = 2 2 2 t −1 t 1 + D0 δ1 − δ2 + 2 m20 2 t + universal parts + loop corrections. (5) 181 Here t = tan β, and D0 ∼ = 1−(mt /200(GeV) sin β)2 ∼ = 0.25. One sees that the universal m20 term is quite small, and one can easily choose the δi to make the coefficient of m20 negative. A reduction of µ2 increases the higgsino content of the neutralino, and thus increases the χ̃10 –χ̃10 –Z coupling. In [12], it was shown that this allowed the opening of a new region of allowed relic density at high m1/2 and high tan β. We consider first the simple case where only δ2 is nonzero and choose δ2 = 1. Fig. 4 shows σχ̃ 0 −p as a func1 tion of m1/2 for this case when tan β = 40, µ > 0, mh > 111 GeV and A0 = m1/2 . The lower line corresponds to the usual stau–neutralino co-annihilation corridor. The upper dashed curves show the new allowed band arising from increased early universe annihilation through the Z s-channel pole. It is quite broad and has a large scattering cross section. The curves terminate at low m1/2 due to the b → sγ constraint, and we have terminated the curves at the high end when m0 or m1/2 exceed 1 TeV. The vertical lines are the high m1/2 endpoints due to the lower bound on aµ of Eq. (1). One sees that the parameter space is significantly reduced, though there is still a large Z-channel band remaining. Increasing mh increases the lower bound of m1/2 . For mh > 117 GeV we find the co-annihilation (solid line) now begins at m1/2 = 510 GeV, and the Z channel band begins at 500 GeV due to the mh constraint, leaving a sharply reduced region of parameter space. A second example of new nonuniversal effects is furnished by choosing δ10 (= δ3 = δ4 = δ5 ) to be nonzero (as might be the case for an SU(5) or SO(10) model). We consider here δ10 = −0.7. In this case [12] the τ̃1 –χ̃10 co-annihilation corridor occurs at a much higher value of m0 than in the universal case (i.e., for m0 = 600–800 GeV), and is somewhat broadened. The Z-channel band lies above it and is considerably broader. In Fig. 5 we have plotted σχ̃ 0 −p as a function of m1/2 for the lower side of 1 the co-annihilation corridor (lower curve) and for the upper side of the Z channel band (upper curve) for tan β = 40, µ > 0, A0 = m1/2 and mh > 111 GeV. (Note that while the Z channel lies at a higher m0 in the m0 –m1/2 plane than the co-annihilation corridor, the cross section is still larger since µ2 is reduced.) The curves terminate at the left due to the b → sγ constraint. The vertical lines show the termination at high m1/2 due to the lower bound on aµ , significantly shrinking the allowed parameter space. For mh > 117 GeV, the Higgs mass governs the termination at Fig. 4. σχ̃ 0 −p as a function of m1/2 (mχ̃ 0 ∼ = 0.4m1/2 ) for 1 1 tan β = 40, µ > 0, mh > 111 GeV, A0 = m1/2 for δ2 = 1. The lower curve is for the τ̃1 –χ̃10 co-annihilation channel, and the dashed band is for the Z s-channel annihilation allowed by nonuniversal soft breaking. The curves terminate at low m1/2 due to the b → sγ constraint. The vertical lines show the termination at high m1/2 due to the lower bound on aµ of Eq. (1). A0 = m1/2 and mh > 111 GeV. The lower curve is for the bottom of the τ̃1 –χ̃10 co-annihilation corridor, and the upper curve is for the top of the Z channel band. The termination at low m1/2 is due to the b → sγ constraint, and the vertical lines are the upper bound on m1/2 due to the lower bound of aµ of Eq. (1). Fig. 5. σχ̃ 0 −p as a function of m1/2 for tan β = 40, µ > 0, 1 182 R. Arnowitt et al. / Physics Letters B 505 (2001) 177–183 while b → sγ determines it for tan β = 40. Both are equally constraining for tan β = 30. If we consider the 90% C.L. bound (aµ > 21 × 10−10 ) [24], one finds for A0 = 0 that tan β 10, and for tan β 40 that m1/2 = (290–550) GeV, and m0 = (70–300) GeV. This greatly constrains SUSY particle spectrum expected at accelerators, as can be seen in Table 1. Thus at the 90% C.L. bound on aµ the tri-lepton signal will be unobservable at the Tevatron Run II since tan β and m1/2 are relatively large [25], and the other SUSY particles are also beyond its reach, except for the light Higgs, provided mh 130 GeV [26]. (One would need to triple the Tevatron’s energy to see a significant part of the SUSY mass spectrum.) Only the τ̃1 and ẽ1 would possibly be within the reach of a 500 GeV NLC (and very marginally the χ̃1± ), while all the SUSY particles would be accessible to the LHC. The Brookhaven E821 experiment has a great deal more data that can reduce the error by a factor of about 2. When analysed, this would greatly narrow the predictions made here. One of the interesting features of Fig. 6 is that mSUGRA can no longer accommodate large values of aµSUGRA . If the full E821 data should require a value significantly larger than 40 × 10−10 , this would be a signal for the existence of nonuniversal soft breaking. From Eq. (3) one sees that one can increase aµ by reducing µ, and from Eq. (5) this might be accomplished by nonuniversal soft breaking of the scalar masses (and also from nonuniversal gaugino masses at MG ). Thus the gµ − 2 experiment may give us significant insight into the nature of physics beyond the GUT scale. low m1/2 , and the co-annihilation (lower curve) now begins at m1/2 = 515 GeV, and the Z channel (upper curve) begins at m1/2 = 520 GeV. The above discussion shows that for SUGRA models, and particularly for mSUGRA, the aµ data, when combined with the mh , b → sγ and relic density constraints have begun to greatly limit the SUSY parameter space. Thus the mh and b → sγ constraints determine a lower bound on m1/2 and hence an upper bound on aµSUGRA , while the experimental lower bound on aµ determines an upper bound on m1/2 . The combined aµ and mh bound puts lower bound on tan β for a given value of A0 . This can be seen most clearly in Fig. 6, where the mSUGRA contribution to aµ is plotted as a function of m1/2 for A0 = 0, tan β = 10 (lower curve), tan β = 30 (middle curve) and tan β = 40 (upper curve). Further, most of the allowed m1/2 region lies in the τ̃1 –χ̃10 coannihilation domain (m1/2 350 GeV), and so from Fig. 1 one can see that m0 is approximately determined in terms of m1/2 . In Fig. 6, the mh bound determines the lower limit on m1/2 for tan β = 10, Acknowledgements Fig. 6. mSUGRA contribution to aµ as a function of m1/2 for A0 = 0, µ > 0, for tan β = 10, 30 and 40 (bottom to top) and mh > 111 GeV. This work was supported in part by National Science Foundation grant No. PHY-0070964. Table 1 Allowed ranges for SUSY masses in GeV for mSUGRA assuming 90% C.L. for aµ for A0 = 0. The lower value of mt˜ can be reduced to 1 240 GeV by changing A0 to −4m1/2 . The other masses are not sensitive to A0 χ̃10 χ̃1± g̃ τ̃1 ẽ1 ũ1 t˜1 123–237 230–451 740–1350 134–264 145–366 660–1220 500–940 R. Arnowitt et al. / Physics Letters B 505 (2001) 177–183 References [1] H.N. Brown et al., Muon (g-2) Collaboration, hep-ex/0102017. [2] P. Fayet, in: S. Ferrara, J. 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Pilaftsis b a Physics Division, School of Technology, Aristotle University of Thessaloniki, 54006 Thessaloniki, Greece b Institut für Theoretische Physik und Astrophysik, Universität Würzburg, Am Hubland, 97074 Würzburg, Germany Received 2 January 2001; received in revised form 21 February 2001; accepted 1 March 2001 Editor: G.F. Giudice Abstract A possible discovery of a relatively light charged Higgs boson H + in near future experiments, with a mass MH + 110 GeV, together with the present LEP2 direct limits on the chargino and neutral Higgs sectors, would disfavour the minimal supersymmetric standard model as well as its frequently discussed next-to-minimal supersymmetric extension. We show that a supersymmetric origin can naturally be ascribed to the existence of such a light charged Higgs scalar within the context of the recently introduced minimal nonminimal supersymmetric standard model. 2001 Published by Elsevier Science B.V. Supersymmetry (SUSY) appears to be a compelling ingredient of string theories which are expected to successfully describe the Planck-scale dynamics, thereby aspiring to unify all fundamental forces in nature, including gravity. For these reasons, low-energy realizations of SUSY softly broken at 0.1–1 TeV energies, such as the Minimal Supersymmetric Standard Model (MSSM) and its minimal extensions, are considered to be the best-motivated models [1] of physics beyond the Standard Model (SM). Most interestingly, such low-energy realizations of SUSY exhibit gaugecoupling unification [2] and can solve, at least technically, the problem of perturbative stability of radiative effects between the soft SUSY-breaking scale MSUSY ∼ 1 TeV and the Planck mass MP . These appealing properties of low-energy SUSY might be considered to mainly emanate from the doubling of the particle spectrum of the SM; the theory introduces a new fermion (boson) for each SM boson (fermion), E-mail address: [email protected] (C. Panagiotakopoulos). its so-called superpartner. Superpartners have typical masses of the order of the soft SUSY-breaking scale MSUSY and should be heavier than ∼ 100 GeV, for phenomenological reasons [3]. In addition, within the framework of SUSY, the holomorphicity of the superpotential together with the requirement of cancellation of the triangle gauge anomalies entail that the SM Higgs sector itself must be augmented by at least one Higgs doublet of opposite hypercharge. To be specific, low-energy SUSY models include a minimal set of two Higgs iso-doublets and so necessarily predict the existence of at least one (doublet) charged Higgs boson, H ± , in addition to a number of neutral Higgs particles. As we will see in this note, the mass of H + introduces a new scale into the neutral Higgs-boson mass spectrum and can play a key role in distinguishing among different minimal models of electroweakscale SUSY. In the decoupling limit of a heavy charged Higgs boson, low-energy SUSY models make a definite prediction for the mass of that neutral Higgs boson H , which is predominantly responsible for the spontaneous breaking of the SM gauge group. In other 0370-2693/01/$ – see front matter 2001 Published by Elsevier Science B.V. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 3 4 7 - 1 C. Panagiotakopoulos, A. Pilaftsis / Physics Letters B 505 (2001) 184–190 words, for large values of MH + , e.g., MH + ∼ 1 TeV, the mass of the SM-like H boson reaches a calculable model-dependent maximum [4]. For instance, in the MSSM, with radiative effects included [5], recent computations [6–8] lead to the upper limit: MH 110 (130) GeV for tan β ≈ 2 ( 10), where tan β is the ratio of the vacuum expectation values (VEVs) of the two Higgs doublets. On the other hand, in the frequently discussed extension of the MSSM known as the Next-to-Minimal Supersymmetric Standard Model (NMSSM) [9], the maximum of the corresponding H boson mass may increase by an amount of ∼ 30 GeV, for tan β ≈ 2, while it remains unaffected for large values of tan β [10]. Apart from tan β, however, the Higgs-boson mass spectrum depends very sensitively on the actual value of MH + and the stop-mixing parameter Xt = At − µ/ tan β, where At is the soft SUSYbreaking Yukawa coupling to stops and µ the mixing parameter of the two Higgs-doublet superfields 2 . For example, for MH + 110 GeV, the 1 and H H masses of the lightest CP-even Higgs boson h and the CP-odd scalar A are predicted to be both less than ∼ 80 GeV, almost independently of tan β, provided no unusually large values of |µ| are considered, e.g., for |µ| 2MSUSY . However, such a scenario is disfavoured by the latest LEP2 data [11], since it predicts an enhanced ZhA-coupling, whenever the hZZcoupling is suppressed, and hence would have been detected in the corresponding ZhA channel. As has been explicitly demonstrated in [12], a similar negative conclusion may be reached in the NMSSM as well, for MH + 110 GeV. In this model, the SM-like Higgs boson H always comes out to be lighter than H + , provided the effectively generated µ-parameter lies in the phenomenologically favoured range, |µ| 100 GeV, as is suggested by the non-observation of chargino production at LEP2 [3,13]. 1 Given the difficulty that the MSSM and NMSSM cannot easily accommodate a charged Higgs boson H + lighter than the SM-like neutral Higgs boson H , one may now raise the following question: should such 1 Throughout the Letter we shall not consider possible indirect constraints on the H + -boson mass from b → sγ and other observables involving B mesons, as the derived limits sensitively depend on several other model-dependent parameters of the theory, such as the sign of µ [14] and the low-energy flavour-mixing structure of the squark sector [15]. 185 a light charged particle, with MH + 110 GeV, be observed, e.g., at the upgraded Tevatron collider, is it then possible to ascribe to it a supersymmetric origin within a minimal SUSY extension of the SM? In this note we address this important question in the affirmative within the framework of the recently introduced Minimal Nonminimal Supersymmetric Standard Model (MNSSM) [12,16]. In the MNSSM the µ-parameter is promoted to a chiral singlet superfield S, and all linear, quadratic and cubic operators involving only S are absent from the renormalizable superpotential; S enters through the 2 . The crucial difference between 1 H single term λ SH the MNSSM and the NMSSM lies in the fact that the S 3 does not appear in the renormalcubic term 13 κ izable superpotential of the former. This particularly simple form of the renormalizable MNSSM superpotential may be enforced by discrete R-symmetries, such as Z5R [12,16] and Z7R [12]. These discrete Rsymmetries, however, must be extended to the gravityinduced non-renormalizable superpotential and Kähler potential terms as well. Here, we consider the scenario of N = 1 supergravity spontaneously broken by a set of hidden-sector fields at an intermediate scale. Within this framework of SUSY-breaking, we have been able to show [12] that the above Rsymmetries are sufficient to guarantee the appearance of the potentially dangerous tadpole tS S, with tS ∼ 2 (1/16π 2)n MP MSUSY , at loop levels n higher than 5. As a consequence, we have |tS | 1–10 TeV3 , and therefore the gauge hierarchy does not get destabilized. Notice that the so-generated tadpole tS S together with the soft SUSY-breaking mass term m2S S ∗ S lead to a VEV for S, S = √1 vS , of order MSUSY . 2 The latter gives rise to a µ-parameter at the required electroweak scale, i.e., µ = − √1 λvS ∼ MSUSY , thus 2 offering a natural explanation for the origin of the µparameter. Finally, since the effective tadpole term tS S explicitly breaks the continuous Peccei–Quinn symmetry governing the remaining renormalizable Lagrangian of the MNSSM, the theory naturally avoids the presence of a phenomenologically excluded weakscale axion. The MNSSM predicts, in addition to the charged Higgs scalar H + , five neutral Higgs bosons. Under the assumption of CP invariance, three of the neutral Higgs particles, denoted as H1 , H2 and H3 in order of 186 C. Panagiotakopoulos, A. Pilaftsis / Physics Letters B 505 (2001) 184–190 increasing mass, are CP-even, while the other two, A1 and A2 (with MA1 < MA2 ), are CP-odd. Nevertheless, since the tadpole |tS | naturally takes values of the order of 1–10 TeV3 , the Higgs-boson mass spectrum of the MNSSM simplifies considerably: the heaviest states 2 ≈ M 2 ≈ λt /µ, decouple as H3 and A2 , with MH S A2 3 singlets from the remaining Higgs sector. Then, the masses of H + and A1 satisfy the relation 1 2 2 2 2 λ v − δrem , MA2 1 ≈ Ma2 = MH (1) + − MW + 2 where MW = gw v/2 is the W -boson mass and δrem contains the radiative corrections which may be approximately determined by [12,17,18] δrem ≈ − 3h4t µ2 v 2 , 32π 2 m2˜ + m2˜ t t 1 (2) 2 where t˜1 and t˜2 are the stop mass eigenstates. Notice that the relation (1) is very analogous to the one known from the MSSM. Specifically, the squared mass term Ma2 enters the non-decoupled 2 × 2 CP-even mass matrix the same way as the squared mass of the wouldbe CP-odd Higgs scalar in the MSSM. As opposed to the MSSM, however, the presence of the term 12 λ2 v 2 in Eq. (1) implies that the H + boson can become even lighter than A1 , for λ ∼ gw ; H + can be as light as its experimental lower bound, MH + ∼ 80 GeV [3,11]. As an important consequence, the H + boson can naturally be lighter than the SM-like Higgs boson H . As we will see, this prediction is very unique for the MNSSM. In the MSSM, such a result may be achieved for unconventionally large values of |µ|, in which case δrem in Eq. (2) will start playing a very analogous role as the term 12 λ2 v 2 in Eq. (1) does for the MNSSM. For our phenomenological discussion, we denote with gHi ZZ , gHi W W and gHi Aj Z the strength of the effective Hi W W -, Hi ZZ- and Hi Aj Z-couplings, respectively, normalized to the SM values of the H W W -, H ZZ- and H ZG0 -couplings, where G0 is the would-be Goldstone boson of Z. These SMnormalized couplings the unitarity relaobey effective 3 2 2 2 = 1 and tions: 3i=1 gH i=1 j =1 gHi Aj Z = 1, iV V with V = W, Z. Moreover, as a consequence of a large |tS |, the effective Higgs-to-gauge-boson couplings satisfy the approximate equalities 2 2 ≈ gH , gH 1V V 2 A1 Z 2 2 gH ≈ gH , 2V V 1 A1 Z (3) which are essentially identical to the corresponding complementarity equalities of the MSSM. We should remark that the relations (3) are not valid in the NMSSM, since the states H3 and A2 do not decouple as singlets from the lightest Higgs sector in the latter model. Our study of the MNSSM Higgs-boson mass spectrum in the decoupling limit of a large |tS | utilizes renormalization-group (RG) techniques developed in [6,8,19] for the MSSM case and so improves in several respects earlier considerations in the NMSSM, in which an analogous decoupling limit is lacking. Specifically, in addition to the one-loop stop (t˜) and sbottom (b̃) corrections, our RG improvement consists in including two-loop leading logarithms induced by QCD and top (t) and bottom (b) quark Yukawa interactions. Further, we take into account the leading logarithms originating from gaugino and higgsino oneloop graphs [17], as well as we implement the potentially large two-loop contributions induced by the oneloop t˜- and b̃-squark thresholds in the t- and b-quark Yukawa couplings [8]. In the MNSSM and NMSSM, the SM-normalized effective couplings gHi V V and the CP-even Higgsboson masses MHi satisfy an important sum rule: 3 2 2 gH MH iV V i i=1 3h2 = MZ2 cos2 2β + 12 λ2 v 2 sin2 2β 1 − t2 t 8π 4 4 2 3h v sin β 4αs Xt + t 1+ 2 8π 3π MSUSY 2 X Xt2 × t+ 2t 1− 2 MSUSY 12MSUSY 1 3 2 + h − 32παs 16π 2 2 t 2Xt2 Xt2 2 × 1 − t + t 2 2 MSUSY 12MSUSY Xt6 +O (4) , 6 MSUSY 2 where t = ln(MSUSY /m2t ), and the strong fine structure constant αs , the t-quark Yukawa coupling ht and the SM VEV v are to be evaluated at mt . The mass-coupling sum rule (4) is independent of MH + C. Panagiotakopoulos, A. Pilaftsis / Physics Letters B 505 (2001) 184–190 Table 1 Predictions of the MNSSM [21], using as inputs: Xt = 0, At = Ab , MSUSY = 1 TeV, mB = mW = 0.3 TeV, mg̃ = 1 TeV, λtS /µ = 2 5 TeV2 , MH1 ≈ 95 GeV, MH2 ≈ 115 GeV, gH ZZ ≈ 0.1 and 1 2 ≈ 0.9 gH 2 ZZ MH + [GeV] tan β λ µ [GeV] MA1 [GeV] 70 1.74 0.614 −574 101.8 80 2.82 0.542 −331 95.7 90 4.93 0.488 −224 95.4 100 7.70 0.424 −229 95.9 110 10.86 0.338 −290 96.2 120 14.34 0.205 −410 96.4 125.2 16.35 0. −825 96.5 Table 2 Predictions of the MNSSM [21], using the same inputs as in Table 1, with the exception that Xt = 2.45 TeV and λtS /µ = 1.5 TeV2 MH + [GeV] tan β λ µ [GeV] MA1 [GeV] 60 2.24 0.649 −441 102.8 70 3.28 0.590 −402 97.8 80 5.15 0.541 −397 96.7 90 7.73 0.489 −424 96.7 10.7 0.427 −481 97.1 110 14.0 0.348 −585 97.7 120 17.65 0.249 −810 99.0 100 and makes the definite prediction that the mass of the neutral Higgs boson H with SM coupling to the 2 Z boson, gH ZZ ≈ 1, is completely specified by a model-dependent value determined from the RHS of Eq. (4). It can thus be estimated from Eq. (4) that in the MNSSM and NMSSM, the SM-like Higgs-boson mass can reach a maximum of ∼ 142 GeV, for tan β = 2, λ = 0.65, MSUSY ≈ 1 TeV and Xt ≈ 2.45 TeV. In addition, one should observe that the mass-coupling sum rule holds true for the MSSM case as well, after setting λ = 0. The analytic result of the RHS of Eq. (4) is then in agreement with the one computed in [8], after one follows the suggested RG approach to implementing stop threshold effects on the t-quark Yukawa coupling. 187 It is now very interesting to quote results of variants of the MNSSM that could be probed at LEP2 and especially at the upgraded Tevatron collider in the immediate future. For definiteness, in our numerical estimates, we fix the soft SUSY-breaking squark masses to MSUSY = 1 TeV, and the U(1)Y , SU(2)L and SU(3)c gaugino masses to mB = mW = 0.3 TeV and mg̃ = 1 TeV, respectively. Motivated by the recently observed excess of events for a SM-like Higgs boson of a mass ∼ 115 GeV at LEP2 [20], we choose in Tables 1 and 2 the mass of the second lightest CP-even Higgs boson H2 to be MH2 ≈ 115 GeV, 2 with gH ≈ 0.9. For the lightest Higgs boson H1 , 2 ZZ whose squared effective coupling to the Z boson is 2 2 ≈ 1 − gH ≈ 0.1, we assume a necessarily gH 1 ZZ 2 ZZ lower mass, i.e., MH1 ≈ 95 GeV, compatible with the present LEP2 data [11]. In Table 1 we consider the zero stop-mixing scenario, i.e., Xt = 0, and choose λtS /µ = 5 TeV2 . We find that the mass of the charged Higgs boson may naturally lie below 110 GeV, for reasonable values of the MNSSM parameters. In particular, we obtain λ 0.65, for tan β 2. Interestingly enough, such a range of λ values is also consistent with the requirement of perturbativity of the MNSSM up to the gauge-coupling unification scale MU ∼ 1016 GeV [2,10]. Also, in accordance with our earlier discussion, we observe that the H + boson must be as heavy as 125 GeV in the MSSM limit λ → 0, i.e., heavier than the SM-like Higgs boson H2 . In Ta2 and use the value ble 2 we select λtS /µ = 1.5 TeV√ of maximal stop mixing, Xt ≈ 6 MSUSY , characterized by the fact that the radiative effects given in Eq. (4) get approximately maximized. We arrive at the very same conclusion, namely the H + boson can be lighter than ∼ 110 GeV and so lighter than the SMlike Higgs boson H2 . This particular feature of the MNSSM is also reflected in Fig. 1, where we show numerical values of H1 - and H2 -boson masses and of their squared effective couplings to the Z boson as functions of µ, for three variants of the MNSSM from Table 1: MH + = 80, 100 and 120 GeV. We observe that the aforementioned LEP2-motivated scenario of a SM-like Higgs boson may be accounted for by a wide range of µ values. An interesting alternative emerges if one of the two non-decoupled CP-even Higgs bosons, e.g., H2 , has 2 a mass MH2 ≈ 115 GeV with gH ≈ 1, while the 2 ZZ other one, H1 , does not couple to the Z boson but has 188 C. Panagiotakopoulos, A. Pilaftsis / Physics Letters B 505 (2001) 184–190 2 Fig. 1. Numerical predictions for (a) MH1 and MH2 , and (b) gH 1 ZZ 2 and gH ZZ , as functions of µ in the MNSSM. 2 Fig. 2. Numerical values of (a) MH1 and MH2 , and (b) gH and 1 ZZ 2 gH ZZ , as functions of µ in the MNSSM. 2 gA ≈ 1, and vice versa. Such an alternative is eas1 H1 Z ily compatible with the LEP2 data, as long as the mass inequality constraint, MH1 + MA1 170 GeV [11], is met. Assuming λtS /µ = 2 TeV2 , the above scenario may be realized for a wide range of charged Higgs-boson masses between 60 and 110 GeV, and for both zero and maximal stop mixing. For instance, for Xt = 0, such a kinematic dependence insensitive to MH + may be obtained for tan β = 2.5, λ = 0.623 and µ ≈ −283 GeV, while for Xt = 2.45 TeV, one may choose tan β = 5, λ = 0.645 and µ ≈ −393 GeV. In Fig. 2, we display the dependence of the H1 - and H2 boson masses and of their squared effective couplings to the Z boson as a function of the µ-parameter, for MH + = 80, 100 and 120 GeV in the aforementioned variant of the MNSSM with Xt = 0. We observe again that the H + boson can be lighter than ∼ 110 GeV and therefore lighter than the SM-like Higgs boson. In addition, we notice that there exists a SM-like Higgs boson for a very wide range of µ values and, only for a very short interval of µ, the H1 and H2 bosons interchange their couplings to the Z boson, while they are degenerate in mass. Finally, we should reiterate the fact that analogous possibilities are present in the MSSM for large values of |µ|. In agreement with our earlier observation, we find that the H + boson can be as light as 100 GeV, with MH2 ≈ 115 GeV, MH1 ≈ 2 2 82.3 GeV, MA1 ≈ 92.6 GeV and gH ≈ gH 2 ZZ 1 A1 Z 2 2 C. Panagiotakopoulos, A. Pilaftsis / Physics Letters B 505 (2001) 184–190 ≈ 1, if the large value of |µ|, µ ≈ −3.97 TeV, together with tan β = 12.3 and Xt = 1 TeV, is employed. In the NMSSM the situation is different. The charged Higgs-boson mass MH + exhibits a strong monotonic dependence on the µ-parameter; |µ| gets rapidly smaller for smaller values of MH + . This generic feature of the NMSSM may mainly be attributed to the fact [12] that no analogous decoupling limit due to a large |tS | exists in this model. In particular, unlike the MNSSM case, no actual use of the presence of the contribution 12 λ2 v 2 in the Higgs-boson mass matrices can be made in the NMSSM. In fact, we find that it is always MH + 110 GeV, for phenomenologically favoured values of µ, i.e., for |µ| 100 GeV [3,13], 2 assuming that the theory stays perturbative up to MU . If the H + boson becomes heavier than the neutral SM-like Higgs boson H , the phenomenological distinction between the NMSSM and MNSSM is getting more difficult and additional experimental information would be necessary, such as the testing of the complementarity coupling relations of Eq. (3). To summarize: the renormalizable low-energy sector of the MNSSM in the decoupling limit due to a large |tS | has effectively one parameter more than the corresponding one of the (CP-conserving) MSSM, namely, the coupling λ. In fact, in the MNSSM the natural size of the higher-loop generated tadpole parameter |tS | is of order 1–10 TeV3 . For unsuppressed values of λ, tS leads to masses of the order of 1 TeV for the heaviest CP-even and CP-odd Higgs scalars H3 and A2 , so these states decouple as heavy singlets giving rise to an active low-energy Higgs sector consisting only of doublet-Higgs fields, closely analogous to the one of the MSSM. The latter should be contrasted with the NMSSM case, where no analogous decoupling limit due to a large |tS | exists in this model. Most strikingly, the MNSSM may also predict a light charged Higgs boson, which can be even lighter than the SM-like Higgs boson H . We should stress again that in the light of the present LEP2 data, such a prediction cannot be naturally obtained in the MSSM or NMSSM. In the same vein, we note that it would be 2 If this last constraint on the µ-parameter is lifted, then charged Higgs-boson masses as low as 90 GeV might be possible in the NMSSM [22]. 189 very interesting to study as well as identify the compelling low-energy structure of other SUSY extensions of the SM that could lead to the inverse mass hierarchy MH + MH . From our discussion, however, it is obvious that the MNSSM truly represents the simplest and most economic non-minimal supersymmetric model proposed in the literature after the MSSM. In conclusion, it is very important that the upgraded Tevatron collider has the physics potential to probe the exciting hypothesis of a light charged Higgs boson in top decays t → H + b [23] and analyze its possible consequences within the framework of the MNSSM. Acknowledgements We wish to thank Manuel Drees for clarifying comments regarding Ref. [22] which led to the second footnote in our paper. 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Quirós, A. Riotto, Nucl. Phys. B 439 (1995) 466, Erratum. [20] ALEPH Collaboration, R. Barate et al., Phys. Lett. B 495 (2000) 1; L3 Collaboration, M. Acciarri et al., Phys. Lett. B 495 (2000) 18. [21] Numerical values are obtained by the Fortran code mnssm which is available from http://pilaftsi.home.cern.ch/pilaftsi/. [22] M. Drees, E. Ma, P.N. Pandita, D.P. Roy, S.K. Vempati, Phys. Lett. B 433 (1998) 346. [23] T. Affolder et al., CDF Collaboration, http://www-cdf.fnal. gov/physics/preprints/cdf5124_charged_higgs_prd.ps. 26 April 2001 Physics Letters B 505 (2001) 191–196 www.elsevier.nl/locate/npe Near-horizon conformal structure of black holes Danny Birmingham a , Kumar S. Gupta b , Siddhartha Sen b,1 a Department of Mathematical Physics, University College Dublin, Belfield, Dublin 4, Ireland b Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Calcutta 700 064, India Received 12 February 2001; received in revised form 6 March 2001; accepted 13 March 2001 Editor: P.V. Landshoff Abstract The near-horizon properties of a black hole are studied within an algebraic framework, using a scalar field as a simple probe to analyze the geometry. The operator H governing the near-horizon dynamics of the scalar field contains an inverse square interaction term. It is shown that the operators appearing in the corresponding algebraic description belong to the representation space of the Virasoro algebra. The operator H is studied using the representation theory of the Virasoro algebra. We observe that the wave functions exhibit scaling behaviour in a band-like region near the horizon of the black hole. 2001 Elsevier Science B.V. All rights reserved. 1. Introduction The relation between the physics of black holes and conformal field theory has been explored recently in a variety of contexts [1–3]. In particular, the nearhorizon symmetry structure of general black holes in arbitrary dimensions (including the Schwarzschild case) has been studied [2,3]. By imposing suitable boundary conditions at the horizon, it was shown that the relevant algebra of surface deformations contains a Virasoro algebra in the (r − t)-plane. This analysis is based on an extension of the Brown–Henneaux algebra of three-dimensional anti-de Sitter gravity [4]. In the latter case, purely classical considerations lead to the existence of an asymptotic symmetry algebra containing two copies of the Virasoro algebra. E-mail addresses: [email protected] (D. Birmingham), [email protected] (K.S. Gupta), [email protected] (S. Sen). 1 On leave from: School of Mathematics, Trinity College Dublin, Ireland. In a separate line of development, it was found that the dynamics of particles or scalar fields near the horizon of a black hole is associated with a Hamiltonian containing an inverse square potential [5–7]. Since the scalar field can be viewed as a tool to probe the near-horizon geometry of the black hole, its dynamics should reveal any underlying symmetry of the system. Indeed, such a Hamiltonian was shown to have conformal symmetry quite some time ago [8], and this idea has been further explored recently [5,6,9]. In this Letter, we provide a synthesis of the ideas appearing in the above approaches within an algebraic framework. Since our essential interest is in the nearhorizon geometry of the black hole, it is useful to restrict attention to a very simple probe. Thus, we consider the time-independent modes of a scalar field in the black hole background. In particular, we study the case of a such a field in the background of a Schwarzschild black hole. The Klein–Gordon operator H governing the dynamics of the probe contains an inverse square potential term [7]. It is shown that H can be written in a factorized form, 0370-2693/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 3 5 4 - 9 192 D. Birmingham et al. / Physics Letters B 505 (2001) 191–196 which leads to an algebraic description of the system in terms of the enveloping algebra of the Virasoro algebra. The inverse square interaction term plays a crucial role in obtaining this result. It may be noted that previous works in this direction did not treat the interaction term algebraically. Incorporating this term within the algebraic framework leads to a structure that contains approximately half of all the Virasoro generators. The requirement of a unitary representation of the resulting algebra allows us to include the remaining generators. We then describe the spectrum of H in terms of the wedge representations of the Virasoro algebra [10]. It should be noted that the inverse square term also plays an important role in determining the selfadjoint extensions of the Klein–Gordon operator [7]. In general, the corresponding wave functions violate scaling in the near-horizon region. However, we show that scaling behaviour is present in a small band-like region near the horizon, for certain choices of the selfadjoint extension. These self-adjoint extensions thus play a crucial role in providing a consistent picture of the whole analysis. The existence of this band region is reminiscent of the stretched horizon picture of black holes, see for example [11], and also [12]. This Letter is organized as follows. In Section 2, we study the example of a scalar field probing the nearhorizon properties of the Schwarzschild black hole. The operator governing the dynamics of the timeindependent modes is written in a factorized form. It is shown that the resulting factors lead to an algebraic description in terms of the enveloping algebra of the Virasoro generators. Section 3 discusses the properties of this algebra in terms of the wedge representations of Ref. [10]. This leads to an algebraic description of the spectrum of the time-independent Klein–Gordon operator. The near-horizon scaling behaviour at the quantum level is discussed in Section 4. We conclude in Section 5 with a brief discussion regarding the application of our results to more general black holes. independent modes of the scalar field. The Klein– Gordon operator governing the near-horizon dynamics can then be written as [7] H =− In this section, we consider the case of a scalar field probing the near-horizon geometry of a Schwarzschild black hole. We shall restrict the analysis to the time- (2.1) where a is a real dimensionless constant, and x ∈ [0, ∞] is the near-horizon coordinate. For the Schwarzschild background, we have a = − 14 . For the moment, however, we can consider a general value of a. The essential point to note is that the operator H can be factorized as H = A+ A− , (2.2) where A± = ± and 1 b= ± 2 b d + , dx x √ (2.3) 1 + 4a . 2 (2.4) We note that a = − 14 is the minimum value of a for which b is real. For real values of b, A+ and A− are formal adjoints of each other (with respect to the measure dx), and consequently H is formally a positive quantity (there are some subtleties to this argument arising from the self-adjoint extensions of H which will be discussed later). When a < − 14 , b is no longer real and A+ and A− are not even formal adjoints of each other. However, H can still be factorized as in Eq. (2.2), but it is no longer a positive definite quantity. It can still be made self-adjoint [13], but remains unbounded from below; this case has been analyzed in [14]. Let us now define the operators Ln = −x n+1 Pm = 2. Algebraic formulation of the near-horizon dynamics a d2 + , dx 2 x 2 1 , xm d , dx n ∈ Z, m ∈ Z. (2.5) (2.6) In terms of these operators, A± and H can be written as A± = ∓L−1 + bP1 , (2.7) H = (−L−1 + bP1 )(L−1 + bP1 ). (2.8) D. Birmingham et al. / Physics Letters B 505 (2001) 191–196 Thus, L−1 and P1 are the basic operators appearing in the factorization of H . Taking all possible commutators of these operators between themselves and with H , we obtain the following relations [Pm , Pn ] = 0, [Lm , Pn ] = nPn−m , (2.9) (2.10) c 3 m − m δm+n,0 , 12 (2.11) [Pm , H ] = m(m + 1)Pm+2 + 2mL−m−2 , (2.12) [Lm , Ln ] = (m − n)Lm+n + [Lm , H ]= 2b(b − 1)P2−m − (m + 1) × (L−1 Lm−1 + Lm−1 L−1 ). (2.13) Eq. (2.11) describes a Virasoro algebra with central charge c. Note that the algebra of the generators defined in Eq. (2.5) would lead to [Lm , Ln ] = (m−n)Lm+n . However, this algebra is known to admit a non-trivial central extension. Moreover, for any irreducible unitary highest weight representation of this algebra, c = 0 [15]. For these reasons, we have included the central term explicitly in Eq. (2.11). Eqs. (2.9)–(2.11) describe the semidirect product of the Virasoro algebra with an abelian algebra satisfied by the shift operators {Pm }. Henceforth, we denote this semidirect product algebra by M. Note that L−1 and P1 are the only generators that appear in H . Starting with these two generators, and using Eqs. (2.12) and (2.13), we see that the only operators which appear are the Virasoro generators with negative index (except L−2 ), and the shift generators with positive index. Thus, Lm with m 0 and Pm with m 0 do not appear in the above expressions. In the next section, we will discuss how these quantities are generated. Although the algebra of Virasoro and shift generators has a semidirect product structure, the operator H , however, does not belong to this algebra. This is due to the fact that the right-hand side of Eq. (2.13) contains products of the Virasoro generators. While such products are not elements of the algebra, they do belong to the corresponding enveloping algebra. The given system is thus seen to be described by the enveloping algebra of the Virasoro generators, together with the abelian algebra of the shift operators. This algebraic system has been extensively studied in the literature [10]. 193 3. Representation We wish to discuss the representation theory of the algebra M, and the implications for the quantum properties of the Klein–Gordon operator H . The eigenvalue equation of interest is H |ψ = E|ψ, (3.1) with the boundary condition that ψ(0) = 0. We are especially interested in the bound state sector of H . As we have seen, the operator H can be expressed in terms of certain operators that belong to the algebra M. This observation allows us to give a description of the states of H in terms of the representation spaces of M. We first recall the relevant aspects of the representation theory of M. Following [10], we introduce the space Vα,β of densities containing elements of the form P (x)x α (dx)β , where α, β are complex numbers, in general. Here, P (x) is an arbitrary polynomial in x and x −1 , where x is now treated as a complex variable. It may be noted that the algebra M remains unchanged even when x is complex. It is known that Vα,β carries a representation of the algebra M. The space Vα,β is spanned by a set of vectors, ωm = x m+α (dx)β , where m ∈ Z. The Virasoro generators and the shift operators have the following action on the basis vectors ωm , Pn (ωm ) = ωm−n , (3.2) Ln (ωm ) = −(m + α + β + nβ)ωn+m . (3.3) The representation Vα,β is reducible if α ∈ Z and if β = 0 or 1; otherwise it is irreducible. The requirement of unitarity of the representation Vα,β leads to several important consequences. In any unitary representation of M, the Virasoro generators must satisfy the condition L†−m = Lm . In the previous section, we saw that L−2 and Lm for m 0 did not appear in the algebraic structure generated by the basic operators appearing in the factorization of H . However, the requirement of a unitary representation now leads to the inclusion of Lm for m > 0. The remaining generators now appear through appropriate commutators, thus completing the algebra M. Unitarity also constrains the parameters α and β, which must now satisfy the conditions β + β̄ = 1, (3.4) α + β = ᾱ + β̄, (3.5) 194 D. Birmingham et al. / Physics Letters B 505 (2001) 191–196 where ᾱ denotes the complex conjugate of α. Finally, the central charge c in the representation Vα,β is given by c(β) = −12β 2 + 12β − 2. (3.6) The above representation of M can now be used to analyze the eigenvalue problem of Eq. (3.1). We would like to have a series solution to the differential Eq. (3.1), and consequently choose an ansatz for the wave function |ψ given by |ψ = ∞ cn ωn . (3.7) n=0 Furthermore, the operator H , as written in Eq. (2.2), has a well-defined action on |ψ. From Eq. (3.3), it may be seen that L−1 (ωn ) = −(n + α)ωn−1 , (3.8) which is independent of β. Therefore, it appears that an eigenfunction of H may be constructed from elements of Vα,β for arbitrary β. However, the unitarity conditions of Eqs. (3.4), (3.5) put severe restrictions on β, as we shall see below. The indicial equation obtained by substituting Eq. (3.7) in Eq. (3.1) gives α = b, or (1 − b). (3.9) To proceed, we analyze the cases (i) a − 14 , and (ii) a < − 14 separately. (i) a − 14 This is the main case of interest as it includes the value of a for the Schwarzschild background. It follows from Eqs. (2.4) and (3.9) that b and α are real. The unitarity condition of Eq. (3.4) now fixes the value of β = 12 , and the corresponding central charge is given by c = 1. It may be noted that relation of the central charge calculated here to that appearing in the calculation of black hole entropy depends on geometric properties of the black hole in question. We do not address this issue here. Thus, we see that for the Schwarzschild black hole, we have identified the relevant representation space as V1/2,1/2. (ii) a < − 14 In this case, we can write a = − 14 − µ2 where µ ∈ R. It follows from Eq. (2.4), that b = 12 ± iµ. Eq. (3.9) then gives α = 12 ± iµ, or − 12 ∓ iµ. Let us take the case when α = 12 + iµ, the other cases being similar. From Eqs. (3.4) and (3.5), we find β = 12 − iµ. The value of the corresponding central charge is given by c = 1 + 12µ2 . The operator H in this case can be made self-adjoint but its spectrum remains unbounded from below [13,14]. The algebraic description, however, always leads to a well-defined representation. We return now to the eigenvalue problem for the differential operator H , and focus attention on the Schwarzschild background, for which a = − 14 . As already mentioned, we are interested in the bound state sector of H . These states have negative energy and satisfy Eq. (3.1) with energy −E, where E > 0. The wave function satisfies the boundary condition ψ(0) = 0. This may seem contradictory to the statement in Section 2, which claimed that H as written in Eq. (2.2) is a positive quantity. The resolution of this apparent paradox is as follows. It is known that the operator H admits a one-parameter family of self-adjoint extensions labelled by a U (1) parameter eiz , where z is real [7,16,17]. For a = − 14 , there is an infinite number of bound states for a given self-adjoint extension z. In all these cases, A+ and A− are not adjoints of each other, and consequently H is not a positive quantity. The eigenfunctions and eigenvalues of H in this case are given by [7] √ ψn (x) = Nn x K0 En x , (3.10) z π En = exp (1 − 8n) cot , (3.11) 2 2 where n is an integer, Nn is a normalization factor, and K0 is the modified Bessel function. We have thus shown how to obtain the spectrum of H using the representation of the algebra M. In Section 4, we shall analyze the properties of this spectrum in the near-horizon region. 4. Scaling properties As we have seen, the Virasoro algebra plays an important role in determining the spectrum of H . Since this operator is associated with a probe of the near-horizon geometry, one might expect that the corresponding wave functions would exhibit certain scaling behaviour in this region. D. Birmingham et al. / Physics Letters B 505 (2001) 191–196 Firstly, let us recall that the horizon in this picture is located at x = 0. However, the wave functions ψn vanish at x = 0, and, therefore, do not exhibit any nontrivial scaling. Nevertheless, it is of interest to examine the behaviour of the wave functions near the horizon. For x ∼ 0, the wave functions have the form √ ψn = Nn x A − ln En x , (4.1) where A = ln 2 − γ , and γ is Euler’s constant [18]. While the logarithmic term, in general, breaks the scaling property, one notices that it vanishes at the point √ x0 ∼ 1/ En , where the wave functions exhibit a scaling behaviour. The entire analysis so far, including the existence of the Virasoro algebra, is valid only in the near-horizon region of the black hole. Therefore, consistency of the above scaling behaviour requires that x0 belongs to the near-horizon region. The minimum value of x0 is obtained when En is maximum. When the parameter z appearing in the self-adjoint extension of H is positive, the maximum value of En is given by z π cot . E0 = exp (4.2) 2 2 However, when z is negative, the maximum value of En is obtained when n → ∞. In this case, x0 → 0 where, as we have seen before, the wave function vanishes and scaling becomes trivial. We, therefore, conclude that 1 x0 ∼ √ , E0 z > 0, (4.3) is the minimum value of x0 . It remains to show that x0 given by Eq. (4.3) belongs to the near-horizon region. We first note that we are free to set z to an arbitrary positive value. Thus, we consider z > 0 such that cot 2z 1; this is achieved by choosing z ∼ 0. For all such z, we find that x0 is small but nonzero, and thus belongs to the near-horizon region. In effect, we can use the freedom in the choice of z to restrict x0 to the near-horizon region. We now a band-like region ∆ = √ consider √ [x0 − δ/ E0 , x0 + δ/ E0 ], where δ ∼ 0 is real and positive. The region ∆ thus belongs to the nearhorizon region of the black hole. At a point x in the region ∆, the leading behaviour of ψn is given by √ z . ψn = Nn x A + 2πn cot (4.4) 2 195 Thus, all the eigenfunctions of H exhibit a scaling be√ haviour, i.e., ψn ∼ x, in the near-horizon region ∆. It should be stressed that this analysis is made possible by utilizing the freedom in the choice of z. The parameter z, which labels the self-adjoint extensions of H , thus plays a crucial role in establishing the selfconsistency of this analysis. We conclude this section with the following remarks: 1. A particular choice of z is equivalent to a choice of domain for the differential operator H . Physically, the domain of an operator is specified by boundary conditions. A specific value of z is thus directly related to a specific choice of boundary conditions for H . Thus, we see that the system exhibits nontrivial scaling behaviour only for a certain class of boundary conditions. These boundary conditions play a conceptually similar role to the fall-off conditions as discussed in Refs. [2,3]. 2. The analysis above provides a qualitative argument which suggests that the scaling behaviour in the presence of a black hole should be observed within a region ∆. Although ∆ belongs to the near-horizon region of the black hole, it does not actually contain the event horizon. Our picture is thus similar in spirit to the stretched horizon scenario of Refs. [11,12]. 5. Conclusion In this Letter, we have analyzed the near-horizon properties of the Schwarzschild black hole, using a scalar field as a simple probe of the system. We restricted attention to the time-independent modes of the scalar field, and this allowed us to obtain a number of interesting results regarding the near-horizon properties of the black hole. It is possible that more sophisticated probes of general field configurations may lead to additional information. The factorization of H , leading to the algebraic formulation of Section 2, is a process which appears to be essentially classical. However, the central charge in the algebra M goes beyond the classical framework, as it arises from the requirement of a non-trivial representation. As discussed, the algebra appearing in Eqs. (2.9)–(2.13) does not at first contain all the Virasoro generators. The requirement of unitarity of the representation leads to the inclusion of all the 196 D. Birmingham et al. / Physics Letters B 505 (2001) 191–196 generators. It is thus fair to say that the full Virasoro algebra appears in our framework only at the quantum level. The operator H does not belong to M but is contained in the enveloping algebra of the Virasoro generators. The enveloping algebra is the natural tool that is used to obtain representations of M. Thus, even though H is not an element of M, it nevertheless has a well-defined action in any representation of M. It is this feature that makes the algebraic description useful. In Section 3, we summarized some results from the representation theory of M. The operator H is now treated at the quantum level, and the corresponding eigenvalue problem is studied using the representations of M. Unitarity again plays a role in restricting the space of allowed representations. It is interesting to note that for all values of the coupling a − 14 , the value of the central charge in the representation of M is equal to 1. Other black holes which have a in this range would exhibit a universality in this regard. As mentioned in Section 3, the relationship of c calculated here to that appearing in the entropy calculation of a particular black hole would depend on other factors which are likely to break the universality. If a Virasoro algebra is associated with the nearhorizon dynamics, then some reflection of it should appear in the spectrum of H . In particular, we can expect that the wave functions of H in the nearhorizon region should exhibit scaling behaviour. Such a property was indeed found in a band-like region near the horizon. It is interesting to note that this band excludes the actual horizon. This is similar in spirit to the stretched horizon scenario of black hole dynamics. The parameter z describing the self-adjoint extensions of H is restricted to a set of values in this process. This implies that the near-horizon wave functions exhibit scaling behaviour only for a certain class of boundary conditions. It is important to note that boundary conditions also played a crucial role in proving the existence of a Virasoro algebra in Refs. [2,3]. This feature provides a common thread in these different approaches towards the problem. It is known that the near-horizon dynamics of various black holes is described by an operator of the form H [5,7], for different values of a − 14 . Any such operator can be factorised as in Eq. (2.2) and the above analysis will also apply to these black holes. It has been claimed in [2,3] that a Virasoro algebra is associated with a large class of black holes in arbitrary dimensions. It seems plausible that the near-horizon dynamics of probes in the background of these black holes would be described by an operator of the form of H . Acknowledgements K.S.G. would like to thank A.P. Balachandran for discussions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] A. Strominger, JHEP 9802 (1998) 009. S. Carlip, Phys. Rev. Lett. 82 (1999) 2828. S.N. Solodukhin, Phys. Lett. B 454 (1999) 213. J.D. Brown, M. Henneaux, Commun. Math. Phys. 104 (1986) 207. P. Claus, M. Derix, R. Kallosh, J. Kumar, P.K. Townsend, A.V. Proeyen, Phys. Rev. Lett. 81 (1998) 4553. G.W. Gibbons, P.K. Townsend, Phys. Lett. B 454 (1999) 187. T.R. Govindarajan, V. Suneeta, S. Vaidya, Nucl. Phys. B 583 (2000) 291. 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Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1970. 26 April 2001 Physics Letters B 505 (2001) 197–205 www.elsevier.nl/locate/npe Solving the hierarchy problem with brane crystals Steven Corley, David A. Lowe Department of Physics, Brown University, Providence, RI 02912, USA Received 27 January 2001; accepted 27 February 2001 Editor: M. Cvetič Abstract The brane world scenario advocated by Arkani-Hamed et al. transmutes the hierarchy problem into explaining why extra dimensions have sizes much larger than the fundamental scale. In this Letter we discuss possible solutions to this problem by considering the compactified dimensions to be populated by a large number of branes in a crystal lattice. The experimental consequences of this scenario are described, including the presence of large energy gaps in the spectrum of Kaluza–Klein modes. 2001 Published by Elsevier Science B.V. 1. Introduction Around two years ago Arkani-Hamed et al. [1] pointed out that it was consistent with known experiments for extra dimensions to exist with sizes of order a millimeter. The motivation for this observation came from string theory, where additional “curled up” dimensions are required for the consistency of the theory. String theory also allows for the existence of D-branes, which give a way to restrict the Standard Model fields to a three-dimensional slice of the higher dimensional space. Without this additional entrapment of the Standard Model fields to a brane, large extra dimensions would be in immediate contradiction with atomic physics. The hierarchy problem becomes re-expressed as explaining why the size of the extra dimensions r0 is much larger than the fundamental length scale. Denoting the fundamental scale by M∗ (which we can take to be of order 1 TeV), and the four-dimensional E-mail addresses: [email protected] (S. Corley), [email protected] (D.A. Lowe). Planck scale MPl = 1019 GeV, one finds 2 MPl = r0n M∗n+2 , (1.1) for n flat extra dimensions with size r0 . For n = 1, r0 is required to be of cosmological scales, which is immediately ruled out. The n = 2 case requires r0 of order one millimeter. There remains a hierarchy between r0 and 1/M∗ ≈ 10−19 m. Current rounds of experiments [2,3] are beginning to place direct constraints on the simplest scenarios. Short-range gravity experiments now probe down to 200 µm strongly constraining the n = 2 case. Accelerator experiments constrain M∗ 1 TeV [3]. The strongest constraints arise from astrophysical considerations. Production of Kaluza–Klein modes in supernova SN 1987A places a bound M∗ 30 TeV for n = 2 [1]. One of the main theoretical challenges in implementing some of the proposed large extra dimension scenario’s is how to stabilize the size of the extra dimensions, without introducing additional fine tuning problems. This problem should be more readily addressed in the large extra dimension scenario, versus the traditional approach of compactification at the fun- 0370-2693/01/$ – see front matter 2001 Published by Elsevier Science B.V. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 3 4 4 - 6 198 S. Corley, D.A. Lowe / Physics Letters B 505 (2001) 197–205 damental scale, because the analysis may be carried out at the classical level. A variety of solutions have been proposed [4–9] in the warped compactification scenarios, where by warped we mean not only the Randall–Sundrum warped metric scenarios [10–13] but also cases where some other field, e.g., a scalar as in [8], has a non-trivial profile in the extra dimensions. Another proposed solution in the unwarped compactification context of [1] is to have a large number of branes N (one of which would be our world) interacting in such a way that they form a crystal lattice in the internal dimensions [14]. The interbrane separation could be the size of the fundamental length scale, but the size of the extra dimensions would nevertheless be large enough to solve the hierarchy problem for a large enough number of branes. While it is still not clear how to realize such a brane lattice crystal from a more fundamental theory such as string theory, these models are nevertheless of interest from a phenomenological point of view, especially in light of the fact that they could be tested experimentally in the near future. One is still left with the problem of explaining the large integer N . We take the point of view that replacing a fine tuned continuous parameter by a large integer parameter is an improvement, as one can set the integer N once and for all using initial conditions, and under suitable circumstances, this integer will be stable with respect to time evolution. Since the hierarchy is set by a conserved number N , it is automatically stable with respect to radiative corrections. In the following we begin by discussing the approach to brane crystals of Arkani-Hamed et al. [14]. They propose a number of scenarios for stabilizing the extra dimensions. One particularly compelling example does this without introducing additional fine tuning, aside from the usual problem with the fourdimensional cosmological constant. A problem with this scenario is the presence of unbalanced charge on a compact space. Balancing the charge leads us to consider a scenario where neutral non-BPS branes interact via a nearest neighbor potential. We show this does lead to a natural solution of the hierarchy problem. The crystal potential leads to a distinctive experimental signature for this scenario — the existence of large energy gaps in the Kaluza–Klein spectrum. Such energy gaps have also been noticed in the Randall– Sundrum scenario brane lattices discussed in [15,16]. 2. Brane crystal review We begin by reviewing the model considered in [14]. They consider a 3-brane embedded in a universe with 3 large spatial dimensions and n small spatial dimensions. The system is described by a bulk action Sbulk = − d 4+n x − det g4+n × M∗2+n R + Λ − Lmatter + · · · (2.1) and a brane action Sbrane = − d 4 x − det g4 f 4 + · · · , (2.2) where Lmatter is the Lagrangian of the bulk matter fields and the ellipses denote higher derivative terms which may be dropped at low enough energies. Here g4 denotes the induced metric on the brane, g4+n denotes the bulk metric, Λ is the bulk cosmological constant and f 4 is the brane tension. Interaction terms between the branes are not included. In the next section we discuss scenarios where such interaction terms are relevant. The metric is assumed to take the form r n 0 dt 2 − R 2 gij dx i dx j − r 2 gI J dx I dx J , ds 2 = r (2.3) where R = R(t) is the scale factor of the three large dimensions and r = r(t) that of the n small dimensions, with r0 = r(0). 1 Also we use lower case Latin indices i, j, . . . to denote the 3 large spatial dimensions and upper case Latin indices I, J, . . . for the n small spatial dimensions. Inserting this form of the metric into the bulk (2.1), and brane (2.2) actions and integrating over the spatial coordinates results in 2 Ṙ n(n + 2) ṙ 2 3 n+2 n + S = dt R M∗ r0 −6 R 2 r r 2n 0 − (2.4) Vtot(r) , r 1 [14] do not include the conformal factor r −n multiplying the large dimensions. We find it more convenient however to include it as it will lead to diagonalized kinetic terms for r and R in the action below and is the ordinary conformal factor appearing in Kaluza– Klein reductions. S. Corley, D.A. Lowe / Physics Letters B 505 (2001) 197–205 199 after an integration by parts. obtaining this re In √ 3 x det g sult we have assumed that d I J = 1 and n d x det gij = 1. The potential Vtot (r) is given by normalized. The mass can then be read from the action and is given by Vtot (r) = Λr n − κn(n − 1)M∗n+2 r n−2 + f 4 , m2r = (2.5) where the term proportional to κ arises from the curvature of the n dimensional space. For an n-sphere κ = 1 and for an n-torus κ = 0. A similar term could be added for the three large spatial dimensions, but for large R would be negligible so we drop it. For static solutions the equations of motion are simply given by Vtot (r0 ) = 0, Vtot (r0 ) = 0, (2.6) assuming that R(t) = R0 is constant. Note that with the potential (2.5) r0 is fixed entirely in terms of the bulk interaction. For the above potential (2.5) these equations constrain Λ assuming that r0 is chosen to solve the hierarchy problem (1.1). Specifically let’s assume that the brane tension is set by the higherdimensional fundamental scale M∗ so that f 4 ≈ M∗4 . From (2.6), assuming the internal space is an n-sphere, it follows that 4/n M∗n+2 n+4 M∗ ≈ M∗ , Λ≈ (2.7) MPl r02 where we have used (1.1). For N branes with equal tensions, the f 4 term in (2.5) is replaced by Nf 4 . Solving the Vtot(r0 ) = 0 equation leads to MPl 2(n−2)/n N≈ (2.8) . M∗ This varies from 1 for n = 2 to 1020 for n = 6. Note one is still left with an extra fine tuning problem, in order that the bulk cosmological constant satisfy the relation (2.7). Also note the brane number N plays no role in fixing the size of the extra dimensions — this is entirely determined by fine tuning Λ. N is fixed only by requiring the four-dimensional cosmological constant vanish. For a single brane, stability of this solution follows (r ) > 0 where r solves the from the condition Vtot 0 0 equations of motion (2.6). This is straightforward to see by expanding the action to quadratic order in the small perturbation δr where r = r0 + δr and then rescaling δr so that the kinetic term is canonically (r ) r02 Vtot 1 0 . 2 n(n + 2) MPl (2.9) Evaluating this for (2.5) gives mr ≈ 1/r0 . Note the static equations of motion (2.6) and stability condition do not depend on the specific form of Vtot(r), and yield strong constraints on the parameters of more general potentials. This analysis presumes that derivative couplings of the radion to higher spin Kaluza–Klein modes may be neglected, which is not true in general. However this should not change the qualitative conclusions. The analysis also neglects the Hamiltonian constraint, which would take the form 2 Ṙ n(n + 2) ṙ 2 −6 + R 2 r r 2n 0 + Vtot (r) = 0. r M∗n+2 r0n (2.10) This tells us if we really considered perturbations independent of the three noncompact spatial dimensions, we would generate an non-zero energy density everywhere in space, which would lead to expansion or contraction of R. This is easily remedied by generalizing to perturbations localized in the noncompact spatial dimensions. 3. Interbrane forces We now consider in more detail the effect of an interbrane potential on the above analysis. We continue to work in an approximation where the compactification is not warped, i.e., the (4 + n)dimensional metric does not depend on the internal coordinates x I . This presumes the brane separation will be stabilized at a size parametrically larger than the fundamental length scale 1/M∗ , so that treating gravity at the classical level is sufficient. We also continue to treat perturbations homogeneous in the spatial directions, with only time dependence, with the metric ansatz (2.3). 200 S. Corley, D.A. Lowe / Physics Letters B 505 (2001) 197–205 Let us generalize the brane action to N branes, with a Born–Infeld type action Sbrane = − N d 4x I ∂ XJ det gµν − gI J ∂µ X(k) ν (k) k=1 Vbrane r X(l) − X(k) , (3.1) × fk4 + l<k with an interbrane potential Vbrane . Here we have assumed the metric is g = g(t) induced on each brane is the same, and we use X(k) = X(k) (t) to denote worldvolume fields corresponding to the brane positions in the transverse space. Indices µ, ν label the coordinates (t, x i ). The brane potential has been chosen to depend on the proper distance separating the branes. This leads to the factor of r in potential term. In writing the Born–Infeld action we have used the worldvolume diffeomorphism invariance to fix the µ gauge X(k) = x µ . Note that had we included spatial dependence of the fields, the brane potential would take a complicated non-local form, that is difficult to write down explicitly. A useful analogy for the branes interacting with the bulk gravitational field is gravitational waves interacting with a resonant gravitational wave detector, [17], as we will comment further below. A natural candidate for Vbrane is a simple Coulomb coupling. This leads to the most interesting brane crystal scenario studied in [14] with “non-extensive bulk cosmological constant”, where the hierarchy problem was solved without the additional fine tuning associated with the bulk cosmological constant. The radion was stabilized in the infrared using a negative curvature term in the internal space, and the Coulomb force was used to provide a short distance stabilizing force. The difficulty with this picture is that a collection of like charged branes on a compact space carries infinite vacuum energy, since the electric flux has nowhere to end. To remedy this, one could consider brane configurations with zero charge per unit cell. Of course, once branes with opposite sign are present there will be attractive forces. For supersymmetric D-branes in string theory, oppositely charged branes will annihilate. Furthermore, to our knowledge, there are no known neutral and stable branes. The description of D-brane charges as living in K-theory groups [18] however predicts the existence of stable non-BPS branes which carry charge in a finite, or torsion, group. This charge is not associated with a gauge symmetry and there is therefore no Gauss law preventing us from considering N such branes on a compact space. It remains an open question as to whether such branes could be used to construct a stable lattice configuration. For the moment we take a phenomenological point of view and assume that a stable lattice can be constructed. These objects will then not experience a Coulomb interaction. The Van der Waals interaction is one natural interaction between such objects, arising from the interaction of induced electric dipole moments, falling off like 1/r 2n . To obtain a stabilizing potential, this must be combined with a hardcore repulsive interaction. Taking our motivation from molecular crystals, a possible potential would be the n-dimensional version of the Lennard–Jones potential Vbrane (rX) = M∗4 v(M∗ rX), 4n γ 2n β − , v(x) = x x (3.2) where β and γ are both O(1). One could also imagine an ionic lattice of branes, with screened Coulomb interactions. An importance difference with the Coulomb force example is that now Vbrane will scale like the number of branes N , rather than N 2 since nearest neighbor interactions will be dominant. The precise form of the potential will not be important for what follows. We now want to show that the size of the internal space is fixed in terms of the number of branes N , rather than by using the bulk quantities κ and Λ, which generally introduce extra fine tuning problems. We shall therefore set κ and Λ to zero, the former implying that the compact extra dimensions are flat and for simplicity we take their geometry to be the torus (S1 )n . Consider an interbrane potential of the form Vbrane r X(k) − X(l) = M∗4 v X(k) − X(l) rM∗ , (3.3) where v(x) is not fine tuned. The fundamental scale M∗ sets the scale of the interaction. We assume v(x) is short ranged, so only nearest neighbor interactions S. Corley, D.A. Lowe / Physics Letters B 505 (2001) 197–205 are dominant. The static equations of motion are then roughly given by v(x0 ) = O(1) and v (x0 ) = 0. These have solutions x0 = O(1), or more explicitly α , &X ≈ (3.4) rM∗ where &X denotes the coordinate separation between nearest neighbor branes and α is a constant of order 10 or so, which we will discuss momentarily. Using the fact that the coordinate periodicity around any of the S 1 ’s of our extra dimensions is 1 and summing up the &X’s along one of these dimensions yields the static value of r, r0 ≈ α N 1/n . M∗ (3.5) We find therefore that the size of the internal dimensions is set by the number of branes along with the fundamental scale. The value of r0 however was already fixed in (1.1) and thus yields the necessary number of branes 1 1 MPl 2 = n 1032 . N≈ n (3.6) α M∗ α α = 1 corresponds to one brane per fundamental volume in the internal space, saturating the number of branes. We have therefore required that α is of order 10 or larger (but not so large that we have another fine tuning problem) in order that classical gravity be a good approximation. The effective potential induced for r takes the form Vtot (r) = Nf 4 + NVbrane r/N 1/n , (3.7) where integrating out the brane coordinates sets the coordinate distance between neighboring branes to 1/N 1/n . Vanishing of the four-dimensional cosmological constant requires one fine tuning, corresponding to Vtot = 0, but note no additional fine tuning is needed. The mass of the radion may be obtained using (2.9), which gives mr = M∗ /α n/2 . (3.8) We have so far taken vanishing bulk cosmological constant. This is expected at tree-level if supersymmetry is unbroken in the bulk. However, if supersymmetry is broken on the branes then a cosmological constant will appear at one loop and could change the results. If the breaking takes place at the fundamental 201 scale M∗ on the branes then the induced mass splittings in the bulk are given by a tree-level gravitational effect [14] &m2 ≈ N M∗4 M∗2 = , 2 αn MPl (3.9) where the last expression was obtained by evaluating r0 at (3.5). By dimensional analysis, this induces a cosmological constant Λquantum ≈ (&m2 )(4+n)/2 so that the potential gets a contribution of n M∗4+n n x0 r → M∗4 N , Vquantum(r) ≈ (4+n)n/2 α α (4+n)/2 (3.10) where x0 was defined above. Therefore at x0 ≈ α the induced cosmological constant contribution to the potential is subdominant and our original estimates above still apply. We expect the brane crystal scenario will only work when the number of extra dimensions n 3. This follows from [19] where it is shown that classical crystal lattices do not exhibit long-range order in dimensions two or less. For M∗ ≈ 1 TeV, (1.1) places r0 ≈ 10−7 m for n = 3. 4. Experimental consequences To understand the experimental consequences of our scenario we must investigate the spectrum of the theory as well as the couplings to Standard Model fields. To do this it is convenient to first fix the (4 + n)-dimensional diffeomorphism invariance. For linearized perturbations hMN about a flat metric g̃MN (with M, N labeling the (4 + n)-dimensional space) an infinitesimal diffeomorphism generated by a vector ξM acts on the metric as hMN → hMN + ∇M ξN + ∇N ξM , (4.1) where we have decomposed the metric into a background part g̃MN and a fluctuation hMN as gMN = g̃MN + hMN . (4.2) I ’s this will act as On the X(k) XI → XI − ξ I . (4.3) 202 S. Corley, D.A. Lowe / Physics Letters B 505 (2001) 197–205 Part of the gauge symmetry can be fixed by demanding ∂ M h̄MN = 0, (4.4) where 1 h̄MN = hMN − g̃MN h, (4.5) 2 and indices are being raised and lowered with the background metric. This gauge choice does not fix the diffeomorphism invariance corresponding to vectors satisfying the (4 + n)-dimensional wave equation ∂ M ∂M ξN = 0. We begin by expanding the Born–Infeld piece of the brane action for a general perturbation d 4 x det Gµν = d 4 x det g̃µν × 1 + 12 g̃ µν nµν + 18 g̃ γ β g̃ µν − g̃ µγ g̃ νβ − g̃ µγ g̃ νβ nγ β nµν + ··· , (4.6) where Gµν is the full induced metric on the brane, and ∂XI ∂XI ∂XI ∂XJ + h + g̃ . I ν I J ∂x ν ∂x µ ∂x µ ∂x ν (4.7) Since hµν picks up a nontrivial second-order contribution under diffeomorphisms of the form ξ µ = 0, ξ I = 0 nµν = hµν + hµI hµν → hµν + hµI ∂ν ξ I + hνI ∂µ ξ I , (4.8) as may be seen by expanding (4.1), it is convenient to write hµν = jµν + hµI hIν , so jµν will then be invariant under such diffeomorphisms. This redefinition also makes clear that (4.6) gives rise to a mass term for the four-dimensional vector fields hµI . A different approach to seeing the vector fields become massive is discussed in [1]. However since we are tuning the fourdimensional cosmological constant to zero, the overall coefficient of (4.6) will vanish, and the vectors will be massless (as is the graviton). To examine terms arising from the brane potential, we again restrict to perturbations independent of the spatial directions. We gauge fix the fluctuations of I to zero, as explained in the brane coordinates X(k) more detail in Appendix A. At quadratic order in the fluctuations, then only the radion modes h̄I J couple to I in the action. the brane coordinates X(k) The modes independent of the internal dimensions will therefore be a massless graviton j¯µν , a set of massless vector fields h̄Iµ , and a set of massive radion scalars h̄I J with mass given by (3.8), following through the same calculation. Each of these modes will be at the bottom of a tower of Kaluza–Klein states which are standing waves in the internal dimensions. These may be treated in the same way as Bloch waves [20]. For the low lying modes, the effects of interactions may be neglected (at least for sufficiently large α). This is precisely analogous to the case of gravitational waves propagating through a resonant detector, where one needs to go to next order in the equations of motion to see the effect of gravitational interactions on the response of the detector [17]. This gives rise to a typical Kaluza–Klein spectrum for the spin-two and vector modes mk = |k/r0 |, while for the radion modes m2k = m2r + (k/r0 )2 . It is also interesting to calculate the energy band gap at the edge of the Brillouin zone boundary, where we have standing waves commensurate with the lattice spacing of the crystal. In general, interaction effects will become large there. For the radion modes, we can get a reasonable estimate of this band gap by taking into account only the interaction of the radion through the brane potential. For plane wave modes propagating in the I th direction, the equations of motion are the same as that of an electron moving in a periodic array of delta function potentials. This is a special limit of the Kronig–Penney model [21]. The wavefunction takes the form r(x) = eikx u(x), (4.9) xI where x is the in question, and u(x) is periodic under lattice translations. Solving the equations of motion for u(x) a linear combination of e±iKx yields cos(ka) = cos(Ka) + 1 sin(Ka) , 2α n−2 Ka (4.10) where a is the lattice spacing r0 /N 1/n . Here Ka is to be identified with αm/M∗ , where m is the mass of the mode. For the first Brillouin zone boundary k = π/a. Solving this equation yields the band gap &m2 ∼ M∗2 /α n . This is the same form as the mass gap of the radion near k = 0. S. Corley, D.A. Lowe / Physics Letters B 505 (2001) 197–205 203 Fig. 1. Illustration of expected band structure in the spectrum of Kaluza–Klein states. The mass squared is plotted versus the wavevector in the internal dimension k times the lattice spacing a ∼ α/M∗ . The orders of magnitude of the scales of the band width and band gaps are indicated. Higher order interactions also lead to a band gap for the spin-two and vector modes at the Brillouin zone boundary. Estimating the energy difference between a standing wave with nodes on the branes versus a standing wave with peaks on the branes leads to the same calculation as in (3.9). We therefore expect the band gap &m2 to be of roughly the same order of magnitude as for the radion modes. The picture of the band structure that emerges is illustrated in Fig. 1. The Standard Model fields are coupled to the induced metric on the brane via the usual covariant couplings. Expanding these terms about the background metric we find that the coupling between the bulk metric fluctuations discussed above and the Standard Model fields will be suppressed by 1/MPl . The analysis of the phenomenological constraints of [1] will therefore carry over to the brane crystal model unchanged. Acknowledgements We thank A. Houghton, R. Myers, R. Pelcovits and L. Randall for helpful discussions. This research is supported in part by DOE grant DE-FE0291ER40688Task A. Appendix A. Gauge fixing the brane coordinates As noted above, the linearized Einstein equations in covariant gauge ∂ M h̄MN have a residual gauge freedom associated with diffeomorphisms satisfying ∂ M ∂M ξN = 0. This freedom can be used to set the fluctuations of the transverse brane coordinates to zero given some assumptions. Specifically the brane 204 S. Corley, D.A. Lowe / Physics Letters B 505 (2001) 197–205 coordinates transform under a diffeomorphism as µ µ I I x → X(k) x − ξ I x µ, x J , X(k) (A.1) where we have assumed that ξ µ = 0. To set the brane coordinate fluctuation to zero therefore only requires fixing ξ I on the brane, and it’s extension off the brane can be chosen at our convenience. In this case we choose it’s extension off the brane so that it solves the (4 + n)-dimensional wave equation. In general this will not be possible. This follows by noting that the ξ I live in the bulk spacetime and therefore must satisfy the periodicity conditions of the extra dimensions. As a result the ξ I cannot have arbitrary dependence on the x µ coordinates if they are to satisfy the wave equation. To be explicit consider a mode decomposition of ξ I where one of the modes satisfies ∂ µ ∂µ ξ I = m2 ξ I . For ξ I to satisfy the (4 + n)-dimensional wave equation we then need ∂ J ∂J ξ I = m2 ξ I . For generic values of m there will be no solution to this equation that satisfies the required periodicity conditions in the extra dimensions. Rather the 4-dimensional mass m will be quantized according to m = q12 + · · · + qn2 /r0 for arbitrary integers qi . There is no such constraint on the dependence of the brane coordinates on x µ however because they are functions only of the x µ . We assume for simplicity nevertheless that the brane coordinate fluctuations can be gauged away in this manner. In other words we assume that the mode decomposition of the brane coordinates contains only fluctuations with the quantized masses given above therefore allowing us to gauge them away. Appendix B. Alternate derivation of linearized equations Given these gauge conditions described in Appendix A, we now give a more general derivation of the linearized equations of motion describing the metric fluctuations. We could try to write down a potential term describing the interactions between the branes and then gauge fixing as described above, but this turns out to be somewhat subtle. We therefore argue using symmetry considerations. The bulk contribution comes only from the Einstein–Hilbert term in the action (2.2). In the gauge (4.4) it is well known that this contributes only ∂ P ∂P h̄MN to the linearized equations of motion. From the brane terms in the action we expect the linearized equations of motion to contain a sum of δ-function terms in the extra dimensions corresponding to the fixed brane positions with coefficients determined by symmetry and dimensional analysis. Specifically we find M4 a(k) h̄µν + b(k) g̃µν h̄4 ∂ M ∂M h̄µν = 2∗ MPl (k) I , × δ (n) XI − X(k) 4 M I c(k) hµI δ (n) XI − X(k) ∂ M ∂M hµI = 2∗ , MPl (k) ∂ M ∂M h̄I J = M∗4 d(k)h̄I J + e(k) g̃I J h̄n 2 MPl (k) I × δ (n) XI − X(k) , (B.1) where h̄4 = g̃ µν h̄µν , h̄n = g̃ I J h̄I J , and the coefficients a(k), . . . , e(k) are all O(1). The overall factor 2 on the right-hand sides of all equations of M∗4 /MPl is easy to understand by going to a coordinate sysI = r0 XI . In these coordinates M∗ is the only tem X dimensionful parameter so that M∗2−n would have to be the overall coefficient following from dimensional analysis. Going back to the XI then yields the above coefficient. Decomposing the fluctuations h̄MN into eigenstates of the 4-dimensional wave operator yields ∂ M ∂M h̄MN = −r0−2 ∂I ∂I + m2 h̄MN (B.2) for a mode with 4-dimensional mass m. It is now straightforward to estimate the spectrum of metric fluctuations following the discussion around (4.9). In particular taking a plane wave ansatz for the metric fluctuations one recovers the relation (4.10) from which we find that the first excited state and mass gap energies will be or order m2 , &m2 ≈ M∗2 /α n , respectively. The discussion so far applies for all three equations of motion in (B.1), so in particular it implies that the lowest energy fluctuation of hµν would have four-dimensional mass of order M∗ /α n/2 . This is of course unacceptable if we are to recover Newtonian gravity on our brane. For the hµν equation therefore we must tune the a(k) and b(k) coefficients so that we have a massless fluctuation, or massless fourdimensional graviton. This corresponds to fine tuning S. Corley, D.A. Lowe / Physics Letters B 505 (2001) 197–205 the four-dimensional cosmological constant to zero. Note however that this fine tuning will not in general imply that the low lying vector fluctuations hµI or radion fluctuations hI J will be massless. [7] [8] [9] [10] References [11] [1] N. Arkani-Hamed, S. Dimopoulos, G. Dvali, Phys. Lett. B 429 (1998) 263; N. Arkani-Hamed, S. Dimopoulos, G. Dvali, Phys. Rev. D 59 (1999) 086004; I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos, G. Dvali, Phys. Lett. B 436 (1998) 257. [2] G.L. Smith, C.D. Hoyle, J.H. Gundlach, E.G. Adelberger, B.R. Heckel, H.E. Swanson, Short-range tests of the equivalence principle, Phys. Rev. D 61 (2000) 022001; C.D. Hoyle, U. Schmidt, B.R. Heckel, E.G. Adelberger, J.H. Gundlach, D.J. Kapner, H.E. Swanson, Submillimeter tests of the gravitational inverse square law: a search for ‘large’ extra dimensions, hep-ph/0011014. [3] G. Landsberg, Mini-review on extra dimensions, hep-ex/ 0009038. [4] W.D. Goldberger, M.B. Wise, Phys. Rev. Lett. 83 (1999) 4922, hep-ph/9907447. [5] I. Antoniadis, C. Bachas, Phys. Lett. B 450 (1999) 83, hep-th/ 9812093. [6] N. Arkani-Hamed, S. Dimopoulos, J. March-Russell, Logarithmic unification from symmetries enhanced in the submil- [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] 205 limeter infrared, in: M.A. Shifman (Ed.), The Many Faces of the Superworld, 1999, pp. 627–648, hep-th/9908146. G. Dvali, Phys. Lett. B 459 (1999) 489, hep-ph/9905204. N. Arkani-Hamed, L. Hall, D. Smith, N. Weiner, Phys. Rev. D 62 (2000) 105002, hep-ph/9912453. A.G. Cohen, D.B. Kaplan, Phys. Lett. B 470 (1999) 52, hep-th/ 9910132. L. Randall, R. Sundrum, Phys. Rev. Lett. 83 (1999) 3370, hepph/9905221. L. Randall, R. Sundrum, Phys. Rev. Lett. 83 (1999) 4690, hepph/9906064. M. Gogberashvili, Hierarchy problem in the shell universe model, hep-ph/9812296. M. Gogberashvili, Europhys. Lett. 49 (2000) 396, hep-ph/ 9812365. N. Arkani-Hamed, S. Dimopoulos, J. March-Russell, Stabilization of sub-millimeter dimensions: the new guise of the hierarchy problem, hep-th/9809124. S. Nam, JHEP 04 (2000) 002, hep-th/9911237. N. Kaloper, Crystal manyfold universes in AdS space, Phys. Lett. B 474 (2000) 269, hep-th/9912125. See for example, C.W. Misner, K.S. Thorne, J.A. Wheeler, Gravitation, Freeman, San Francisco, 1980. E. Witten, D-branes and K-theory, JHEP 9812 (1998) 019, hep-th/9810188. N.D. Mermin, Crystalline order in two dimensions, Phys. Rev. 176 (1968) 250. See for example, C. Kittel, Introduction to Solid State Physics, Wiley, 1995. R. de L. Kronig, W.G. Penney, Proc. R. Soc. London A 130 (1931) 499. 26 April 2001 Physics Letters B 505 (2001) 206–214 www.elsevier.nl/locate/npe Twisted Kac–Moody algebras and the entropy of AdS3 black hole Sharmanthie Fernando a , Freydoon Mansouri b a Physics Department, Northern Kentucky University, Highland Heights, KY 41099, USA b Physics Department, University of Cincinnati, Cincinnati, OH 45221, USA Received 26 October 2000; received in revised form 14 January 2001; accepted 9 March 2001 Editor: M. Cvetič Abstract We show that an SL(2, R)L × SL(2, R)R Chern–Simons theory coupled to a source on a manifold with the topology of a disk correctly describes the entropy of the AdS3 black hole. The resulting boundary WZNW theory leads to two copies of a twisted affine Kac–Moody algebra, for which the respective Virasoro algebras have the same central charge c as the corresponding untwisted theory. But the eigenvalues of the respective L0 operators are shifted. We show that the asymptotic density of states for this theory is, up to logarithmic corrections, the same as that obtained by Strominger using the asymptotic symmetry of Brown and Henneaux. 2001 Published by Elsevier Science B.V. 1. Introduction The entropy of the AdS3 black hole [1,2], has been investigated from a variety of points of view. Some of the more prominent approaches to this problem have been compared and contrasted by Carlip [3]. In this work we will address this problem in the framework of pure gravity in 2 + 1 dimensions. Within this framework, a direct method of obtaining the entropy of the BTZ black hole was given by Strominger [4], in which use is made of the earlier work of Brown and Henneaux [5]. Using their results, he demonstrated that the asymptotic symmetry of the BTZ black hole is generated by two copies of the Virasoro algebra with central charges cL = cR = 3l , 2G E-mail addresses: [email protected] (S. Fernando), [email protected] (F. Mansouri). (1) where l is the radius of curvature of the AdS3 space, and G is Newton’s constant. Then, assuming that the ground state eigenvalue ∆0 of the Virasoro generator L0 vanishes, he obtained the Bekenstein– Hawking expression for the entropy. As pointed out by Strominger [4], in this derivation one must take for granted the existence of a quantum gravity theory with appropriate symmetries. In the absence of such a quantum theory, there will be no practical way of computing either ∆0 or the value of the classical central charge given by Eq. (1) from first principles. Other approaches to the entropy problem make use of the Chern–Simons theory representation of gravity in 2 +1 dimensions [6,7]. One common feature among them is that to account for the microscopic degrees of freedom of the black hole, the free Chern– Simons theory is formulated on a manifold with boundary [8–10]. Although the significance of the boundary in these works differ, they all lead to WZNW theories [11]. More recently, these scenarios have been further refined, improved, and extended [8,12– 15]. One important feature of a typical conformal 0370-2693/01/$ – see front matter 2001 Published by Elsevier Science B.V. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 3 7 1 - 9 S. Fernando, F. Mansouri / Physics Letters B 505 (2001) 206–214 field theory obtained in this way is that its central charge varies between the rank and the dimension of the gauge group. The relevant gauge groups for the AdS3 black hole are two copies of the group SL(2, R), so that in the corresponding Virasoro algebras the central charges vary in the range 1 c 3. On the other hand, the values of the central charges given by Eq. (1) are very large and seem to be unrelated to Kac–Moody algebras arising from relevant gauge groups. Thus, it appears that in the Chern–Simons approach one reaches an impasse in providing a quantum mechanical basis for the classical results of Brown and Henneaux. In this work, we describe a way to resolve this apparent contradiction by interpreting the classical asymptotic Virasoro algebra of Brown and Henneaux [5] as an “effective” symmetry characterized by an “effective central charge” in the sense defined by Carlip [3]. Then, rather than naively comparing central charges, we derive the consequences of this effective theory, including its “effective central charge” from yet another approach which makes use of Chern–Simons theory but which is physically very different from the ones mentioned above. To begin with, in contrast to previous works, in our approach the Chern–Simons theory is coupled to a source. Then, since the BTZ black hole is a solution of source-free Einstein’s equations [1,2], it is clear that the manifold M on which the Chern–Simons theory is defined cannot be identified with space–time. Instead, as shown in previous work [16,17], the classical black hole space–time can be constructed from the information encoded in the manifold M. In particular, this information supplied, mass, angular momentum, and the all important discrete identification group [1,2] which distinguishes the black hole from anti-de-Sitter space. One important advantage of this point of view is that the manifold M is specified by its topology (no metric). As a result, for a manifold with the topology of, say, a disk, the “size” of M and the location of the boundary relative to the source does not enter into the formalism, and a conformal field theory constructed on its boundary is independent of where that boundary is. In other words, it is unnecessary to specify whether the boundary refers to a horizon or to asymptotic infinity. Just as in obtaining the classical features of the black hole space–time [16,17], the coupling to a source turns out to be essential in arriving at a micro- 207 scopic description of the black hole entropy. In particular, it results in a conformal field theory on the boundary with two copies of a twisted affine Kac–Moody algebra. In the corresponding Virasoro algebra, the value of the central charge remains the same as the theory without a source, but the eigenvalues of the operator L0 are shifted and are nonvanishing. Taking these features as well as the subtleties that arise from the noncompactness of SL(2, R) into account, we find that the asymptotic density of states for this microscopic theory agree with that given by Strominger [4] if we identify the Brown–Henneaux values for the central charge [5] with the effective central charge ceff of our theory. 2. Chern–Simons action and boundary effects For a simple or a semi-simple Lie group, the Chern– Simons action has the form k Ics = (2) Tr A ∧ dA + 23 A ∧ A , 4π M where Tr stands for trace and A = Aµ dx µ . (3) We require the (2 + 1)-dimensional manifold M to have the topology R × Σ, with Σ a two-manifold and R representing the time-like coordinate x 0 . Moreover, we take the topology of Σ to be trivial in the absence of sources, with the possible exception of a boundary. Then, subject to the constraints 1 F b [A] = ij ∂i Abj − ∂j Abi + b cd Aci Adj = 0 (4) 2 the Chern–Simons action for a simple group G will take the form k Ics = dx 0 d 2 x − ij ηab Aai ∂0 Abj + Aa0 Fa , 2π Σ R (5) where i, j = 1, 2. We want to explore the properties of the Chern– Simons theory coupled to a source for the group SL(2, R)L × SL(2, R)R on a manifold with boundary. Since the gauge group is semi-simple, the theory breaks up into two parts, one for each SL(2, R), where by SL(2, R) we mean its infinite cover. So, 208 S. Fernando, F. Mansouri / Physics Letters B 505 (2001) 206–214 to simplify the presentation, we will study a single SL(2, R). Much of what we discuss in this and the next section hold for any simple Lie group, G. Also, to establish our notation, we consider first the theory in the absence of the source. The main features of a Chern–Simons theory on a manifold with boundary has been known for sometime [11,18]. Here, with M = R × Σ, we identify the two-dimensional manifold Σ with a disk D. Then, the boundary of M will have the topology R × S 1 . We parametrize R with τ and S 1 with φ. In this parametrization, the Chern–Simons action on a manifold with boundary can be written as k k Tr AdA + 23 A3 + Aφ Aτ . Scs = (6) 4π 4π M ∂M The surface term vanishes in the gauge in which Aτ = 0 on the boundary. In this action, let A = Ã + Aτ ∂ and d = dτ ∂τ + d̃. Then, the resulting constraint equations for the field strength take the form F̃ = 0. (7) They can be solved exactly by the ansatz [11,18] Ã = −d̃U U −1 , (8) where U = U (φ, τ ) is an element of the gauge group G. Using this solution, the Chern–Simons action given by Eq. (5) can be rewritten as SWZNW 3 k = Tr U −1 dU 12π M k + Tr U −1 ∂φ U U −1 ∂τ U dφ dτ. 4π ∂M This implies an infinite number of conserved currents: Jφ = −kU −1 ∂φ U = Jφa Ta . (12) Here, Ta are the generators of the algebra g of the group G, and Jφ is a function of φ only because ∂τ Jφ = 0. If we expand Jφ in a Laurent series, we obtain Jφ = ΣJn z−n−1 , (13) where z = exp(iφ). As usual, Jn satisfy the Kac– Moody algebra a b c Jn , Jm = fcab Jm+n (14) + kng ab δm+n,0 . The corresponding energy–momentum tensor for the action SWZNW can be computed using the Sugawara– Sommerfield construction. For example, for the gauge group SL(2, R), 1 gab : Jφa (z)Jφb (z) : (k − 2) 1 a = Σ : Jn−m Jma : z−n−2 k−2 = ΣLn z−n−2 , Tφφ = (15) where (9) ∂M We thus arrive at a WZNW action and can take over many result already available in the literature for this model. As in any WZNW theory, the change in the integrand of this action under an infinitesimal variation δU of U is a derivative. We interpret this to mean that U = U (φ, τ ), i.e., it is independent of the third (radial) coordinate of the bulk. In other words, the information encoded in the disk depends only on its topology and is invariant under any scaling of the size of the disk. The above Lagrangian is invariant under the following transformations of the U field [18]: U (φ, τ ) → Ω̄(φ)U Ω(τ ), where Ω̄(φ) and Ω(τ ) are any two elements of G. To obtain the conserved currents, let U → U + δU . The corresponding variation of the action leads to SWZNW → SWZNW + δSWZNW , where k δSWZNW = (11) ∂τ U −1 ∂φ U δU. 2π (10) 1 a Σ : Jn−m (16) Jma : . k−2 The Ln operators satisfy the following Virasoro algebra: c [Ln , Lm ] = (n − m)Ln+m + n n2 − 1 δn+m,0 , 12 (17) with c the central charge. For SL(2, R), it is given by 3k c = k−2 . We note that for large negative values of k, the value of c approaches 3 which is the dimension of the group. We also note that this boundary WZNW theory has one, not the more usual two, Virasoro algebra. It will be shown below that when the Chern– Simons theory is coupled to a source on a manifold Ln = S. Fernando, F. Mansouri / Physics Letters B 505 (2001) 206–214 with the topology of a disk, the central charge of the Virasoro algebra of the corresponding modified WZNW theory remains the same as that in the sourcefree theory discussed above. 209 This Lagrangian is also invariant under the following transformation: U (φ, τ ) → Ω̄(φ)U Ω(τ ), (24) where Ω(τ ) commutes with λ. Varying the action under the above symmetry transformation, we get 3. The coupling of a source δStotal = δSWZNW + δSsource , Next, we couple a source to the Chern–Simons action on the manifold M with disk topology, which, as in the previous section, has the boundary R × S 1 . In general, we take the source to be a unitary representation of the group G. To be more specific, let us consider a source action given by [11,18] Ssource = dτ Tr λω(τ )−1 (∂τ + Aτ )ω(τ ) . (18) Here λ = λi Hi where Hi are elements of the Cartan subalgebra H of G. We will take λa to be appropriate weights. The quantity ω(τ ) is an arbitrary element of G. The above action is invariant under the transformation ω(τ ) → ω(τ )h(τ ), where h(τ ) commutes with λ. Now the total action on M is, k k Stotal = Tr AdA + 23 A3 + Aτ Aφ 4π 4π + dτ Tr λω(τ )−1 (∂τ + Aτ )ω(τ ) . (19) The new constraint equation takes the form, k F̃ (x) + ω(τ )λω−1 (τ )δ 2 (x − xp ) = 0, 2π (20) where xp specifies the location of the source, heretofore taken to be at xp = 0. The solution to the above equation is given by Ã = −d̃ Ũ Ũ −1 , (21) where [18] 1 −1 Ũ = U exp ω(τ )λω (τ )φ . k (22) The new effective action on the boundary ∂M is then 1 Stotal = SWZNW + (23) Tr λU −1 ∂τ U . 2π ∂M where δSsource = 1 2π Tr −U −1 δU U −1 ∂τ U, λ . (25) (26) Hence, the requirement that δStotal = 0 will give rise to the conservation equation [20] ∂τ −kU −1 ∂φ U + U −1 ∂τ U, λ = 0. (27) The first term in this expression has the same structure as the current Jφ of the source free theory. Hence, requiring that U (φ, τ ) = U (φ + τ ), we can write the new current Jˆφ in terms of the current in the absence of the source as Jˆφ = e k (φ+τ ) Jφ e− k (φ+τ ) . λ λ (28) It is easy to check that ∂τ Jˆφ = 0. (29) With the new currents at our disposal, the next step is to see how this modification affects the properties of the corresponding conformal field theory. In this respect, we note from Eq. (28) that our new currents Jˆφ are related to the currents Jφ in the absence of the source by a conjugation with respect to the elements of the Cartan subalgebra H of the group G. This kind of conjugation has been noted in the study of Kac–Moody algebras [19,21,22]: the algebra satisfied by the new currents fall in the category of twisted affine Kac–Moody algebras. So, to understand how the coupling to a source modifies the structure of the source-free conformal field theory, we follow the analysis of reference [19] and express the Lie algebra of the group G of rank r in the Cartan–Weyl basis. Let H i be the elements of the Cartan subalgebra and denote the remaining generators by E α . Then, with label a = (i, α), i α i j H , E = αi E α , H , H = 0, (30) 210 S. Fernando, F. Mansouri / Physics Letters B 505 (2001) 206–214 (α, β)E α+β if α + β is a root, E , E = 2α −2 (α.H ) (31) if α = −β, 0 otherwise. In this expression, 1 i, j r, and α, β are roots. Now we can rewrite the affine Kac–Moody algebra g of the source free theory of the last section in this basis as follows: i j Hm , Hn = kmδ ij δm,−n , i α , Hm , Enα = α i Em+n (32) α β α , Enα Em α+β (α, β)Em+n if α + β is a root, −2 = 2α (α.Hm+n + kmδm,−n ) if α = −β, 0 otherwise. (33) We also note from the last section that in the absence of the source the element L0 of the Virasoro algebra of the source-free theory and the currents Jna have the following commutation relations: L0 , Jna = −nJna . (34) It can be seen from Eq. (28) that the new currents can be viewed as an inner automorphism of the algebra g in the form ζ(J ) = γ J γ −1 . The effect of this on the component currents can be represented by eiχ.H . (35) As a result of this inner automorphism on elements of the algebra g, we obtain a modified algebra ĝ the elements of which in the Cartan–Weyl basis are given by [19] ζ E α = eiχ.α E α . ζ H i = H i, (36) If the map ζ is endowed with the property that ζ N = 1, then we must have Nχ.α = 2nπ , where n is an integer for all roots α g: e iχ.α =e 2π in N , (37) where n is a positive integer N − 1. As far as the currents obtained from the Chern–Simons theory coupled to a source are concerned, all possible values of N are allowed. As we will see in the next section, the choice of a particular value of N requires additional physical input. We also note that the automorphism ζ divides a suitable combination of the generators of ĝ into eigenspaces ĝ(m) . Thus, the basis of ĝ consists of the elements Hmi and Enα where m Z and n (Z + χ.α 2π ). These operators satisfy a Kac–Moody algebra which has formally the same structure as that of g but with rearranged (fractional) values of the suffices. Hence the algebra ĝ can be viewed as the “twisted” version of the algebra g. Since the automorphism which relates the two algebras is of inner variety [19], we must look for features, if any, that distinguish the algebra ĝ from its untwisted version g. These features depend on the extent to which we can undo the twisting. To this end, we introduce a new basis for ĝ α Ênα = En+ χ.α , 2π Ĥni = Hni + k i χ δn,0 . 2π (38) The new operators, Ênα and Ĥni satisfy the same commutation relations as the elements of the untwisted affine Kac–Moody algebra g. The corresponding conformal field theories are not identical, however. This can be seen most easily if we express the Virasoro generators L̂n of the twisted theory in terms of untwisted generators: L̂m = Lm − 1 i i k i i χ Hn + χ χ δn,0 . 2π 4π 2 (39) In particular, we get for L̂0 , L̂0 = L0 − 1 i i k i i χ H0 + χ χ . 2π 4π 2 (40) Thus the eigenvalues ∆ˆ of the operator L̂0 are shifted relative to the eigenvalues ∆ of L0 . But, as can be verified directly, the value of the central charge c remains unchanged [19,21,22]. More specifically, we have ˆ µ = ∆| ˆ ∆, ˆ µ, L̂0 |∆, (41) where µ is a weight and 1 i i k i i ∆ˆ = ∆ − (42) χ µ + χ χ . 2π 4π 2 So, for the highest (lowest) weight states, we get 1 i i 1 χ µ0 + (43) kχ 2 . 2π 4π 2 With minor exceptions, most of the derivation of our twisted Kac–Moody algebras from the Chern–Simons theory applies to any gauge group. But the relation ∆ˆ 0 = ∆0 − S. Fernando, F. Mansouri / Physics Letters B 505 (2001) 206–214 between the irreducible representations of an affine Kac–Moody algebra and its Lie subalgebra imposes restrictions on the value of the central term k. For example, for SU(2), the value of k is restricted to the nonnegative values [19]. But for discrete unitary representations of SL(2, R) with a lowest weight, the quantity k is restricted to [23] k < −1. (44) It follows that in this case large negative values of k are allowed. We will take advantage of this feature in the application of this formalism to entropy of the AdS3 black hole in the next section. 4. The entropy of the AdS3 black hole As pointed out in the introduction, in the derivation of the entropy of the AdS3 black hole by Strominger [4], use was made of the expression for the central charges of the two asymptotic Virasoro algebras obtained by Brown and Henneaux [5] using classical (nonquantum) arguments. They are given by 3l , (45) 2G where l is the radius of curvature of the AdS3 space, and G is Newton’s constant. The presence of such a symmetry indicates that there is a conformal field theory at the asymptotic boundary [24]. It was shown by Strominger that the BTZ solution satisfies the Brown–Henneaux boundary conditions so that it possessed an asymptotic symmetry of this type. So, he identified the degrees of freedom of the black hole in the bulk with those of the conformal field theory at the infinite boundary. Then, using Cardy’s formula [25] for the asymptotic density of states, he showed that for l G the entropy of this conformal field theory is given by cL = cR = 2πr+ , (46) 4G in agreement with Bekenstein–Hawking formula. Here, the quantity r+ is the outer horizon radius of the black hole. An important assumption in this derivation was that the ground state eigenvalue ∆0 of the operator L0 vanishes. The formula by Cardy [25] for the asymptotic density of states, leading to the above expression for S= 211 entropy is given by ρ(∆) ≈ exp 2π c∆/6 , (47) where ρ(∆) is the number of states for which the eigenvalue of L0 is ∆. The result holds when ∆ is large and the lowest eigenvalue ∆0 vanishes. From the analysis of the previous section, it is clear that in the conformal field theory arising from a Chern–Simons theory coupled to a source the eigenvalue ∆ˆ 0 does not vanish, so that the above Cardy formula must be appropriately modified. In such a case, the asymptotic density of states for large ∆ is given by [3]: ρ(∆) ≈ exp 2π (c − 24∆0 )∆/6 ρ(∆0 ) = exp 2π ceff ∆/6 ρ(∆0 ). (48) Thus, it is the latter formula which must be used in the application of our formalism to the black hole entropy. It is important to note that the expression for the asymptotic density of states given by Eq. (48) rests on the existence of a consistent conformal field theory with a well defined partition function. For Kac–Moody algebras based on compact Lie groups this can be established rigorously. But for Kac– Moody algebras based on noncompact groups such as SL(2, R), no general proof exists. So, all the conformal field theories based on SL(2, R), which have been made use of in connection with the AdS3 black hole, including the present work, share this common weakness. We want to show that the results obtained from such a microscopic analysis are in agreement with those given by Strominger [4]. In so doing, we will rely heavily on our previous results which dealt with the understanding of the macroscopic features of the BTZ black hole [16,17]. We recall from these references that the unitary representations of SL(2, R) which are relevant to the description of the macroscopic features of the black hole are the infinite-dimensional discrete series which are bounded from below. These irreducible representations are characterized by a label F which can be identified with the lowest eigenvalue of the SL(2, R) generator which is being diagonalized. In the literature of the SL(2, R) Kac–Moody algebra [23], this (lowest weight) label, which is convenient for the description of this series, is often referred to as −j . Thus, the Casimir eigenvalues in the 212 S. Fernando, F. Mansouri / Physics Letters B 505 (2001) 206–214 two notations are related according to j (j + 1) = F (F − 1). (49) On the other hand, in the description of the black holes in terms of a Chern–Simons theory with gauge group SL(2, R)L × SL(2, R)R , the mass and the angular momentum of the black hole are related to the Casimir invariants of j±2 of this gauge group as follows [16,17]: 1 j±2 = (lM ± J ). 2 So, for positive mass black holes we must have (50) 1 (51) 1 + 1 + 2(lM ± J ) . 2 In particular [16,17], for the ground state F± ≈ 1, we must have 2(lM0 + J0 ) 1. We will identify this quantity with the ratio of the two scales of the theory: F± = G . (52) l Next, consider the determination of the Chern– −1 Simons couplings k± . These were referred to as a± in Ref. [16,17]. Since the gauge group SL(2, R)L × SL(2, R)R is semi-simple, the couplings k+ and k− are independent. Also, since the two SL(2, R) groups play a parallel role in our approach, we will focus on determining one of them, say, k+ . The other one can be obtained in a similar way. To this end, we recall from references [16,17] that in our approach, the manifold M on which the Chern–Simons theory is defined is not space–time. This means that from the data encoded in M we must be able to obtain all the features of the classical black hole space–time. One of these is the discrete identification group of the black hole [1,2]. This was obtained within our framework by considering the holonomies around the source in M. In this respect, we note that the Cartan sub-algebra of SL(2, R) is one-dimensional, and the corresponding weight is F as discussed above. Then, using nonabelian Stokes theorem [26] and Eq. (20), the holonomies can be evaluated. To get the correct discrete identification group, this implies that [16,17] 1 G 2F+ = ± (r+ + r− ) = ±2 2 (lM0 ± J0 ), (53) k+ l l 2(lM0 + J0 ) ≈ where r+ and r− are, respectively, the outer and the inner horizon radii of the black hole. A similar analysis can be carried out for k− . Using the values of F and 2(lM0 + J0 ) for the ground state given above, we find for both couplings k+ = k− = ± l . G (54) The sign of k± is not fixed by holonomy considerations alone. But from the discussion of the last section it is clear that we must choose the negative sign for both since SL(2, R) is noncompact. Our determination of k± is to be compared with other approaches in one or both k’s are taken to be positive. In one approach [14], e.g., the manifold M was taken to be space–time, and the free Chern– Simons action led to the classical black hole solution in M. In that case, the signs of k+ and k− are opposite each other, so that one of them would have to be positive. This appears to be inconsistent with what we know about SL(2, R) Kac–Moody algebras. Next, consider the ground state eigenvalue ∆ˆ 0 for one of the two SL(2, R) Virasoro algebras. Since the Cartan subalgebra is one-dimensional, the sums in Eq. (43) consist of one term each, and, from Eq. (37), the quantity χ is given by χ= 2πn . Nα (55) The quantity µ0 in Eq. (43) is clearly the weight of the ground state, i.e., the weight F ≈ 1 described above. We also note that the root α is the weight of the adjoint representation of SL(2, R), so that α = 1. Then, Eq. (44) specialized to the case at hand will take the form 2 n n ˆ . ∆0 = ∆0 − + k (56) N N Also substituting for k from Eq. (54), we get l n 2 n . ∆ˆ 0 = ∆0 − − N G N (57) In this expression, the quantity ∆0 is the ground state eigenvalue of L0 , and its value is not known but is often taken to be zero without a priori justification. For our purposes, it is only necessary that it be small compared to the last term. Let us now compute the effective central charge ceff , defined via Eq. (48), for our theory. It is given by S. Fernando, F. Mansouri / Physics Letters B 505 (2001) 206–214 ceff = c − 24∆ˆ 0 = c − 24∆0 − 24 l n n + 24 N G N 2 . (58) In this expression, l/G 1 whereas 1 c 3, and the n/N is a fraction less than one. Assuming that ∆0 is also relatively small, we get l n 2 ceff ≈ 24 (59) . G N As for the ratio (n/N), we pointed out that its determination more requires physical input. The key point to keep in mind is that its value does not affect the qualitative features of the answer to the AdS3 black hole entropy. The situation here is somewhat similar to the fixing of the ground state eigenvalue ∆0 by Strominger [4]. In that work, the conformal field theory alone does not limit the continuous infinity of the possible values of ∆0 , and the requirement that it vanish has only a posteriori justification. Compared to this, we can make a stronger case for a particular choice of the ratio (n/N). This is because our starting point, i.e., the Chern–Simons theory coupled to a source leads, on the one hand, to the twisted Kac– Moody algebras described above and, on the other hand, to [16,17] the classical BTZ solution [1,2] for which the central charge for the asymptotic Virasoro algebra was given by Brown and Henneaux [5]. It is then necessary that the classical and quantum results which follow from the same Chern–Simons theory be consistent with each other. So, it is reasonable to require that the asymptotic density of states of the above quantum theory, as computed from Cardy–Carlip [3], agree, up to logarithmic terms, with the asymptotic density of states obtained by Strominger [4] using the traditional Cardy formula and the classical Brown– Henneaux value of central charge (45). The simplest way to satisfy this requirement is to set our effective central charge ceff equal to the Brown–Henneaux central charge. This fixes n/N = 1/4. That this requirement makes sense can also be seen by noting that for the Virasoro algebra obtained by Brown and Henneaux, the underlying Kac–Moody algebra is not known, so that there is no direct way of calculating its central charge c or its ground state eigenvalue from a fundamental Kac–Moody algebra. Then, taking the classical theory to be an “effective theory”, we see that we can compute its “effective central charge” and 213 its “effective ground state eigenvalue” from the above quantum theory for a particular choice of inner automorphism. It follows that for this conformal field theory, the expression for the asymptotic density of states given by Eq. (48) reduces to ˆ ˆ ≈ exp 2π ceff ∆/6 ρ(∆ˆ 0 ) ρ(∆) ˆ = exp 2π l ∆/(4G) (60) ρ(∆ˆ 0 ). Modulo a logarithmic correction, this expression is identical with that used by Strominger [4]. This resolves the longstanding controversy in the traditional method of comparing the central charge of a conformal field theory obtained from the Chern–Simons approach with the (effective) classical results of Brown and Henneaux. One of the crucial features of our work which led to this resolution was the recognition that for SL(2, R) Kac–Moody algebras, large negative values of k are allowed. So far, we have dealt with the density of states for one SL(2, R), say, SL(2, R)L . This will contribute an amount SL to the black hole entropy. Clearly, we can repeat this computation for the density of states of SL(2, R)R . Then, to the extent that the logarithmic corrections can be neglected, the black hole entropy S = SL + SR is in agreement with that given by Strominger [4]. More recently, the logarithmic contributions to the black hole entropy have been discussed in the literature [27,28]. One would then have to assess the relative size of our logarithmic term compared to those given in these works. Acknowledgements This work was supported, in part by the Department of Energy under the contract number DOE-FG0284ER40153. We would like to thank Philip Argyres and Alex Lewis for reading the manuscript and suggesting improvements. References [1] M. Bañados, C. Teitelboim, J. Zanelli, Phys. Rev. Lett. 69 (1992) 1849. 214 S. Fernando, F. Mansouri / Physics Letters B 505 (2001) 206–214 [2] M. Bañados, M. Henneaux, C. Teitelboim, J. Zanelli, Phys. Rev. D 48 (1993) 1506. [3] S. Carlip, hep-th/9806026. [4] A. Strominger, J. High Energy Physics 02 (1998) 009, hepth/9712252. [5] J.D. Brown, M. Henneaux, Commun. Math. Phys. 104 (1986) 207. [6] A. Achucarro, P. Townsend, Phys. Lett. B 180 (1986) 35. [7] E. Witten, Nucl. Phys. B 311 (1988) 46; E. Witten, Nucl. Phys. B 323 (1989) 113. [8] S. Carlip, Phys. Rev. D 51 (1995) 632. [9] A.P. Balachandran, L. Chandar, A. Momen, Nucl. Phys. B 461 (1996) 581, gr-qc/9506006. [10] M. Bañados, Phys. Rev. D 52 (1995) 5815. [11] E. Witten, Commun. Math. Phys. 121 (1989) 351. [12] S. Carlip, Phys. Rev. D 55 (1997) 878. [13] M. Bañados, T. Brotz, M.E. Ortiz, hep-th/9802076. [14] M. Bañados, hep-th/9901148. [15] M. Bañados, hep-th/9903178. [16] S. Fernando, F. Mansouri, Int. J. Mod. Phys. A 14 (1999) 505, hep-th/9804147. [17] S. Fernando, F. Mansouri, in: L. Brink, R. Marnelius (Eds.), Proceedings of XXII Johns Hopkins Workshop on Novelties in [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] String Theory, Goteborg, Sweden, August 20–22, 1998, hepth/9901163. S. Elitzur, G. Moore, A. Schwimmer, N. Seiberg, Nucl. Phys. B 326 (1989) 108. P. Goddard, D. Olive, Int. J. Mod. Phys. A 1 (1986) 303. F. Ardalan, S. Fernando, F. Mansouri, Cincinnati preprint UCTP102.00, in: Proceedings of the Sixth International Wigner Symposium, Istanbul, Turkey, August, 1999, in press. N. Sakai, P. Suranyi, Nucl. Phys. B 362 (1989) 655. J.K. Freericks, M.B. Halpern, Ann. Phys. (N.Y.) 188 (1988) 258. J. Balog, L. O’Raifeartaigh, P. Forgác, A. Wipf, Nucl. Phys. B 325 (1989) 225; S. Hwang, Nucl. Phys. B 354 (1991) 100; P.M.S. Petropoulos, Phys. Lett. B 236 (1990) 151. S. Fernando, F. Mansouri, Cincinnati preprint UCTP107.00. J.A. Cardy, Nucl. Phys. B 270 (1986) 186. R.L. Karp, F. Mansouri, J.S. Rno, hep-th/9903221; R.L. Karp, F. Mansouri, J.S. Rno, J. Math. Phys. 40 (1999) 6033, hep-th/9910173. S. Carlip, gr-qc/0005017. J. Jing, M.L. Yan, gr-qc/0005105. 26 April 2001 Physics Letters B 505 (2001) 215–221 www.elsevier.nl/locate/npe Boundary string field theory, the boundary state formalism and D-brane tension S.P. de Alvis Department of Physics, Box 390, University of Colorado, Boulder, CO 80309, USA Received 2 February 2001; accepted 11 February 2001 Editor: M. Cvetič Abstract Recently a boundary string field theory that had been proposed some time ago, was used to calculate correctly the ratios of D-brane (both BPS and non-BPS) tensions. We discuss how this work is related to the boundary state formalism and open string closed string duality, and argue that the latter clarifies why the correct tension ratios are obtained in these recent calculations. 2001 Published by Elsevier Science B.V. 1. Introduction The notion that the effective low energy field theory of string modes is given essentially by the partition function of the corresponding string sigma model, has a long history [1,2]. In particular Witten [2] suggested that (a certain expression derived from) the partition function for the bosonic open string on a disc, with a boundary term giving the contribution of open string fields, was a candidate for open string field theory. This has been called boundary string field theory (BSFT) in the recent literature. This theory has been used to study Sen’s arguments 1 on tachyon condensation [3,4]. The most remarkable aspect of this is that with the quadratic profile for the tachyon, for which one could calculate the partition function exactly [2], exact agreement with the expected ratio of bosonic D-brane tensions [4] was obtained. In the superstring case, where it was postulated that the E-mail address: [email protected] (S.P. de Alvis). 1 For a review see [6]. BSFT was given exactly by the partition function, one also obtained [5] the correct tension ratio [6] between the BPS and non-BPS branes. Now in these calculations the emergence of the right ratio seems to be somewhat mysterious, arising from the constant term in the asymptotic behavior of the Gamma function! The purpose of this note is to relate this calculation to previous calculations of the D-brane tension; in particular to the boundary state formalism and the T-duality argument. Many of the ingredients for our discussion have been presented in papers by other authors. In particular the use of open/closed string duality in the compactified string formalism to calculate the normalization of the boundary state in the bosonic case has been done in [9,10] (see also [11]). The relation of the boundary tachyon coupling RG flow to the change in boundary conditions from Neumann to Dirichlet has been discussed in [12]. What we do here is to put all these ideas together to demonstrate that open/closed string duality leads to the correct D-tension formulae (for BPS and non-BPS branes) and to elucidate the relation to T-duality and the BSFT method. 0370-2693/01/$ – see front matter 2001 Published by Elsevier Science B.V. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 3 1 4 - 8 216 S.P. de Alvis / Physics Letters B 505 (2001) 215–221 2. Boundary state formalism, open/closed string duality The boundary state formalism was used to discuss open closed string duality in a series of papers by the authors of [7] and in [8]. We will follow closely the development given in the former reference and that in [13] where the boundary state formalism was first applied to non-BPS branes. Let us first quote some standard formulae in order to establish our conventions. The string action (in flat space) is given by (setting α = 2) 1 S= dw d w̄ ∂w Xµ ∂w̄ Xµ 4π + ψ µ ∂w̄ ψµ + ψ̃ µ ∂w ψ̃µ . (2.1) We will work in the double Wick rotated formalism as in [13] in order to avoid introducing the ghost sector. So we will take µ = 0, 1, . . . , 7 to be directions transverse to the light cone directions which are taken to be µ = 8, 9. (i.e., we have Wick rotated X9 → iX9 , X0 → −iX0 ). The disadvantage is that we are really now discussing D-instantons rather than D-branes and we are confined to p 7 but the results obviously will apply to D-branes after Wick rotating back. The procedure should be completely equivalent to working with ghosts and is simply a much less cumbersome way of getting at the right results which in fact should actually be valid for all p 9. We will also put the system on a torus with radii R µ since one of our objects is to demonstrate the relation to T-duality. Closed string channel The closed string solution is nµ τ − wµ R µ σ Rµ 1 µ −im(σ −iτ ) α µ eim(σ +iτ ) + α̃m +i , e m m m=0 (2.2) µ Xµ = x0 − 2i ψ(w) = ψrµ eir(σ +iτ ) , r Z +ν ψ̃(w̄) = ψ̃rµ e−ir(σ −iτ ) . r Z +ν The (anti) commutation relations are (2.3) µ ν µ ν , α̃n = mδ µν δm,−n , αm , αn = α̃m µ ν αm , α̃n = 0, µ ν µ ν ψr , ψs = ψ̃r , ψ̃s = δr,−s δ µν , µ ν ψr , ψ̃s = 0. (2.4) The closed string Hamiltonian is H = L0 + L̄o ∞ n2 1 2 2 µ µ = + R + α̃mµ αm αmµ + α̃m w 2 4 R n=1 1 µ µ r:ψ−r ψrµ : + r:ψ̃−r ψ̃rµ : + Cc + 2 r=Z +ν (2.5) with Cc = −1 in the NSNS sector and Cc = 0 in the RR sector. In the closed string channel we need to calculate the amplitude Z c (l) = Dp|e−2πlHc |Dp (2.6) for the emission and absorption of a closed string state between two (or the same) Dp branes. The latter are constructed in terms of boundary states |Bp, η that satisfy the following boundary conditions [7,8]. (αm + α̃m )µ |Bp, η = 0, (ψr + iηψ̃−r )µ |Bp, η = 0, µ = 0, . . . , p, (αm − α̃m )µ |Bp, η = 0, (ψr − iηψ̃−r )µ |Bp, η = 0, µ = p + 1, . . . , 7, (2.7) where η = ± for the two spin structures. The solution to these conditions is ∞ 1 µ i ν |Bp, ηi = gp exp α Tµν α̃−n n −n n=1 µ ν + iη ψ−r Tµν ψ̃−r |Bp, η0i . (2.8) r=Z + − 12 Here the index i goes over NSNS and RR sectors and T is a diagonal 8 × 8 matrix with −1 for the Neumann (µ = 0, . . . , p) directions and +1 for the Dirichlet (µ = p + 1, . . . , 7) directions. The ground state is an eigenstate of momentum with eigenvalue zero in the N directions and is an eigenstate of position in the D S.P. de Alvis / Physics Letters B 505 (2001) 215–221 217 directions. Also we have [7] Hamiltonian is 1 RR g = gpNSNS ≡ gp . 4i p ∞ n2 1 µ 2 + (wR) + α−m αmµ H = Lo = 2 R2 2 4 η(2il) , (2.9) Bp ± |e−2πlHc |Bp∓NSNS = gp2 f (R) Bp ± |e θ01 (0, 2il)4 , η(2il)12 −2πlHc θ10 = −16gp2 f (R) η(2il)12 Bp ± |e −2πlHc (2.10) , θ11 (0, 2il)4 = 0, η(2il)12 (2.11) (2.12) where (R µ )2 f (R) ≡ θoo 0, il 2 µ 2 × θoo 0, il µ 2 . (R ) (2.13) µ⊥ In the above we have put ∂ = {µ = 0, . . . , p}, ⊥ = {µ = p + 1, . . . , 9}. Open string channel The solution with N boundary conditions is (omitting the space time index) cos nσ n e−mτ αm , X = x + 4i τ + i2 R m (2.14) m=0 where m is an integer characterizing the Kaluza–Klein momentum while that for D boundary conditions is sin nσ e−mτ αm , m µ⊥ (0, it)4 θαβ , η(it)12 (2.17) where t = 2l1 . Using the modular transformation properties of the theta functions we get µ √ √ # + #⊥ l R / 2 1 Zαβ = √ 4 √8 µ 2 l ⊥R / 2 × f (R) X = x + 2wR + i2 o = trα e−2πt Ho eiπβF Zαβ 2 i4t = θoo 0, µ 2 θoo 0, it R µ (R ) × |Bp∓RR = −16gp2 f (R) (2.16) where Co = − 12 /0 in NS/R sectors. Calculating now in the open string channel we have µ |Bp±RR (0, 2il)4 m=1 r=Z +ν Bp ± |e−2πlHc |Bp±NSNS θ00 (0, 2il) = gp2 f (R) 12 ⊥ 1 µ + r :ψ−r ψrµ : + Co , 2 Using these formulae we can calculate the amplitude Z in the various sectors and we get θαβ (0, it)4 . η(it)12 (2.18) Now in conformal field theory we would have just equated this to the corresponding closed string expression (2.9) if we had only the eight non-light cone directions. However in string theory even in the light cone gauge the zero modes go over ten directions so that in the above # + #⊥= 10 rather than 8. So the actual equation is dt o c Z (t) = dl Zβα (l). (2.19) 2t αβ This then gives 2 after restoring α , R µ α 4−p 2 gp = ⊥ µ . 25 ∂R (2.20) Actually the properly projected closed string sectors requires that we take the appropriate left and right GSO projections so that the correct boundary (2.15) m=0 where w is the winding number. Note that for simplicity we are taking the two branes to be coincident. The dt dl 1 2t = 2l one gets an extra factor of 2l inside the l integral that cancels the extra factor of l in the numerator of (2.18). 2 Note that since 218 S.P. de Alvis / Physics Letters B 505 (2001) 215–221 states are, 3 while that with the BPS state is (see (2.23)) 1 |BpNSNS = √ (|Bp+NSNS − |Bp+NSNS ). (2.21) 2 1 g|Dp = √ g|BpNSNS = igp . (3.3) 2 The coupling to the graviton should be proportional to the tension of the brane and so the above is just Sen’s result [15] (see also [16]) that the non-BPS tension is √ 2 times the corresponding BPS tension. Let us now derive the exact formula for the tension of the branes. On the one hand we have the formula (2.20) for gp and on the other we have argued above that it must be proportional to the tension. Since gp is dimensionless we may therefore write, R0 · · · Rp α 4−p 1/2 gp = Rp+1 · · · R9 25 Thus we have dlBp|e−2ilHc |BpNSNS = dt trNS−R e−2πt Ho . (2.22) The open string channel is not GSO projected and so there is an open string tachyon. Thus the boundary state (2.21) in fact represents the non-BPS D-brane (for odd (even) p in type IIA (IIB)). 4 The BPS D brane states are given by 1 |Dp = √ (|BpNSNS + |BpRR ), 2 (2.23) + |Bp−RR ) is the where |BpRR = GSO projected RR state. Then we have dlDp|e−2ilHc |DpNSNS 1 = dt trNS−R 1 + (−1)F e−2πt Ho (2.24) 2 √1 (|Bp+RR 2 so that the open string tachyon is projected out. Because of the zero modes and the GSO projection in the RR sector the construction is consistent only for p even (odd) in the IIA (IIB) theory. 3. D-brane tension Let us compute the overlap of the BPS and non-BPS boundary states with the one graviton/dilaton state (with zero momentum). (For definiteness we will take the tensor component in say the 00 direction which is longitudinal for all p.) |g = ψ− 1 ψ̃− 1 |0NSNS (3.1) The overlap with the non-BPS state gives √ g|BpNSNS = i 2gp (3.2) 2 2 3 See, for example, [14] for a recent review. 4 For an explanation of why the other values of p do not give non-BPS states see, for instance, [14]. = CTp p 2πRµ . (3.4) µ=0 So we have p Tp gp Rp µ=0 2πRµ = =√ p−1 Tp−1 gp−1 α µ=0 2πRµ (3.5) giving, Tp 1 √ . (3.6) = Tp−1 2π α This is exactly the formula obtained from T-duality [17,18] and is to be expected since the passage from N boundary conditions to D boundary conditions can be effected by T-duality. Indeed the R dependence in gp reflects that as noted in [10], since for each such switch of boundary conditions R → α /R in (2.20). The absolute normalization can be fixed as in [18] by defining the coupling constant g to be the ratio of the F 1 string to the D string, i.e., writing T1 = g −1 2πα . Then 5 we get C= gα 5 √ 2π R0 · · · R9 and Tp = g −1 √ p+1 . (2π)p α (3.7) 5 The original formula derived by Polchinski [19] had a factor of the gravitational coupling in it since it was derived by comparing the string calculation to the low energy effective action. The formula in (3.7) was derived in [18]. The comparison between the two fixes the gravitational coupling in terms of the string scale and the string coupling. The latter also follows from the Dirac quantization rule. S.P. de Alvis / Physics Letters B 505 (2001) 215–221 If we remain within a particular theory (say IIA) then we can start with p = 9 (which in this case is non-BPS brane) and then change boundary conditions in one direction to get the (BPS) 8-brane etc. What the discussion of the above two paragraphs shows is that as one goes down in p the value of the normalization constant (after accounting world volume factor) √for the 1 when we go from effectively changes by 2 √ 2π α non-BPS to BPS D-branes. 219 regular in the upper half plane (or interior of the disc) m−1/2 X(σ, τ ) = x0 + m>0 × xm e−imσ + x̄m eimσ e−mτ . (4.3) This state is normalized and satisfies the completeness relation, [dx][d x̄]|x, x̄x, x̄| = 1, [dx] = dxm . (4.4) m>1 4. Relation to BSFT In this section we will show following [7] how the boundary state can be written as a path integral. In particular we will show that the normalization coefficient will be given by the integral over the modes of the sigma model field on the boundary of a disc of the classical action. This then relates the previous calculation to that of [5] (see also [20,21]). The idea is to first construct the boundary state corresponding to having N boundary conditions in all directions. The boundary state with some D directions is then going to be obtained by adding a ‘tachyon’ term that will result in RG flow to a new fixed point that will correspond to D boundary conditions as in [12]. Let us first just consider the bosonic sector and focus on one coordinate. At τ = 0 (and confining ourselves to the winding number zero sector) we expand |m|−1/2 am e−imσ + ãm eimσ , X(σ, 0) = x0 + m=0 (4.1) where we have written (as in [7]) for later convenience √ √ αm = −i m am , α−m = −i m a−m , etc. (4.2) † ˆ † Define also x̂m = am + ãm , x̄ m = am + ãm , m > 0. The eigenstate of these operators which is also an eigenstate of total momentum with eigenvalue zero is, |x, x̄ = ∞ † † † e− 2 x̄m xm −am ãm +am xm +x̄m ãm |0, 1 m=1 where am |0 = ãm |0 = 0, m > 0, p̂|0 = 0. Note that the first term in the exponential is simply the bulk bosonic action evaluated with the solution that is The boundary state is then written as |Ψ, b = [dx][d x̄]e−S(x,x̄) |x, x̄, (4.5) where S is a boundary action. When the latter is zero we have a state with N boundary conditions, i.e., † † |Ψ, b0 = [dx][d x̄]|x, x̄ = (4.6) eam ãm |0. m>0 For the fermion in the NS sector we have (at τ = 0) the expansions, ψr eirσ , ψ̃(σ, 0) = ψr eirσ , ψ(σ, 0) = r r (4.7) where rZ − 1/2. The Majorana conditions give ψ−r = −ψr† , ψ̃−r = ψ̃r† . The fermionic position eigenstate [7] is 6 |θ, θ̄; ± = exp −i θ̄θr ± iψr† ψ̃r† r>0 √ √ + i 2 ψr† θr ∓ 2 θ̄r ψ̃r† |0 (4.8) with ψr |0 = ψ̃r |0 = 0. This state satisfies the boundary conditions √ ( 2 θ̄r − ψr† ∓ i ψ̃r )|θ, θ̄ ; ± = 0, √ (i 2 θr − ψr ± i ψ̃r† )|θ, θ̄; ± = 0. (4.9) The analog of the bosonic boundary state (4.5) is |Ψ, θ = (4.10) d θ̄r dθr e−S(θ) |θ, θ̄. r 6 We’ve redefined the θ coordinate in [7] in order to be able to write the amplitude as a classical action. 220 S.P. de Alvis / Physics Letters B 505 (2001) 215–221 Now let us put in the tachyon boundary term [12] 1 dσ T 2 + θ µ ∂µ T ∂σ−1 θ ν ∂ν T , ST = 8π τ =0 where θ = θ (σ, τ ) = θr eirσ + θ̄r e−irσ e−rτ (4.11) r>0 is defined in terms of the boundary coordinates θr introduced above and is regular in the upper half w (= σ + iτ ) plane (or in the interior of the disc |z| < 1 in terms of the coordinate z = e−iw ). If we use this expansion for θ we get for the classical bulk fermionic action 1 θ̄r θr , Sψ = dσ dτ (θ ∂w θ + θ ∂w̄ θ ) = i 2π r>0 (4.12) as in [20]. This is the first term in the exponential in the definition of the boundary state (4.8) and it was in order to get this agreement with the bulk action that we redefined θ from that given in [7]. Let us now introduce a linear tachyon profile as in [5] and for simplicity take it to be along one coordinate direction. So T = yX. πR uµ µ 2 µ F (uµ ) e− 4 (x0 ) C (2πR ) µ µ⊥ (4.16) −πR in the limit u⊥ → ∞. C is essentially the u (and hence p) independent constant determined earlier. The ratio of infinite products in the above F (u) has been evaluated (after √ regularization) in [5,20,21] and takes the value 2π4u uΓ (u)2 /Γ (2u). As shown in these papers the correct tension ratio is obtained from this formula. What we have demonstrated above is that this is a consequence of the fact that what is evaluated there is essentially the normalization constant gp of the boundary state. Acknowledgement (4.13) Using the expansions (4.3) and (4.11) we then get u u 2 x0 + 2 m−1 xm x̄m + θ̄r θr , (4.14) ST = 4 r m>0 where With this tachyon profile the boundary state (the product of (4.5) and (4.10)) can be easily evaluated since the integrals are Gaussian. We find 7 |B, uNSNS ∞ ∞ 1 + ur = u 1+ m 1 m=1 r= 2 † † 2 2 − 1− 1+u/m am ãm −i 1− 1+u/r ψr† ψ̃r† πR × dx0 e− 4 x0 |x0 . u 2 I wish to thank Joe Polchinski for a discussion. This work is partially supported by the Department of Energy contract No. DE-FG02-91-ER-40672. r>0 u = y2. ×e Clearly this boundary state is such that it satisfies Neumann boundary conditions (appropriate to the directions) for u = 0 and Dirichlet conditions (appropriate to the ⊥ directions) as u → ∞ (see (2.7)). Thus the state (2.21) or the first term of (2.23) may be written alternatively as products of (4.15) with u = 0 in the directions and u → ∞ in the ⊥ directions. The overlap with the one graviton state (3.3) is now given by (calling the ratio of the infinite products F (u) |0 (4.15) −πR 7 Ignoring a u independent infinite product of 2’s and π ’s. References [1] E.S. Fradkin, A.A. Tseytlin, Quantum string theory effective action, Nucl. Phys. B 261 (1985) 1; O.D. Andreev, A.A. Tseytlin, Partition function representation for the open superstring effective action: cancellation of mobius infinities and derivative corrections to Born–Infeld Lagrangian, Nucl. Phys. B 311 (1988) 205. [2] E. Witten, On background independent open string field theory, Phys. Rev. D 46 (1992) 5467, hep-th/9208027; E. Witten, Some computations in background independendent off-shell string theory, Phys. Rev. D 47 (1993) 3405, hepth/9210065. [3] A.A. Gerasimov, S.L. Shatashvili, On exact tachyon potential in open string field theory, JHEP 0010 (2000) 034, hepth/0009103. [4] D. Kutasov, M. Marino, G. 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Polchinski, Dirichlet-branes and Ramond–Ramond charges, Phys. Rev. Lett. 75 (1995) 4724, hep-th/9510017. [20] P. Kraus, F. Larsen, Boundary string field theory of the D-bar system, hep-th/0012198. [21] T. Takayanagi, S. Terashima, T. Uesugi, Brane–antibrane action from boundary string field theory, hep-th/0012210. 26 April 2001 Physics Letters B 505 (2001) 222–230 www.elsevier.nl/locate/npe Supersymmetry breaking in 5-dimensional space–time with S 1/Z2 compactification Masud Chaichian a , Archil B. Kobakhidze a,c , Mirian Tsulaia b,c a HEP Division, Department of Physics, University of Helsinki and Helsinki Institute of Physics, FIN-00014 Helsinki, Finland b Bogoliubov Laboratory of Theoretical Physics, JINR, RU-141980 Dubna, Russia c Andronikashvili Institute of Physics, GAS, GE-380077 Tbilisi, Georgia Received 6 February 2001; accepted 28 February 2001 Editor: P.V. Landshoff Abstract We consider supersymmetric models in 5-dimensional space–time compactified on S 1 /Z2 orbifold, where N = 2 supersymmetry is explicitly broken down to N = 1 by the orbifold projection. We find that the residual N = 1 supersymmetry is broken spontaneously by a stable classical wall-like field configurations which can appear even in the simple models discussed. We also consider some simple models of bulk fields interacting with those localized on the 4-dimensional boundary wall where N = 1 supersymmetry can survive in a rather non-trivial way. 2001 Published by Elsevier Science B.V. 1. Introduction The remarkable success in the understanding of non-perturbative aspects of string theories gives a new insights into the particle phenomenology. One of the phenomenologically most promising approach has been proposed by Hořava and Witten within the 11-dimensional supergravity compactified on S 1 /Z2 orbifold that is, on an interval of the length R bounded by mirror hyperplanes [1]. This theory gives the strongly-coupled limit of the heterotic string theory with two sets of E8 super-Yang–Mills theories residing on each of the two 10-dimensional hyperplanes of the orbifold and the supergravity fields living in the full 11-dimensional bulk. The important property of this model is that when R is increased, the 11-dimensional Planck mass decreases as R −1 , while the E8 gauge coupling remains fixed. This allows to E-mail address: [email protected] (M. Chaichian). achieve unification of gauge and gravitational couplings at a grand unification scale 1016 GeV [2] inferred in turn from the low-energy values of gauge couplings which are measured with very higher accuracy at Z-peak. Further, the above construction has initiated even more dramatic reduction of the fundamental higher-dimensional Planck mass down to the TeV scale with a millimeter size extra dimensions [3] and the models with TeV scale unification [4]. Obviously, in order to get a realistic phenomenology one has to compactify 6 of the 10 remaining dimensions, transverse to R on a 6-dimensional volume. After such a compactification one obtains 5-dimensional theory on an interval with mirrorplane boundaries [5] which can be described as a 5dimensional supergravity field theory, with some possible additional bulk supermultiplets, coupled to matter superfields residing on the boundary walls. If R is the largest dimension in this set-up then one can ignore the finite volume of the 6-dimensional compactified 0370-2693/01/$ – see front matter 2001 Published by Elsevier Science B.V. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 3 2 7 - 6 M. Chaichian et al. / Physics Letters B 505 (2001) 222–230 space. The minimal supersymmetric Standard model (MSSM) in such a field theoretical limit has been constructed in [6] and we below closely follow to this construction. One of the most important issue of the MSSM phenomenology is the mechanism of supersymmetry breaking and the origin of soft masses. A commonly accepted scenario is to break supersymmetry either spontaneously or dynamically in the hidden sector. This breaking then shows up in the visible sector due to either gauge or gravitational interactions. In the Hořava–Witten theory compactification matter could be at a strong coupling regime on one boundary, and could break supersymmetry on this boundary dynamically [1] through the gaugino condensation. Then the supersymmetry breaking effects can be transmitted to the other boundary by gravitational [7] or gauge [8] fields propagating in the 11- (or 5)dimensional bulk. More recently it was also shown [9] that supersymmetry breaking from the one boundary to another can be mediated through the superWeyl anomaly [9]. Thus, in these models, fields living on one of the boundaries play the role of the hidden sector for the fields living on another boundary. A distinct higher-dimensional source of supersymmetry breaking is provided by the Scherk–Schwarz mechanism [10] where non-trivial boundary conditions for the fields along the compactified dimensions are responsible for the supersymmetry breaking. This mechanism has been studied recently within the framework of large extra dimensions as well [11]. In the present Letter we investigate the possibility of the breaking of a rigid supersymmetry in 5-dimensional field-theoretic limit of the Hořava– Witten compactification [1]. We will show that there could be a new source of supersymmetry breaking that relied on the Dvali–Shifman mechanism of supersymmetry breaking [12]. Particularly, we will argue that in a wide class of models with bulk supermultiplets under a certain boundary condition imposed there appear classical stable wall-like field configurations that break the residual N = 1 supersymmetry spontaneously, while the initial N = 2 supersymmetry is explicitly broken down to N = 1 due to the orbifold projection. We will also give some simple examples where N = 1 supersymmetry can survive in a rather non-trivial way. 223 2. Supersymmetry in 5 dimensions compactified on S 1 /Z2 In this section we introduce various supersymmetric multiplets in 5-dimensional space–time subject to the S 1 /Z2 compactification. N = 1 supersymmetric 5-dimensional multiplets can be easily deduced from the N = 2 four-dimensional ones (see, e.g., [13]). Throughout the Letter capitalized indices M, N = 0, . . . , 4 will run over 5-dimensional space–time, while those of lower-case m, n = 0, . . . , 3 will run over its 4-dimensional subspace; i = 1, 2 and a = 1, 2, 3 will denote SU(2) spinor and vector indices, respectively. We work with metric with the most negative signature ηMN = diag(1, −1, −1, −1, −1) and take the following basis for the Dirac matrices: 0 σm −iI2 0 , , ΓM = (1) 0 iI2 σ̄ m 0 where σ m = (I2 , σ ) = σ̄m (I2 is a 2×2 identity matrix) and σ (σ 1 , σ 2 , σ 3 ) are the standard Pauli matrices. Symplectic-Majorana spinor is defined as a SU(2)doublet Dirac spinor χ i subject to the following constraints: χ i = cij C χ̄ j T , where c = −iσ 2 = 0 1 (2) −1 0 and C = c c (3) are 2×2 and 4×4 charge conjugation matrices, respectively. A symplectic-Majorana spinor (2) can be decompose into the 4-dimensional chiral fermions as: i χL i χ = (4) , χRi i where two-component chiral fermions χL,R are related to each other according to equation j∗ i = cij cχ̄R(L). χL(R) (5) 2.1. Hypermultiplet The 5-dimensional off shell hypermultiplet H = hi , ψ, F i ) consist the scalar field hi (i = 1, 2) being a doublet of SU(2), an SU(2)-singlet Dirac fermion ψ = (ψL , ψR )T and SU(2)-doublet F i , being an auxiliary field. These fields form two N = 1 four-dimensional chiral multiplets H1 = (h1 , ψL , F 1 ) and 224 M. Chaichian et al. / Physics Letters B 505 (2001) 222–230 H2 = (h2 , ψR , F 2 ). The supersymmetry transformation laws are: √ δξ hi = 2 cij ξ̄ j ψ, √ √ δξ ψ = −i 2 Γ M ∂M hi cij ξ j − 2 F i ξ i , √ δξ F i = i 2 ξ̄ i Γ M ∂M ψ (6) while the corresponding 5-dimensional Lagrangian has the form: + M i (5) ∂ h Lhyper = ∂M hi + i F . + i ψ̄Γ M ∂M ψ + F i (7) In order to project the above structure down to a 4-dimensional N = 1 supersymmetric theory on the boundary wall one should define the transformation properties of fields entering in the hypermultiplet under the discrete Z2 orbifold symmetry. The Z2 acts on the fifth coordinate as x 4 → −x 4 . A generic bosonic field ϕ(xm , x4 ) transforms like ϕ x m , x 4 = Pϕ x m , −x 4 (8) while the fermionic one η(x m , x 4 ) transforms as: η x m , x 4 = Piσ 3 Γ 4 η x m , −x 4 , (9) where P is an intrinsic parity equal to ±1. One can assign the eigenvalues of the parity operator to the fields considered as in Table 1, so the bulk Lagrangian is invariant under the action of P. Then on the wall located at x4 = 0 the transformations (6) are reduced to the following N = 1 supersymmetry transformations of the even-parity states generated by parameter ξL1 : √ δξ h1 = 2 ξL1T cψL , √ √ δξ ψL = i 2 σ m cξL1∗ ∂m h1 − 2 ξL1 F 1 + ∂4 h2 , √ δξ F 1 + ∂4 h2 = i 2 ξL1+ σ̄ m ∂m ψL (10) Table 1 An intrinsic parity P of various fields. We define the supersymmetry transformation parameter ξL1 to be even (P = 1), while the parameter ξL2 to be odd (P = −1) Parity, P + + + Hypermultiplet h1 ψL F1 Vector multiplet Am λ1L X3 − A5 − − − h2 ψR F2 Σ λ2L X1,2 being the usual N = 1 supersymmetry transformations for the chiral multiplet. Thus what we have on the boundary wall is the simplest non-interacting massless Wess–Zumino model. Note, that an effective auxiliary field for the chiral multiplet contains the derivative term ∂4 h2 which is actually even under the Z2 orbifold transformation. Thus the expectation value of ∂4 h2 plays the role of the order parameter of supersymmetry breaking on the boundary wall. 2.2. Vector supermultiplet Now let us consider a 5-dimensional SU(N) Yang– Mills supermultiplet V = (AM , λi , Σ, Xa ). It contains a vector field AM = AMα T α , a real scalar Σ = Σ α T α , an SU(2)-doublet gaugino λi = λiα T α and an SU(2)triplet auxiliary field Xa = Xaα T α all in the adjoint representation of the gauge SU(N) group. Here α = 1, . . . , N runs over the SU(N) indices and T α are the generators of SU(N) algebra [T α , T β ] = if αβγ T γ with Tr[T α , T β ] = 12 δ αβ . This N = 2 supermultiplet consists of an N = 1 four-dimensional vector V = (Am , λ1L , X3 ) and a chiral supermultiplets Φ = (Σ + iA4, λ2L , X1 + iX2 ). Under the N = 2 supersymmetry transformations the fields of the vector supermultiplet V transform as: δξ Σ = i ξ̄ i λi , δξ AM = i ξ̄ i Γ M λi , ij δξ λi = σ MN FMN − Γ M DM Σ ξ i − i Xa σ a ξ j , δξ Xa = ξ̄ i (σ a )ij Γ M DM λj − i Σ, ξ̄ i (σ a )ij λj , (11) where once again the symplectic Majorana spinor ξ i is the parameter of supersymmetric transformations, DM is the usual covariant derivative, DM Σ(λi ) = ∂M Σ(λi ) − i[AM , Σ(λi )], and σ MN = 14 [Γ M , Γ N ]. As in the case of the hypermultiplet we define an intrinsic parity P for the fields in V (see Table 1), thus projecting it down to a 4-dimensional N = 1 supersymmetric vector multiplet residing on the orbifold boundary. Let ξL1 be the supersymmetry parameter of N = 1 supersymmetric transformations on the boundary. Then on the boundary at x4 = 0, the supersymmetric transformations (11) for the even-parity (P = 1) states reduces to: δξ Am = iξL1+ σ̄ m λ1L + h.c., δξ λ1L = σ mn Fmn ξL1 − i X3 − ∂4 Σ ξL1 , M. Chaichian et al. / Physics Letters B 505 (2001) 222–230 δξ X3 − ∂4 Σ = ξL1+ σ̄ m Dm λ1L + h.c. (12) These are indeed the transformation laws for an N = 1 four-dimensional vector multiplet V = (Am , λ1L , D) with an auxiliary field D = X3 − ∂4 Σ [8]. Once again the derivative term ∂4 Σ enters into the effective auxiliary field on the boundary. Finally, the bulk Lagrangian invariant under the above supersymmetric transformations looks as: 1 Tr(FMN )2 2g52 1 + 2 Tr(DM Σ)2 + Tr λ̄iΓ M DM λ g5 2 + Tr Xa − Tr λ̄[Σ, λ] . (13) (5) LYang–Mills = − The explicit breaking of N = 2 supersymmetry down to the N = 1 by the orbifold projection discussed in this section is rather transparent from analyzing general N = 2 supersymmetry algebra i j Q , Q = cij Γ M CPM + Ccij Z, (14) where Z is a central charge. The relation (14) is invariant under the Z2 orbifold transformations: Qi −→ i(σ 3 )ij Γ 4 Qj , Z −→ −Z, Pm −→ Pm , P4 −→ −P4 . (15) Then modding out the Z2 orbifold symmetry Q1R = Q2L = 0, Z = 0, P4 = 0, we obtain from (14): i j Q , Q = cij Γ m CPm , (16) (17) Taking now into account that supercharges should satisfy chirality condition given by (16) and Eq. (5), we finally arrive to the familiar N = 1 supersymmetric algebra in 4 dimensions: 1 1 Q , Q = Γ m CPm . (18) Note, however, that since (14) is operatorial equation the right hand side of (17) can be modified by the terms P4 ± Z when acting on the parity odd state as it was the case for the models considered in this section. 225 3. Supersymmetry breaking 3.1. Free fields in the bulk We begin our discussion of supersymmetry breaking from the simplest supersymmetric models considered in the previous section. The models similar to those described above are often used to construct phenomenologically viable theories, such as MSSM on the 4-dimensional boundary [6]. The remaining N = 1 supersymmetry on the boundary can be broken through the Scherk–Schwarz mechanism [10] by requiring that MSSM superpartners satisfy non-trivial boundary conditions which in turn result in the softbreaking masses [6,11]. However, there could exist a distinct source of supersymmetry breaking that relied on the Dvali– Shifman mechanism [12]. This mechanism is based on the fact that any field configuration which is not BPS state and breaks translational invariance breaks supersymmetry totally as well. Such a stable non-BPS configurations with a purely finite gradient energy can also appear in a compact spaces (or, more generally, in spaces with a finite volume) if there exist moduli forming a continuous manifold of supersymmetric states. This is indeed the case in the 5-dimensional models considered in the previous section. To be more specific consider first the case of the pure hypermultiplet in the 5-dimensional bulk. The corresponding Lagrangian (11) describes a system of free massless fields. Thus, it seems that any constant value of these fields will be a ground state of the model and, moreover, these ground states will preserve supersymmetry. However this is not the case for the parity-odd fields, in particular for the complex scalar field h2 , since h2 = const does contradict the boundary condition given by (8) with P(h2 ) = −1. Thus the boundary condition singles out the trivial configuration h2 = 0 among all constant states. Besides this trivial configuration however, there could actually be stable non-trivial configurations as well. One of them is the constant phase configuration that linearly depends on the fifth coordinate: h2 = 1x 4 , (19) where 1 is an arbitrary constant which can be chosen to be real. The configuration (19) is odd under the Z2 orbifold transformation as it should be and 226 M. Chaichian et al. / Physics Letters B 505 (2001) 222–230 obviously breaks translational invariance in x4 direction. However, the configuration (19) does not satisfy the ordinary periodicity condition on a S 1 circle, h2 (x m , x 4 +2R) = h2 (x m , x 4 ), but rather the modified one, which we define as: h2 x m , x 4 + 2R = h2 x m , x 4 + 21R. (20) Then the configuration h2 interpolates from 0 to 21R when one makes a full circle around the compactified dimension. 1 Clearly, the Lagrangian density L(5) hyper (5) (7) remains single-valued and periodic, Lhyper (x 4 + (5) 2R) = Lhyper (x 4 ), so the theory with boundary condition (20) will be consistent as it was in the case of the ordinary periodic boundary conditions. Thus, if we assume that h2 and its superpartner ψR are defined modulo 21R on S 1 /Z2 space, then the configuration (19) will be perfectly compatible with S 1 /Z2 orbifold symmetries. Stability of the above configuration (19) can be straightforwardly checked by performing finite deformation h2 + δh2 , where δh2 → 0 at infinity. Then the variation of an energy functional: 2 2 ∂h 5 2 2 E = d x ∂m h + 4 (21) ∂x is indeed zero for the configuration (19): δE (22) = −∂4 ∂ 4 h2 = 0. δh2∗ Being stable, we can treat the configuration (19) as a possible vacuum state of the model. While (19) is multiply defined, the vacuum energy density is a constant given by the purely gradient energy E = 1 2 . One can see that to see that this vacuum configuration spontaneously breaks remaining N = 1 supersymmetry on the boundary wall. Indeed, the effective F -term on the boundary (see (10)) is nonzero, F 1 + ∂4 h2 = 1 = 0, indicating the spontaneous breaking of N = 1 supersymmetry. Despite of the fact that the supersymmetry is completely broken all fields in the model remain massless, 1 Similar but BPS configurations on a non-simply connected compact spaces in lower dimensions have been considered in [14] (see also [15]) and further explored in [16]. Contrary, non-BPS and thus supersymmetry breaking configurations in models with twisted boundary conditions are discussed in [17]. so it looks such as the Fermi–Bose degeneracy is still present. The reason for such a degeneracy is following. All four real components of h2 and h1 are massless, because all of them are the Goldstone bosons: one mode corresponds to the spontaneously broken translational invariance and another three correspond to the complete spontaneous breaking of the global SU(2) invariance. The massless fermion ψL is a goldstino of the spontaneously broken N = 1 supersymmetry on the boundary wall. Obviously, if one gauges the model, all these massless states will give rise to the masses of the corresponding gauge fields (graviphoton, gravitino and SU(2) gauge fields) through the (super)Higgs mechanism. Now let us turn to the Z2 -even scalar field h1 from the H1 = (h1 , ψL , F 1 ) chiral supermultiplet. It is obvious, that all x 4 -dependent configurations of h1 will be unstable and only those being the trivial constant can be realized as a vacuum states. These vacuum states are supersymmetry preserving. Beside the trivial homogenous configurations, however, there can exist also x 4 -independent stable configuration with a winding phase: h1 = 1r sin θ eiϕ , (23) 1 2 2 2 3 2 where r = (x ) + (x ) + (x ) and ϕ and θ are azimuthal and zenith angles in {x 1 , x 2 , x 3 } plane. The configuration (23) is indeed a solution of the equation of motion: −∂M ∂ M h1 1 ∂h1 ∂h1 1 ∂ ∂ r2 + 2 sin θ ≡ 2 ∂r ∂θ r ∂r r sin θ ∂θ 2 1 2 1 2 ∂ h ∂ h1 ∂ h 1 + − = 0, + ∂(x 4 )2 ∂(x 0 )2 r 2 sin2 θ ∂ϕ 2 (24) and compatible with S 1 /Z2 orbifold symmetries. The solution (23) for any hyperplane θ = const = πn reminds the ordinary global cosmic string configuration, except that the modulus of the scalar field h1 never assumes a constant value. Obviously, this configuration also breaks supersymmetry. Actually, there can be the cases where both configurations (19) and (23) are present simultaneously. The solution similar to the x 4 -dependent constant phase configuration (19) can be obtained as well for the Z2 -odd scalar field Σ + iA5 as well when the case M. Chaichian et al. / Physics Letters B 505 (2001) 222–230 when vector supermultiplet lives in the bulk and the non-trivial boundary condition similar to (20) for the Σ + iA5 is assumed. This is the case not only for the Abelian (non-interacting ) vector supermultiplet but also for the non-Abelian Yang–Mills supermultiplet. For the later case the interaction terms also do not contribute to the vacuum energy, and this is indeed satisfied for the vacuum configuration where all fields except of Σ have a zero vacuum expectation value. Before proceeding further, let us make a comment on the case of massive non-interacting hypermultiplet. Namely, if ordinary periodic boundary conditions are assumed, one can add a mass term to the Lagrangian L(5) hyper (7): j + i + ij j i ij L(5) (25) + h c F mass = m ψ̄ψ + h c F since it transforms as a full derivative under the supersymmetric transformations. The condition of vanishing F -terms leads to hi = 0. So we have no more continuous manifold of degenerate supersymmetric states the presence of which was so crucial in the massless case discussed above. However, in the case of S 1 /Z2 compactification L(5) mass is Z2 -odd, while L(5) Z -even. Thus Z -parity actually forbids mass 2 hyper 2 (5) term Lmass (25). Note, that even in the case of compactification on a simple circle S 1 non-trivial boundary condition (20) forbids the existence of the mass term as well, since it is multiply-defined in this case. To conclude, the stable spatially extended configurations (19) and/or (24) most likely appear in models with free bulk superfields and if so they inevitable break supersymmetry completely. 3.2. Interacting bulk fields Let us briefly discuss the theories with non-trivial interactions in the bulk in connection with the supersymmetry breaking mechanism. From the above discussion we conclude that the existence of a moduli forming a continuous manifold of degenerate supersymmetric states is a necessary ingredient for the considered supersymmetry breaking mechanism to work. Thus, one can ask: can some non-trivial interactions which presumably appear in realistic theories remove the degeneracy in the case of free supermultiplets above and in this way protect supersymmetry? Indeed, one can expect that a certain interaction removing the 227 degeneracy can drive the classical field configurations to be supersymmetry preserving with vanishing F and D terms. It might also happen that non-vanishing potential energy (F -term) exactly cancel the gradient energy (D-term) as it is the case for the BPS configurations [14,15,18]. However, these models with N = 1 supersymmetry in four dimensions straightforwardly applicable to the case of N = 2 supersymmetry. The point is that the framework of N = 2 supersymmetry is more restrictive than of N = 1, so many interactions allowed by N = 1 can not be straightforwardly extended in the case of N = 2. Moreover, the orbifold symmetries seem to put further restrictions. Interacting supersymmetric gauge theories in five dimensions have been discovered relatively recently [19]. The most general Lagrangian (with up to two derivatives and four fermions) on the Coulomb branch is ∂F (Φ) + 2V 1 Im d 4θ Φ e L= 8π ∂Φ ∂ 2 F (Φ) 2 + d 2θ (26) W , ∂Φ 2 where F (Φ) is a holomorphic function. The N = 1 chiral superfield Φ along with the N = 1 vector superfield V forms N = 2 vector supermultiplet V, and W being the standard gauge field strength corresponding to V . The prepotential F (Φ) can be at most cubic [19]: F (Φ) = 4π 2 c 3 Φ + Φ . 3 g52 (27) The first term in (27) produces just the kinetic terms in the Lagrangian (26), while the second one generates the non-trivial interaction terms. However, since the second term of the prepotential (27) is odd under the Z2 orbifold transformations one should take c = 0, to keep Z2 invariance of the Lagrangian (26) [20]. It is unlikely that this term can be generated at a quantum level (at least perturbatively) as it is the case when the fifth dimension compactifies on a simple circle [21]. Thus, we are left with theories described by the Lagrangians of type (13) with, possibly, some additional hypermultiplets charged under the gauge group. In this case one can find the field configurations which completely breaks supersymmetry in a complete analogy to the cases that we have discussed above. Note, however, that the Lagrangians (7) and (13) are (and therefore the total Lagrangian which includes 228 M. Chaichian et al. / Physics Letters B 505 (2001) 222–230 the interaction with the bulk fields) not N = 2 supersymmetric actually but rather they are N = 1 supersymmetric under the constraints imposed by orbifold boundary conditions. So, if one can considers the interactions of the bulk fields with those localized on the boundary, these interactions will be explicitly N = 1 supersymmetric. As we will see in the next section one can keep N = 1 supersymmetry unbroken in such a cases. 4. Keeping supersymmetry on the boundary wall In this section we will consider some simple models of the bulk fields interacting with a superfields localized on the 4-dimensional boundary wall where the N = 1 supersymmetry can survive. First is the model of bulk hypermultiplet interacting with the boundary N = 1 chiral superfield and another is the model of U (1) gauge supermultiplet in the bulk with Fayet– Iliopoulos (FI) D-terms on the boundary. The situation that appears in these models is similar to the one discussed in [18] where the different non-compact 4-dimensional models are considered. through the superpotential WΦH1 (Φ, H1 ): ∂WΦH1 ∂WΦH1 = − F + + h.c. L(4) F H Φ 1 ΦH1 ∂h1 ∂φ + fermionic terms, (30) where FH1 = F 1 + ∂4 h2 . The equations of motion for the F -terms resulting form the Lagrangian (29) are: F 2 = 0, ∂WΦH1 ∂WΦ + , FΦ+ = ∂φ ∂φ ∂WΦH1 F 1+ = δ(x 4 ), ∂h1 while the equation of motion for h2 is: ∂4 ∂4 h2 + F 1 = 0. (31) (32) Now, if F 1 = 0, then the degeneracy in h2 is actually removed and the configuration h2 can satisfy the following equation: ∂4 h2 + F 1 = 0. (33) 4.1. Bulk fields interacting with boundary fields For ∂WΦH1 /∂h1 = α = const we get from (33): −α, x 4 > 0, h2 = −αε(x 4 ) ≡ (34) α, x 4 < 0. Let us consider the N = 1 chiral superfield Φ = (φ, χL , FΦ ) localized on the 4-dimensional boundary x 4 = 0. The boundary Lagrangian has the usual form of a 4-dimensional chiral model built from an N = 1 supersymmetric chiral superfields Φ: Thus, if equation FΦ = 0 is satisfied additionally (that can be easily justified in general), N = 1 supersymmetry remains unbroken. In the case of zero F 1 = 0 the degeneracy in h2 is restored and we come back to the case of free fields considered above (see Eq. (19)). + m + + m L(4) Φ = (∂m φ) (∂ φ) + iχL i σ̄ ∂m χL + FΦ FΦ ∂WΦ FΦ + h.c. − ∂φ 1 ∂ 2 WΦ T χL cχL + h.c. , − (28) 2 ∂φ∂φ where WΦ (Φ) is a superpotential. Then the total Lagrangian has the form: (4) (5) (4) Lhyper + LΦ + LΦH1 δ(x 4 ), (29) where L(4) ΦH1 (Φ, H1 ) describes the interactions between the chiral superfields Φ = (φ, χL , FΦ ) and H1 = (h1 , ψL , F 1 + ∂4 h2 ) on the boundary x 4 = 0 4.2. U(1) in the bulk with FI term Now we are going to consider the model of N = 1 super-Maxwell theory in five dimensions with FI D-term. The FI term can also lift the degeneracy of supersymmetric states. If we forget for a moment about the orbifold compactification, we can straightforwardly add FI term −2ηa Xa (35) to the Lagrangian (13) for the U (1) supermultiplet, where ηa is a SU(2)-triplet of constants. However, Z2 orbifold symmetry actually forbids such term because the auxiliary fields X1,2 and X3 have an opposite M. Chaichian et al. / Physics Letters B 505 (2001) 222–230 orbifold parity (see Table 1). The only FI term we can add in the simplest case considered here is that localized on the boundary wall −2η X3 − ∂4 Σ δ(x 4 ). (36) Then the situation becomes much similar to the case of bulk hypermultiplet interacting with boundary superfield discussed just above. Indeed, the degeneracy in Σ is lifted for non-zero value X3 = g52 ηδ(x 4 ), since now ∂4 Σ = X3 . (37) Thus the configuration Σ is given by (34) with α = −g52 η now, and supersymmetry is indeed unbroken again. 229 phenomenological models is worth to investigate. We hope to touch these issues in future publications. Acknowledgements We are indebted to A. Pashnev for useful discussions and to M. Zucker for kind communication. The work of M.C. and A.B.K. was supported by the Academy of Finland under the Project No. 163394 and that of M.T. by the Russian Foundation of Basic Research under the Grant 99-02-18417. M.T. and A.B.K. are grateful to the Abdus Salam ICTP High Energy Section where the part of this work was done. References 5. Conclusions and outlook The 5-dimensional supersymmetric models with S 1 /Z2 orbifold compactification are often considered as a phenomenologically valuable low-energy limit of the Hořava–Witten theory. Here we considered the question of supersymmetry breaking in theories of such kind. In particular, we argue that in a wide class of semi-realistic models there typically exist the spatially extended field configurations which are not supersymmetric. On the other hand, we also considered some explicit examples of models where bulk fields interact with boundary ones and show that supersymmetry can be preserved in a rather non-trivial way. 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Phys. 104 (2000) 835, hepph/0010288. 26 April 2001 Physics Letters B 505 (2001) 231–235 www.elsevier.nl/locate/npe Dilaton tadpoles and mass in warped models Antonios Papazoglou Theoretical Physics, Department of Physics, Oxford University 1 Keble Road, Oxford, OX1 3NP, UK Received 7 February 2001; accepted 7 March 2001 Editor: P.V. Landshoff Abstract We review the brane world sum rules of Gibbons et al. for compact five-dimensional warped models with identical fourgeometries and bulk dynamics involving scalar fields with generic potential. We show that the absence of dilaton tadpoles in the action functional of the theory is linked to one of these sum rules. Moreover, we calculate the dilaton mass term and derive the condition that is necessary for stabilizing the system. 2001 Elsevier Science B.V. All rights reserved. 1. Introduction Recently Gibbons, Kallosh and Linde [1] derived an infinite set of sum rules for five-dimensional models with a compact periodic extra dimension and identical four geometries. These constraints were an immediate consequence of the equations of motion and served as consistency checks of several recent constructions. An interesting result was that the Goldberger–Wise (GW) mechanism [2] of stabilizing the two three-brane Randall–Sundrum (RS1) model [3] has to include the backreaction on the metric in order to agree with a specific constraint, something done in the DeWolfe– Freedman–Gubser–Karch mechanism [4]. A particular sum rule the attention µthat attracted of [1] was the dy W Tµ − 2T55 = 0, where W (y)2 stands for the warp factor. This constraint was firstly derived by [5] as a condition of vanishing of the four-dimensional cosmological constant. The interesting point was that this combination of the energy– E-mail address: [email protected] (A. Papazoglou). momentum tensor components appeared in a condition for the absence of dilaton 1 tadpoles in the action functional of the theory in the paper by Kanti, Kogan, Olive [6]. The condition of [6] √ and Pospelov µ reads dy G(5) Tµ − 2T55 ] = 0. It was, however, pointed out by [1] that the two constraints were not µ identical because the Tµ − 2T55 combination was integrated powers of the warp factor √ with different √ since G(5) = W 4 g. A closer inspection reveals that these two constraints are actually identical given the assumptions made in [6]. In more detail, the condition in [6] was derived for matter dominated branes were the effect of the warp factor is negligible. Then, for W ≈ 1 the two constraints coincide. In this Letter we will iterate the calculation of [6], including the full effect of the warp factor. In that case, the condition of [6] for the absence of dilaton 1 Here we use the term “dilaton” to denote the modulus corre- sponding to the fluctuation of the overall size of the system. We use instead the term “radions” for the moduli associated with the position of freely moving branes along the extra dimension (not on orbifold fixed points). 0370-2693/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 3 5 8 - 6 232 A. Papazoglou / Physics Letters B 505 (2001) 231–235 tadpoles is modified and the new condition coincides with an other sum rule of [1]. Furthermore, having the quadratic action functional for the dilaton, it is straightforward to read off its mass. We find a generic formula relating the dilaton mass with the sum of the tensions of the branes and the curvature of the fourgeometries. Demanding that this mass is not tachyonic we can derive the necessary condition for stabilizing the overall size of the system. This is in accordance with the result found in [1] that the GW stabilization mechanism of the RS1 model has to include the backreaction on the metric. 2. Review of sum rules At first it would be instructive to review the sum rules presented in [1]. We will concentrate as in [1] in the case where the background metric of the fivedimensional spacetime can be written in the form: ds = W (y) gµν (x) dx dx + dy , 2 2 µ ν 2 (1) with gµν (x) a general background four-dimensional metric and W (y) a generic warp factor. We should stress here that this is not the most general choice of metric in five dimensions as we have explicitly assumed that the all four-dimensional sections have the same geometry. We can now consider an arbitrary number of minimally coupled scalar bulk fields Φ I (x, y) with internal metric GI J and arbitrary bulk potential V (Φ) (which includes bulk cosmological constant), coupled to an again arbitrary number of branes with brane potential λi (Φ) (which again includes the brane tensions). The action describing the above system is the following: √ S = d 4 x dy −G 1 × 2M 3 R − GI J ∂M Φ I ∂ M Φ J − V (Φ) 2 (i) −G λi (Φ)δ(y − yi ) √ (2) , − −G i (i) where G µν is the induced metric on the brane and M the fundamental 5D scale. The Einstein equations arising from the above metric can be written in the form: 1 4 4M 3 Rµµ = − Tµµ − T55 , (3) 3 3 1 2 4M 3 R55 = − Tµµ + T55 , (4) 3 3 where the energy–momentum tensor components are: Tµµ = −∂µ Φ · ∂ µ Φ − 2Φ · Φ − 4V (Φ) λi (Φ)δ(y − yi ), −4 (5) i 1 1 T55 = − ∂µ Φ · ∂ µ Φ + Φ · Φ − V (Φ), (6) 2 2 with the indices in the above formulas raised and lowered by Gµν = W (y)2 gµν (x) and where dot product denotes construction with the internal metric GI J . Since we are interested on a background configuration, the ∂µ Φ · ∂ µ Φ terms can be dropped. The Ricci tensor is easily calculated to be: Rµµ = W −2 Rg − 12W 2 W −2 − 4W W −1 , R55 = −4W W −1 . (7) (8) If we now consider the function (W a ) with a an arbitrary real number, its integral around the compact extra dimension is zero. Using (3), (4), (7), (8) we arrive at an infinite number of constraints [1]: dy W a Tµµ + (2a − 4)T55 = 4M 3 (1 − a)Rg dy W a−2 . (9) As it is obvious, these constraints are a natural consequence of the equations of motion. It is straightforward to see for example that they are satisfied in the RS1 model [3] and the bigravity/multigravity models [7,8]. We will single out three constraints which we will be important for the subsequent discussion, namely, the ones for a = 0, 1, 2: µ 5 3 dy Tµ − 4T5 = 4M Rg dy W −2 , (10) dy W Tµµ − 2T55 = 0, (11) dy W 2 Tµµ = −4M 3 Rg dy. (12) One could also use the above constraints for noncompact models, but should be careful that the above A. Papazoglou / Physics Letters B 505 (2001) 231–235 derivation makes sense. For the finite volume flat one three-brane Randall–Sundrum model (RS2) [9], all constraints are valid for a 0. For the infinite volume Gregory–Rubakov–Sibiryakov model (GRS) [10] only the a = 0 constraint is valid and for the infinite volume Karch–Randall model (KR) [11] all constraints are valid for a 0. 3. Dilaton in warped backgrounds We now consider the perturbation related to the overall size of the compact system, namely the dilaton. The general form of the metric for the physical radion perturbations that do not mix with the graviton(s) is given in [12]. For the dilaton, the ansatz is rather simple and can be written in the form [13] (see also [14]): −2 ds 2 = e−W (y) γ (x)W (y)2 gµν (x) dx µ dx ν 2 + 1 + W (y)−2 γ (x) dy 2 . (13) Substituting the above metric in the action (2) (see Appendix A for analytic formulas), integrating out total derivatives, throwing out γ -independent parts and keeping terms up to quadratic order, we get: S= √ d x dy g 4 1 3 −2 µν 6M W g γ,µ γ,ν 2 1 2 + L1 γ − L2 γ , 2 − (14) with 3 L1 = 2M 3 4W 2 + 16W W + W 2 Φ · Φ 2 2 2 + W V (Φ) + 2W λi (Φ)δ( y − yi ), (15) i L2 = 2M 3 W −2 Rg − 32W 2 W −2 + 32W W −1 λi (Φ)δ( y − yi ). + 5Φ · Φ + 4 (16) i At this point, let us work out the integral over the extra dimension of the tadpole term L1 of the Lagrangian. This gives: 233 1 dy W 2 dy L1 = − 2 2 W µ 5 3 W +4 × Tµ − 2T5 − 16M , W2 W (17) where we used the energy–momentum tensor components found in the previous section with respect to the unperturbed background metric (5), (6). We can further simplify this quantity if we use Eqs. (3), (4), (7), (8) which hold for the background metric. The resulting expression is: 1 dy L1 = (18) dy W 2 Tµµ + 4M 3 W −2 Rg , 6 which is exactly zero because of the a = 2 constraint (12). This result should have been expected since the perturbation (13) is bound to extremize the effective potential when one evaluates the action using the background equations of motion. However, it is interesting and rather unexpected that the absence of the tadpole term is linked to this particular sum rule of [1]. It is worth mentioning here that in the case that the warp factor is effectively constant (W ≈ 1), as it was assumed in [6], the condition that the expression (17) vanishes, is identical with the a = 1 constraint (11). Our next task is to read off the mass of the dilaton from the action functional. For this reason we define the canonically normalized dilaton field with mass dimension one γ̄ 2 = 6M 3 dy W −2 γ 2 ≡ Aγ 2 . Then the mass of the canonical dilaton γ̄ is: 1 m2 = (19) dy L2 . A After a lot of simplifications using the relations (3)–(8) we obtain: 1 m2 = − dy 10M 3 W −2 Rg + Φ · Φ 3A +4 λi (Φ)δ(y − yi ) . (20) i We can further simplify the expression using the a = 0 constraint (10) and get a more suggestive result: 1 m2 = (21) dy Φ · Φ − 2M 3 W −2 Rg A or equivalently, 1 λi (Φ) + 3M 3 Rg dy W −2 . (22) m2 = − A i 234 A. Papazoglou / Physics Letters B 505 (2001) 231–235 From the second expression it is clear that we cannot have a massive dilaton if the sum of the effective tensions of the branes λi (Φ) is exactly zero and at the same time they are kept flat. This is the same conclusion that appeared in [1] regarding the GW mechanism in the RS1 scenario. Moreover, the absence of tachyonic mass would guarantee the stabilization of the overall size of any system with the above characteristics. By Eqs. (21), (22) we get two equivalent conditions: dy Φ · Φ − 2M 3 W −2 Rg > 0, (23) (24) λi (Φ) + 3M 3 Rg dy W −2 < 0. It would be interesting to see what happens with higher than quadratic terms in the dilaton potential and the possible role that the other sum rules of [1] play. Moreover, one could work out the same calculation for the other moduli in these configurations, the radions [12], and see if/how the results are modified. Finally, a much more general investigation is needed to obtain the sum rules and their role for the dilaton and radion potentials in models in which the fourdimensional geometries are not identical as it happens with cosmological solutions (see, e.g., [16]). These are important issues for understanding the dynamics of the dilaton/radions in the extra dimensional models and will be addressed in an other publication [17]. i If one wishes to have flat branes, then the sum of brane tensions should be negative or equivalently one should have a non-constant (in y) scalar field configuration. In the case that the above expressions (and thus the mass) vanish, one should look for the higher orders of the effective potential to examine the stability of the system. Finally, we should note that the formulas (21), (22) are valid even for non-compact models whenever the dilaton mode is normalizable. This happens for example in the KR model [11] and one can find from the above expressions the mass of the dilaton, in agreement with [15]. In that case, the dilaton mode cannot be attributed to the fluctuation of the overall size of the system, but can be understood to be a remnant mode if we start with the compact “++” system [8] and send one of the branes to infinity (i.e., after decompactifying the system). 4. Conclusions We have showed that in a general warped metric with identical four-geometries and arbitrary bulk dynamics involving minimally coupled scalar fields, the absence of dilaton tadpoles is related to one particular sum rule of [1]. Moreover, we have calculated the dilaton mass as a function of the sum of the brane tensions and the leftover curvature of the branes. The result agrees with the observation made by [1] that one could not have a massive dilaton for flat four-geometries and zero net brane tensions. Acknowledgements We would like to thank Gary Gibbons, Stavros Mouslopoulos, Luigi Pilo and Graham G. Ross for helpful discussions. We are grateful to Ian I. Kogan and Keith A. Olive for valuable discussions. We are indebted to Panagiota Kanti for important comments and careful reading of the manuscript. This work is supported by the Hellenic State Scholarship Foundation (IKY) No. 8017711802. Appendix A In this appendix we list the Ricci tensor components, the Ricci scalar and the action obtained by the metric: −2 ds 2 = e−W (y) γ (x) W (y)2 gµν (x) dx µ dx ν 2 + 1 + W (y)−2 γ (x) dy 2 . (A.1) The spacetime components of the five-dimensional Ricci tensor are: gµν −2 W γ Rµν = Rg µν + 2 W −6 γ gµν − γ,κ γ ,κ 2 (1 + W −2 γ ) 1 1 − W −2 γ − W −4 γ,µ γ,ν 2 1 + W −2 γ + W −4 γ Dµ ∂ν γ (1 + W −2 γ ) A. Papazoglou / Physics Letters B 505 (2001) 231–235 3 + 4W −2 γ 1 + W −2 γ W W −2 , − gµν e−W γ (1 + W −2 γ ) − gµν e−W −2 γ W2 and the (55) component: −2 R55 = eW γ 1 + W −2 γ W −6 γ,µ γ ,µ −2 − eW γ 1 + W −2 γ W −4 γ − 4 1 + W −2 γ W −1 W − 4 1 + W −2 γ W −4 W 2 γ . Finally, the Ricci scalar is: References (A.2) (A.3) 1 + 3W −2 γ R=e W Rg + e W −4 γ 1 + W −2 γ 1 W −2 γ 1 − 3W −2 γ + e W −6 γ,µ γ ,µ 2 1 + W −2 γ W −2 γ W −2 γ −2 W −1 W 12W −2 W 2 + 20W −4 W 2 γ − . −2 (1 + W γ ) (1 + W −2 γ ) (A.4) In the above expressions the indices are raised and lowered by gµν . The action (2) then becomes: √ S = d 4 x dy g −2 × 2M 3 e−W γ W 2 1 + W −2 γ Rg −8 1 + 52 W −2 γ + 32 W −4 γ 2 g µν Dµ ∂ν γ −2 + e−2W γ −8W W 3 − 12W 2 W 2 − 20W 2 γ + e−W −2 γ W4 1 −2 e−2W γ Φ · Φ 2 (1 + W −2 γ ) −2 − e−2W γ W 4 1 + W −2 γ V (Φ) −2W −2 γ 4 −e W λi (Φ)δ(y − yi ) . − i 235 (A.5) [1] G. Gibbons, R. Kallosh, A. Linde, hep-th/0011225. [2] W.D. Goldberger, M.B. Wise, Phys. Rev. Lett. 83 (1999) 4922, hep-ph/9907447; W.D. Goldberger, M.B. Wise, Phys. Lett. B 475 (2000) 275, hep-ph/9911457. [3] L. Randall, R. Sundrum, Phys. Rev. Lett. 83 (1999) 3370, hepph/9905221. [4] O. DeWolfe, D.Z. Freedman, S.S. Gubser, A. Karch, Phys. Rev. D 62 (2000) 046008, hep-th/9909134. [5] U. Ellwanger, Phys. Lett. B 473 (2000) 233, hep-th/9909103. [6] P. Kanti, I.I. Kogan, K.A. Olive, M. Pospelov, Phys. Rev. D 61 (2000) 106004, hep-ph/9912266. [7] I.I. Kogan, S. Mouslopoulos, A. Papazoglou, G.G. Ross, J. Santiago, Nucl. Phys. B 584 (2000) 313, hep-ph/9912552; S. Mouslopoulos, A. Papazoglou, JHEP 0011 (2000) 018, hepph/0003207; I.I. Kogan, G.G. Ross, Phys. Lett. B 485 (2000) 255, hepth/0003074; I.I. Kogan, S. Mouslopoulos, A. Papazoglou, G.G. Ross, Nucl. Phys. B 595 (2001) 225, hep-th/0006030. [8] I.I. Kogan, S. Mouslopoulos, A. Papazoglou, Phys. Lett. B 501 (2001) 140, hep-th/0011141. [9] L. Randall, R. Sundrum, Phys. Rev. Lett. 83 (1999) 4690, hepth/9906064. [10] R. Gregory, V.A. Rubakov, S.M. Sibiryakov, Phys. Rev. Lett. 84 (2000) 5928, hep-th/0002072. [11] A. Karch, L. Randall, hep-th/0011156. [12] L. Pilo, R. Rattazzi, A. Zaffaroni, JHEP 0007 (2000) 056, hepth/0004028. [13] C. Charmousis, R. Gregory, V.A. Rubakov, Phys. Rev. D 62 (2000) 067505, hep-th/9912160. [14] J. Bagger, D. Nemeschansky, R.-J. Zhang, hep-th/0012163. [15] Z. Chacko, P.J. Fox, hep-th/0102023. [16] P. Binétruy, C. Deffayet, D. Langlois, Nucl. Phys. B 565 (2000) 269, hep-th/9905012; P. Binétruy, C. Deffayet, U. Ellwanger, D. Langlois, Phys. Lett. B 477 (2000) 285, hep-th/9910219; P. Kanti, K.A. Olive, M. Pospelov, Phys. Rev. D 62 (2000) 126004, hep-ph/0005146; P. Kanti, K.A. Olive, M. Pospelov, Phys. Lett. B 481 (2000) 386, hep-ph/0002229. [17] P. Kanti, I.I. Kogan, K.A. Olive, A. Papazoglou, in preparation. 26 April 2001 Physics Letters B 505 (2001) 236–242 www.elsevier.nl/locate/npe Covariant perturbation theory and the Randall–Sundrum picture Ezequiel Alvarez a , Francisco D. Mazzitelli b a Instituto Balseiro, Centro Atómico Bariloche, 8400 Bariloche, Argentina b Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón I, 1428 Buenos Aires, Argentina Received 27 October 2000; received in revised form 20 December 2000; accepted 21 February 2001 Editor: H. Georgi Abstract The effective action for quantum fields on a d-dimensional spacetime can be computed using a non-local expansion in powers of the curvature. We show explicitly that, for conformal fields and up to quadratic order in the curvature, the non-local effective action is equivalent to the d + 1 action for classical gravity in AdSd+1 restricted to a (d − 1)-brane. This generalizes previous results about quantum corrections to the Newtonian potential and provides an alternative method for making local a non-local effective action. The equivalence can be easily understood by comparing the Kallen–Lehmann decomposition of the classical propagator with the spectral representation of the non-local form factors in the quantum effective action. 2001 Published by Elsevier Science B.V. 1. The analysis of the physical effects of quantum fields on the background geometry requires the calculation of the effective action. This is a complicated object even for free fields. With the exception of a few highly symmetric background metrics, it cannot be computed exactly. Moreover, in order to study problems like black hole evaporation or the physics of the early universe, it is necessary to compute the effective action for an arbitrary metric, that should be fixed at the end by minimizing the effective action. A useful approach for the approximate computation of the effective action is the so-called covariant perturbation theory [1]. In this approach, that can be understood as a summation the Schwinger–DeWitt expansion, the effective action is written in powers of curvatures. This approximation contains non-local terms that include important physical information like gravitational particle creation and the leading long distance quantum corrections to general relativity. For conformal fields, in two spacetime dimensions the quadratic term in the covariant perturbation theory reproduces the (exact) Polyakov action. It is possible to derive Hawking radiation from it [2]. In four dimensions this has still not been done, and indeed it is a very difficult task because Hawking radiation is contained in the cubic terms of the expansion. E-mail addresses: [email protected] (E. Alvarez), [email protected] (F.D. Mazzitelli). 0370-2693/01/$ – see front matter 2001 Published by Elsevier Science B.V. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 3 2 2 - 7 E. Alvarez, F.D. Mazzitelli / Physics Letters B 505 (2001) 236–242 237 The four-dimensional quadratic effective action has been used to compute the leading long distance 1/r 3 corrections to the Newtonian potential [3] (16π)2 GB4 GM 1− , V (r) = − (1) r r3 where the constant B4 depends on the spin and number of quantum matter fields. These corrections have been computed by other methods a long time ago [4]. For related works see [5]. In a recent paper, Duff and Liu [6] proved that the same kind of corrections to the gravitational potential do appear in the Randall–Sundrum brane-world proposal [7]. When a 3-brane is inserted into AdS5 , and for classical matter fields in the brane, the classical metric in five dimensions restricted to the brane reproduces the classical Newtonian potential plus the 1/r 3 corrections. The coefficient of 1/r 3 that appears in this scenario coincides with the coefficient due to closed loops of N = 4 superconformal U (N) Yang–Mills theory in the four-dimensional theory. This is in tune with the AdS/CFT correspondence [8]. In this Letter we will extend the results of Ref. [6]. We will prove that, up to quadratic order in the curvature and for free conformal fields, the non-local d-dimensional effective action coincides with the restriction of the gravitational action in AdSd+1 to a (d − 1)-brane. The results are valid for d > 2. We stress that we are not trying to check the consistency between the AdS/CFT and the brane-world relations, as in Ref. [6]. Our aim is to provide an alternative representation for the non-local d-dimensional effective action. 2. For a scalar field in curved spacetimes the effective action is given by Γ = 12 ln det(O/µ2 ), where O = −g µν ∇µ ∇ν + m2 + ξ R is the operator of the classical field equation, µ is an arbitrary parameter with dimensions of mass, m is the mass of the scalar field, and ξ is the coupling to the scalar curvature. The conformal d−2 coupling in d dimensions is ξ = ξc = 14 d−1 . Using heat kernel techniques [9] it is possible to obtain the Schwinger–DeWitt expansion for the effective action 1 Γ =− 2(4π)d/2 ∞ 0 ds s 1+ d2 e−m 2s (−s)l l0 l! dd x √ g al (x). (2) The Schwinger–DeWitt coefficients al are functions of the curvature and its covariant derivatives. When integrating out term by term the expression above, an expansion in inverse powers of the mass is obtained. The expansion is valid for slowly varying metrics that satisfy R m2 (R denotes components of the curvature tensor). The expansion is local, and adequate for the analysis of the divergences of the theory, which are contained in the terms with l less or equal to the integer part of d/2. However, it misses very important physical effects (like particle creation), and it is not adequate for massless quantum fields. It is possible to perform a partial summation of the Schwinger–DeWitt expansion by keeping terms up to a given order in the curvature. The idea was introduced in Ref. [10] and further developed in Refs. [1,11]. The effective action for a massless scalar field in d spacetime dimensions, up to quadratic order in the curvature, can be written as [11] Γ = Γlocal + Γnonloc, where Γlocal = (3) k √ µν d−2 α (l) R(d) ✷l R(d) + β (l)Rµν(d) ✷l R(d) d d x −g −Λ(d) + M(d) R(d) − (4) l=0 and Γnonloc = −α dd x √ µν −g aR(d)f (✷)R(d) + bRµν(d)f (✷)R(d) . (5) 238 E. Alvarez, F.D. Mazzitelli / Physics Letters B 505 (2001) 236–242 2 = 1/16πG Here gµν , x µ , M(d) and Λ(d) are the d-dimensional metric, coordinates, Planck mass (M(4) Newton ) and cosmological constant, R(d) and Rµν(d) the Ricci tensor and scalar, respectively, √ π −d/2 , α = (4π) 8Γ ((d − 1)/2) d 1 a = (ξ − ξc )2 − , b= , 8(d − 1)2 (d + 1) 2(d 2 − 1) and f (✷) = d −✷ 2 −2 −✷ d 2 (−1) , d even, ln 4 4µ2 d −✷ 2 −2 d−1 2 , (−1) π 4 d odd. The summation in Eq. (4) runs up to k, the integer part of d/2 − 2. These terms are needed to renormalize the theory. After renormalization, the coefficients α (l) , β (l) and Λ(d) might take arbitrary values. For simplicity we will take Λ(d) = 0 in what follows. The results for the effective action can be extended for fields of arbitrary spin [11]. For example, for a massless Dirac field in four dimensions, the effective action is six times the result for a conformally coupled scalar field [12]. For gµν = ηµν + hµν , and in the harmonic gauge (i.e., hµν, ν = 12 hα α ,µ ), Eq. (3) can be rewritten as −1 (2) Γ = (6) d d x hµν (∆−1 )µν ρσ hρσ , 4 where (∆ −1 µν ) 1 µν µ ν ηρ ησ − η ηρσ + α✷2 f (✷) bηρµ ησν + aηµν ηρσ 2 d−2 ρσ = −M(d) ✷ + k ✷l+2 α (l) ηµν ηρσ + β (l) ηρµ ησν . (7) l=0 If we add to the theory classical matter described by an energy–momentum tensor T µν , the spacetime metric satisfies hµν = ∆αβ µν Tαβ . (8) In the low-energy approximation the quantum correction in Eq. (7) can be treated as a small perturbation. Using β µ −A ηαβ ηµν + B1 ηµα ην , it is straightforward to check that, that the inverse of Aηµν ηρσ + Bηρ ησν is given by B(B+A·d) up to leading order, −1 1 ηαβ ηµν ∆αβ µν = d−2 ηµα ηνβ − d −2 M(d) ✷ α 1 d2 − 1 α β 2 αβ − (ξ − ξc ) 8 − η ηµν f (✷) ηµ ην − 2(d−2) d −1 (d − 2)2 2(d 2 − 1)M (d) + k̃ j =0 gj(1) ✷j ηµα ηνβ + gj(2) ✷j ηαβ ηµν . (9) E. Alvarez, F.D. Mazzitelli / Physics Letters B 505 (2001) 236–242 239 The constants gj(i) depend on d, M(d) , α (l) and β (l) . The summation here and in what follows is only for d 5; k̃ is equal to k when d is odd and to k − 1 when d is even. The meaning of Eq. (9) is very simple: the first term corresponds to the classical propagation while the second contains the quantum corrections and is easily traced back to the non-local part of the action, Eq. (5). The analytic terms (proportional to ✷j ), will not contribute in the large distance/low-energy limit (see below). 3. We will now prove that a propagator similar to Eq. (9) describes the classical propagation on a brane inserted into AdSd+1 [13]. If the d-dimensional spacetime is thought as a (d − 1)-brane in a (d + 1)-dimensional theory then the classical action reads √ d−1 √ d+1 X −G M(d+1)R(d+1) − Λ(d+1) + Lmatter − d d x −g τ. d (10) Here XI = (y, x ρ ) are the (d + 1)-dimensional coordinates, GI J the metric in d + 1 dimensions, and τ the brane tension. The Lmatter term may include a matter source in the brane as well as in the bulk. The brane geometry is chosen such that y is the coordinate in the bulk and x ρ are coordinates along the brane (which is located at y = 0), then small fluctuations to the metric are represented by ds 2 = dy 2 + e−2|y|/L ηµν + hµν (x ρ , y) dx µ dx ν , (11) d−1 /Λ(d+1) . where L = −d(d − 1)M(d+1) We are only interested in hµν (x ρ , y = 0) when the matter source is located on the brane. In this situation, it has been shown that the effective propagator on the brane is given by [13] α β √ 1 d −2 1 α β 1 1 αβ αβ αβ ηµ ην − η ηµν − d−1 ∆KK −✷ ηµ ην − η ηµν , ∆ µν = − (12) d−1 ✷ d −2 d −1 LM(d+1) M(d+1) where ∆KK √ √ −1 Kd/2−2 ( −✷L) −✷ = √ √ . −✷ Kd/2−1 ( −✷L) (13) Again Eq. (12) has a simple interpretation: the first term describes the zero mode graviton localized on the brane, while the second term corresponds to the√continuum Kaluza–Klein graviton modes. At large distances, corresponding to −✷L 1, Eq. (13) can be expanded to give, up to the first term nonanalytic-in-✷, d k̃ −✷ 2 2 −2 d (e) 2 l + c (e) 2 L ✷L c (−1) ln −✷L2 , d even, l d/2−2 √ 4 1 ∆KK −✷ ≈ l=0 (14) d L k̃ 2 l d−1 −✷ 2 2 −2 (o) (o) L cl ✷L + πcd/2−2 (−1) 2 , d odd. 4 l=0 The coefficients cl(i) can be easily obtained, but we will not need the explicit expression in what follows. Now we compare Eqs. (12) and (9). The classical terms in both propagators coincide if we choose the coupling 1 constants such that d−2 d−1 = d−2 . In order to have agreement between the leading non-analytic terms, the LM(d+1) M(d) d−1 coupling must be conformal, i.e., ξ = ξc . Moreover, we must have cd/2−2Ld−3 /M(d+1) = α(2(d 2 − 1)M(d) )−1 . These equations relate the values of the d + 1 cosmological constant and Planck mass with Planck mass in d dimensions. Had we considered a different free field content on the brane (Ns fields of spin s) the only difference would have been a different relation between the values of these parameters. (e,o) 2(d−2) 240 E. Alvarez, F.D. Mazzitelli / Physics Letters B 505 (2001) 236–242 It is not necessary to require agreement between the terms analytic in ✷ since, as shown below, they are (2) (1) 1 not relevant in the low-energy limit. However, it is worth noting that with the choice gj = − d−1 gj and d−1 gj(1) = −cj(e,o)L2j +1 /M(d+1) the analytic terms also coincide. This would imply particular values for the constants (l) (l) α and β in the non-local effective action Eq. (5), all of them determined by M(d+1) and Λ(d+1) . We will now show that analytic terms are not relevant in the low-energy limit. To illustrate this point we compute the quantum corrections to the d-dimensional Newtonian potential, − 12 h00 (x). We assume a classical µ x ). Here x are the space coordinates, mass M fixed at the origin of coordinates, namely T µν (x) = δ0 δ0ν M δ (d−1) ( x = (x1 , x2 , . . . , xd−1 ). With this in mind, using Eqs. (8) and (9), the quantum corrected Newtonian potential reads 1 M 1 d = 3, 2 r, B3 M(3) −1 00 h (r) = V (r) = (1) j (2) j (d−1) 1 1 M k̃ M 2 ( x ), d 4, Ad M d−2 r d−3 + Bd 2(d−2) r 2d−5 − 2 j =0 gj ✷ + gj ✷ δ M(d) M(d) x |. As anticipated, the analytic terms proportional to ✷j produce where Ad and Bd are constants and r = | quantum corrections localized at the origin. They are therefore irrelevant at large distances. In four dimensions, the Newtonian potential reads G 1 2 GM 1+ 1 + 45 ξ − , V (r) = − (15) r 6 45πr 2 and agrees with previous results for ξ = 1/6 [6,12]. If we consider N0 scalar fields and N1/2 Dirac fields, the Newtonian potential becomes GM G 1 2 V (r) = − (16) 1+ N0 1 + 45 ξ − + 6N1/2 . r 6 45πr 2 4. Non-local effective actions have been previously localized through the introduction of auxiliary fields. For example, in two dimensions, Polyakov’s action √ 1 1 SP = − (17) d 2 x −g R R 96π ✷ can be made local by introducing an auxiliary field ψ and the local action √ 1 Slocal = − d 2 x −g (−ψ✷ψ + 2ψR). 96π (18) On shell for the auxiliary field, both actions SP and Slocal are equivalent. In four dimensions, the effective action that reproduces the conformal anomaly is the so-called Reigert’s action [14]. The non-local part of the Reigert’s action is, schematically, 1 2 SR = d 4 x R2 (19) R , ∆4 where R denotes components of the Riemann tensor, and ∆4 is the fourth-order operator 1 2 ∆4 = ✷2 − 2R µν ∇µ ∇ν + R✷ − ∇ µ R∇µ . 3 3 1 Note that for d = 3 spacetime is flat outside matter, hence there is no gravitational force. The term proportional to B comes from the 3 quantum correction. E. Alvarez, F.D. Mazzitelli / Physics Letters B 505 (2001) 236–242 Reigert’s action can be made local [14,15] by the introduction of auxiliary scalar fields Slocal = d 4 x −ψ∆4 ψ + 2ψR2 . 241 (20) The localization based in the introduction of auxiliary fields works only when the form factors in the non-local effective action are the inverse of polynomials in ✷ and ∇µ . Here the form factors does not satisfy this property. An extra dimension is needed to make local the action. The mathematical reason for this can be understood as follows. The non-analytic form factors can be represented in the form of spectral integrals [1]. For example, in three and four dimensions the form factors can be written as ∞ −✷ 1 1 ln − dλ = , µ2 λ − ✷ λ + µ2 0 (−✷) −1/2 2 = π ∞ 0 dλ 1 . λ2 − ✷ (21) Similar expressions can be found for other dimensions. Note that the non-analytic functions of ✷ are written as 1 1 . or λ2 −✷ integrals that involve massive propagators λ−✷ On the other hand, the restriction of a massless d + 1 propagator on a (d − 1)-brane also admits an analogous representation [16]. Indeed, let us consider the metric ds 2 = dy 2 + w2 (y)gµν (x) dx µ dx ν . The D’Alambertian operator can be written as ✷d+1 = ✷ ✷ ∂2 w ∂ ≡ 2 + ✷y , + + d 2 2 w ∂y w w ∂y (22) where ✷ is the d-dimensional D’Alambertian associated to gµν . We introduce the eigenfunctions θλ(i) (y), that satisfy ✷y θλ(i) = − wλ2 θλ(i) . It can be easily shown [16] that the 1 , restricted to a fixed slice y = const, admits the following representation massless propagator ∆ = ✷d+1 ∆(x, y, x , y) = (i) 2 θ (y) λ i,λ 1 . ✷−λ (23) This is analogous to the Kallen–Lehman decomposition in quantum field theory, with a weight function µ(λ, y) = (i) 2 i |θλ (y)| . The similarity between the form factors in the non-local quantum effective action and the restriction of the classical propagator on a brane is now clear (compare Eqs. (21) and (23)). Roughly speaking, in this Letter we have shown that the weight function in AdSd+1 spacetime reproduces the spectral representation of the d-dimensional form factor for conformal fields. It is possible that, by taking a different metric in the bulk, one could reproduce the non-local effective action for non-conformal fields. Alternatively, a different quantum field theory on the brane could reproduce the AdSd+1 propagator beyond leading order. The equivalence shown in this Letter could be useful as a tool for computations of the effects of quantum fields on the spacetime metric, since it may be technically more easy to work with an extra dimension than with a non-local effective action. Acknowledgements This work was supported by Universidad de Buenos Aires, CONICET (Argentina) and CNEA (Argentina). We would like to thank G. Giribet and J. Russo for useful conversations. 242 E. Alvarez, F.D. Mazzitelli / Physics Letters B 505 (2001) 236–242 References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] A.O. Barvinsky, G.A. Vilkovisky, Nucl. Phys. 333 (1990) 471. V.P. Frolov, G.A. Vilkovisky, in: M.A. Markov, P.C. West (Eds.), Quantum Gravity, Plenum, London, 1984. D. Dalvit, F.D. Mazzitelli, Phys. Rev. D 50 (1994) 1001. M.J. Duff, Phys. Rev. D 9 (1974) 1837. J.F. Donoghue, Phys. Rev. Lett. 72 (1994) 2996; H. Hamber, S. Liu, Phys. Lett. B 357 (1995) 51; I. Muzinich, S. Vokos, Phys. Rev. D 52 (1995) 3472. 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Klusoň Department of Theoretical Physics and Astrophysics, Faculty of Science, Masaryk University, Kotlářská 2, 611 37, Brno, Czech Republic Received 21 December 2000; received in revised form 21 February 2001; accepted 26 February 2001 Editor: M. Cvetič Abstract In this short note we would like to present a simple topological matrix model which has close relation with the noncommutative Chern–Simons theory. 2001 Published by Elsevier Science B.V. Keywords: Matrix models 1. Introduction In recent years there was a great interest in the area of the noncommutative theory and its relation to string theory. In particular, it was shown in the seminal paper [1] that the noncommutative theory can be naturally embedded into the string theory. It was also shown in the recent paper [2] that there is a remarkable connection between noncommutative gauge theories and matrix theory. For that reason it is natural to ask whether we can push this correspondence further. In particular, we would like to ask whether other gauge theories, for example Chern–Simons theory, can be also generalised to the case of noncommutative ones. It was recently shown [3] that this can be done in a relatively straightforward way in the case of Chern–Simons theory. It is then natural to ask whether, in analogy with [2], there is a relation between topological matrix models [4] and Chern– Simons noncommutative theory. It was suggested in many papers [6,7,11] that Chern–Simons theory could play profound role in the E-mail address: [email protected] (J. Klusoň). nonperturbative formulation of the string theory, Mtheory. On the other hand, one of the most successful (up to date) formulation of M-theory is the matrix theory [5], for review, see [12–16]. We can ask the question whether there could be some connection between matrix models and Chern–Simons theory. This question has been addressed in interesting papers [6,7], where some very intriguing ideas have been suggested. Noncommutative Chern–Simons theory could also play an important role in the description of the quantum hall effect in the framework of string theory [17, 18]. All these works suggest plausible possibility to describe some configurations in the physics of the condense systems in terms of D-branes, which can be very promising area of research. On the other hand, some ideas of physics of the condense systems could be useful in the nonperturbative formulation of the string theory. In summary, on all these examples we see that it is worth to study the basic questions regarding to the noncommutative Chern–Simons theory and its relation to the matrix theory and consequently to the string theory. In this Letter we will not address these exciting ideas. We will rather ask the question whether some 0370-2693/01/$ – see front matter 2001 Published by Elsevier Science B.V. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 3 3 7 - 9 244 J. Klusoň / Physics Letters B 505 (2001) 243–248 from of the topological matrix model can lead to the noncommutative Chern–Simons theory. Such a model has been suggested in [8] and further elaborated in [4]. We will show that the simple topological matrix model [4] cannot lead (as far as we know) to the noncommutative Chern–Simons theory. For that reason we propose a simple modification of this model when we include additional term containing the information about the background structure of the theory. Without including this term in the action we would not be able to obtain noncommutative version of the Chern–Simons theory. It is remarkable fact that this term naturally arises in D-brane physics from the generalised Chern–Simons term in D-brane action in the presence of the background Ramond–Ramond fields [19]. For that reason we believe that our proposal could really be embedded in the string theory and also could have some relation with M-theory. 3. Matrix model of Chern–Simons theory It was argued in [4,8] that we can formulate the topological matrix model which has many properties as the Chern–Simons theory [8]. The action for this model was proposed in the form S = µ1 ···µD Tr Xµ1 · · · XµD . 2. Brief review of Chern–Simons theory In this section we would like to review the basic facts about Chern–Simons actions and in particular their extensions to noncommutative manifolds. We will mainly follow [3]. The Chern–Simons action is the integral of the 2n + 1 form C2n+1 over spacetime manifold 1 which satisfies dC2n+1 = Tr F n+1 , of the noncommutative geometry [9]. In this short article we will not discuss the operator formalism in more details since it is well know from the literature. (See [9] and reference therein.) The definition of the noncommutative Chern–Simons action is very straightforward in the operator formalism as was shown in [3], where the whole approach can be found. We will see the emergence of the noncommutative Chern–Simons action in the operator formalism in the next section where this action naturally arises from the modified topological matrix model [4]. (2.1) where the wedge operation ∧ between forms F is understood. The action is defined as δ δS2n+1 = (2.2) C2n+1 = (n + 1)F n , δA δA with the conventions A = Aµ dx µ , F = dA − iA ∧ A 1 = ∂µ Aν − ∂ν Aµ − i[Aµ , Aν ] dx µ ∧ dx ν . (2.3) 2 The extension of this action to the case of the noncommutative background is straightforward [3]. The easiest way to see this is in terms of the operator formalism 1 In this article we will consider the Euclidean spacetimes only. (3.1) It is easy to see that this model can be defined in the odd dimensions only: S = µ1 ···µD Tr XµD Xµ1 · · · XµD−1 = (−1)D−1 µD µ1 ···µD−1 Tr XµD Xµ1 · · · XµD−1 = (−1)D−1 S, (3.2) so we have D − 1 = 2n ⇒ D = 2n + 1. The equations of motion obtained from (3.1) are δS = µµ1 ···µD Xµ1 · · · XµD = 0, δXµ µ = 0, . . . , 2n. (3.3) It was argued in [4] that there are solutions corresponding to D-branes. However, it is difficult to see whether these solutions corresponding to some physical objects since we do not know how to study the fluctuations around these solutions. For example, for D = 3 we obtain from the equation of motion for µ = 0, 1, 2 0 1 2 0 1 2 X , X = 0, X , X = 0. (3.4) X , X = 0, We see that the only possible solutions correspond to separate objects where the matrices X are diagonal or solution X1 = 0 = X2 with any X0 . We do not know any physical meaning of the second solution. For that reason we propose the modification of the J. Klusoň / Physics Letters B 505 (2001) 243–248 topological matrix model which, as we will see, has a close relation with the noncommutative Chern– Simons theory [3]. We propose the action in the form D + 1 µ1 n X · · · XµD S = (2π) µ1 ···µD Tr (−1)n/2 2D (n−1)/2 D + 1 µ1 µ2 µ3 µD θ X ···X + (−1) , 4(D − 2) (3.5) where D = 2n + 1 and the numerical factors (−1)n/2 , (−1)(n−1)/2 arise from the requirement of the reality of the action. The other factors (D + 1)/(2D), (D + 1)/(4(D − 2)) were introduced to have a contact with the work [3]. In the previous expression (3.5) we have also introduced the matrix θ µν which characterises given configuration. The equations of motion have a form (−1)n/2 (n + 1)µµ1 ···µ2n Xµ1 Xµ2 · · · Xµ2n n+1 µµ1 µ2 ···µ2n + (−1)(n−1)/2 2 × θ µ1 µ2 Xµ3 · · · Xµ2n = 0. (3.6) We would like to find solution corresponding to the noncommutative Chern–Simons action. From the fact that we have odd number of dimensions we see that one dimension should correspond to the commutative one. In order to obtain Chern–Simons action in the noncommutative spacetime we will follow [10] and compactify the commutative direction X0 . For that reason we write any matrix as µ µ XI J = Xij mn , I = m × M + i, J = n × M + j, (3.7) where Xij is M × M matrix with M → ∞ and also m, n go from −N/2 to N/2 and we again take the limit N → ∞. In other words, the previous expression corresponds to the direct product of the matrices X = A ⊗ B ⇒ Xxy = Aij Bkl , x = i × M + k, y = j × M + l, (3.8) with M × M matrix B. We impose the following constraints on the various matrices [10] i Xij mn = Xiji m−1,n−1 , a = 1, . . . , 2n, 0 Xij mn = Xij0 m−1,n−1 , m = n, 0 Xij nn = 2πRδij + Xij0 n−1,n−1 , (3.9) 245 where R is a radius of compact dimension. These constraints (3.9) can be solved as [10] i Xij mn = Xiji 0,m−n = Xiji m−n , 0 Xij mn = 2πRmδmn ⊗ δij + Xij0 m−n . (3.10) We then immediately obtain 0 i X , X ij mp = 2πRmδmn Xiji np − Xiji mn 2πRnδnp 0 i i 0 Xkj np − Xik Xkj np + Xik mn mn 0 i i = 2πR(m − p) Xij m−p + X , X ij m−p , (3.11) µ (Xij )0m µ = (Xij )m . We see that the commutator Xi has a form of the covariant derivative where X0 with any [10] where the first term correspond to the ordinary derivative −i∂0 with respect to the dual coordinate x̃0 which is identified as x̃0 ∼ x0 + 2π/R. The second term is the commutator of the gauge field X0 = A0 with any matrix. We could then proceed as in [10] and rewrite the action in the form of the dual theory = 1/R, but defined on the dual torus with the radius R for simplicity we will use the original variables. Using this result we will write X0 = KC0 as (C0,ij )mn = p0,mn ⊗ 1M×M + (A0,ij )mn , (3.12) where the acting of p0 on various matrices is defined in (3.11) and where the numerical factor K will be determined for letter convenience. For illustration of the main idea, let us consider matrix model defined in D = 2n + 1 = 3 dimensions. Let us consider the matrix θ µν in the form 0 0 0 µν θ = 1mn ⊗ 0 (3.13) 0 θ 1N×N , 0 −θ 1N×N 0 where 1N×N is a unit matrix with N going to infinity. Then the equations of motion, which arise from (3.5), are i012X1 X2 + i021X2 X1 + 12 012 θ 12 + 021 θ 21 = 0, i102X0 X2 + i120X2 X0 = 0, i201X0 X1 + i210X1 X0 = 0, (3.14) The second and the third equation gives the condition [X0 , Xi ] = 0 which leads to the solution A0 = 0 and [p0 , Xi ] = 0. These equations, together with the first 246 J. Klusoň / Physics Letters B 505 (2001) 243–248 one, can be solved as i j x , x = iθ ij . Xi = δmn ⊗ xji k , where we have used (3.15) Xi Thanks to the presence of the unit matrix δmn , commutes with p0 ⊗ 1M×M and so is the solution of the equation of motion (3.14). Following [2], we can study the fluctuations around this solution with using the ansatz X0 = ω12 C0 = ω12 p0 ⊗ 1M×M + (A0,ij )mn , Xi = θ ij Cj , Ci = 1N×N ⊗ pi + (Ai,ij )mn , pi = ωij x j , i = 1, 2, ωij = θ −1 ij , (3.16) It is easy to see that this configuration corresponds to the noncommutative Chern–Simons action in D = 3 dimensions [3]. More precisely, let us introduce formal parameters dx µ , = dx µ1 ∧ · · · ∧ dx (3.22) Now it is easy to see that (3.19) is a correct action for the fluctuation fields A. The equations of motion arising from (3.19) are −2i(d + A) ∧ (d + A) + 2ω = 0. (3.23) Looking at (3.22) it is easy to see that the configuration A = 0 is a solution of equation of motion as it should be for the fluctuating field. With using = i∂µ Aν dx µ ∧ dx ν = id · A, µD (3.17) and the matrix valued one form C = Cµ dx µ = d + A, ω0i = 0. d ∧ A + A ∧ d = [pµ , Aν ] dx µ ∧ dx ν µ = 0, . . . , 2, dx µ ∧ dx ν = −dx ν ∧ dx µ , µ1 ...µD d ∧ d = pµ pν dx µ dx ν 1 = [pµ , pν ] dx µ ∧ dx ν = −iω, 2 1 ω = ωµν dx µ ∧ dx ν , ωij = θ −1 ij , 2 (3.18) where Cµ is given in (3.16). Then the action describing the fluctuations around the classical solution (3.15) has a form √ S = 2π det θ Tr − 2i3 C ∧ C ∧ C + 2ω ∧ C . (3.19) We rewrite this action in the form which has a closer contact with the commutative Chern–Simons theory. Firstly we prove the cyclic symmetry of the trace of the forms Tr A1 ∧ · · · ∧ AD µ 1 ∧ · · · ∧ dx µD = Tr A1µ1 A2µ2 · · · AD µD dx 2 1 µ2 = Tr Aµ2 · · · AD ∧ dx µD ∧ dx µ1 µD Aµ1 dx 2 = Tr A ∧ · · · ∧ AD ∧ A1 , (3.20) where we have used the fact that D is odd number so that dx µ1 commutes with even numbers of dx. Then the expression (3.19) is equal to √ S = −2π det θ × Tr iA ∧ (d ∧ A + A ∧ d) + i2 3 A∧A∧A , (3.21) (3.24) we obtain the derivative d· that is an analogue of the exterior derivative in the ordinary commutative geometry. In this case the action has a form √ S = 2π det θ Tr A ∧ d · A − 2i3 A ∧ A ∧ A , (3.25) which is the standard Chern–Simons action in three dimensions. We observe that this action differs from the action given in [3] since there is no the term ω ∧ A in our action. This is a consequence of the presence of the second term in (3.5) that is needed for the emergence of noncommutative structure in the Chern–Simons action. On the other hand, from the fact that similar matrix structure arises in the study of quantum hall effect in D-brane physics [18] we believe that our proposal of topological action could have relation to the string theory and M-theory. As usual, this action can be rewritten using in terms of the integral over spacetime with ordinary multiplication replaced with star product [9]. Generalisation to the higher dimensions is straightforward. The equations of motion (3.6) give 1 iµν Xµ Xν + µν θ µν = 0 ⇒ Xµ , Xν = iθ µν , 2 µ, ν = 1, . . . , 2n. (3.26) J. Klusoň / Physics Letters B 505 (2001) 243–248 We restrict ourselves to the case of θ of the maximal rank. For simplicity, we consider θ in the form 0 ... ... ... ... 0 0 θ1 0 ... 0 0 0 0 −θ1 0 . . . . . . µν θ = . (3.27) ... ... ... ... ... ... 0 ... ... ... 0 θn 0 . . . . . . 0 −θn 0 As in 3-dimensional case we introduce the matrix ω defined as follows ωij = θ −1 ij , i, j = 1, . . . , 2n, ωi0 = ω0i = 0. (3.28) and we define C = Cµ dx µ = d + A, µ = 0, . . . , 2n, where the dimension x 0 is compactified as above. And finally various Cµ are defined as X = θ Cj , n ωi C0 , X0 = i ij ωi = −θi−1 , (3.29) i=1 with as Cµ same as in (3.16). Then the action has a form √ n + 1 2n+1 C S2n+1 = (2π)n det θ Tr (−1)n (−1)n/2 2n + 1 n+1 ω ∧ C 2n−1 . + (−1)n−1 (−1)(n−1)/2 2n − 1 (3.30) In order to obtain more detailed description of the δ δ = δA , we can action we will follow [3]. Since δC 2 write √ δS2n+1 = (2π)n det θ (−1)n (−1)n/2 (n + 1)C 2n δC + (−1)n−1 (−1)(n−1)/2(n + 1)ω ∧ C 2n−2 √ = (2π)n det θ (n + 1)(F − ω)n + (n + 1)ω ∧ (F − ω)n−1 , (3.31) where we have used the fact that C 2 = −iω + i(−idA − iAd − iA2 ) = −iω + iF . Since ω and F are both two forms and ω is a pure number from the point of view of the noncommutative geometry we immediately see that F and ω commute so that we can 2 We will write C n instead of C ∧ · · · ∧ C. 247 write n n (F − ω) = (−ω)n−k F k . k n (3.32) k=0 Following [3] we introduce the other form of the Lagrangian δ L̃2k+1 = (k + 1)F k . δC Then we can rewrite (3.31) as √ δ S2n+1 − (2π)n det θ δC n n + 1 (−ω)n−k L̃2k+1 × k+1 (3.33) k=0 n + 1 n−k − (−1) ∧ L̃2k+1 =0 ω n k+1 k=0 √ ⇒ S2n+1 = (2π)n det θ n n−k n + 1 × Tr ωn−k L̃2k+1 (−1) k+1 n−1 n−k−1 1 k=0 n−1 n−k−1 1 n + 1 + (−1) n k+1 k=0 × ωn−k ∧ L̃2k+1 . (3.34) As a check, for n = 1 we obtain from (3.34) √ S3 = (2π) det θ Tr −2ω ∧ L̃1 + L̃3 + 2ω ∧ L̃1 √ = 2π det θ Tr L̃3 , (3.35) and using δ L̃3 = 2F = −2i dA + Ad + A2 δA ⇒ L̃3 = −iA ∧ d ∧ A 2 − id ∧ A ∧ A − i A ∧ A ∧ A, (3.36) 3 we obtain √ S3 = 2π det θ Tr A ∧ d · A − 2i3 A ∧ A ∧ A , (3.37) which, as we have seen above, is a correct form of the noncommutative Chern–Simons action in three dimensions. 248 J. Klusoň / Physics Letters B 505 (2001) 243–248 4. Conclusion Acknowledgements In this short note we have shown that simple modification of the topological matrix model [4] could lead to the emergence of the noncommutative Chern– Simons action [3]. In order to obtain this action we had to introduce the antisymmetric matrix θ expressing the noncommutative nature of the spacetime. It is crucial fact that we must introduce this term into the action explicitly which differs from the case of the standard matrix theory [2], where different configurations with any values of the noncommutative parameters arise as particular solutions of the matrix theory. It is also clear that we can find much more configurations than we have shown above. The form of these configurations depend on ω. It is possible to find such a θ which leads to the emergence of lower dimensional Chern–Simons actions and also which leads to the emergence of point-like degrees of freedom in the Chern–Simons theory. For example, we can consider θ in the form We would like to thank Rikard von Unge for many helpful discussions. This work was supported by the Czech Ministry of Education under Contract No. 144310006. θ µν = 1mn ⊗ A= 0k×k 0 0 0 0 0 0 0 A , −A 0 0 . θ 1N×N (4.1) This corresponds to the configuration describing Chern–Simons action with the presence of k point-like degrees of freedom — “partons”. We could analyse the interaction between these partons and gauge fields in the same way as in matrix theory (For more details see [13,15] and reference therein.) It is possible that this simple model could have some relation to the holographic model of M-theory [11]. In particular, we see that the partons arise naturally in our approach. On the other hand, the similar analysis as in [11] could determine θ , i.e., requirements of the consistency of the theory could choose θ in some particular form. In short, we hope that the approach given in this Letter could shine some light on the relation between the matrix models and Chern–Simons theory. References [1] N. Seiberg, E. Witten, String theory and noncommutative geometry, J. High Energy Phys. 09 (1999) 032, hepth/9908142. [2] N. Seiberg, A note on background independence in noncommutative gauge theories, matrix models and tachyon condensation, hep-th/0008013. [3] A.P. Polychronakos, Noncommutative Chern–Simons terms and the noncommutative vacuum, hep-th/0010264. [4] I. Oda, Background independent matrix models, hepth/9801051. [5] T. Banks, W. Fischer, S. Shenker, L. Susskind, M-theory as a matrix model: a conjecture, Phys. Rev. D 55 (1997) 5112, hep-th/9610043. [6] L. Smolin, M theory as a matrix extension of Chern–Simons theory, hep-th/0002009. [7] L. Smolin, The cubic matrix model and duality between strings and loops, hep-th/0006137. [8] L. Smolin, Covariant quantisation of membrane dynamics, hep-th/9710191. [9] L. Alvarez-Gaume, S.R. Wadia, Gauge theory on a quantum phase space, hep-th/0006219. [10] W. Taylor, D-brane field theory on compact spaces, Phys. Lett. B 394 (1997) 283, hep-th/9611042. [11] P. Hořava, M-theory as a holographic theory, Phys. Rev. D 59 (046004) (1999), hep-th/9712130. [12] T. Banks, Matrix theory, Nucl. Phys. Proc Suppl. 67 (1998) 180, hep-th/9710231. [13] W. Taylor, Lectures on D-branes, gauge theory and M(atrices), hep-th/9801182. [14] T. Banks, TASI lectures on matrix theory, hep-th/9911068. [15] W. Taylor, The M(atrix) model of M-theory, hep-th/0002016. [16] A. Konechny, A. Schwarz, Introduction to M(atrix) theory and noncommutative geometry, hep-th/0012145. [17] J. Brodie, L. Susskind, N. Toumbas, How Bob Laughlin tamed the giant graviton from Taub-NUT space, hep-th/0010105. [18] L. Susskind, The quantum hall fluid and non-commutative Chern–Simons theory, hep-th/0101029. [19] R.C. Myers, Dielectric D-branes, J. High Energy Phys. 9912 (1999) 022, hep-th/9910053. 26 April 2001 Physics Letters B 505 (2001) 249–254 www.elsevier.nl/locate/npe Quantum mechanics on a noncommutative brane in M(atrix) theory V.P. Nair a,b a Physics Department, City College of the CUNY, New York, NY 10031, USA b Physics Department, Rockefeller University, New York, NY 10021, USA Received 19 December 2000; received in revised form 25 February 2001; accepted 27 February 2001 Editor: M. Cvetič Abstract We consider the quantum mechanics of a particle on a noncommutative two-sphere with the coordinates obeying an SU(2)algebra. The momentum operator can be constructed in terms of an SU(2) × SU(2)-extension and the Heisenberg algebra recovered in the smooth manifold limit. Similar considerations apply to the more general SU(n) case. 2001 Published by Elsevier Science B.V. In this Letter we shall consider the question of setting up the quantum mechanics of a particle on a brane configuration in the matrix model of M-theory [1,2]. It is by now clear that the matrix model can successfully describe many of the expected features of Mtheory. Smooth brane configurations and solutions of M-theory can be obtained in the large N -limit of appropriate (N × N)-matrix configurations [2,3]. Now, brane solutions in M(atrix) theory are examples of noncommutative manifolds, specifically those with an underlying Lie algebra structure. The relationship between the matrix description and M-theory and strings suggests that noncommutative manifolds with an underlying Lie algebra structure (or their specializations into cosets) would be the most interesting ones from a physical point of view. Therefore we shall focus on such manifolds, although one can, of course, consider the question of quantum mechanics on more general noncommutative manifolds as well. E-mail address: [email protected] (V.P. Nair). There is, by now, an enormous number of papers dealing with noncommutative geometry. One line of development has to do with spectral actions and the use of the Dirac operator to characterize the manifold, motivated by quantum gravity [4]. Quantization of such actions has also been attempted [5]. The majority of recent papers deals with noncommutative manifolds with an underlying canonical structure and the construction of field theories on these spaces [6]. There has also been some recent work on manifolds with an underlying Lie algebra structure, including the definition of a star product and the construction of gauge fields which take values in the enveloping algebra [7]. The topic of the present Letter fits within the general milieu of these ideas and investigations, but we also have a very specific theoretical context, namely, brane solutions in M(atrix) theory. If the world is a brane [8], and if it is realizable as a solution in M(atrix) theory, then the quantum mechanics of a particle on a brane is clearly of more than mathematical interest. Consider a particular brane solution in M-theory, say, the noncommutative spherical membrane. In this 0370-2693/01/$ – see front matter 2001 Published by Elsevier Science B.V. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 3 3 8 - 0 250 V.P. Nair / Physics Letters B 505 (2001) 249–254 case the brane has the topology of S 2 × S 1 , where the S 2 is the noncommutative part described by matrices and S 1 denotes the compactified 11th dimension. The two-sphere is given in terms of three matrix coordinates which may be taken as r Qa = √ (1) ta , j (j + 1) where r is a fixed number which is the radius of the sphere and ta , a = 1, 2, 3, are the generators of SU(2) in the (2j + 1)-dimensional matrix representation. As the dimension (2j + 1) → ∞, we get a smooth manifold which is S 2 . This limit can be very explicitly understood by the representation T+ = T1 + iT2 = z2 ∂z − Nz, T− = T1 − iT2 = −∂z , T3 = z∂z − 12 N. (2) A basis of states on which these act is given by |α, α = 0, 1, . . . , N , with z|α = zα . The inner product is given by 2 1 d z g|f = (N + 1) (3) ḡ f. π (1 + zz̄)N+2 The matrix elements of Ta with the N + 1 states |α give the standard matrix version of ta , viz., (ta )αβ = α|Ta |β. By partial integration, we can see that Ta can be replaced, in matrix elements, by T+ = λφ + z2 ∂z , T− = λφ̄ − ∂z , T3 = λφ3 + z∂z , (4) where λ = 12 (N + 2) and 2z̄ 1 − zz̄ 2z , φ̄ = , φ3 = . (5) 1 + zz̄ 1 + zz̄ 1 + zz̄ As N → ∞, the λφ-terms in the above expressions for Ta dominate and we find ta → λφa . Thus Qa → rφa , with φa φa = 1. The membrane is described by the continuous coordinates z, z̄. At finite N , the two-sphere is described by the N +1 states |α which may be thought of as approximating the sphere by N + 1 points, none of which has sharply defined coordinates. Translations of |α can be achieved by the use of T± . However, this is not what we want. As N → ∞, T± go over to φ, φ̄ and correspond to the mutually commuting coordinates φ= z, z̄. They do not play the role of momenta conjugate to those coordinates obeying the Heisenberg algebra. We need to identify the momenta which lead to the Heisenberg algebra as N → ∞. Since the latter does not have finite-dimensional matrix representations, it is also clear that we should expect a modified algebra at finite N . Ultimately, from the point of view of noncommutative spaces, one keeps N finite, the limit being taken only to show agreement with the smooth manifold limit. The classical dynamics of a particle moving on a sphere gives a clue to the choice of a momentum operator or generator of translations. In the classical case, we may write the momentum as Pa = (1/q 2)abc qb Jc , where qa is the coordinate and Jc is the angular momentum operator, here taken as the fundamentally defined quantity. (If we reduce Jc in terms of qb , and pb conjugate to it, we find Pa = (δab − qa qb /q 2 )pb , which are the correct translation generators consistent with qa qa = 1.) Absorbing r into the definition of Qa , a possible choice of Pa is then − 12 (λ/Q2 )abc (Qb Jc + Jc Qb ), where we have symmetrized Qa , Ja to form a hermitian combination. The operators Qa , Ja obey the algebra [Qa , Qb ] = i abc Qc , λ (6) [Ja , Qb ] = iabc Qc , [Ja , Jb ] = iabc Jc . (7) Notice that i 1 abc (Qb Jc + Jc Qb ) = abc Qb Jc + cbk Qk 2 2 = abc Qb Jc − iQa = abc Qb (Jc − λQc ) = λabc Qb Kc , where λKa = Ja − λQa . Further i [Ka , Kb ] = abc Kc , λ [Ka , Qb ] = 0. (8) (9) Therefore, rather than starting with Ja , we might as well consider the mutually commuting SU(2) × SU(2)-algebra of Qa , Ka and define the momentum operator as abc Kb Qc , Pa = λ (10) Q2 K 2 V.P. Nair / Physics Letters B 505 (2001) 249–254 where Q2 = Qa Qa , K 2 = Ka Ka . Obviously, [Pa , Q2 ] = [Pa , K 2 ] = 0 so that there is no ordering ambiguity in the definition of Pa . Wehave chosen to divide by the symmetric expression Q2 K 2 eventhough the classical expression had q 2 . As we shall see below, Q2 ≈ K 2 in the continuous manifold limit. Also the parameter λ will be related to Q2 , K 2 below. The commutation rules for Pa become i 1 [Pa , Qb ] = δab Q · K − (Qa Kb + Qb Ka ) 2 2 2 Q K abc Pc , − 2λ Q·K [Pa , Pb ] = iabc 2 2 Jc . (11) Q K Ja = λ(Qa + Ka ) are the generators of the diagonal SU(2) subgroup. The smooth manifold limit can be understood by considering large representations for Qa and Ka , and analyzing representations of the diagonal SU(2) of Ja . Labelling the corresponding spins by lower case letters, we find λ2 Q2 = q(q + 1), λ2 K 2 = k(k + 1), J 2 = j (j + 1) and 2λ2 Q · K = j (j + 1) − q(q + 1) − k(k + 1). If we take q, k very large and thecombined spin j to be small and fixed, and λ2 = Q2 K 2 ≈ q(q + 1) ≈ k(k + 1), we find that the algebra (6), (9), (11) reduces to [Qa , Qb ] ≈ 0, Qa Qb , [Pa , Qb ] ≈ −i δab − Q2 Jc [Pa , Pb ] ≈ −iabc 2 . (12) Q We also have Qa Qa ≈ 1. Eqs. (12) are the Heisenberg algebra restricted toa smooth two-sphere of unit radius. (Taking λ2 = Q2 K 2 /r 2 , we can get a radius equal to r.) The emergence of the continuous coordinates and the large λ-expansion can be seen in more detail as follows. We write a general SU(2)-valued (2 × 2)matrix as x2 x g = 1 − 2 + i σ · , (13) r r where σa are the Pauli matrices. We then find that σa g −1 dg = i Eab dx b , 2 dg g −1 = i where Eab = σa Eab dx b , 2 251 (14) 2 δab (r 2 − x 2 ) + xa xb − , x + √ abc c r2 r2 − x2 ab = Eba . E (15) The above equations define the frame fields on SU(2). ab are given by The inverses to Eab , E 1 −1 Eab = abc xc + δab r 2 − x 2 , 2 −1 = E −1 . E (16) ab ba The quantities −1 ∂ , Qa = i E ka ∂xk −1 ∂ Ka = −iEka ∂xk (17) obey mutually √ commuting SU(2) √ algebras. Further, since [xa , r 2 − x 2 ∂b ] − [xb , r 2 − x 2 ∂a ] = 0, we see that we can shift Qa by xa and Ka by −xa and still obtain the same algebra. In other words, we can define ∂ i , Qa = xa + abc xc + δab r 2 − x 2 2λ ∂xb ∂ i abc xc − δab r 2 − x 2 Ka = −xa + , 2λ ∂xb ∂ . Ja = λ(Qa + Ka ) = −iabc xb (18) ∂xc This is in a form suitable for the large λ-expansion for SU(2) × SU(2), with the combined total spin being small. As λ → ∞, the xa -terms are dominant in the expressions for Qa , Ka and we get Qa → xa , Ka → −xa . The algebra (6), (9), (11) reduces to xa xb [Pa , Qb ] ≈ −i δab − 2 , x Jc [Pa , Pb ] ≈ −iabc 2 , (19) x x 2 = xa xa = r 2 is a constant in this limit. The φ’s given in (5) are a particular parametrization of the xa ’s subject to xa xa being constant. In taking the limit as above we have retained S 2 topology for the smooth manifold. It is important to realize that since we are dealing with Q’s which obey 252 V.P. Nair / Physics Letters B 505 (2001) 249–254 a Lie algebra, Q2 is fixed for any representation and hence we will not get a flat Heisenberg algebra. A way to obtain the flat space algebra would be to take the radius r to be very large and then restrict the operators to a small neighbourhood on the sphere. This will lead to a flat two-dimensional Heisenberg algebra as r → ∞. For example, we can expand around xa = (0, 0, r). It is interesting to see how this works out directly in terms of the operators Qa , Ka . The neighbourhood of xa = (0, 0, r) corresponds to Q3 and −K3 being large. Since Q3 ∼ r and λ ∼ k/r, we see that [Q1 , Q2 ] ∼ ir 2 /k and so, the commutativity of coordinates in the large k-limit requires that r 2 ∼ k δ with δ < 1 as k → ∞. On the other hand, we also have [P1 , P2 ] ∼ 1/r 2 and the vanishing of this requires δ > 0. The simplest and symmetrical choice is to take δ = 12 or r ∼ k 1/4 . We define eigenstates of Q3 , K3 by K3 |m, n = (−k + m)|m, n, Q3 |m, n = (k − n)|m, n (20) Restricting to small neighbourhood of large Q3 , −K3 means that the integers m, n can be considered to be small compared to k. In this case, introducing raising and lowering operators α † , α for n and β † , β for m, we find r α + α† , Q1 = √ 2k ir † α −α , Q2 = √ 2k i k P1 = − α − α† + β † − β , r 2 1 k α + α† + β † + β , P2 = − (21) r 2 1/4 and, as usual, α|m, n = where we can take r = r0 k√ √ n|m, n − 1, β|m, n = m|m − 1, n, etc. The flat space Heisenberg algebra is now easily verified. In the usual procedure of quantization, starting with a set of classical coordinates qa , one introduces the momenta and the phase space, thereby doubling the number of variables. The quantum theory is then defined by one irreducible representation of the Heisenberg algebra equivalent to the standard Schrödinger representation. The restriction to irreducibility is equivalent to the requirement that the wavefunctions depend only on half of the phase space variables, the coordinates qa , for example. This is the so-called polarization condition. For a given wavefunction, this allows the determination of the momenta as pa ψ = −i(∂ψ/∂qa ). In our case, starting with Qa , obeying the SU(2)algebra (6), we introduce the SU(2) × SU(2)-algebra (6), (9) of Qa , Ka . The set Qa , Ka can be considered as the analogue of the phase space. The analogue of the polarization condition implies that we must choose an irreducible representation of Qa , Ka . As we have seen already, the smooth manifold limit corresponds to q, k → ∞. For a given irreducible representation, labelled by the spin values (q, k), there are several representations possible for the angular momentum Ja = λ(Qa + Ka ), the lowest possible j -value being |q − k|. The difference |q − k| may be interpreted as the strength of a magnetic monopole at the center of the sphere, or a uniform magnetic field through the sphere. (I thank Polychronakos for discussions clarifying this point.) In the absence of any magnetic monopole field, we can take q = k. Generalization to SU(n) More general brane solutions require the consideration of N -dimensional representations of SU(n), n > 2, with N → ∞ eventually. The generalization of our considerations to SU(n) is straightforward. Basically one has to consider an SU(n) × SU(n)-algebra i fabc Qc , λ i [Ka , Kb ] = fabc Kc , λ [Ka , Qb ] = 0. [Qa , Qb ] = (22) The momentum operator can then be defined by Pa = Kb Qc nλ fabc . 2 Q2 K 2 (23) Pa is a derived quantity, with Qa , Ka defining the basic algebra, as in the case of SU(2). The commutator of Pa with Qb can be evaluated without too much trouble, eventhough it is more involved than in the case of SU(2). The following identity for the the structure constants is useful for this calculation. Let t a be hermitian (n × n)-matrices which form a basis of the Lie algebra of SU(n) with [t a , t b ] = V.P. Nair / Physics Letters B 505 (2001) 249–254 ba . The This equation shows explicitly that Eab = E left and right translation generators are then if abc t c , Tr(t a t b ) = 12 δ ab . We can then write famc fbkc + fbmc fakc = ∂4 ∂x a ∂x b ∂y m ∂y k F, F = − Tr [t · x, t · y][t · x, t · y] . (24) The traces can be evaluated using the identity t ·x t ·y +t ·yt ·x = x ·y + 2 dabc xa yb tc , n (25) where dabc = Tr{(ta tb + tb ta )tc }. Eq. (24) then becomes the identity [Va , Vb ] = ifabc Vc , [Aa , Ab ] = ifabc Vc . + 8 dabc dmkc − 4 damc dbkc − 4 dakc dbmc . (26) With the help of this identity, the commutator of Pa with Qb is now obtained as i δab K · Q − 12 (Ka Qb + Kb Qa ) [Pa , Qb ] = K 2 Q2 i in + fabc Pc + Km Qn 2λ K 2 Q2 × (2dabc dmnc − damc dbnc − dbmc danc ). (27) The calculation of [Pa , Pb ] is more involved. It does not seem to be very illuminating for our discussion. It is also possible to develop expressions for Qa , Ka , which are analogues of Eqs. (13)–(18), in terms of an (n2 − 1)-vector xa which parametrizes SU(n). We can write the variation of a group element g ∈ SU(n) ab dx b . as g −1 dg = ita Eab dx b and dg g −1 = ita E ab are transposes of each The quantities Eab and E other. For example, if we use an exponential parametrization g = exp(it · x), we can write Eab = −1 ∂ , La = i E ka ∂x k −1 ∂ Ra = −iEka (29) ∂x k with La g = −ta g, Ra g = gta . These obey the Lie algebra relations [ξa , ξb ] = ifabc ξc , ξ = L, R. In terms of Va ≡ (La + Ra ) and Aa ≡ (La − Ra ), this becomes [Va , Ab ] = ifabc Ac , famc fbkc + fbmc fakc 4 2 δab δmk − (δam δbk + δak δbm ) = n n 1 253 dα 2 Tr ta e−iαt ·x tb eiαt ·x , (30) −1 + Since Aa involves the symmetric combination Eka −1 , the last of these relations is unaltered by shifting E ka the Aa by xa , i.e., [Aa + xa , Ab + xb ] = ifabc Vc . Further, eiθ·V g = e−it ·θ geit ·θ , showing that xa transforms as a vector under the action of Va . The operators Qa and Ka can then be defined as 1 (Va + Aa ), 2λ 1 Ka = −xa + (Va − Aa ). (31) 2λ These can be used as the starting pont for a large λ-expansion around some chosen value of xa . Qa = xa + Acknowledgements This work was supported in part by the National Science Foundation grant PHY-9605216 and a PSCCUNY-30 award. I thank Professor Bunji Sakita for useful discussions. I also thank Professor N. Khuri and the Rockefeller University for hospitality during part of this work and I. Giannakis for a critical reading of the manuscript. References 0 ab = E 1 0 dα 2 Tr ta eiαt ·x tb e−iαt ·x . (28) [1] T. Banks, W. Fischler, S. Shenker, L. Susskind, Phys. Rev. D 55 (1997) 5112. [2] For a recent review, see W. Taylor, Lectures at the NATO School, Iceland, 1999, hep-th/0002016. 254 V.P. Nair / Physics Letters B 505 (2001) 249–254 [3] M. Berkooz, M.R. Douglas, hep-th/9610236; T. Banks, N. Seiberg, S. Shenker, Nucl. Phys. B 497 (1997) 41; D. Kabat, W. Taylor, Adv. Theor. Math. Phys. 2 (1998) 181; S.-J. Rey, hep-th/9711081; J. Castelino, S. Lee, W. Taylor, Nucl. Phys. B 526 (1998) 334; V.P. Nair, S. Randjbar-Daemi, Nucl. Phys. B 533 (1998) 333; W. Taylor, M. van Raamsdonk, hep-th/9910052. [4] A. Connes, Noncommutative Geometry, Academic Press, 1994; For some reviews, see A.H. Chamseddine, J. Fröhlich, hepth/9307012; G. Landi, hep-th/9701078; A.P. Balachandran et al., Nucl. Phys. Proc. Suppl. 37 C (1995) 20. [5] C. Rovelli, Phys. Rev. Lett. 83 (1999) 1079; M. Hale, gr-qc/0007005. [6] There are well over 200 recent papers on this; citations to the following papers will give an overall view of the field; A. Connes, M.R. Douglas, A. Schwarz, hep-th/9711162; N. Seiberg, E. Witten, hep-th/9908142. [7] B. Jurco, P. Schupp, hep-th/0001032; J. Madore, S. Schraml, P. Schupp, J. Wess, hep-th/0001203; B. Jurco, P. Schupp, J. Wess, hep-th/0005005; B. Jurco, S. Schraml, P. Schupp, J. Wess, hep-th/0006246. [8] L. Randall, R. Sundrum, Phys. Rev. Lett. 83 (1999) 4690. 26 April 2001 Physics Letters B 505 (2001) 255–262 www.elsevier.nl/locate/npe The AdS/CFT correspondence and topological censorship G.J. Galloway a , K. Schleich b , D.M. Witt c , E. Woolgar d a Department of Mathematics, University of Miami, Coral Gables, FL 33124, USA b Department of Physics and Astronomy, University of British Columbia, Vancouver, BC V6T 1Z1, Canada c Department of Physics and Astronomy, University of British Columbia, Vancouver, BC V6T 1Z1, Canada d Department of Mathematical Sciences and Theoretical Physics Institute, University of Alberta, Edmonton, AB T6G 2G1, Canada Received 8 February 2001; accepted 26 February 2001 Editor: M. Cvetič Abstract In this Letter we consider results on topological censorship, previously obtained by the authors in Phys. Rev. D 60 (1999) 104039, in the context of the AdS/CFT correspondence. These, and further, results are used to examine the relationship of the topology of an asymptotically locally anti-de Sitter spacetime (of arbitrary dimension) to that of its conformal boundary-atinfinity (in the sense of Penrose). We also discuss the connection of these results to results in the Euclidean setting of a similar flavor obtained by Witten and Yau in Adv. Theor. Math. Phys. 3 (1999). 2001 Published by Elsevier Science B.V. PACS: 4.20.Gz; 4.20.Bw; 4.50.+h; 11.25.Sq 1. Introduction The AdS/CFT correspondence, first proposed by Maldacena [3], asserts the existence of a correspondence between string theory (or supergravity) on an asymptotically locally anti-de Sitter spacetime and an appropriate conformal field theory on the boundaryat-infinity. This conjectured correspondence has been supported by calculations which, for example, show a direct connection between black hole entropy as calculated classically and the number of states of the conformal field theory on the boundary-at-infinity; cf. [4] for a comprehensive review. Thus, the AdS/CFT correspondence conjecture provides new insight into the E-mail addresses: [email protected] (G.J. Galloway), [email protected] (K. Schleich), [email protected] (D.M. Witt), [email protected] (E. Woolgar). old puzzle of black hole entropy in the context of string theory. Moreover, it is believed that this conjecture, if true, may hold answers to other long-standing puzzles in gravity. It is natural to consider what implications the topology of an asymptotically locally anti-de Sitter spacetime has for the AdS/CFT correspondence. In general, the topology of an asymptotically locally anti-de Sitter spacetime can be rather complicated. Indeed, there are well known examples that admit black holes and wormholes of various topologies [5– 10]. Furthermore, one can show that there exist initial data sets with very general topology that evolve as locally anti-de Sitter spacetimes [11]. However, if the AdS/CFT correspondence conjecture is valid, one would expect the topology to be constrained in a certain manner: there should be some correspondence between the topology of an asymptotically locally anti-de Sitter spacetime and that of its boundary- 0370-2693/01/$ – see front matter 2001 Published by Elsevier Science B.V. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 3 3 5 - 5 256 G.J. Galloway et al. / Physics Letters B 505 (2001) 255–262 at-infinity. For if the topology of an asymptotically locally anti-de Sitter spacetime were arbitrary, how could a conformal field theory that only detects the topology of its boundary-at-infinity correctly describe its physics? Witten and Yau [2] have recently addressed this issue in the context of a generalized Euclidean formulation of the AdS/CFT correspondence [12]. In this formulation one considers a complete connected Riemannian manifold M n+1 which admits a conformal compactification (analogous to the spacetime notion of Penrose [13]), with conformal boundary (or conformal infinity) N n . In [2], Witten and Yau show in this context that if M is an Einstein manifold of negative Ricci curvature, and if the conformal class of N admits a metric of positive scalar curvature then the nth homology of M vanishes, and, in particular, N must be connected. As discussed in [2,14], this resolves the potential problem of the coupling (via the bulk M) of seemingly independent conformal field theories (corresponding to the components of N ), in the case that the conformal class of N admits a metric of positive scalar curvature. The results obtained by Witten and Yau [2] in the Euclidean setting do not directly address the relationship of the topology of an asymptotically locally antide Sitter space to that of the boundary-at-infinity in the context of spacetimes, i.e., Lorentzian manifolds, the standard arena for the AdS/CFT correspondence conjecture. Their results, however, are reminiscent of some results previously obtained by the authors [1] in the spacetime (Lorentzian) setting as a consequence of topological censorship. The aim of the present Letter is to discuss some of these latter results in the context of the AdS/CFT correspondence, and, also, to describe the connection of these results to those of Witten and Yau. Some new results concerning the relationship of the topology of an asymptotically locally anti-de Sitter spacetime to that of the boundary-at-infinity are also presented. Topological censorship is a basic principle of spacetime physics, which expresses the notion that the topology of the region of spacetime outside all black holes and white holes should, in some sense, be simple. According to topological censorship, in a spacetime with appropriate asymptotic structure obeying natural energy and causality conditions, any causal curve with initial and final end points on the boundary- at-infinity I can be continuously deformed to a curve that lies in I itself. Thus observers passing through the interior of such a spacetime, who remain outside all black holes and white holes, detect no topological structure not also present in the boundary-atinfinity. This result was first proved for asymptotically flat spacetimes by Friedman, Schleich and Witt [15]. More recently it has been extended by the present authors [1] to asymptotically locally anti-de Sitter spacetimes. As shown in [1], topological censorship is a powerful tool for studying the topology of the socalled domain of outer communications (the region outside all black holes and white holes) and the topology of black holes in asymptotically locally anti-de Sitter spacetimes in 3 + 1 dimensions. Some of these results remain valid in arbitrary spacetime dimension n + 1, n 2, and, as discussed below, have a direct relevance to the AdS/CFT correspondence. (The proof of topological censorship, itself, is valid in arbitrary dimension n + 1, n 2.) In Section 2, we introduce some basic concepts and present a statement of topological censorship for asymptotically locally anti-de Sitter spacetimes. In Section 3, we discuss the relationship of topological censorship to Witten and Yau’s connectedness of the boundary result. In Section 4 we discuss how topological censorship constrains the topology of the domain of outer communications. Results both in arbitrary dimension, and in specific low dimensions of interest are considered. 2. Topological censorship in (n + 1)-dimensional asymptotically locally anti-de Sitter spacetimes Let M n+1 be an (n + 1)-dimensional spacetime (i.e., connected time oriented Lorentzian manifold), with metric gab . Recall [18], the timelike future (resp., timelike past) of A ⊂ M, is denoted by I + (A, M) (resp., I − (A, M)), and consists of all points in M that can be reached from A by a future (resp. past) directed timelike curve in M. The causal future and past of A in M, denoted J ± (A, M), are defined in an analogous way using causal, rather than timelike, curves. We use Penrose’s notion of conformal infinity [13] to describe what is meant by “asymptotically locally anti-de Sitter” (“ALADS” for short). We will say that M is an ALADS spacetime provided there exists G.J. Galloway et al. / Physics Letters B 505 (2001) 255–262 Fig. 1. The Penrose diagram for Schwarzschild–AdS spacetime. a spacetime-with-boundary M , with Lorentz metric , such that (a) the boundary I = ∂M is timelike, gab i.e., is a Lorentzian manifold in the metric induced , (b) M is the interior of M , and hence from gab are related by M = M ∪ I, and (c) gab and gab 2 gab = Ω gab , where Ω is a smooth function on M satisfying (i) Ω > 0 on M and (ii) Ω = 0 and dΩ = 0 along I. 1 The conditions on the conformal factor Ω are standard, and guarantee that the physical metric gab falls off at a reasonable rate as one approaches the boundary-at-infinity I. Universal antide Sitter spacetime (of dimension n + 1) [4,18] is the canonical example of an ALADS spacetime. It conformally imbeds into the Einstein static universe R × S n , so that its closure M is R × B n , where B n is a closed hemisphere of S n , and I = R × S n−1 . Another useful example to have in mind is the Schwarzschild– anti-de Sitter black hole spacetime, see Fig. 1. This has a causal structure similar to extended Schwarzschild spacetime, except that the boundary-at-infinity, which consists of two components each having topology R × S 3 , is timelike, rather than null. To simplify our presentation a little, we shall assume in our definition of an ALADS spacetime that I is spatially closed, i.e., that each component Iα of I admits a compact spacelike hypersurface Sα . From a further assumption made below, Iα will be homeomorphic to R × Sa . We do not, however, make any 1 We are using a very weak form of ALADS. Usually, it is required that the vacuum Einstein equations with negative cosmological constant hold asymptotically, in a certain prescribed sense, which then forces I to be timelike [16]. 257 assumptions about the topology of Sα , e.g., that it be spherical. This is what is meant by “locally” in the definition of an ALADS spacetime. Indeed many of the interesting examples in the literature have nontrivial (i.e., nonspherical) topology at infinity, see for example [9]. Topological censorship requires some form of causal regularity, related to the cosmic censorship hypothesis that there be no singularities visible from infinity. Recall [18], a spacetime N is said to be globally hyperbolic iff N is strongly causal (i.e., there are no closed, or “almost closed” causal curves in N ), and the “causal intervals” J + (p, N) ∩ J − (q, N) are compact for all p, q ∈ N . Note that this definition still makes sense even if N is a spacetime-with-boundary; the sets J + (p, N) ∩ J − (q, N) may then meet the boundary, which is permitted. For example, the spacetime-withboundary R × B n , where B n is a closed hemisphere of S n , is globally hyperbolic in this sense. Thus, although universal anti-de Sitter spacetime is not globally hyperbolic, the closure of its conformal image in the Einstein static universe, as a spacetime-withboundary, is. The domain of outer communications D of an ALADS spacetime M, which represents the region outside of all black holes and white holes, is the region of spacetime that can communicate with infinity, both to the future and past. In mathematical terms, D = [I − (I, M ) ∩ I + (I, M )] ∩ M. Then D = I − (I, M ) ∩ I + (I, M ) = D ∪ I is a spacetime-withboundary (not necessarily connected), with timelike boundary I. We say that an ALADS spacetime M is causally regular provided D = D ∪ I is globally hyperbolic, as a spacetime-with-boundary. The compactness of the sets J + (p, D ) ∩ J − (q, D ) rules out the presence of any naked singularities in D. Finally, topological censorship requires an energy condition, such as the null energy condition (NEC): Rab Xa Xb 0 for all null vectors Xa , where Rab is the Ricci tensor of spacetime. In fact, to prove topological censorship, it is sufficient to require a weaker, averaged version of the NEC, 2 but for simplicity 2 It is sufficient to assume the ANEC, together, with the null generic condition. To avoid the null generic condition, which is not satisfied in some models, one may use a modified form of the ANEC; see [1] for further details. 258 G.J. Galloway et al. / Physics Letters B 505 (2001) 255–262 we will state things in terms of the NEC here. Although stated in a geometric form, the NEC can be interpreted physically by invoking the Einstein equations to relate the Ricci tensor to its sources. In particular, if the Einstein equations with cosmological constant hold, Rab − 12 Rgab + Λgab = 8πTab , then, as gab Xa Xb = 0 for any null vector X, we have, Rab Xa Xb = 8πTab Xa Xb . Hence, the NEC depends only on the stress energy tensor, and is insensitive to the sign of the cosmological constant. In particular, the NEC is satisfied in vacuum spacetimes with negative cosmological constant. We may now state the following version of topological censorship, which follows from Theorem 2.2 in [1]. Theorem 1. Let M n+1 , n 2, be a causally regular ALADS spacetime satisfying NEC. Let I0 be a connected component of the boundary-at-infinity I. Then every causal curve in M = M ∪ I with initial and final end points on I0 is fixed end point homotopic to a curve in I0 . As we describe in Section 4, Theorem 1 can be formulated in terms of fundamental groups. This enables one to study the topology (e.g., homology) of the domain of outer communications by standard algebraic topological techniques. In the next section we describe a result closely related to Theorem 1, which can be interpreted as a Lorentzian analogue of the Witten–Yau connectedness of the boundary result. 3. Causal disconnectedness of disjoint components of the boundary-at-infinity Theorem 1 is a consequence of the following basic result (cf., Theorem 2.1 in [1]). n 2, be a causally regular Theorem 2. Let ALADS spacetime satisfying NEC. Then distinct components of the boundary-at-infinity I cannot communicate, i.e., if I0 and I1 are distinct components of I then J + (I0 , M ) ∩ J − (I1 , M ) = ∅. M n+1 , Simply put, no causal curve can extend from one component of infinity to another. Schwarzschild–anti- de Sitter spacetime provides a clear illustration of this fact, see Fig. 1. Theorem 2 is related to the fact in black hole theory [18, Proposition 9.2.8], and may be proved in a similar fashion, that so-called outer trapped surfaces cannot be seen from infinity. Theorem 1 follows from Theorem 2 by constructing a covering space of M in which all curves not homotopic to curves on I0 are unwound. Any causal curve with endpoints on I0 not fixed endpoint homotopic to a curve in I0 will begin and end on different components of the boundary-at-infinity in this covering space. But since this covering space is itself an ALADS spacetime satisfying the conditions of Theorem 2, such a curve cannot exist. Hence, Theorem 1 follows. The implications of Theorem 2 for the AdS/CFT correspondence are immediate. In an ALADS spacetime satisfying reasonable physical conditions, any component of the boundary-at-infinity cannot communicate with any other component. Thus any field operator evaluated at a point of one component of I will commute with any field operator evaluated on any other component. Thus conformal field theories defined on disjoint components of the boundary-atinfinity do not interact dynamically. Clearly however, one can set up correlations in the initial vacuum states of the conformal field theories. However, any such correlations are not dynamic. It is in this sense that our Lorentzian result may be viewed as an analogue of the Euclidean connectedness of the boundary result obtained by Witten and Yau [2,14]. Now, consider an observer in, for example, Schwarzschild–anti-de Sitter spacetime, who attempts to travel from one component of infinity to another; the observer ends up entering the black hole region. This is the typical situation: Distinct components of I are “screened apart” by black holes (and/or white holes). We have the following corollary. Corollary 3. Let M n+1 , n 2, be a causally regular ALADS spacetime satisfying NEC. If I is not connected then M contains a black hole and/or a white hole. Proof. Let {Iα }α∈A denote the components of the boundary-at-infinity I; hence I is the disjoint union of the Iα ’s. Let Dα denote the domain of outer communications with respect to the component Iα , G.J. Galloway et al. / Physics Letters B 505 (2001) 255–262 Dα = [I − (Iα , M )∩I + (Iα , M )]∩M. It follows from Theorem 2 that the Dα ’s are mutually disjoint, Dα ∩ Dβ = ∅, α = β, and that the (full) domain of outer communications D = [I − (I, M ) ∩ I + (I, M )] ∩ M is the disjoint union of the Dα ’s. The black hole region is the region of spacetime from which an observer cannot escape to infinity. Mathematically, it is the region M \ I − (I, M ). Time dually, the white hole region is the region M \ I + (I, M ). The union of these two regions is the region M \ D. Since spacetime is assumed to be connected, but D is not connected, D cannot be all of M, M \ D = ∅. Hence, there must be a black hole and/or a white hole in M. ✷ Recall that the Euclideanization procedure for transforming (via Wick rotation) a spacetime into a Riemannian manifold involves a single component of the domain of outer communications. The above proof shows that the timelike boundary-at-infinity of a single component of the domain of outer communications is connected, in formal consistency with the Witten–Yau connectedness of the boundary result. 4. The topology of the domain of outer communications In this section we study the topology of the domain of outer communications, emphasizing the manner and extent to which it is controlled by the topology at infinity. We begin by describing, as mentioned in Section 2, how Theorem 1 can be expressed in terms of fundamental groups. Theorem 1, when taken in conjunction with a certain covering space argument, can be shown to imply the following stronger result. Theorem 4. Let M n+1 , n 2, be a causally regular ALADS spacetime satisfying NEC. Let I0 be a connected component of the boundary-at-infinity I. Then any curve in D0 = D0 ∪ I0 , with initial and final end points on I0 is fixed end point homotopic to a curve in I0 . Here, D0 = [I − (I0 , M ) ∩ I + (I0 , M )] ∩ M is the domain of outer communications with respect to I0 . Theorem 4 removes the qualifier in Theorem 1 that 259 the curve in D0 be causal. (Note that a causal curve with end points on I0 is necessarily contained D0 .) The proof is a slight modification of the proof of Proposition 3.1 in [1]. Now, fix a point p0 ∈ I0 , and consider loops in D0 based at p0 . Theorem 4 implies that any loop in D0 based at p0 can be continuously deformed to a loop in I based at p0 . Recalling that the inclusion map i : I0 → D0 induces a natural homomorphism of fundamental groups i∗ : Π1 (I0 ) → Π1 (D0 ), the last comment can be expressed in more formal terms as follows, cf., Proposition 3.1 in [1]. Corollary 5. Let M n+1 , n 2, be a causally regular ALADS spacetime satisfying NEC. Let I0 be a connected component of the boundary-at-infinity I. Then the group homomorphism i∗ : Π1 (I0 ) → Π1 (D0 ) induced by inclusion is surjective. Corollary 5 says that, at the fundamental group level, the topology of D0 can be no more complicated than the topology of I0 . In particular, if I0 is simply connected (e.g., I0 ≈ R × S n ), then so is D0 , thus generalizing the result of [17]. This supports the notion of holography: the topology of infinity determines to some extent the topology of the bulk. Corollary 5 is a natural Lorentzian analogue of Theorem 3.3 in [2], but requires no curvature assumptions on the boundary-atinfinity. Further information about the topology of the domain of outer communications can be obtained by considering certain spacelike hypersurfaces slicing through it. This method was used extensively in [1] to study the topology of the domain of outer communications and the topology of black holes in (3 + 1)dimensional ALADS spacetimes. Some of the results obtained there extend to arbitrary dimension n + 1, n 2. Here we give a brief description of the basic methodology, and consider some results that hold in arbitrary dimension, as well as some results that are dimension specific. Some limitations of the methodology are also discussed. Let M be a causally regular ALADS spacetime, and, as above, let D0 be the domain of outer communications with respect to a component I0 of the boundary-at-infinity I. Consider a spacelike hypersurface-with-boundary V (dim V = n) in D0 = D0 ∪ I0 , whose boundary Σ∞ is an (n − 1)-dimensional 260 G.J. Galloway et al. / Physics Letters B 505 (2001) 255–262 spacelike surface contained in I0 . Using the fact that D0 is globally hyperbolic (as a manifold-withboundary), V can be chosen so that D0 is homeomorphic to R × V , D0 ≈ R × V . (This is accomplished by extending the notion of a Cauchy surface [18] to spacetimes with timelike boundary, and applying a suitable analogue of [18, Proposition 6.6.8].) By restricting this homeomorphism to I, it also follows that I ≈ R × Σ∞ . Such a spacelike hypersurface V for Schwarzschild–anti-de Sitter spacetime is depicted in Fig. 1. 3 Now, let V denote the closure of V in the full spacetime-with-boundary M = M ∪ I. If there are black holes present, V will meet the black hole event horizon (the boundary of the black hole region). We assume that V is a compact, orientable n-manifoldwith-boundary, with interior V \Σ∞ , and with boundary ∂V = ΣH ∪ Σ∞ , where ΣH is a compact (n − 1)dimensional surface contained in the event horizon. We allow ΣH to have multiple (k 0, say) components; each component of ΣH corresponds to a black hole in the “time slice” V . In the Schwarzschild–AdS example depicted in Fig. 1, V ≈ [0, 1]×S n−1 , and ΣH and Σ∞ are (n − 1) spheres. Since D0 ≈ R × V , the topology of D0 is completely determined by the topology of V , which, in turn, is completely determined by the topology of V . Moreover, the relationship between the fundamental groups of I0 and D0 , as described in Corollary 5, descends to the fundamental groups of Σ∞ and V . Proposition 6. Let M n+1 , n 2, be a causally regular ALADS spacetime satisfying NEC. Let V be the compact spacelike hypersurface-with-boundary in M = M ∪ I with boundary ∂V = ΣH ∪ Σ∞ , as described above. Then the homomorphism of fundamental groups i∗ : Π1 (Σ∞ ) → Π1 (V ) induced by inclusion is surjective. Again, this means that every loop in V can be continuously deformed to a loop in Σ∞ . Proposition 6 illustrates at the spatial level how the topology at infinity controls the topology of the bulk. 3 In certain circumstances it is useful to modify the procedure outlined here, by constructing V with respect to certain globally hyperbolic subregions of D0 ; cf. [1, Section 4]. The study of the topology of the domain of outer communications with respect to the component I0 of the boundary-at-infinity I has now been reduced to the study of the topology of the spacelike slice V . We now briefly describe a few of the main results concerning the topology of V . Detailed proofs and further results may be found in [19]. Theorem 7. Let M n+1 , n 2, be a causally regular ALADS spacetime satisfying NEC, and let V be as in Proposition 6. Then the (n − 1)-homology of V is given by, Hn−1 (V , Z) = Zk , where k 0 is the number of components of ΣH . Theorem 7, which was proved in 3 + 1 dimensions in [1], is a consequence of Proposition 6, together with standard techniques and results in algebraic topology. Although proved in a completely different way, and not requiring a curvature condition on Σ∞ , Theorem 7 may be viewed as a spacetime analogue of Theorem 3.4 in [2]. In fact, in the absence of black holes (k = 0), the conclusions are formally the same. Theorem 7 has a natural geometrical/physical interpretation. The k components of ΣH determine k linearly independent elements of Hn−1 (V ). But since Hn−1 (V ) = Zk , these components must span all of Hn−1 (V ). Hence all of the (n − 1)-homology of V is generated by its boundary components, and in this sense Hn−1 (V ) is as simple as possible. Any topological structure in the interior of V that would generate another independent element of Hn−1 (V ) cannot exist. In particular, V cannot contain any wormholes. A wormhole in V would correspond to a handle grafted to V , which would introduce an (n − 1)-sphere that does not bound in V . This intuitive observation can be formulated in a precise way as follows. Corollary 8. Let M n+1 , n 2, be a causally regular ALADS spacetime satisfying NEC, and let V be as in Proposition 6. Then there exists no closed n-manifold N with b1 (N) > 0 such that V = U # N . In the above, # denotes the operation of connected sum, and b1 denotes the first Betti number. In the case of a wormhole we would have N = S 1 × S n−1 , and hence b1 (N) > 0. The proof of Corollary 8 involves an application of the Mayer–Vietoris sequence. G.J. Galloway et al. / Physics Letters B 505 (2001) 255–262 We conclude with some comments concerning results in specific dimensions. It turns out, as shown in [1], that in 3 + 1 dimensions Proposition 6 is sufficient to completely determine the homology of V . Moreover, we establish in [1] a basic topological relationship between the 2-surfaces ΣH and Σ∞ . We show that the genus of ΣH (or, if it has more than one component, the sum of the genera of its components) is bounded above by the genus of Σ∞ . Hence the topology of the black holes is controlled by the topology at infinity. Of special relevance to the AdS/CFT correspondence conjecture are results for ALADS spacetimes in 2 + 1 and 4 + 1 dimensions. In 2 + 1 dimensions one has the following. Theorem 9. Let M 2+1 be a causally regular ALADS spacetime satisfying NEC, and let V be as in Proposition 6. Then the 2-dimensional hypersurface V is either B 2 (a disk) or I × S 1 . Remark. The assumption made at the outset that V is orientable is not needed in Theorem 9; orientability follows from Proposition 6 and the fact that the boundary of V , being one-dimensional, is necessarily orientable. Proof. Since, as follows from Proposition 6, i∗ : H1 (Σ∞ ) → H1 (V ) is surjective, the rank of the free part of H1 (V ) cannot be greater than that of H1 (Σ∞ ), i.e., b1 (V ) b1 (Σ∞ ). In the case n = 2, Σ∞ is a 1-manifold so b1 (Σ∞ ) 1, and thus b1 (V ) 1. From the classification of 2-manifolds, V must be a closed 2-manifold minus a disjoint union of disks. The first Betti number of such manifolds is b1 = 2g +k, where g is the genus and k +1 the number of disjoint disks; it follows that g = 0. Since V must have at least one boundary, the only possible topologies for V are B 2 or I × S 1 . ✷ Thus, topological censorship gives a topological rigidity theorem in (2 + 1)-dimensional gravity: V for the domain of outer communications of each component of I in a (2 + 1)-dimensional black hole spacetime will have product topology. The case of (2 + 1)-dimensional asymptotically flat spacetimes can be similarly treated to produce the 261 same conclusions as Theorem 9. It follows that there are no asymptotically flat geons in 2 + 1 dimensions. In the case of (4 + 1)-dimensional spacetimes, although Proposition 6 can be used to obtain some information about the topology of V , it is not enough to fix the topology of V . This may be illustrated by considering the restricted case for which Σ∞ is simply connected. It then follows that V is a simply connected manifold-with-boundary. This is a fairly significant restriction; however one will have an infinite number of such manifolds. One obtains these simply by taking the connected sum of V with any closed simply connected 4-manifold. One can readily show that the connected sum of two such manifolds leaves Hk unchanged except for H2 . There are an infinite number of closed simply connected 4-manifolds characterized by their Hirzebruch signature and Euler characteristic. Furthermore, the restriction that V is simply connected is not enough to deduce the topology of the boundaries Σi even in this simple case. It is well known that all closed 3-manifolds are cobordant to S 3 . In fact one can construct a cobordism with trivial fundamental group [20]. Thus, at least by the methods discussed here, the topology of the interior of a (4 + 1)-dimensional ALADS spacetime is constrained but not completely characterized by the topology of the boundary-at-infinity. We mention in closing that the geodesic methods used to prove topological censorship can be adapted to the Euclidean setting to improve the results of Witten and Yau, cf. [21]. Acknowledgements We would like to thank G. Semenoff and E. Witten for useful conversations. This work was partially supported by the Natural Sciences and Engineering Research Council of Canada and by the National Science Foundation (USA), Grant No. DMS-9803566. References [1] G. Galloway, K. Schleich, D. Witt, E. Woolgar, Phys. Rev. D 60 (1999) 104039. [2] E. Witten, S.-T. Yau, Adv. Theor. Math. Phys. 3 (1999). [3] J. Maldacena, Adv. Theor. Math. Phys. 2 (1998) 231. 262 G.J. Galloway et al. / Physics Letters B 505 (2001) 255–262 [4] O. Aharony, S. Gubser, J. Maldacena, H. Ooguri, Y. Oz, Phys. Rep. 323 (2000) 184. [5] M. Banados, C. Teitelboim, J. Zanelli, Phys. Rev. Lett. 69 (1992) 1849. [6] J. Lemos, Phys. Lett. B 352 (1995) 46. [7] S. Aminneborg, I. Bengtsson, S. Holst, P. Peldán, Class. Quantum Grav. 13 (1996) 2707. [8] R. Mann, Class. Quantum Grav. 14 (1997) 2927. [9] D. Brill, J. Louko, P. Peldán, Phys. Rev. D 56 (1997) 3600. [10] C. Cadeau, E. Woolgar, gr-qc/0011029. [11] G. Mess, Lorentz spacetimes of constant curvature, MSRI Preprint 90-05808 (1990). [12] E. Witten, Adv. Theor. Math. Phys. 2 (1998) 253. [13] R. Penrose, in: C.M. DeWitt, B.S. DeWitt (Eds.), Relativity, Groups and Topology, Gordon and Breach, New York, 1964. [14] E. Witten, Talk at the ITP Conference on New Dimensions in Field Theory and String Theory, November, 1999, http: //online.itp.ucsb.edu/online/susy_ c99/witten/. [15] J. Friedman, K. Schleich, D. Witt, Phys. Rev. Lett. 71 (1993) 1486. [16] A. Ashtekar, A. Magnon, Class. Quantum Grav. 1 (1984) L39. [17] G. Galloway, Class. Quantum Grav. 12 (1995) L99. [18] S. Hawking, G. Ellis, The large scale structure of spacetime, Cambridge University Press, Cambridge, 1973. [19] G. Galloway, K. Schleich, D. Witt, E. Woolgar, hepth/9912119. [20] D. Rolfsen, Knots and Links, Publish or Perish, Berkeley, 1976. [21] M. Cai, G.J. Galloway, Adv. Theor. Math. Phys. 3 (1999). 26 April 2001 Physics Letters B 505 (2001) 263–266 www.elsevier.nl/locate/npe Quantum fields in anti-de-Sitter spacetime and degrees of freedom in the bulk/boundary correspondence Henrique Boschi-Filho, Nelson R.F. Braga Instituto de Física, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, 21945-970 Rio de Janeiro, RJ, Brazil Received 21 September 2001; accepted 13 November 2001 Editor: L. Alvarez-Gaumé Abstract The quantization of a scalar field in anti-de-Sitter spacetime using Poincaré coordinates is considered. We find a discrete spectrum that is consistent with a possible mapping between bulk and boundary quantum states. 2001 Elsevier Science B.V. All rights reserved. The holographic principle asserts that the degrees of freedom of a quantum system with gravity can be represented by a theory on the boundary [1–3]. The presence of gravity makes it possible to define a mapping between theories defined in manifolds of different dimensionality. One interesting realization of the holographic principle can be done in a space of constant negative curvature, the anti-de-Sitter (AdS) spacetime. Such a realization was proposed by Maldacena in the form of a conjecture [4] on the equivalence (or duality) of the large N limit of SU(N) superconformal field theories in n dimensions and supergravity on anti-deSitter spacetime in n + 1 dimensions (AdS/CFT correspondence). Then, using Poincaré coordinates in the AdS bulk, Gubser, Klebanov and Polyakov [5] and Witten [6] found prescriptions for relating theories that live in the bulk and on the boundary, where the AdS solutions play the role of classical sources for the boundary field correlators. E-mail addresses: [email protected] (H. Boschi-Filho), [email protected] (N.R.F. Braga). Despite the fact that field quantization in AdS in terms of global coordinates has been known for a long time [7,8], the corresponding formulation in Poincaré coordinates and thus a comprehensive picture of holography in terms of bulk quantum fields is still lacking. The aim of the present Letter is to investigate a quantum theory for a scalar field in the AdS bulk in terms of Poincaré coordinates. We will see that the dimensionality of the phase space is such that a mapping between this theory and states on the boundary is possible. This conclusion essentially depends on the fact that the AdS space in Poincaré coordinates should be compactified, as it happens in the usual global coordinates, in order to include appropriate boundary conditions at infinity and find a consistent quantization. Although the Poincaré coordinates extend to infinity we will need to introduce a finite radius (cutoff) R corresponding to the fact that we cannot represent the whole compactified AdS space used in the AdS/CFT correspondence (including the infinity) into just one single set of Poincaré coordinates. Naturally, we can take R large enough to describe as much of the entire AdS space as we want. We are going to see in the following that this result is in agreement with the counting of 0370-2693/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 2 4 4 - 1 264 H. Boschi-Filho, N.R.F. Braga / Physics Letters B 505 (2001) 263–266 degrees of freedom in the bulk/boundary correspondence. The anti-de-Sitter spacetime of n + 1 dimensions can be represented as the hyperboloid (Λ = constant) X02 2 + Xn+1 − n Xi2 = Λ2 (1) i=1 in a flat (n + 2)-dimensional space with metric 2 ds 2 = −dX02 − dXn+1 + n dXi2 . (2) i=1 The so-called global coordinates ρ, τ, Ωi for AdSn+1 can be defined by [9,10] X0 = Λ sec ρ cos τ, Xi = Λ tan ρΩi , n Ωi2 = 1, i=1 Xn+1 = Λ sec ρ sin τ, (3) correlators should be first taken at some small z that then goes to zero [10–13]. In the same way, we will consider the boundary to be at some small z = δ. So, let us consider at z = δ a hypersurface of area A corresponding to variations x 1 · · · x n−1 in the space coordinates: n−1 Λ x 1 · · · x n−1 A = (6) δ and calculate the volume generated by this surface moving z from δ to ∞, finding A . (7) n−1 This is the expected result that the volume is proportional to the area in the bulk/boundary correspondence for a fixed . In order to count the degrees of freedom we can split V in pieces of equal volume corresponding to cells whose boundaries are hypersurfaces located at V = Λ with ranges 0 ρ < π/2 and 0 τ < 2π. Poincaré coordinates z, x , t can be introduced by zj = 1 2 z + Λ2 + x 2 − t 2 , 2z Λx i , Xi = z 1 2 z − Λ2 + x 2 − t 2 , Xn = − 2z Λt Xn+1 = , z with j = 1, . . . , − 1. Note that the last cell extends to infinity. These volume cells can be mapped into the area ∆A by dividing it also in parts. This way, one finds a one to one mapping between degrees of freedom of bulk and boundary. This analysis shows us that despite the fact that the variable z has an infinite range, the volume, and thus the associated degrees of freedom, corresponding to a finite surface, are finite. One could, in a simplified way, think of the system as being “in a box” in terms of degrees of freedom, with respect to z. We could have changed z to a variable that measures the volume, say ζ = 1/δ n−1 − 1/zn−1 , to explicitly find this compact role of the radial coordinate z but this would not be better than just going to global coordinates, Eqs. (3). However, we want to see from the point of view of the Poincaré coordinates, where the AdS/CFT correspondence takes its more natural form, how does this compactified character of the radial coordinate manifests itself. Then, let us consider a massive scalar field φ in the AdSn+1 spacetime described by Poincaré coordinates with action 1 √ I [φ] = (9) d n+1 x g ∂µ φ∂ µ φ + m2 φ 2 , 2 X0 = (4) where x has n − 1 components and 0 z < ∞. In this case the AdSn+1 measure with Lorentzian signature reads ds 2 = Λ2 2 dz + (d x)2 − dt 2 . (z)2 (5) Then the AdS boundary described by usual Minkowski coordinates x , t corresponds to the region z = 0 plus a “point” at infinity (z → ∞). In order to gain some insight into the form of the spectrum associated with quantum fields to be defined in the AdS/CFT framework, let us discuss an essential point of the correspondence: the mapping between the degrees of freedom of the bulk volume and those of the boundary hypersurface. The metric is singular at z = 0, so the prescriptions [5,6] for calculating field δ , √ 1 − j/ n−1 (8) H. Boschi-Filho, N.R.F. Braga / Physics Letters B 505 (2001) 263–266 √ where we take x 0 ≡ z, x n+1 ≡ t, g = (x 0 )−n−1 and µ = 0, 1, . . . , n + 1. The classical equation of motion reads √ 1 ∇µ ∇ µ − m2 φ = √ ∂µ g ∂ µ φ − m2 φ = 0 g (10) and the solutions can be found [14,15] in terms of x n/2 +i k· z × Bessel functions using the ansatz φ = e−iωt χ(z). √ Taking ω2 > k2 and defining u = ω2 − k2 and ν = 12 n2 + 4m2 , we have two independent solutions Φ ± = e−iωt +i k·x zn/2 J±ν (uz), (11) if ν is not integer. If ν is integer one can take Φ + and Φ − = e−iωt +i k·x zn/2 Yν (uz) (12) as independent solutions. On the other hand, if k2 > ω2 the solution is x n/2 = e−iωt +i k· Φ (13) z Kν (qz), where q = k2 − ω2 (the second solution in this case is proportional to Iν (qz) which is divergent as z → ∞). As discussed in Refs. [14,15], Φ + are the only normalizable solutions in the range 0 < z < ∞. They are thus the natural candidates for the role of quantum fields, if we want to be able to take the limit of δ → 0 at the end. One could then naively think of just adding all possible solutions Φ + and thus building up a quantum field like u)Φ + (k, u) + c.c. du d n−1 k f (k, (14) However, from our previous analysis of degrees of freedom, we expect to find a discrete spectrum associated with the radial coordinate z. Such a discretization would be in accordance with the results coming from the quantization in global coordinates [7,8,14,15]. One can understand why this discretization also takes place in Poincaré coordinates by considering a simpler situation: the stereographic mapping of the surface of a sphere on a plane. One can map the points of a sphere on a plane plus a point at infinity. However, looking at the sphere one sees that this compact manifold has discrete sets of eigenfunctions but looking at the plane how can we realize that the spectrum of eigenfunctions in the radial direction 265 would be discrete? In close analogy with the case of finite volumes ∆V for z → ∞ in AdS discussed above, here in the case of the sphere if we calculate the area of the plane taking the metric induced by the sphere into account we would find a finite value (equal to the area of the sphere). So, the radial coordinate on the plane looks also like a “compact” one (in the sense of degrees of freedom or whatever we associate with area cells) in the same way as the z coordinate of AdS. The extra point at infinity corresponds to the fact that we should impose the condition that going to infinity in any direction would led to the same point. This condition would mean that either the functions on the plane have no angular dependence or they vanish as the radial coordinate tend to zero. These conditions would not yet lead to a discrete spectrum (in the radial direction). This problem is simply related to the fact that the point at infinity, which has zero measure, is not represented on the plane. If instead of using just one plane, we project the sphere on two different planes we would be able to represent all the points of the sphere. We could even choose one of the mappings to cover “as much of the surface of the sphere” as we want. As long as we map it into two disks of finite radius (with an appropriate matching boundary condition), one would then clearly see that the spectrum of eigenfunctions is discrete. Now coming back to the AdS case, a consistent quantization in this space in global coordinates [7,8] requires the introduction of boundary conditions at the surface corresponding to ρ = π/2 in order to have a well defined Cauchy problem. So, one must consider a compactified AdS including ρ = π/2 in order to find a consistent theory. The limit z → ∞ in Poincaré coordinates (4) corresponds to a point that in global coordinates sits in the hypersurface ρ = π/2. Thus, this hypersurface is not completely represented in just one set of Poincaré coordinates. In the same way as in the case of the sphere, we can solve this problem mapping the compactified AdS in two sets of Poincaré coordinates. We can simply stop at z = R in one set and map the rest, including the point at infinity, in a second set. We can take R arbitrarily large so that we can map as much of the compactified AdS spacetime as we want in just one set. In this region 0 z R we can introduce as quantum fields 266 H. Boschi-Filho, N.R.F. Braga / Physics Letters B 505 (2001) 263–266 Φ(z, x, t) = ∞ p=1 zn/2 Jν (up z) d n−1 k ν+1 (up R) (2π)n−1 Rwp (k)J +i k· x −iwp (k)t + c.c. , × ap (k)e (15) = u2p + k2 and up are such that where wp (k) Jν (up R) = 0. Imposing that the operators a, a † satisfy the commutation relations a † (k ) = 2(2π)n−1 wp (k)δ pp ap (k), p × δ n−1 (k − k ), ap (k ) = ap† (k), a † (k ) = 0 ap (k), p (16) we find, for example, for the equal time commutator of field and time derivative ∂Φ (z , x , t) Φ(z, x , t), ∂t x − x ). = izn−1 δ(z − z )δ( (17) Now considering again the field (15) we realize that the discretization of the spectrum makes it possible to map the phase space up , k into the momentum space of a field theory defined on the boundary, in the same way as we can map an infinite but enumerable set of lines into just one line. Taking, for simplicity, AdS3 where k has just one component, the phase space in the bulk would be an enumerable set of lines each one corresponding to the continuous values of −∞ < k < ∞ and one fixed value of p. One can map these lines into just one line, corresponding to some momentum, say κb on the boundary, by dividing the line of κb into segments of finite size. This kind of mapping would not be possible if the spectrum were not discrete, as one cannot define a one to one mapping between a plane and a line. So, the discretization of the spectrum is a necessary ingredient for the holographic mapping to hold. In conclusion, we have obtained a quantum scalar field in the AdS bulk that exhibits a discrete spectrum associated with the radial Poincaré coordinate. This result was obtained taking into account the compactification of AdS that in Poincaré coordinates corresponds to adding a point at infinity. This discretization is in agreement with the counting of degrees of freedom suggested by the holographic principle. However, it is in contrast to the continuous spectrum found in Ref. [16]. Acknowledgements The authors were partially supported by CNPq, FINEP and FUJB — Brazilian research agencies. We also thank Mauricio Calvão, Juan Mignaco, Cassio Sigaud and Arvind Vaidya for interesting discussions. References [1] G. ’t Hooft, Dimensional reduction in quantum gravity, in: A. Aly, J. Ellis, S. Randjbar-Daemi (Eds.), Salam Festschrifft, World Scientific, Singapore, 1993, gr-qc/9310026. [2] L. Susskind, J. Math. Phys. 36 (1995) 6377. [3] L. Susskind, E. Witten, The holographic bound in anti-de Sitter space, SU-ITP-98-39, IASSNS-HEP-98-44, hep-th/9805114. [4] J. Maldacena, Adv. Theor. Math. Phys. 2 (1998) 231. [5] S.S. Gubser , I.R. Klebanov, A.M. Polyakov, Phys. Lett. B 428 (1998) 105. [6] E. Witten, Adv. Theor. Math. Phys. 2 (1998) 253. [7] S.J. Avis, C.J. Isham, D. Storey, Phys. Rev. D 18 (1978) 3565. [8] P. Breitenlohner, D.Z. Freedman, Phys. Lett. B 115 (1982) 197; P. Breitenlohner, D.Z. Freedman, Ann. Phys. 144 (1982) 249. [9] O. Aharony, S.S. Gubser, J. Maldacena, H. Ooguri, Y. Oz, Phys. Rep. 323 (2000) 183. [10] J.L. Petersen, Int. J. Mod. Phys. A 14 (1999) 3597. [11] W. Mueck, K.S. Viswanathan, Phys. Rev. D 58 (1998) 041901. [12] D.Z. Freedman, S.D. Mathur, A. Matusis, L. Rastelli, Nucl. Phys. B 546 (1999) 96. [13] H. Boschi-Filho, N.R.F. Braga, Phys. Lett. B 471 (1999) 162. [14] V. Balasubramanian, P. Kraus, A. Lawrence, Phys. Rev. D 59 (1999) 046003. [15] V. Balasubramanian, P. Kraus, A. Lawrence, S.P. Trivedi, Phys. Rev. D 59 (1999) 1046021. [16] M.M. Caldarelli, Nucl. Phys. B 549 (1999) 499. 26 April 2001 Physics Letters B 505 (2001) 267–274 www.elsevier.nl/locate/npe Quantum mechanics on the noncommutative plane and sphere V.P. Nair a,b , A.P. Polychronakos a,b,c,1 a Physics Department, City College of the CUNY, New York, NY 10031, USA b The Graduate School and University Center, City University of New York, New York, NY 10016, USA c Physics Department, Rockefeller University, New York, NY 10021, USA Received 19 December 2000; received in revised form 25 February 2001; accepted 27 February 2001 Editor: M. Cvetič Abstract We consider the quantum mechanics of a particle on a noncommutative plane. The case of a charged particle in a magnetic field (the Landau problem) with a harmonic oscillator potential is solved. There is a critical point with the density of states becoming infinite for the value of the magnetic field equal to the inverse of the noncommutativity parameter. The Landau problem on the noncommutative two-sphere is also solved and compared to the plane problem. 2001 Published by Elsevier Science B.V. 1. Introduction Noncommutative spaces can arise as brane configurations in string theory and in the matrix model of M-theory [1]. Fluctuations of branes are described by gauge theories and thus, motivated by the existence of noncommutative branes, there has recently been a large number of papers dealing with gauge theories, and more generally field theories, on such spaces [2]. However, there has been relatively little work exploring the quantum mechanics of particles on noncommutative spaces. Since the one-particle sector of field theories, which can be treated in a more or less self-contained way in the free field or weakly coupled limit, leads to quantum mechanics, the brane connection suggests that a more detailed study of this topic E-mail addresses: [email protected] (V.P. Nair), [email protected] (A.P. Polychronakos). 1 On leave from Theoretical Physics Department, Uppsala University, Sweden and Physics Deptartment, University of Ioannina, Greece. should be useful. This is the subject of the present Letter. Some of the algebraic aspects of quantum mechanics on spaces with an underlying Lie algebra structure were considered in Ref. [3]. The noncommutative plane can be defined in terms of a projection to the lowest Landau level of dynamics on the commuting plane [4]; some features of particle dynamics in terms of a similar construction were contained in [5]. The spectrum of a harmonic oscillator on the noncommutative plane was derived in [6], the equation of motion of a particle in an external magnetic field was discussed in [7], and the case of a general central potential was recently discussed in [8]. In this Letter, we will analyze the algebraic structures in more detail. We will solve the problem of a charged particle in a magnetic field (the Landau problem) with an oscillator potential on the noncommutative plane. There is an interesting interplay of the magnetic field B and the noncommutativity parameter θ , with a critical point at Bθ = 1 where the density of states becomes infinite. We also solve the Landau problem on the noncommu- 0370-2693/01/$ – see front matter 2001 Published by Elsevier Science B.V. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 3 3 9 - 2 268 V.P. Nair, A.P. Polychronakos / Physics Letters B 505 (2001) 267–274 tative sphere, for which the basic algebraic structure turns out to be SU(2) × SU(2). We also show how the results on the plane can be recovered in the limit of a large radius for the sphere. 2. The noncommutative plane We start with the quantum mechanics of a particle on the noncommutative two-dimensional plane. For single particle quantum mechanics, we need the Heisenberg algebra for the position and momentum operators. The two-dimensional noncommutative plane is described by the coordinates x1 , x2 which obey the commutator algebra [x1 , x2 ] = iθ where θ is the noncommutativity parameter. With the momentum operator pi , i = 1, 2, we may write the full Heisenberg algebra as [x1 , x2 ] = iθ, [xi , pj ] = iδij , [p1 , p2 ] = 0. (1) The fact that x1 and x2 commute to a constant may suggest that they can themselves serve as translation operators. However, this is not adequate to obtain the last of the relations (1); one needs independent operators. A realization of the momentum operators, for example, would be 1 p1 = (x2 + k1 ), θ 1 p2 = (−x1 + k2 ) (2) θ with [k1 , k2 ] = −iθ and [ki , xj ] = 0. In this case, (x1 , x2 ) and (k2 , k1 ) obey identical commutation rules and are mutually commuting. pi are thus constructed from two copies of the x-algebra. We may use the realization (2) of the pi to solve specific quantum mechanical problems. However, before turning to specific examples, some comments about the pi -operators are in order. In the usual quantum mechanics with commuting x’s, a single irreducible representation for the x-algebra would be given by xi = ci for fixed real numbers ci . Coordinate space is spanned by an infinity of irreducible representations of the x-algebra. Additional independent operators pi are needed to obtain a single irreducible representation, now for the augmented set of operators. The pi ’s connect different irreducible representations of the x-algebra. In order to recover this structure for small θ , we need the independent set of operators ki in (2). Single particle quantum mechanics may also be viewed as the one-particle sector of quantum field theory, in the free field or very weakly coupled limit, with the Schrödinger wave function obeying essentially the free field equation. Since quantum field theories on noncommutative manifolds have already been defined and investigated to some extent, this may seem to give a quick and simple way to write down one-particle quantum mechanics. The case of a nonrelativistic Schrödinger field suffices to illustrate the point. The field Φ(x) is a function of the noncommuting coordinates xi . The action for this field in an external potential and coupled to a gauge field may be written as (Di Φ)† (Di Φ) † † −Φ VΦ , S = dt Tr Φ iD0 Φ − 2m (3) where Dµ Φ = ∂µ Φ + ΦAµ . Even though we have indicated the derivative as ∂µ Φ, it should be emphasized that, since Φ is noncommutative, even classically, translations must be implemented by taking commutators with an operator conjugate to x. This is implicit in the definition of ∂µ Φ. Further, in (3), the gauge fields act on the right of the field Φ and the potential on the left. This ensures that the action of the gauge field and the potential commute and allows an unambiguous separation of these two types of interaction terms. The one-particle wavefunction is the matrix element of Φ between the vacuum and one-particle states. The equation of motion for (3) is iD0 Φ + 1 Di (Di Φ) − V Φ = 0 2m (4) and taking the appropriate matrix element, we see that the Schrödinger equation has a similar form, with the qualification that the action of derivatives is defined via commutators with an operator conjugate to x. With the algebra of the xi ’s in (1), we can see that translations of the argument of Φ may be achieved using just the xi ’s themselves by writing [9] ij xj ij xj 1 −i∂i Φ = [ij xj , Φ] = Φ −Φ . θ θ θ (5) V.P. Nair, A.P. Polychronakos / Physics Letters B 505 (2001) 267–274 In other words, by using the adjoint action of ij xj , we can obtain translations on functions of xi . It is then easy to check that [∂1 , ∂2 ] = 0. Translations thus involve the left and right actions of the xi ’s on Φ, which are mutually commuting actions. Since we do not usually take commutators of operators with the wavefunction in quantum mechanics, it is preferable, in going to the one-particle case, to replace the right action of the x’s on Φ formally by a left action by xiR Φ = Φxi . (6) This can be done in more detail as follows. If we realize the x-algebra on a Hilbert space H, then Φ is where H an element of H ⊗ H, is the dual Hilbert space and we can write Φ = mn Φmn |m n| in terms to the of a basis {|m}. Mapping the elements of H corresponding elementsof H in the standard way, we can introduce Φ = mn Φmn |m ⊗ |n. The right action of x’s on Φ is then mapped onto the left action on Φ as given above. Note that [xiR , xj R ]Φ = −Φ[xi , xj ] = −iθ ij Φ. We can thus identify −ij xj R as ki and we obtain the realization given in (2). The one-particle limit of field theory thus naturally leads to the structure of two mutually commuting copies of the x-algebra. We see that from both points of view, namely, of one-particle quantum mechanics as defined by an irreducible representation of the Heisenberg algebra generalized to include noncommutativity of coordinates, or as defined by the one-particle limit of field theory, we are led to the algebraic structure (1, 2). This result is consistent with the discussion of quantum mechanics on the noncommuting two-sphere given in Ref. [3]. In that case also, one had two mutually commuting copies of the x-algebra, which was SU(2). The momentum operator was then constructed from the SU(2) × SU(2) algebra in a way analogous to the realization (2). The present results for the plane may in fact be obtained, as we shall see later, for a small neighborhood of the sphere, in the limit of large radius. A concrete and simple example which illustrates the general discussion so far is provided by the harmonic oscillator on the noncommutative plane. It is not any more difficult to solve the more general case of a charged particle in a magnetic field (the Landau problem) with a quadratic (or oscillator) potential and so we shall treat this case below. The fact that we have 269 a magnetic field B can be incorporated by modifying the commutation rule for the momenta to [p1 , p2 ] = iB. In other words, B measures the noncommutativity of the momenta. The interplay of B and θ can thus lead to some interesting behavior. Denoting the position and momentum operators by ξi , i = 1, . . . , 4, ξ = (x1 , x2 , p1 , p2 ), the commutation rules are [ξi , ξj ] = iPij , 0 θ 1 0 −θ 0 P = −1 0 0 0 −1 −B 0 1 . B 0 (7) The Hamiltonian for the oscillator with magnetic field is H= 1 2 p1 + p22 + ω2 x12 + x22 . 2 (8) It is obviously invariant under rotations in the plane. The angular momentum, being the generator of these rotations, takes the form 1 B 2 L= x1 + x22 x1 p2 − x2 p1 + 1 − θB 2 θ 2 2 (9) + p1 + p2 . 2 We observe that it acquires θ -dependent corrections compared to the commutative case. The algebra (7) has many possible realizations. The ‘minimal’ one in terms of two independent sets of canonical coordinates and momenta (x̄i , p̄i ) satisfying standard Heisenberg commutation relations would be x1 = x̄1 , p1 = p̄1 + B x̄2 , x2 = x̄2 + θ p̄1 , p2 = p̄2 . (10) We prefer, however, to use a realization as close to (2) as possible to maintain contact with noncommutative field theory. Using the realization (2) for the momenta, we find [k1 , k2 ] = i(B − (1/θ )). Because of this, the cases B < 1/θ and B > 1/θ should be treated differently. Consider first the case B < 1/θ . In this case, we can define x1 = lα1 , x2 = lβ1 , 1 p1 = β1 + qα2 , l 1 p2 = α1 − qβ2 , l (11) 270 V.P. Nair, A.P. Polychronakos / Physics Letters B 505 (2001) 267–274 where l 2 = θ and q 2 = (1/θ ) − B. αi , βi form a set of canonical variables, i.e., [αi , βj ] = iδij . The Hamiltonian for the oscillator with the magnetic field is given by 1 2 p1 + p22 + ω2 x12 + x22 , 2 1 1 = ω2 l 2 + 2 α12 + β12 + q 2 α22 + β22 2 l 2q (α1 β2 + α2 β1 ) . + (12) l H= We can now make a Bogolyubov transformation on this by expressing αi , βi in terms of a canonical set Qi , Pi by writing Q1 P2 α1 Q2 P1 α2 (13) + sinh λ . = cosh λ β1 P1 Q2 β2 P2 Q1 Choosing tanh 2λ = − 2ql 1 + ω2 l 4 + q 2 l 2 (14) the Hamiltonian (12) becomes H= 1 2 Ω+ P12 + Q21 + Ω− P22 + Q22 , (15) where Ω± = 2 1 2 1 ω θ − B + 4ω2 ± ω2 θ + B . 2 2 (16) Eq. (15) shows that the spectrum is given by that of two harmonic oscillators of frequencies Ω+ and Ω− . The case of B > 1/θ can be treated in a similar way. With q 2 = B − (1/θ ), we can write x1 = lα1 , x2 = lβ1 , 1 p1 = β1 + qα2 , l 1 p2 = − α1 + qβ2 . l (17) In terms of the αi , βi , the Hamiltonian becomes 1 1 2 2 2 H= ω l + 2 α1 + β12 + q 2 α22 + β22 2 l 2q + (18) (α2 β1 − α1 β2 ) . l The required Bogolyubov transformation is Q1 P2 α1 Q2 P1 α2 + sin λ . = cos λ β1 P1 −Q2 β2 P2 −Q1 (19) The required choice of λ is given by tan 2λ = 2ql . 1 + ω2 l 4 − q 2 l 2 (20) H can then be written as in (15) with 2 1 2 1 ω θ − B + 4ω2 + ω2 θ + B . (21) Ω± = ± 2 2 We again have two oscillators of frequencies Ω± . We see from the above results that there is a critical value of the magnetic field or θ given by Bθ = 1. Ω− vanishes upon approaching this value from either side. The Hamiltonian is independent of P2 , Q2 . Thus the number of states for fixed energy will become unbounded, since all the states generated by P2 , Q2 are now degenerate. This large degeneracy can also be seen from a semiclassical estimate of the number of states for fixed energy. Going back to (7), we see that det P = (1 − Bθ )2 . The phase volume is thus given by 1 dµ = √ dx1 dx2 dp1 dp2 det P 1 dx1 dx2 dp1 dp2 . = |1 − θ B| (22) Surfaces of equal energy in phase space are ellipsoids defined by E = 12 (p12 + p22 + ω2 x12 + ω2 x22 ). A semiclassical estimate of the number of states with energy less that E is given by the volume inside this surface divided by (2π)2 . We obtain 2 V 1 E N= (23) = . (2π)2 2|1 − Bθ | ω The criticality of the point θ B = 1 is once again clear; the density of states is infinite at this point. When ω2 = 0 we have the pure Landau problem. In this case Ω+ = B, Ω− = 0 for B > 0, or Ω+ = 0, Ω− = |B| for B < 0 and we have the standard, infinitely degenerate Landau levels as in the commutative case. The density of states per unit area, denoted by ρ, however, is now modified to 1 B . ρ= (24) 2π 1 − θ B V.P. Nair, A.P. Polychronakos / Physics Letters B 505 (2001) 267–274 To demonstrate this, observe that the magnetic translations, defined as the operators performing translations on xi and commuting with the Hamiltonian, are now 1 (p1 − Bx2 ), 1 − θB 1 (p2 + Bx1 ). D2 = (25) 1 − θB These are the operators responsible for the infinite degeneracy of the Landau levels and their commutator determines the density of degenerate states on the plane. Di commute with xj in the standard way, [xi , Dj ] = iδij , but their mutual commutator is now D1 = B (26) 1 − θB which reproduces the result (24). We observe that for the critical value of the magnetic field B = 1/θ the density of states on the plane becomes infinite. The same result can also be obtained in the semiclassical way of the previous paragraph, where now we calculate the phase space volume of a circle E = 12 (p12 + p22 ) in momentum space times a domain of area A in coordinate space. The result is [D1 , D2 ] = −i N= EA V = 2 (2π) 2π|1 − θ B| (27) which is compatible with (24) upon filling the lowest n Landau levels such as E = n|B|. It is also interesting to calculate the magnetic length in this case, that is, the minimum spatial extent of a wavefunction in the lowest Landau level. This can be achieved by putting both oscillators Ω+ and Ω− in their ground state: the one with nonvanishing frequency excites Landau levels while the one with vanishing frequency creates annular states on the plane for each Landau level. In this state we have Pi2 = Q2i = 12 and Pi = Qi = 0. Using (11), (13) and (14) we can calculate xi2 for B < 1/θ as 1 2 − Bθ x12 + x22 = l 2 cosh2 λ + sinh2 λ = |B| (28) while for B > 1/θ we obtain from (17) and (19) 2 x1 + x22 = l 2 cos2 λ + sin2 λ = θ. (29) So we see that for subcritical magnetic field the magnetic length is more or less as in the commutative 271 case √ while for overcritical one it assumes the value l = θ which is the minimal uncertainty on the noncommutative plane. We conclude by noting that the oscillator frequency ω and magnetic field B appearing in the Hamiltonian are distinct from the corresponding ‘kinematical’ quantities that appear in the equations of motion. Expressing the equations of motion in terms of xi and its time derivatives we obtain ẍi = B + θ ω2 ij ẋi − (1 − θ B)ω2 xi . (30) We recognize the Lorentz force and the spring force with effective magnetic field and spring constant = B + θ ω2 , B ω̃2 = (1 − θ B)ω2 . (31) The spectral frequencies Ω± in terms of the kinematical parameters become identical to the corresponding noncommutative ones, namely 2 + 4ω̃2 + 1 B . Ω± = ± 12 B (32) 2 In this parametrization the noncommutativity of space manifests only through the density of states and spatial correlation functions. Interestingly, for the critical value B = 1/θ the oscillator ω transmutates entirely = θ ω2 + θ −1 . into a magnetic field B 3. The noncommutative sphere We now turn to the quantum mechanics of a particle on the noncommutative two-sphere. To set the stage, we first give a review of the commutative sphere with a magnetic monopole at the center. The observables of the theory consist of the particle coordinates xi and the angular momentum generators Ji , i = 1, 2, 3. Their algebra is [xi , xj ] = 0, [Ji , xj ] = iij k xk , [Ji , Jj ] = iij k Jk (33) while the Hamiltonian is taken to be 1 2 (34) J . 2x 2 The algebra (33) has two Casimirs, which can be chosen to have fixed values, say, H= 272 V.P. Nair, A.P. Polychronakos / Physics Letters B 505 (2001) 267–274 x 2 = a 2, n x · J = − a. (35) 2 The first one is simply the square of the radius of the sphere. In the second one, n can be identified as the monopole number. Indeed, in the presence of a monopole field the angular momentum acquires a term in the radial direction proportional to the monopole number which makes the second Casimir nonvanishing. The interpretation as a magnetic field can be independently justified by deriving the equations of motion of xi using the Hamiltonian (34) 2 1 J 2 − (n/2)2 J − (n/2)2 ẍi = − xi + xi 2 a4 a4 nxk (36) + ij k ẋj 3 . 2a The first term is a centripetal force, due to the motion on a curved manifold; the kinematical angular momentum squared is seen to be J 2 − (n/2)2 . The second term is a Lorentz force, corresponding to a radial magnetic field Bi = (n/2)x̂i /a 2 . The monopole number, then, is 4πa 2 B (37) = n. 2π It is interesting that the magnetic field does not appear as a parameter in the Hamiltonian, not even as a modification of the Poisson structure (as in the planar case), but rather as a Casimir of the algebra of observables. We now turn to the noncommutative sphere. The quantum mechanics of a particle on a noncommutative sphere was discussed in [3]. The structure of observables is similar, with the difference that the coordinates do not commute but rather form an SU(2) algebra. Specifically, N= [Ri , Rj ] = iij k Rk , [Ji , Rj ] = iij k Rk , [Ji , Jj ] = iij k Jk , R2 (38) is a Casimir, as before, but the magnetic Casimir is deformed to R · J − 12 J 2 . This operator structure is realized in terms of an SU(2) × SU(2)-algebra with corresponding mutually commuting generators Ri , Ki . In terms of these, the angular momentum is Ji = Ri + Ki . We have two Casimir operators, R 2 = r(r + 1) and K 2 = k(k + 1), and we can label an irreducible representation by the maximal spin values (r, k). The magnetic Casimir becomes 12 (R 2 − K 2 ). If the radius of the sphere is denoted by a as before, we can identify the coordinates xi as a xi = √ (39) Ri . r(r + 1) The commutative case can be obtained as the limit in which both r and k become very large, but with their difference k − r = n/2 being fixed (so that the angular momentum J 2 remain finite). In that limit, the magnetic Casimir becomes 2 n k+r +1 2 1 − r, (40) 2 R − K = (r − k) 2 2 n becomes the monopole number. We can, therefore, identify the integer n = 2(k − r) as a quantized ‘monopole’ number in the noncommutative case. The Hamiltonian of the particle can again be taken proportional to the square of the angular momentum: γ H = 2J2 (41) 2a with γ some coefficient depending on the Casimirs. In the limit of a commutative sphere γ should become 1 in order to reproduce the standard results. In general, however, there is no a priori reason to fix a specific value for γ and, as we shall demonstrate, a different choice must be made in order to recover the limit of the noncommutative plane. The energy spectrum of the particle is clearly |n| |n| , + 1, . . . , j + k. 2 2 (42) Both the energy and angular momentum have a finite spectrum, reflecting the fact that the Hilbert space is finite dimensional. Comparison to the noncommutative plane can be made by scaling appropriately the parameters of the model. We should take the radius a in (39) to infinity and consider a small neighborhood, say, around the ‘north pole’ R3 = r, with x1 , x2 being the relevant coordinates. From the definition (39) of xi , we then identify the noncommutativity parameter as θ ≈ a 2 /r. So the scaling of the parameter r is E= r= γ j (j + 1), 2R 2 a2 , θ a → ∞. j= (43) V.P. Nair, A.P. Polychronakos / Physics Letters B 505 (2001) 267–274 Only the low-lying states of J 2 and H should be considered in this limit, with j = |n| 2 + l, l = 0, 1, 2, . . . . Since R3 ≈ r for the states of interest in this limit, we must also have K3 ≈ −k = −r − n/2 so that j ≈ |n|/2. This also means that J3 ≈ −n/2. k = r + n/2 and γ should then scale appropriately to obtain the planar operator algebra of observables for a particle on the noncommutative plane in the presence of a magnetic field. The operators ij Jj /a (i, j = 1, 2) generate translations of xi in the planar limit, i.e., i δij R3 ≈ iδij . xi , a1 j k Jk = √ r(r + 1) (44) So one might be tempted to identify them with the momentum operators in that limit. In the presence of a magnetic field, however, we understand that these should instead become the magnetic translations Di , since they both commute with the Hamiltonian. Their commutator 1 i n (45) [J2 , −J1 ] = 2 J3 ≈ −i 2 a2 a 2a should then reproduce the result (26) for the plane. This leads to the identification [D1 , D2 ] = n= 2Ba 2 1 − θB (46) which fixes the scaling of n and k. It remains to identify the momenta pi = ẋi . From the Hamiltonian (41) we obtain ẋi = γ ij k Kj Rk . R r(r + 1) √ (47) The commutator of xi and pj = ẋj then becomes [xi , pj ] = iγ (Ki Rj − Kk Rk δij ). r(r + 1) (48) In the planar limit K3 and R3 dominate over K1,2 and R1,2 . Therefore, the above commutator becomes, for i, j = 1, 2, [xi , pj ] ≈ − iγ k K3 R3 δij ≈ iγ δij 2 r r (49) (we also set r(r + 1) ≈ r 2 ). To reproduce the canonical commutators on the plane we must set γ= r r = k r+ n 2 = 1 − θB (50) 273 which fixes the scaling of γ . We can now calculate the commutator of momenta i nr [p1 , p2 ] = 2 2 K · R(K3 + R3 ) ≈ i 2 = iB (51) k a 2a k which is, indeed, the correct planar commutator. Finally, the spectrum of the Hamiltonian becomes |n| γ |n| +l +l+1 E= 2 2a 2 2 2 γ |n| γn ≈ 2 + l + 12 8a 2a 2 B2a2 + |B| l + 12 . = (52) 2(1 − θ B) Apart from a zero-point shift of order a 2 , we have agreement with the Landau level spectrum of the noncommutative plane. The above spectrum, but without the zero-point shift, is also reproduced by the lowlying states of the operator H = 12 pi2 , thus establishing the full correspondence with the plane. The density of states on each Landau level can also be calculated. For a given energy eigenvalue corresponding to j = |n|/2 + l there are 2j + 1 degenerate states. The space density of these