Download Are mirror worlds opaque?

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.
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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.
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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.
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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.
References
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[12] M.D. Cortina-Gil et al., Phys. Lett. B 401 (1997) 9.
[13] F. Auger et al., ENPE 99, Seville, Spain, June 21–26, AIP
Conf. Proc. 495 (1999) 13;
A. Lagoyannis et al., submitted for publication.
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[16] A.M. Sawkes et al., Phys. Rev. C 13 (1976) 451.
[17] M.C. Mermaz, Phys. Rev. C 50 (1994) 2620.
[18] M.D. Cortina-Gil et al., Nucl. Phys. A 641 (1998) 263.
[19] Y. Blumenfeld et al., NIM Phys. Res. A 421 (1999) 471.
[20] A. de Vismes et al., in preparation;
A. de Vismes, PhD Thesis, University of Caen, France, 2000.
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[22] R.F. Carlson et al., Nucl. Instrum. Methods 123 (1975) 509.
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S. Kox et al., Phys. Rev. 35 (1987) 1678.
M.-D. Cortina-Gil, private communication.
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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. This disagreement on the separation energy of the 1/2− state
is common with other shell-model calculations [19,20]
and might be due to other correlations and other states
formed from the excitation of two or more particles
from the p-shell into the sd-shell. However, the empirical WBT and WBP interactions are determined under
We acknowledge the help of T. Aumann, F. Deák,
Á. Horváth, K. Ieki, Y. Iwata, Y. Higurashi, Á. Kiss,
J. Kruse, V. Maddalena, H. Schelin, Z. Seres, and
S. Takeuchi during the experiment, M. Steiner and
J. Stetson for producing the radioactive beams, and
P. Danielewicz for discussions. This work was supported by the National Science Foundation under
grants PHY-9528844 and PHY-9605207.
References
[1] T. Aumann et al., Phys. Rev. Lett. 84 (2000) 35, and references
therein.
[2] M. Thoennessen et al., Phys. Rev. C 59 (1999) 111.
[3] R.A. Kryger et al., Phys. Rev. C 37 (1993) R2439.
[4] M. Zinser et al., Phys. Rev. Lett. 75 (1995) 1719.
[5] M. Zinser et al., Nucl. Phys. A 619 (1997) 151.
[6] K.K. Seth et al., Phys. Rev. Lett. 58 (1987) 1930.
[7] H.G. Bohlen et al., Z. Phys. A 330 (1988) 227.
[8] W. von Oertzen et al., Nucl. Phys. A 588 (1995) 129c.
[9] A.A. Ogloblin, Z. Phys. A 351 (1995) 355.
[10] K.M. Watson, Phys. Rev. 88 (1952) 1163.
[11] B.M. Sherrill et al., Nucl. Instrum. Methods B 70 (1992) 298.
[12] J.J. Kruse et al., to be published.
[13] P.D. Zecher et al., Nucl. Instrum. Methods A 401 (1997) 329.
[14] L. Chen, Ph.D. Thesis, Michigan State University, March
2000.
[15] L. Chen et al., to be published.
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[16] G.F. Bertsch et al., Phys. Rev. C 57 (1998) 1366.
[17] R.H. Stokes, P.G. Young, Phys. Rev. Lett 18 (1967) 611;
R.H. Stokes, P.G. Young, Phys. Rev. 178 (1969) 2024.
[18] E.K. Warburton, B.A. Brown, Phys. Rev. C 46 (1992) 923.
[19] N.A.F.M. Poppelier, L.D. Wood, P.W.M. Glaudemans, Phys.
Lett. 157B (1985) 120.
[20] S. Pieper, private communication.
[21] G. Audi, A.H. Wapstra, Nucl. Phys. 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
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[23]
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[26]
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H.A. Enge, S.B. Kowalsky, in: Proc. 3rd Int. Conf. on Magnet
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1999, unpublished.
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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
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and the hard charm component is expected to play a
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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.
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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
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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.
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Physics Letters B 505 (2001) 47–58
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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
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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.
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[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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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[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;
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[15] J. Bijnens, A. Bramon, F. Cornet, Phys. Rev. Lett. 61 (1988)
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Ll. Amettler et al., Phys. Rev. D 45 (1992) 986.
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See also, R. Alkofer, C.D. Roberts, Phys. Lett. B 369 (1996)
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[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.
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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. Furthermore the authors thank
Ph. de Forcrand for comments on the first version of
the manuscript and, in particular, for drawing our attention to Ref. [38].
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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.
We are thus encouraged to believe that in 4d simulations analytic continuation of imaginary chemical
potential results would give physically relevant results if a good ansatz for the µ-dependence is available, and would allow to go closer to Tc determining,
e.g., the free energy density and the spatial correlation lengths there. Furthermore, it appears that at high
enough temperatures it is even sufficient to determine
only the coefficients of the first terms depending on µ,
which amounts simply to susceptibility measurements
at µ = 0 (see, e.g., [38,39] and references therein).
Acknowledgements
M.L. thanks Chris Korthals Altes for a useful
discussion. This work was partly supported by the
TMR network Finite Temperature Phase Transitions
in Particle Physics, EU contract no. FMRX-CT970122. The work of A.H. was supported in part by UK
PPARC grant PPA/G/0/1998/00621.
147
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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± , . . . ;
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(ii) rules out J P = 1+ , 2+ if no
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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.
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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.
Acknowledgement
The author is supported by the BK21 program of
Ministry of Education.
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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.) Most significant, however is that an accelerator at the Tripler energy appears to be technologically feasible.
167
Acknowledgements
We thank Muge Karagoz for her participation in
the earlier stage of our analysis. This work was
supported in part by DOE grant Nos. DE-FG0395ER40917, DE-FG03-95ER40924 and NSF grant
No. PHY-9722090.
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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.).
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26 April 2001
Physics Letters B 505 (2001) 177–183
www.elsevier.nl/locate/npe
Muon g − 2, dark matter detection and accelerator physics
R. Arnowitt, B. Dutta, B. Hu, Y. 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
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26 April 2001
Physics Letters B 505 (2001) 184–190
www.elsevier.nl/locate/npe
Light charged Higgs boson and supersymmetry
C. Panagiotakopoulos a , A. 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. The work of A.P. is supported
in part by the Bundesministerium für Bildung und
Forschung under the contract number 05HT9WWA9.
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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.
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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]
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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]
à = −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
à = −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.
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hep-th/9804147.
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Proceedings of XXII Johns Hopkins Workshop on Novelties in
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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.
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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
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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.
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5. Conclusions and outlook
The 5-dimensional supersymmetric models with
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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)
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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
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26 April 2001
Physics Letters B 505 (2001) 243–248
www.elsevier.nl/locate/npe
Matrix model and Chern–Simons theory
J. 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
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[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.
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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.
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G. Landi, hep-th/9701078;
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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.
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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
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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.
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[13] R. Penrose, in: C.M. DeWitt, B.S. DeWitt (Eds.), Relativity,
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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.
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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