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Four-quark spectroscopy within the hyperspherical formalism J. Vijande 1,2, N. Barnea 3, A. Valcarce 2 1Theoretical Physics Department, Universidad de Valencia 2Nuclear Physics Group, Universidad de Salamanca 3The Racah Institute of Physics, The Hebrew University. PRD73 (2006) 054004 24 May, 2017 Four-quark spectroscopy within the hyperspherical formalism 1 • Motivation • Hyperspherical formalism • Constituent quark model • Four-body results (cccc) • Summary 24 May, 2017 Four-quark spectroscopy within the hyperspherical formalism 2 Motivation Why multiquarks? | Meson B 0 | qq | gg | qqg | qqqq ... Most of the meson spectra can be described assuming that the qq component is not only the dominant one, but also the only one. However, there are some exceptions like mesons with exotic quantum numbers or those with properties which are difficult to explain (like the light scalars, the X(3872) or the opencharmed mesons). • The hyperspherical harmonic method, widely applied and tested in nuclear physics for N ≤ 7, has only been applied to quark physics for N = 3. •Therefore, its generalization would be ideally suited for the study of the properties of multiquark systems for all quantum numbers. 24 May, 2017 Four-quark spectroscopy within the hyperspherical formalism 3 Hyperspherical harmonic formalism • The idea is to generalize the simplicity of the spherical harmonic expansion for the angular functions of a single particle motion to a system of particles. r LK YLM , K , L , M , Sym 3 N 4 • The main disadvantage of this method is that the number of HH needed to obtaing a good convergence is very large. So, one needs to construct HH functions of proper symmetry for any value of K. This has been overcome by means of a HH formalism based on the symmetrization of the N−body wave function with respect to the symmetric group using the Barnea and Novoselsky algorithm (Ann. Phys. 256, 192). 24 May, 2017 Four-quark spectroscopy within the hyperspherical formalism 4 Color-Spin basis with well defined C-parity • 1) 2) 3) The difficult part is to construct a symmetrized color-spin basis for the N-body system with well defined C-parity. – Spin part → One make use of the SU(2) Clebsh-gordan coefficients. – Color part → One could be temped to use the SU(3) Clebsh-gordan coefficients, however this is not feasible. To construct the color part we have used a method based on an algorithm by Novoselsky, Katriel and Gilmore (J. Math. Phys. 29, 1368), obtaining states with well defined permutational symmetry, spin and color proyection. Evaluating the quadratic casimir SU(3) operator one can determine the specific representation of each state, picking only those belonging to the SU(3) singlet. Evaluating the charge conjugation operator one can choose only those states with well defined C-parity, c = ±1. 24 May, 2017 Four-quark spectroscopy within the hyperspherical formalism 5 Technical details 6500 4He ccc c 6400 E (MeV) 6300 6200 6100 • Usually unbound systems 6000 0 5 10 15 20 K • Confinement color structure PRC61 (2000) 054001 Conf cc Confine/ Deconfine 24 May, 2017 Four-quark spectroscopy within the hyperspherical formalism 6 Improving convergence We extrapolate the energies using E K E K K where E(K=∞), α, and β are fitted 24 May, 2017 (K0,Kf ) E(K=∞) 0−+ (3,21) 7018 (5,21) 7007 (7,21) 7004 (9,21) 7003 (11,21) 7000 (13,21) 7000 (15,21) 6993 (17,21) 6990 Four-quark spectroscopy within the hyperspherical formalism [MeV] 7 Comparison with other approaches Bhaduri Model (ccnn) S=1 I=0 L=0 4120 • HH expansion (This work) •Variational [FBS 35 (2004) 175] • HO basis [Z. Phys. C 57 (1993) 273] 4080 EPJA19 (2004) 383. (ccnn) (S,I) Variational (li=0) HH (li=0) (0,1) 4155 4154 (1,0) 3927 3926 (1,1) 4176 4175 (2,1) 4195 4193 E (MeV) 4040 4000 3960 3920 3880 0 5 10 15 20 25 K 24 May, 2017 Four-quark spectroscopy within the hyperspherical formalism 8 Constituent quark model • This model has been applied to the description of the barion-barion interaction and the meson and baryon spectra. • Confinement: Unquenched lattice QCD 1 r Coulomb 1 e c c c c r One Pion Exchange Color magnetic • One gluon exchange: c c Tensor Sigma One and spin - orbitExchange One Eta Exchange • Goldstone Boson exchange: One Kaon Exchange 24 May, 2017 Four-quark spectroscopy within the hyperspherical formalism 9 Multiquark spectrosopy 4600 4400 Since we are going to study cccc systems we need to describe properly the spectra of the charmonium 4200 E (MeV) 4000 3800 3600 3400 3200 3000 2800 24 May, 2017 c0-+ J/(1--) c(0++) c(1++) c(2++) Four-quark spectroscopy within the hyperspherical formalism hc(1+-) (2--) 10 Two-meson thresholds Bound M (ccc c ) M 1 (cc ) M 2 (cc ) Unbound M (ccc c ) M1 (cc ) M 2 (cc ) cc ccc c cc • Quantum number conservation 24 May, 2017 Four-quark spectroscopy within the hyperspherical formalism 11 L=0 Four-body results 7400 Exotic Exotic Non-exotic 7100 E (MeV) 6800 6500 6200 5900 0++ 24 May, 2017 0-+ 1++ 1+-+- 1---- ++ 2++ 22-+-+ 22---- Four-quark spectroscopy within the hyperspherical formalism 00+-+- 00---- 11-+-+ 22+-+- 12 Four-quark state Vs. Quark mass. 200 200 P-wave thresholds S-wave thresholds 100 2+− MeV (MeV) 100 1+− 0 0+− 0 2++ -100 -100 0 1000 2000 3000 0 1000 2000 3000 mq (MeV) mq (MeV) E E(ccc c ) T (M1M 2 ) 24 May, 2017 Four-quark spectroscopy within the hyperspherical formalism 13 • Estimation in the light sector (Work in progress). 2++ (I=2) 1−+ (I=2) M (MeV) ≈ 1500 2900 PDG (Exp) X(1600) 1600 ± 100 (I=1) 1(1400) 1376 ± 17 (I=1) 1(1600) 1653 ± 17 • Only four set of quantum numbers seems promising to be detected mq < 1 GeV mq ≈1.7 GeV mq > 3 GeV 2++ Bound Bound Bound 0+− Small Width Almost bound Bound 1+− Unbound Almost bound Bound 2+− Unbound Small Width Small Width 24 May, 2017 Four-quark spectroscopy within the hyperspherical formalism 14 Summary and outlook • We have generalized a numerical technique widely used in nuclear physics to study the two-quark two-antiquark systems. • We have applied it to the L=0 cccc multiquarks, obtaining that only three(four) states are good candidates to be observed, the 2++, the 0+−, the 1+−, and maybe the 2+−. • Improve the method to: • Include isospin degree of freedom. • Consider non identical quarks. • We want to apply the method to • L different from 0. • Heavy-light tetraquarks. QQq q • Hidden-charm/bottom tetraquarks. QqQ q • General heavy tetraquarks. Q1qQ2q • Light sector (scalar, exotics….) qqq q 24 May, 2017 Four-quark spectroscopy within the hyperspherical formalism 15 24 May, 2017 Four-quark spectroscopy within the hyperspherical formalism 16 Two-body spectroscopy 11200 2000 More than 110 states reproduced 11000 1800 10800 1600 10600 1400 E (MeV) E (MeV) 10400 1200 10200 1000 10000 800 9800 600 Light I=1 9600 400 Bottomonium 9400 200 9200 0 24 May, 2017 --(1 ) -b+(0-+) + 0 2 ++ -b0--(0++) b1--(1++) b2(2 (2 +- ) ++ ) b1(1 ) a2(2 ) (1 ) 3 Four-quark spectroscopy within the hyperspherical formalism a1(1++) 17