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Chiral Quark Model Study of Ud¯S¯S Tetraquark States*

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Chiral Quark Model Study of Ud¯S¯S Tetraquark States* 1 31 ò 1 9 Ï p U Ô n Ø Ô n Vol. 31, No. 9 2007 c 9 HIGH ENERGY PHYSICS AND NUCLEAR PHYSICS Sep., 2007 Chiral Quark Model Study of ud¯s¯s Tetraquark States * ZHANG Zong-Ye1 WANG Wen-Ling2;1) HUANG Fei3 YU You-Wen1 LIU Feng2 1 (Institute of High Energy Physics, CAS, Beijing 100049, China ) 2 (Institute of Particle Physics, Huazhong Normal University, Wuhan 430079, China) 3 (CCAST (World Laboratory), Beijing 100080, China) Abstract The energies of the low-lying isoscalar and isovector ud¯s¯s configurations with spin-parity J P = 0+, 1+, and 2+ are calculated in the chiral SU(3) quark model and the extended chiral SU(3) quark model by using the variational method. The model parameters are determined by the same method as in our previous work, and they still can satisfactorily describes the nucleon-nucleon scattering phase shifts and the hyperon-nucleon cross sections. The s-channel annihilation interaction is fixed by the masses of K and K∗ mesons, and the configuration mixing is considered. The results show that the ud¯s¯s configuration with I = 0 and J P = 1+ lies ∗ ∗ ∗ lower than the K K threshold, and furthermore, this state has a very small KK component, thus it can be treated as a possible tetraquark candidate. Key words tetraquark state, quark model, chiral symmetry 1 Introduction In the past few years, the chiral SU(3) quark model and the extended chiral quark model have proven to be quite successful in reproducing the bind- Since Jaffe predicted the H particle (uuddss) in ing energy of deuteron, the nucleon-nucleon and kaon- 1977[1], the research on multi-quark states has al- nucleon scattering phase shifts, and the nucleon- ways been an attractive topic for nearly three decades hyperon cross sections[10, 11]. In this paper, we use in both theoretical and experimental studies. But these two models to study the structures of the ud¯s¯s up to now, there has been no convinced evidence configurations. The model parameters are determined of their existence in experiments. The Θ particle, by the same method as in our previous work[15|18]. first reported by LEPS Collaboration in 2003[2], has The s-channel quark-antiquark annihilation interac- motivated amounts of theoretical and experimental tion is fixed by the masses of K and K∗ mesons, and studies for pentaquarks and further the multi-quark the configuration mixing is considered. The results states. Nevertheless its existence is still questioned show that the ud¯s¯s configuration with I = 0 and till now. J P = 1+ lies lower than the K∗K∗ threshold, and fur- Besides dibaryon and pentaquark, the possible thermore, this state has a very small KK∗ component, ud¯s¯s tetraquark is another interesting multi-quark thus it can be treated as a possible tetraquark candi- system, and much work has been devoted to this is- date. sue in the past few years[3|9]. Since the studies of the possible ud¯s¯s tetraquark state are presently model 2 Formulation dependent, it seems important and necessary to in- vestigate this state via different approaches. The chiral SU(3) quark model and the extended Received 30 March 2007 * Supported by National Nature Science Foundation of China (10475087) 1) E-mail: [email protected] 887 | 891 888 p U Ô n Ø Ô n ( HEP & NP ) 1 31 ò chiral SU(3) quark model have been widely described SU(3) quark model by taking fchv=gchv as 0 and 2=3, [12|15] in the literatures , and the details can be found respectively. Here fchv is the coupling constant for in these references. Here we just give the salient fea- tensor coupling of the vector meson fields. tures of these two models. Table 1. Model parameters. The meson masses m 0 m In these two models, the total Hamiltonian of the and the cutoff masses: σ = 980MeV, κ = 980MeV, m = 980MeV, mπ = 138MeV, m = ud¯s¯s systems can be written as K 495MeV, mη = 549MeV, mη0 = 957MeV, mρ = 770MeV, mK∗ = 892MeV, m! = H = T −T +V +V¯¯ + V ¯ ; (1) i G 12 34 ij m Λ Xi iX=1;2 782MeV, φ = 1020MeV, and = 1100MeV. j=3;4 χ-SU(3) QM Ex. χ-SU(3) QM where TG is the kinetic energy operator for the center- I II III of-mass motion, and V12, V3¯4¯ and Vi¯j represent the qq, fchv=gchv = 0 fchv=gchv = 2=3 q¯q¯ and qq¯ interactions, respectively, bu/fm 0.5 0.45 0.45 mu/MeV 313 313 313 OGE conf ch V12 = V12 +V12 +V12 ; (2) ms/MeV 470 470 470 g2 0.766 0.056 0.132 OGE u where V12 is the OGE interaction, the confine- 2 gs 0.846 0.203 0.250 conf ment potential V12 is taken as the linear form, and gch 2.621 2.621 2.621 ch gchv 2.351 1.973 V12 represents the effective quark-quark potential in- mσ/MeV 595 535 547 duced by the quark-chiral field coupling. The de- c auu/(MeV/fm) 87.5 75.3 66.2 tailed expressions of these potentials can be found c aus/(MeV/fm) 100.8 123.0 106.9 c in Refs. [12|15]. ass/(MeV/fm) 152.2 226.0 196.7 c0 auu/MeV −77.4 −99.3 −86.6 V3¯4¯ in Eq. (1) represents the antiquark-antiquark c0 aus/MeV −72.9 −127.9 −109.6 interaction, c0 ass /MeV −83.3 −174.20 −148.7 OGE conf ch V3¯4¯ = V3¯4¯ +V3¯4¯ +V3¯4¯ ; (3) The S-wave ud¯s¯s wave functions can be written OGE conf as: where V3¯4¯ and V3¯4¯ can be obtained by replacing ∗ ∗ c • c c • c OGE conf ch 3¯ 3 λ1 λ2 with λ3¯ λ4¯ in V12 and V12 , and V3¯4¯ has j1i = fudg1fs¯s¯g1 P + 0 the same form as V ch. I = 1; J = 0 =) 8 12 < 6 6¯ j2i = fudg0fs¯s¯g0 0 Vi¯j in Eq. (1) represents the quark-antiquark in- : 3¯ 3 teraction, j3i = [ud]0fs¯s¯g1 P + 1 I = 0; J = 1 =) 8 OGE conf ch ann 6 6¯ Vi¯j = V ¯ +V ¯ +V ¯ +V ¯ ; (4) < ij ij ij ij j4i = [ud]1fs¯s¯g0 1 OGE conf : P ¯ where Vi¯j and Vi¯j can be obtained by replacing + 3 3 I = 1; J = 1 =) j5i = fudg1fs¯s¯g1 ∗ 1 c • c c • c OGE conf ch λ1 λ2 with −λi λ¯j in V12 and V12 , and Vi¯j can I = 1; J P = 2+ =) j6i = fudg3¯fs¯s¯g3 be obtained from the G parity transformation: 1 1 2 ch Gk ch;k Where f g and [ ] represent the flavor symmetry and Vi¯j = (−1) Vij ; (5) Xk the antisymmetry, respectively. The superscript is with (−1)Gk being the G parity of the kth meson. the representation of the color SU(3) group, and the ann Vi¯j denotes the s-channel quark-antiquark annihi- subscript is the spin quantum number. Making a re- lation interaction. For the ud¯s¯s system, u(d)¯s can coupling calculation, we can express these wave func- ∗ only annihilate into K and K mesons, tions as: ∗ ann K K 3¯ 3 V ¯ = V +V : (6) j1i ≡ fudg fs¯s¯g = ij ann ann 1 1 0 [13] Their expressions can be found in the literature . 1 1 ((us¯)1(ds¯)1)1 − ((us¯)1(ds¯)1)1 − All the model parameters are tabulated in Table 1, 2 0 0 0 r12 1 1 0 where the first set is for the chiral SU(3) quark model, 1 1 ((us¯)8(ds¯)8)1 + ((us¯)8(ds¯)8)1 ; (7) the second and third sets are for the extended chiral r2 0 0 0 r6 1 1 0 1 9 Ï ÜmðµÃ§.e ud¯s¯s o§ïÄ 889 6 6¯ 3¯ 3 j2i ≡ fudg0fs¯s¯g0 0 = j5i ≡ fudg1fs¯s¯g1 1 = 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ((us¯) (ds¯) ) + ((us¯) (ds¯) ) + ((us¯)0(ds¯)1)1 + ((us¯)1(ds¯)0)1 − r6 0 0 0 r2 1 1 0 r6 r6 1 8 8 1 1 8 8 1 1 1 1 1 ((us¯) (ds¯) ) − ((us¯) (ds¯) ) ; (11) ((us¯)8(ds¯)8) + ((us¯)8(ds¯)8) ; (8) r3 0 1 1 r3 1 0 1 r12 0 0 0 2 1 1 0 1 j6i ≡ fudg3¯fs¯s¯g3 = ((us¯)1(ds¯)1)1 − ¯ 1 1 2 r 1 1 2 j3i ≡ [ud]3fs¯s¯g3 = 3 0 1 1 2 8 8 1 1 1 ((us¯)1(ds¯)1)2 ; (12) ((us¯)1(ds¯)1)1 − ((us¯)1(ds¯)1)1 + r3 r12 0 1 1 r12 1 0 1 where (us¯)(ds¯) represents 1=2[(us¯)(ds¯)+(ds¯)(us¯))] 1 1 p ((us¯)1(ds¯)1)1 − ((us¯)8(ds¯)8)1 + for Eqs. (7), (8), (11) and (12), and denotes r6 1 1 1 r6 0 1 1 1=2[(us¯)(ds¯)−(ds¯)(us¯)] for Eqs. (9) and (10). 1 1 p ((us¯)8(ds¯)8)1 − ((us¯)8(ds¯)8)1 ; (9) r6 1 0 1 r3 1 1 1 3 Results and discussions We calculate the energies for six low configura- j4i ≡ [ud]6fs¯s¯g6¯ = 1 0 1 tions of ud¯s¯s system in the chiral quark models. The ∗ 1 1 1 1 1 1 1 1 size parameter for K and K is taken to be 0:4fm, − ((us¯)0(ds¯)1) + ((us¯)1(ds¯)0) + r6 1 r6 1 which is smaller than that for baryons.
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