1 31 ò 1 9 Ï p U Ô n † Ø Ô n Vol. 31, No. 9 2007 c 9  HIGH ENERGY PHYSICS AND Sep., 2007

Chiral Model Study of ud¯s¯s 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 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 - J P = 0+, 1+, and 2+ are calculated in the chiral SU(3) 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 scattering phase shifts and the -nucleon cross sections. The s-channel annihilation interaction is fixed by the masses of K and K∗ , 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 - 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 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 , 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 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, mω = 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 |1i = {ud}1{s¯s¯}1 P + 0 the same form as V ch. I = 1, J = 0 =⇒   12  6 6¯ |2i = {ud}0{s¯s¯}0 0 Vi¯j in Eq. (1) represents the quark-antiquark in-   3¯ 3 teraction, |3i = [ud]0{s¯s¯}1 P + 1 I = 0, J = 1 =⇒   OGE conf ch ann 6 6¯ Vi¯j = V ¯ +V ¯ +V ¯ +V ¯ , (4)  ij ij ij ij |4i = [ud]1{s¯s¯}0 1  OGE conf  P ¯ where Vi¯j and Vi¯j can be obtained by replacing + 3 3 I = 1, J = 1 =⇒ |5i = {ud}1{s¯s¯}1 ∗ 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+ =⇒ |6i = {ud}3¯{s¯s¯}3 be obtained from the G parity transformation: 1 1 2  ch Gk ch,k Where { } 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) |1i ≡ {ud} {s¯s¯} = 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 Ümð µÃ§Ž.e ud¯s¯s o§ŽïÄ 889

6 6¯ 3¯ 3 |2i ≡ {ud}0{s¯s¯}0 0 = |5i ≡ {ud}1{s¯s¯}1 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 |6i ≡ {ud}3¯{s¯s¯}3 = ((us¯)1(ds¯)1)1 − ¯ 1 1 2 r 1 1 2 |3i ≡ [ud]3{s¯s¯}3 = 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- |4i ≡ [ud]6{s¯s¯}6¯ = 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 . The calcu- 1 1 lated results (without configuration mixing) are given ((us¯)1(ds¯)1)1 − ((us¯)8(ds¯)8)1 + r3 1 1 1 r12 0 1 1 in Table 2. In this table, the first set is for the chiral SU(3) quark model, the second and third sets are for 1 1 ((us¯)8(ds¯)8)1 + ((us¯)8(ds¯)8)1 , (10) r12 1 0 1 r6 1 1 1 the extended chiral SU(3) quark model.

Table 2. Energies (in MeV) of ud¯s¯s six single states in various chiral quark models. χ-SU(3) QM Ex. χ-SU(3) QM Threshold I II III

fchv/gchv = 0 fchv/gchv = 2/3 JP = 0+ 3¯ 3 {ud}1{s¯s¯}1 1744 1683 1681 KK (990)  0 6 6¯ {ud}0{s¯s¯}0 1753 1834 1808 KK (990)  0 JP = 1+ 3¯ 3 ∗ [ud]0{s¯s¯}1 1641 1682 1675 KK (1387)  1 6 6¯ ∗ [ud]1{s¯s¯}0 1722 1784 1770 KK (1387)  1 3¯ 3 ∗ {ud}1{s¯s¯}1 1771 1754 1745 KK (1387)  1 JP = 2+ 3¯ 3 ∗ ∗ {ud}1{s¯s¯}1 1821 1872 1852 K K (1784)  2 In Ref. [8], Cui et al. argued that the strong at- while they are repulsive in the other four configura- tractive color-magnetic interaction could reduce the tions. In the extended chiral SU(3) quark model, energies of the ud¯s¯s systems, and they found a I = 0 the OGE is largely reduced and the color-magnetic and J P = 1+ ud¯s¯s tetraquark state with a mass attractions are almost replaced by ρ exchange. Fur- around 1347MeV. In our chiral SU(3) quark model thermore, both in the chiral SU(3) quark model and calculation, the color-magnetic interactions are at- the extended chiral SU(3) quark model, the σ and P + 3¯ 3 π tractive both in the isovector J = 0 {ud}1{s¯s¯}1 0 exchanges provide more attractive interactions in P + 3¯ 3  P + 3¯ 3 state and the isoscalar J = 1 [ud]0{s¯s¯}1 1 state, the isovector J = 0 {ud}1{s¯s¯}1 0 state and the   890 p U Ô n † Ø Ô n ( HEP & NP ) 1 31 ò

P + 3¯ 3 ∗ ∗ isoscalar J = 1 [ud]0{s¯s¯}1 1 state than in the ration, 1768MeV, is lower than the K K threshold other four configurations. Thus the energies of the (1784MeV), it cannot decay into K∗K∗ final state. 3¯ 3 3¯ 3 {ud}1{s¯s¯}1 0 and [ud]0{s¯s¯}1 1 states are respec- This means that this state will possibly have a nar- tiv ely the lowest one in J P = 0+ and J P = 1+ cases row width, and can be treated as a good candidate in various models. However, due to the high kinetic for the ud¯s¯s tetraquark state. energies, the attractive interactions are not strong enough to reduce the energies of these two states to Table 3. Energies (in MeV) of ud¯s¯s states with configuration mixing considered. be lower than the corresponding meson-meson thresh- χ-SU(3) QM Ex. χ-SU(3) QM olds (see Table 2). I II III Further we consider the configuration mixing be- I = 1,JP = 0+ 1602 1573 1572 tween different states. The results are shown in Ta- 1857 1909 1882 I = 0,JP = 1+ 1577 1623 1618 ble 3. Comparing it with Table 2, one can see that 1768 1833 1817 the configuration mixing can shift the energies over I = 1,JP = 1+ 1771 1754 1745 60MeV for most configurations. And we also find that I = 1,JP = 2+ 1821 1872 1852 in the chiral SU(3) quark model one energy of the I = 0 and J P = 1+ state is 1768MeV, lower than the 4 Summary threshold of K∗K∗, and the corresponding root mean The structures of ud¯s¯s states with J P = 0+, 1+, square radius is about 0.57fm. The wave function of and 2+ are studied in the chiral SU(3) quark model this state is and the extended chiral SU(3) quark model. We cal- |4i0 = − 0.17((us¯)1(ds¯)1)1 +0.17((us¯)1(ds¯)1)1 + 0 1 1 1 0 1 culate the energies of six low-lying ud¯s¯s configurations

1 1 1 8 8 1 using the variational method. The configuration mix- 0.71((us¯)1(ds¯)1)1 −0.47((us¯)0(ds¯)1)1 + ing is considered, and the model parameters are deter- 8 8 1 8 8 1 0.47((us¯)1(ds¯)0)1 +0.0056((us¯)1(ds¯)1)1 , (13) mined by the same method as in our previous work. where it is clear to see that besides 43.8% part of two With the size parameter for mesons taken to be 0.4 color octet qq¯ pairs, the component of two color sin- fm, the ud¯s¯s configuration with I = 0 and J P = 1+ ∗ ∗ glet qq¯ pairs is 50.4% for K∗K∗ and 5.8% for KK∗, is found to lie lower than the K K threshold, and ∗ which means the K∗K∗ component is dominate and furthermore, this state has a very small KK compo- comparatively the KK component is very small, thus nent, thus it can be treated as a possible tetraquark this state has a few possibility decaying into K and candidate. A dynamical calculation would be done in K∗. Furthermore, since the energy of this configu- future work.

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2007 – 03 – 30 Âv * I[g,‰ÆÄ7(10475087)]Ï 1) E-mail: [email protected]