Structures of (ΩΩ) 0 + and (ΞΩ) 1 + in Extended Chiral SU(3) Quark Model

Commun. Theor. Phys. (Beijing, China) 40 (003) pp. 33 336 c International Academic Publishers Vol. 40, No. 3, September 15, 003 Structures of (ΩΩ) 0 + and (ΞΩ) 1 + in Extended Chiral SU(3) Quark Model ZHANG Zong-Ye, 1 YU You-Wen, 1 and DAI Lian-Rong 1 Institute of High Energy Physics, the Chinese Academy of Sciences, Beijing 100039, China Department of Physics, Liaoning Normal University, Dalian 11609, China (Received March 7, 003) Abstract The structures of (ΩΩ) 0 + and (ΞΩ) 1 + are studied in the extended chiral SU(3) quark model in which vector meson exchanges are included. The effect from the vector meson fields is very similar to that from the one-gluon exchange (OGE) interaction. Both in the chiral SU(3) quark model and in the extended chiral SU(3) quark model, di-omega (ΩΩ) 0 + is always deeply bound, with over one hundred MeV binding energy, and (ΞΩ) 1 + s binding energy is around 0 MeV. An analysis shows that the quark exchange effect plays a very important role for making di-omega (ΩΩ) 0 + deeply bound. PACS numbers: 14.0.Pt, 1.40.Qq, 11.30.Rd Key words: dibaryon, quark model, chiral symmetry 1 Introduction In Refs. [1] [3], using the chiral SU(3) quark model [4,5] we predicted that di-omega (ΩΩ) 0 + is the most interesting dibaryon candidate. It has very large binding energy with quite small root-mean-square radius. To discuss why (ΩΩ) 0 + has so large binding energy and to discuss whether this prediction is very sensitive to the model details, an analysis was also given in Ref. []. It shows that the reason for (ΩΩ) 0 + being deeply bound is that its symmetry property is very special. According to the group classification theory, in all the physical bases of two-baryon s systems, only six of them have the largest component of partition [6] orb in the six-quark orbital space, [6,7] which are ( ) ST =30, ( ) ST =03, ( Σ ) ST =3 1, ( Σ ) ST =0 5, (Ξ Ω) ST =0 1, and (ΩΩ) ST =00. This means that these six states are favorable for forming partition of [6] orb in the orbital space, or in other words, their symmetry property causes a deeply bound state being easily formed. But these six states are all made of two decuplet baryons, and, Σ, Ξ all have very short life time through strong decay besides Ω. Fortunately, there is still one of them that is not allowed to occur in strong decay, which is (ΩΩ) ST =00. Thus (ΩΩ) ST =00 has enough long life time (around 10 10 s) for detecting. On the other hand, since the symmetry property is independent of the model interactions, in this sense the binding energy of (ΩΩ) ST =00 should not be very sensitive to the model details, i.e., when the model interactions between two quarks can reasonably reproduce the experimental data of two baryon systems, including deuteron s binding energy, nucleon-nucleon (N N) scattering phase shifts and hyperon-nucleon (Y N) reaction cross sections, the qualitative property of the binding energy of (ΩΩ) ST =00 would not be very different for these different models. This investigation has been made in Ref. [] by examining the chiral field effects on the structure of (ΩΩ) ST =00, in which three different models, i.e., chiral SU() quark model, σ 0 + pseudo meson exchanges model, and chiral SU(3) quark models are considered, and the results show that the binding energy of (ΩΩ) ST =00 is always several tens MeV to one hundred MeV in these three different models. Recently we extended our chiral SU(3) quark model to include vector chiral fields, [8] and studied the deuteron structure and N N scattering phase shifts. In this extended chiral SU(3) quark model, instead of the one-gluon exchange interaction, the vector meson exchanges play the dominant role in the short range part of the quark-quark interactions. Since dibaryon systems have small sizes, the short range behavior of the interaction must be important for the dibaryon structures. Thus studying the structures of diomega (ΩΩ) ST =00 and some other two-baryon systems, such as (ΞΩ) 1 +, (NΩ) +, and so on, [1,3] in this extended chiral SU(3) quark model to see the effect from vector meson exchanges is very significant. In this paper, we will report the calculating results of the binding energies of (ΩΩ) 0 + and (ΞΩ) 1 +. The paper is arranged as follows. A brief introduction of the extended chiral SU(3) quark model is shown in Sec., and then the results of the binding energies of (ΩΩ) 0 + and (ΞΩ) 1 + in the extended chiral SU(3) quark model and discussions are displayed in Sec. 3. Finally a summary is made in Sec. 4. Brief Introduction of Extended Chiral SU(3) Quark Model In the extended chiral SU(3) quark model, besides the nonet pseudo-scalar meson fields and the nonet scalar me- The project supported by National Natural Science Foundation of China

No. 3 Structures of (ΩΩ) 0 + and (ΞΩ) 1 + in Extended Chiral SU(3) Quark Model 333 son fields, the coupling between vector meson fields and quarks is also considered. With this generalization, now the Hamiltonian of the system is written as and H = i V ij = V conf ij T i T G + i<j V ij, (1) + V OGE ij + V ch ij, () where T i T G is the kinetic energy of the system, and i V ij includes all the interactions between two quarks. Vij conf is the confinement potential, taken as quadratic form, Vij OGE is the one-gluon exchange (OGE) interaction, and Vij ch represents the interactions from chiral field coupling, which in the extended chiral SU(3) quark model includes scalar meson exchange Vij s, pseudo-scalar meson exchange V ps v ij, and vector meson exchange Vij potentials, Vij ch = V sa ( r ij ) + V psa ( r ij ) + V va ( r ij ). (3) Their expressions are given as follows: V sa ( r ij ) = C(g ch, m sa, Λ c ) X 1 (m sa, Λ c, r ij ) (λ a (i)λ a (j)) + V l s s a ( r ij ), (4) m ps a V psa ( r ij ) = C(g ch, m psa, Λ c ) 1m qi m qj and X (m psa, Λ c, r ij )( σ i σ j ) + V tensor ps a ( r ij ), (5) V va ( r ij ) = C(g chv, m va, Λ c ) X 1 (m va, Λ c, r ij ) (λ a (i)λ a (j)) where m v a + C(g chv, m va, Λ c ) 6m qi m qj ( 1 + f chv m qi + m qj + f chv m qi m ) qj g chv M p gchv Mp X (m va, Λ c, r ij ) ( σ i σ j ) (λ a (i)λ a (j)) + V l s v a ( r ij ) + Vv tensor a ( r ij ), (6) C(g, m, Λ) = g Λ m 4π Λ m, (7) X 1 (m, Λ, r) = Y (mr) Λ Y (Λr), (8) m ( Λ ) 3Y X (m, Λ, r) = Y (mr) (Λr), (9) m Y (x) = 1 x e x, (10) and M p is a mass scale, taken as proton mass. The detailed formula expressions can be found in Ref. [8]. Table 1 Model parameters and the corresponding binding energies of deuteron, (ΩΩ) 0 + and (ΞΩ) 1 +. Here m π = 138 MeV, m K = 495 MeV, m η = 548 MeV, m η = 958 MeV, m σ = m ɛ = 980 MeV, m ρ = 770 MeV, m K = 89 MeV, m ω = 78 MeV, m φ = 100 MeV, and Λ = 1100 MeV. SU(3) model Group I Group II m u(d) (MeV) 313 313 313 m s (MeV) 470 470 470 b u(d) (fm) 0.50 0.45 0.45 m σ (MeV) 630 570 585 m κ (MeV) 1430 (980) 1430 (980) 1430 (980) g ch.73.73.73 g chv.35 1.9 f chv /g chv 0 /3 g u(d) (OGE) 0.881 0.77 0.411 g u(d) (OGE) 0.776 0.076 0.169 g s (OGE) 0.764 (0.941) 0.070 (0.549) 0.95 (0.619) g s (OGE) 0.584 (0.885) 0.005 (0.301) 0.087 (0.383) B deu (MeV).8.13.9 B (ΩΩ)0 (MeV) 175.5 (137.4) 140.6 (108.1) 165.6 (19.4) + R ΩΩ (fm) 0.73 (0.80) 0.78 (0.86) 0.74 (0.8) B (ΞΩ)1 + (MeV) 31.6 (17.7) 14.8 (7.3) 0.6 (10.8) R ΞΩ (fm) 1.18 (1.45) 1.57 (1.91) 1.41 (1.73)

334 ZHANG Zong-Ye, YU You-Wen, and DAI Lian-Rong Vol. 40 About the parameters, g ch is the coupling constant for scalar and pseudo-scalar chiral field coupling, which is determined according to the following relation, g NNπ 4π = 9 5 m u MN g ch 4π, (11) and g NNπ/4π is taken to be the experimental value, which is about 14. g chv and f chv are the coupling constants for vector coupling and tensor coupling of the vector meson field, respectively. In principle, these two values can also be determined according to the values of NNρ or NNω coupling constants and the SU(3) relations between quark and baryon levels. Since the data of the coupling constants between nucleon and vector mesons g NNρ and f NNρ or g NNω and f NNω are usually taken from the one-boson exchange models of N N interactions and for different models these coupling constants are different, therefore there are some uncertainties for g chv and f chv. Here we take them as the same values that we used in the study of NN phase shift calculation. [8] The mesons masses m ps, m s, and m v are taken to be the experimental values, while the mass of σ meson (m σ ) is treated as an adjustable parameter. According to the vacuum spontaneously breaking theory, its value should satisfy the following relation, [9] m σ = (m u ) + m π. (1) This means that it can be regarded as almost reasonable when the value of m σ is located in the range of 550 MeV 650 MeV. Here we use the same value that is determined in the N N scattering calculation by fitting the binding energy of deuteron. For the mass of κ, since its experimental value is not fixed uniquely, we take two different values, 980 MeV and 1430 MeV, to see its influence on the binding energies of (ΩΩ) 0 + and (ΞΩ) 1 +. Λ is the cut-off mass, which indicates the chiral symmetry breaking scale. [10,11] All the parameters we used are listed in Table 1 with the corresponding results. 3 Results of Binding Energies of (ΩΩ) 0 + and (ΞΩ) 1 + and Discussions To study the binding energy of two-baryon system on quark level dynamically, we solve the Resonating Group Method (RGM) equation of the Hamiltonian Eqs. () (10). In the RGM calculation, the trial wave function is taken to be Ψ ST = c i Ψ (i) ST ( s i) (13) i with Ψ (i) ST ( s i) = A(φ A ( ξ 1, ξ ) SA T A φ B ( ξ 4, ξ 5 ) SB T B χ( R AB s i ) R CM ( R CM )) ST, (14) where A and B describe two (3q) clusters, and φ, χ, and R represent internal, relative, and center of mass motion wave functions, respectively. s i is the generator coordinate and A is the anti-symmetrization operator, A = 1 P ij, (15) i A, j B where P ij is the permutation operator of the i-th and the j-th quarks. The calculated binding energies of (ΩΩ) 0 + and (ΞΩ) 1 + in the extended chiral SU(3) quark model are listed in Table 1. For comparison, the results of chiral SU(3) quark model (i.e. without vector meson exchanges) [1 3] are also given in the first column of Table 1. The results of the extended chiral SU(3) quark model for two different groups of parameters are shown in another two columns, where the second column is for group I, i.e. without tensor coupling of the vector mesons, f chv /g chv = 0, and the third one for group II with f chv /g chv = /3. As mentioned above, we adjust the mass of σ meson, m σ to fit the binding energy of deuteron. From Table 1, one can see that when the binding energy of deuteron B deu is fitted, m σ is located in the reasonable region 550 MeV 650 MeV for all these three cases. From Table 1, we can see the following points. (i) When the vector meson field coupling is considered, the coupling constant of OGE is largely reduced. The coupling constant of OGE between u(d) quarks, g u, is determined by fitting the mass differences between and N. For both the parameter groups I and II, gu < 0., which is much smaller than the value (0.78) of chiral SU(3) quark model. The coupling constant of OGE between s quarks, g s, is determined by fitting the mass difference between Σ and Λ. Therefore the value of g s is dependent on the mass of κ meson. When m κ is taken to be 1430 MeV, gs < 0.1 for both the parameter groups I and II. If m κ is chosen to be 980 MeV, gs is a little bit larger, which is about 0.3 0.4 (See Table 1). All of them are smaller than those of the chiral SU(3) quark model. This means that in the extended chiral SU(3) quark model, the OGE interaction is quite weak. Instead of the OGE, the vector meson exchanges play a dominant role for the short range interaction between two quarks. So the mechanisms of the quarkquark short range interactions of these two models are totally different. In the chiral SU(3) quark model, it is from OGE, while in the extended chiral SU(3) quark model, it is mainly from vector meson exchanges. (ii) The binding energies of (ΩΩ) 0 + and (ΞΩ) 1 + in the extended chiral SU(3) quark model are quite similar to those of the chiral SU(3) quark model, i.e., (ΩΩ) 0 + is still deeply bound with its binding energy always being larger than 100 MeV, and (ΞΩ) 1 + is always a bound state also, whose binding energy is in the region 8 MeV 0 MeV. These results tell us that when the deuteron s binding energy is fitted, no

No. 3 Structures of (ΩΩ) 0 + and (ΞΩ) 1 + in Extended Chiral SU(3) Quark Model 335 matter whether OGE and/or vector meson exchange control the quark-quark short range interactions, the main properties of (ΩΩ) 0 + and (ΞΩ) 1 + are not changed. (iii) From the binding energy calculations, one can also see that the results are not very sensitive to f chv /g chv. In the case of group I, there is no tensor coupling for the vector meson exchanges, i.e. f chv /g chv = 0, the binding energies of (ΩΩ) 0 + and (ΞΩ) 1 + are a little bit smaller than those of the case of group II. In other words, the tensor coupling of the vector chiral field makes the binding energies of (ΩΩ) 0 + and (ΞΩ) 1 + a little bit larger. (iv) The root mean squared distance between two Ω s of (ΩΩ) 0 +, R ΩΩ, and the root mean squared distance between Ξ and Ω of (ΞΩ) 1 +, RΞΩ, are also calculated. For different models, RΩΩ is always around 0.75 fm 0.85 fm, and RΞΩ is in the region 1. fm 1.9 fm. It shows that the size of (ΩΩ) 0 + is really small, very much similar to a six-quark state. As we discussed in Refs. [] and [3], the quark exchange effect between the two baryons plays a very important role for studying the binding property of the two baryon systems. When the expectation value of the permutation operator of spin flavor color space P σfc ij 0, the quark exchange effect is not important, and when P σfc ij 1/9, the Pauli blocking effect is very strong, and in this case the state is forbidden to form [6] r symmetry in the orbital space. As we pointed out before, there is another important case in which P σfc ij 1/9. This kind of states have the highest symmetry property in the orbital space, and the quark exchange effect makes the two baryon clusters closer. Just because (ΩΩ) 0 + is one of this kind of states, it has a very large binding energy. Here we would like to make a further analysis to see the quark exchange effect on (ΩΩ) 0 + by making a comparison between the results with and without quark exchange effect. In our original calculation in the chiral SU(3) quark model, we did not make this comparison. This is because in that model OGE governs the short range part of the interaction between two quarks, and OGE is a color dependent operator, the direct terms of OGE matrix element between two color singlet baryons are equal to zero, only the exchanged terms exist. When the quark exchange effect is not considered in the calculation, then there would be totally no contribution from OGE. Therefore it is not significant to make this comparison in the chiral SU(3) quark model. But now in the extended chiral SU(3) quark model, instead of the OGE, the vector chiral field coupling contributes the short range interaction between quarks, and the vector chiral field coupling is color-independent, its direct term is dominantly important, thus the differences between the results with and without quark exchange effect are totally from the quark exchange effect, and one can get more information about the quark exchange effect on the structure of two baryons. Table The binding energies of deuteron, (ΩΩ) 0 +, and (ΞΩ) 1 + with and without quark exchange effect. Here m π = 138 MeV, m K = 495 MeV, m η = 548 MeV, m η = 958 MeV, m σ = m ɛ = 980 MeV, m ρ = 770 MeV, m K = 89 MeV, m ω = 78 MeV, m φ = 100 MeV, and Λ = 1100 MeV. Group I with quark exchange without quark exchange (P ij = 0) m u(d) (MeV) 313 313 m s (MeV) 470 470 b u(d) (fm) 0.45 0.45 m σ (MeV) 570 570 (640) m κ (MeV) 1430 1430 g ch.73.73 g chv.35.35 f chv /g chv 0 0 g u(d) g s (OGE) 0.076 0.076 (OGE) 0.005 0.005 B deu (MeV).13 9.7 (.4) B (ΩΩ)0 + B (ΞΩ)1 + (MeV) 140.6 57.8 (34.9) (MeV) 14.8 85.4 (53.9) We choose the case of group I to make this comparison, because in this case the OGE coupling constants both between u(d) quarks and s quarks are almost equal to zero. The results with and without quark exchange effect for

336 ZHANG Zong-Ye, YU You-Wen, and DAI Lian-Rong Vol. 40 group I are listed in Table. From Table, one can see that the quark exchange effect does give repulsion to deuteron and (ΞΩ) 1 +, and give attraction to (ΩΩ) 0 +. When the quark exchange effect is omitted, i.e. P ij = 0, the deuteron binding energy is increased to 9.7 MeV, and the binding energy of (ΩΩ) 0 + goes down by about 80 MeV. If we adjust the mass of σ meson from 570 MeV to 640 MeV to get the correct deuteron binding energy, then the binding energy of (ΩΩ) 0 + becomes 34.9 MeV (see the value in the bracket of Table ). This result convinces us again that the quark exchange effect on (ΩΩ) 0 + is really very important. 4 Conclusions We studied the vector meson exchange effect on two dibaryon systems (ΩΩ) 0 + and (ΞΩ) 1 + on quark level. The results show that (ΩΩ) 0 + is still a deeply bound state, when the short range part of the quark-quark interaction is controlled by the vector meson exchanges, and the binding energy of (ΩΩ) 0 + is about 140 MeV 160 MeV for m κ = 1430 MeV and 110 MeV 130 MeV for m κ = 980 MeV. All of these results are quite similar as those of the chiral SU(3) quark model. We also made a further analysis to discuss the quark exchange effect on the two-baryon system. A comparison between the cases with and without quark exchange effect is made. The binding energy of (ΩΩ) 0 + is reduced by about one hundred MeV, when the quark exchange effect is omitted. Acknowledgments We are in debt to Prof. Shen Peng-Nian for stimulating discussions. References [1] Z.Y. ZHANG, Y.W. YU, and X.Q. YUAN, Nucl. Phys. A670 (000) 178c; YU You-Wen, ZHANG Zong-Ye, and YUAN Xiu-Qing, Commun. Theor. Phys. (Beijing, China) 31 (1999) 1. [] Z.Y. ZHANG, Y.W. YU, C.R. CHING, T.H. HO, and Z.D. LU, Phys. Rev. C61 (000) 06504. [3] Q.B. LI, P.N. SHEN, Z.Y. ZHANG, and Y.W. YU, Nucl. Phys. A683 (001) 487. [4] Z.Y. ZHANG, Y.W. YU, and L.R. DAI, HEP & NP 0 (1996) 363. [5] Z.Y. ZHANG, Y.W. YU, P.N. SHEN, L.R. DAI, et al., Nucl. Phys. A65 (1997) 59. [6] M. Harvey, Nucl. Phys. A35 (1981) 301. [7] F. Wang, J.L. Ping, and T. Goldman, Phys. Rev. C51 (1995) 1648. [8] ZHANG Zong-Ye, YU You-Wen, WANG Ping, and DAI Lian-Rong, Commun. Theor. Phys. (Beijing, China) 39 (003) 569. [9] M.D. Scadron, Phys. Rev. D6 (198) 39. [10] I.T. Obukhovsky and A.M. Kusainov, Phys. Lett. B38 (1990) 14; A.M. Kusainov, V.G. Neudatchin, and I.T. Obukhovsky, Phys. Rev. C44 (1991) 343. [11] A. Buchmann, E. Fernandez, and K. Yazaki, Phys. Lett. B69 (1991) 35; E.M. Henley and G.A. Miller, Phys. Lett. B51 (1991) 453.