Systematic Study of Multi-quark states with group theory method

1. Systematic Study of Multi-quark states with group theory method Fan Wang Dept. of Phys., Nanjing Univ. J.L. Ping and H.X. Huang Dept. of Phys., Nanjing Normal Univ.

2. Outline I. Introduction. II. Fractional parentage expansion coefficients of symmetry bases and transformation coefficients between physical bases and symmetry bases. III. An example, penta-quark calculation; six quark system had been calculated in a similar manner.

3. I. Introduction Hadron spectroscopy only detects QCD interaction in color singlet states. Hadron interaction provides hidden color channel information of QCD interaction in principle. Multi-quark states detect QCD interaction in hidden color channel directly. Hidden color channel includes new physics which is hard to be studied with the hadron degree of freedom if it is not impossible.

6. QCD quark benzene QCD interaction should be able to form a quark benzene consisted of six quarks

7. Why multi-quark is still interested Penta quark might be died, but unquenched quark model has been born where one has the penta quark components within a baryon. Meson-baryon, baryon-baryon scatterings are there, its five, six quark systems. The multi quark states search will be continued, such as four quark states are hot now instead of penta quark.

8. Lattice QCD results of the quark interaction PRL 86(2001)18,90(2003)182001,hep-lat/0407001

9. However lattice QCD seems to be impossible To provide the transition interaction between colorless channel and hidden color channel right now and this interaction is essential for mixing hidden color channels to colorless ones. So we could not but make model assumption and what is our quark delocalization color screening model (QDCSM) did.

10. To study multi-quark states one meets multi channel coupling with many body interaction, so one needs a powerful method to deal with. Group theory method is a power one. The fractional parentage expansion method reduces the matrix elements calculation of a many body Hamiltonian to be two body matrix elements, if only two body interaction is included, and overlap calculations.

11. II.Fractional parentage expansion coefficientsof symmetry bases and transformation coefficients between physical bases and symmetry bases To use the FPE method, the many body states must be the group chain classified states, the symmetry bases (SB). And the corresponding FPE coefficients of SB should be calculated and can be calculated by group theory method; The physical bases (PB) are usually not the SB and should be transformed to SB, the transformation coefficients should be calculated and can be calculated by group theory method.

12. What does systematic mean Physics input is included in the Hamiltonian. Equipped with the FPEC and TC, different physics can be treated with the same set of FPEC and TC. In this sense one has a systematic method to do quark model calculation with non- relativistic and even relativistic quark models. F.Wang, J.L.Ping, T.Goldman, Phys.Rev.C51,1648,(1995).

13. Flow Chart physical bases with TC symmetry bases with FPEC Hamiltonian matrix elements in symmetry bases with TC Hamiltonian matrix elements in physical bases diagonalization the Hamiltonian in physical space stored the TC and FPC computer programized

14. Hard job has been done A new group theory method for calculating the FPEC and TC had been developed in the end of 1970�s and the beginning of 1980�s. J.Q.Chen, J.L.Ping and F. Wang, Group Representation Theory for physicists, (World Scientific, Singapore, 2002). Comprehensive FPEC had been calculated and published. J.Q.Chen et al.,Tables of the Clebsch-Gordan, Racah and Subduction Coefficients of SU(n) Groups (World Sci., Singapore, 1987); Tables of the SU(mn) SU(m)xSU(n) Coefficients of Fractional Parentage (World Sci., Singapore, 1991).

15. III.An example, penta quark calculation Four quark calculation, only SU(2) and SU(3) CGC is needed. The transformation coefficients between symmetry bases and physical bases are simple. For systems with 5 quarks and more one need the full machine of FPE and T methods.

16. Physical bases Jaffe-Wilczek model states will be taken as the physical bases in this discussion, the baryon-meson model states has been taken as the physical bases as well, a different transformation coefficients between this new physical states and symmetry bases has been calculated too. If the space is large enough the results are the same

18. di-quark states

19. physical sates

20. Symmetry bases

21. Transformation

22. FP expansion 4->2+2

23. 4->3+1

24. A sample table

30. Summary I. We have developed a powerful group theory method for multi-quark studies. 2. In general, penta-quark resonances are possible due to hidden color channels coupling. 3. For quark models, which fit the NN experimental data, the parity of ground state of penta-quark is negative. The lowest resonance is around 1.8 Gev. The SU(3) flavor symmetry is broken by large s quark mass.

31. 4. QDCSM and chiral quark models both fit the NN experimental data, they give similar penta-quark spectrum. This shows that the smeson in the meson exchange model can be replaced by QDCS mechanism. The spectrum of chiral soliton model is different from QDCSM and chiral qurak model ones, where the SU(3) flavor symmetry is broken by large s quark mass.

32. Thanks