1. Tetraquark states in Quark Model Jialun Ping Youchang Yang, Yulan Wang, Yujia Zai Nanjing Normal University 中高能核物理大会 November 5-7, 2009, Hefei

2. Outline I. Introduction II. Quark Models and calculation method III. Results IV. Summary and outlook

3. I. Introduction Since 2003, a lot of work focus on mesons with charm quark(s) These states are difficult to be understood as conventional mesons. Explanation: exotic four-quark states, hybrid states with gluonic degrees of freedom, molecules Our goal: looking for tetraquark states

4. Summary of the Charmonium-like XY Z states

5. Charmonium-like states: Z + (4430), Z 1 + (4050), Z 2 + (4250) Belle observed, but BaBar finds no conclusive evidence in their data for the Z + (4430) minimum quark contents: ccud

6. Isospin symmetry breaks? Other charmonium states: minimum quark contents: cc no isospin partner? Annihilation interactions play an important role. 2+4 mixing needed. methods: OGE: q  qqq 3 P 0 calculations are going on.

7. The tetraquark states: QQnn Q=b, c, s, n=u, d No annihilation The minimum quark contents: four quarks Many proposals to explore the states experimentally have been put forward. Boris A. et al, Phys. Lett. B 551,296 (2003). A. Del Fabbro, et al., Phys. Rev. D71, 014008 (2005). D. Janc, et al., Few-Body Systems 35, 175 (2004) ……

8. References J. Carlson, et al., Phys. Rev. D 37, 744(1988) A. V. Manohar, M. B. Wise, Nucl. Phys. B 399, 17(1993). B. Silvestre-Brac and C. Semay, Z. Phys. C 57, 273-282 (1993); 59, 457-470 (1993); 61, 271-275 (1994). S. Pepin, et al., Phys.Lett. B 393, 119 (1997). D. M. Brink, et al., Phys. Rev. D 49, 4665; 57, 6778(1998). D. Janc, M. Rosina, Few-Body Systems 35, 175-196(2004). J. Vijande, et al., Eur. Phys. J. A19, 383-389 (2004); PRD79,074010 (2009) A.Del Fabbro, et al., Phys. Rev. D71, 014008 (2005). bbnn is bound state, ccnn uncertain.

9. Tetraquark states In quark models, Two configurations are used: diquark-antidiquark: qq-qq dimeson and hidden color channels: qq-qq Completeness? All the excited states are included  completeness Over-completeness? configurations mixing, low-lying states are included, calculation tractable  over-completeness Orthogonalization: Eigenfunction method

10. Quark Models Bhaduri, Cohler, and Nogami (BCN) quark model Advantages: simple, powerful Applied to conventional meson, baryon and four-quark system range from light quarks u, d to b with same set of parameters.

11. Chiral quark model

12. Quark delocalization color screening model Hamiltonian is same as ChQM replace σ-meson exchange, introduce color screening

13. Calculation method Gaussian expansion method: high precision numerical method for few body system. E. Hiyama, et al., Prog. Part. Nucl. Phys. 51 223 (2003). Wavefunction:

14. Relative motion coordinates

15. Color, spin, flavor wavefunctions Color Spin Flavor set (a) set (b)

16. Total wavefunctions set (a) Set (b)

17. Variational principle Binding energy

18. Model parameters

19. mesons

20. S-wave QQnn Systematic calculations Diquark-antidiquark configuration Dimeson configuration Configuration mixing

25. Over-completeness Orthogonalization: Eigenfunction method construct the overlap matrix of all the bases, diagonalize the overlap matrix, delete the eigenfunctions with eigenvalue zero, use the remain eigenfunctions to construct the hamiltonian matrix and diagonalize it to obtain the eigenenergies.

26. Configuration mixing

27. QQQQ, QQQn, Qnnn states No bound state is found. Annihilation interactions are not taken into account. The existence of open charm states imply that the annihilation interactions are important.

28. Summary and outlook A systematic calculation of tetraquark states in quark models is done. For QQQQ,QQQn,Qnnn, no bound state is found. (without annihilation interactions) For QQnn, bbnn with (I,J)=(0,1) is always bound in the quark models used. ccnn with (I,J)=(0,1) is bound state with smaller binding energy, ssnn with (I,J)=(0,1) is bound in ChQM after configuration mixing. Configuration mixing introduces more attraction. Orthogonalization with eigenfunction method is used to overcome the problem of over-completeness. 2+4 mixing is important for exotic tetraquark states

29. Thanks !!!