Lattice QCD in China Jianbo Zhang Department of Physics, Zhejiang University Sept. 20, 2006

1. Lattice QCD in ChinaJianbo ZhangDepartment of Physics, Zhejiang UniversitySept. 20, 2006

2. Outline 1. China Lattice QCD Collaboration 2. Selected Topics Charmonium Spectrum Pion-Pion Scattering Phase Shift 3. Future Perspectives

3. 1. China Lattice QCD Collaboration An Introduction

4. Members Faculty Members: Ying Chen Institute of High Energy Physics, CAS Chuan Liu Peking University Yubin Liu Nankai University Xiang-Qian Luo Zhongshan University Jian-Ping Ma Institute of Theoretical Physics, CAS Jianbo Zhang Zhejiang University Graduate Students (not complete): Ming Gong (PKU), Xin Li (PKU), Ji-Yuan Liu (PKU), Xiang-Fei Meng (NKU), Gang Li (IHEP), Yuan-Jiang Zhang (IHEP) , etc.

5. Computers available At present, Roughly 0.6-1.0 million CPUhours are allocated per year.

6. Forthcoming Machines Supercomputing Center of CAS (SCCAS) A 100 Tflopsnew computer is planned and expected to be available in 2008. Shanghai supercomputer Center (SSC) A 100 Tflopsnew computer is expected to be available in 2008.

7. Projects in progress Charmonium spectrum ( excited states, hybrids, etc.) Pion-pion scattering phase shift Lattice QCD at finite temperature and finite density

8. 2. Selected Topics Charmonium Spectrum

9. Motivation A series of heavy meson states of open-charm and closed-charm have been observed recently. X(3872) (most likely , but refuses to fit into the 2P state predictions of non-relativistic quark models ). Y(4260) (likely a hybrid charmonium?) Many model-dependent theoretical interpretation of the newly observed meson states. ( Review: Swanson, Phys. Rep. 429 (2006) 243-305 )

10. Lattice Calculations (Quenched) Tadpole improved Symanzik’s gauge action. Tadpole improved Clover fermions action. Anisotropic lattices.

11. Lattice interpolation field operators The operators are constructed by quark bilinear sandwiched with Gamma matrices and color fields. When calculating the two-point functions, the disconnected diagrams are neglected by assuming the OZI suppression.

12. Data analysis ---Sequential Empirical Bayes Method (Y. Chen et al., hep-lat/0405001) prior Bayes: constrained-curve fitting Empirical: priors are derived from part of data **(‘prior’ means the prior information of parameters) Sequential: states fitted one by one from low to high.

13. Here is an example for the fitting procedure in the vector channel (four-mass-term fit), where t_max =68, t_1=50, t_2=30, t_3=15 Fit model The mass terms are fitted one by one from low to high 1. Choose the t_max. 2. Varying t_1 towards small value, one-mass -term fitting in the interval [t_1,t_max], until the χ2/dof blows up ; 3. Add the second mass term at t_2=t_1-1. Varying t_2, two-mass-term fitting in the interval [t_2,t_max] with the first state constrained by the fitted parameters in Step 2. 4. When χ2/dof blows up again at the last t_2, and add the third state at t_3=t_2-1. 5. Repeat. 6. The latest state is generally taken as a garbage can and therefore not a realistic state. An Example of SEB

14. 0++ Three-mass-term fitting procedure in 0++ channel Red points are data from the simulation, the blue curve is the plot of fit model with fitted parameters.

15. 1++ Three-mass-term fitting procedure in 1++ channel Red points are data from the simulation, the blue curve is the plot of fit model with fitted parameters.

16. 1+- Three-mass-term fitting procedure in 1+- channel Red points are data from the simulation, the blue curve is the plot of fit model with fitted parameters.

17. 1S, 2S and 1P, 2P, 3P states This figure and table on next page illustrate the results from the finest lattice. The continuum extrapolation will be taken later

19. 2P states and X(3872) BGS represents the predictions of Swanson et al quark model. It is difficult to change quark model, as it can reproduce precisely the masses of almost all the known charmonium states (Swanson, hep-ph/0601110). For 2P states, earlier (quenched) lattice QCD predictions (CP-PACS and Chen) of their masses are roughly 100 MeV larger than QM prediction. This may be attributed to their two-mass-term fitting where the contamination of higher states to the first excited states cannot be neglected. Our result for 2P(1++) is consistent with X(3872) in mass.

20. Continuum limit extrapolation performed. It is smaller than the experiment value. The result is in agreement with previous (quenched) works • Hyperfine splitting

21. hybrids (with exotic quantum numbers) • These results are in • agreement with • previous quenched • lattice QCD results.

22. Non-exotic hybrids and conventional charmonium • Masses from the four-mass-term SEB fitting of hybrid-hybrid (HH) and meson- • meson (MM) correlation functions in 1-- and 0-+ channels. • It is understandable that the masses of the ground states are almost the same and • the masses of the first excited states are consistent with each other, because the • operators with the same quantum numbers can overlap to the same hadron states. • The masses of the second excited state of HH are very different from those of MM • No convincing results of masses of non-exotic hybrids can be derived in our work.

23. A summary to the charmonium spectrum study • With SEB, the masses of the first excited states (even the second excited states in some channel) can be reliably derived from charmonium two-point functions. • The masses of 2S charmonium states agrees well with experimental data. • The masses of 2P charmonium states obtained in this work are 3.798( 70), 3.827(50), and 3.799(60) for 0++, 1++and 1+- states, respectively. Given 1++ for X(3872), 2P(1++) is consistent with X(3872) in mass. • Masses of hybrid charmonia with exotic quantum numbers can be derived more soundly, since there are no admixtures of conventional charmonia. However for hybrid charmonia with no-exotic quantum numbers, it still a tough task to separate them from conventional charmonia unambiguously in the present lattice study. • Specifically, we have not observed a clear hybrid states with mass around 4260 MeV in the vector channel.

24. Pion-Pion Scattering Phase Shift • The study of hadron-hadron scattering is helpful for understanding the low-energy structure of QCD. • Luscher’s formula relates the pion-pion phase shift in the continuum to the energy of two-pion system in a finite cubic box, where is modified zeta-function and can be numerically calculated. q and k are related to the two-pion energy by and • at different k can be calculated directly on lattice.

25. We work on the spatially asymmetric lattices , • so that there are more non-degenerate non-zero • three momentum modes. • . Luscher’s formula in a finite cubic box needs to be • modified. For lattice of spatial size (η1Lxη2LxL), • with η1 and η2 equal or great than 1 • . Z00(1, q2,η1,η2) represents the modified zeta • function for the asymmetric box, and can be calculated • numerically.

26. Here is the preliminary results obtained at two beta values. More work is in progress

27. 3. Future Perspectives • Numerical study of lattice QCD with dynamical fermions. • Lattice QCD at finite temperature and finite density. How far we can go depends on the computing resource available. • A way out ---- International collaboration

28. Thank You! 谢谢!