Charmonium prospects from anisotropic lattice study

1. Charmonium prospects from anisotropic lattice study - Using spatial boundary condition - International workshop on “Heavy Quarkonium 2006” June 27-30, 2006 @ Brookhaven National Lab. Hideaki Iida (Yukawa Institute for Theoretical Physics, Kyoto univ.) collaboration with N. Ishii (Tokyo univ.), T. Doi (RIKEN BNL), H. Suganuma and K. Tsumura (Dept. of Phys., Kyoto univ.)

2. Introduction Study of charmonium at high temperatures (Review) ・Based on effective model analyses: T.Hashimoto, O. Miyamura, K. Hirose and T. Kanki, Phys.Rev.Lett.57 (1986) 2123 …Calculation of the mass shift of J/ψ around Tc. T.Matsui and H.Satz, Phys.Lett.B178 (1986) 416 …There occurs J/ψ suppression above Tc. ・Lattice results using Maximal Entropy Method: T. Umeda, K. Katayama, O. Miyamura and H. Matsufuru, Int. J. Mod. A16 (2001) 2115; H. Matsufuru, O. Miyamura, H. Suganuma and T. Umeda, AIP Conf. Proc. CP594 (2001) 258 …J/ψ survives at T～1.1Tc S. Datta, F.Karsch, P. Petreczky and I. Wetzorke, Phys.Rev.D69 (2004) 094507 etc. …J/ψ survives above Tc. (There is a resonance peak still T ～2Tc, then disappears gradually.) M.Asakawa and T. Hatsuda, Phys.Rev.Lett.92 (2004) 012001 …J/ψ survives until T～1.6Tc, then disappears immediately. ・Experiments: SPS (CERN)…observation of anomalies of dilepton spectra (NA50, Pb-Pb collision) RHIC…High luminosity, Au+Au, …

3. Boundary condition dependence Lattice studies suggest J/ψ may survive even above Tc. Question: Is it a compact J/ψ? Isn’t it a cc scattering state? ・ Reproducibility of MEM (especially the width) ・ Narrow width = compact state ? ・ Continuum st. becomes discrete in finite box Our goal is to study whether the cc quasi-bound states above Tc is a compact J/ψ or a cc scattering state. How? → Using the dependence of the energy of the state on the spatial boundary condition

4. Boundary condition dependence We impose a periodic boundary condition or an anti-periodic boundary condition on c and c quark, respectively for x,y,z direction. Compact charmonium No boundary condition dependence If a state is Boundary condition dependence due to the relative momentum of cc cc scattering state Ref) N. Ishii et al. Phys.Rev.D71 (2004) 034001 …Using boundary condition for quarks to distinguish penta-quark and NK scattering state

5. Anisotropic lattice A technical problem in finite temperature QCD At finite temperature, the temporal lattice points Nt becomes the smaller as the temperature becomes the higher. Imaginary time Calculation of hadron masses at high temperatures are difficult. space We use the anisotropic lattice in this study, where the temporal lattice spacing is smaller than the spatial one . . In this work, we use the anisotropy parameter ξ: anisotropic lattice

6. Correlator To enhance the ground state overlap, we use the spatially extended operator with Coulomb gauge. Temporal correlator with extended operator : (Zero momentum projected) Suitable for S-wave In S-wave, is optimal. effective mass If is sufficiently large, the obtained correlator at a temperature is dominated by the ground state. In this time region, has the following form: where represents the mass of the ground state. We define effective mass by the lattice data : If is dominated by the ground st., is independent of t. Then, the effective mass is almost equal to the mass of ground state.

7. Lattice setup Gauge sector (Quenched approx.) Standard Wilson gauge action (anisotropic lattice) :bare anisotropy Parameter set (for gauge configuration)

8. Lattice setup Quark sector O(a) improved Wilson action (anisotropic lattice) clover term (O(a) improvement) Wilson parameter Parameter set (for quarks) (This parameter set reproduces the J/ψ mass at zero temperature ）

9. Spatial boundary condition By changing the spatial boundary condition of c and c, we can distinguish a compact resonance state from a scattering state. ( Note: Temporal boundary condition for quarks and anti-quarks are anti-periodic.) We impose periodic boundary condition or anti-periodic boundary condition for quarks and anti-quarks. Periodic Boundary Condition (PBC) Anti-periodic Boundary Condition (APBC) ☆After zero momentum projection, the total momentum of the system vanishes. However, c and c can have non-vanishing momentum, respectively.

10. S-wave caseJ/ψ(JP=1-), mJ/ψ＝3100MeVηc (JP=0-), mJ/ψ＝2980MeV

11. Spatial boundary condition A compact J/ψ has zero momentum on PBC and APBC after zero momentum projection. Therefore the energy of the state is less sensitive to spatial boundary condition. In contrast, if a state is a cc scattering one, c and c in lowest energy have momentum and , respectively on APBC after zero momentum projection. c c,c c PBCfor cc scattering state APBCfor cc scattering state

12. cc scattering state A compact J/ψ Spatial boundary condition Note 1: This expression is only for S-wave!! Note 2: How is the effect of Yukawa potential? 　　　　 → Less than 20MeV Negligible (Estimated by the potential-model with Yukawa pot. in the finite box on PBC and APBC.)

13. ○：PBC, △：APBC Effective mass plot of J/ψ The fit is done by the cosh type function. T=1.32Tc T=1.11Tc Best fit of PBC and APBC fit range T=2.07Tc T=1.61Tc ・No boundary condition dependence is observed.

14. ○：PBC, △：APBC Effective mass plot of ηc The fit is done by the cosh type function. T=1.32Tc T=1.11Tc T=1.61Tc T=2.07Tc ・No boundary condition dependence is observed.

15. Behavior of J/ψ mass Compact J/ψ⇒ Scattering state ⇒ …Almost no boundary condition dependence of the energy. J/ψ is a compact state above Tc (～2Tc).

16. Behavior of ηc mass Compact ηc⇒ Scattering state ⇒ …Almost no boundary condition dependence of the energy. ηc is a compact state above Tc (～2Tc).

17. P-wave caseχc1 (JP=1+), mχc1=3510MeV

18. Calculation in χc1 channel (JP=1+) • χc1…P-wave state →due to the centrifugal potential, the wave function tends to spread. ⇒ It is sensitive to vanishing of linear potential and appearance of Debye screening effect. Dissociation temperature of χc1 would be lower than that of J/ψ and ηc.

19. Threshold of P-wave state PBC: (BCx,BCy,BCz)=(P,P,P) • In the P-wave case: → Lowest quanta: (nx,ny,nz)=(0,0,1) The highest-threshold APBC: (BCx,BCy,BCz)=(A,A,A) → Lowest quanta: (0,0,0) Threshold is lower than that in PBC case BC is different in the direction Hybrid boundary condition (HBC): (BCx,BCy,BCz)=(P,P,A) → Lowest quanta: (0,0,0) The lowest-threshold Largest between PBC and HBC

20. Gaussian type and spherical extension of the operator may not be suitable for P-wave state. Difficulty with the optimization of the operator → We examined the extension radius ρ=(0-0.5)fm Point-source, Point-sink Extended, Point Extended, Extended

21. Effective mass in χc1 channel Extended-source, Point-sink (ρ=0.2fm) PBC APBC HBC HBC2 T=1.11Tc HBC2: (BCx,BCy,BCz)=(A,A,P) between PBC and HBC No plateau region even at T=1.1Tc!

22. Analysis ofχc1 channel with maximally entropy method (K. Tsumura (Kyoto Univ.)) • Maximally entropy method (MEM) A method to solve an inverse problem: B = K A [M. Asakawa, Y. Nakahara and T. Hatsuda, Prog. In Part. And Nucl. Phys 46 (2001) 459.] Information we want to know Obtained image Mapping function which “dirty” the information …We can obtain B from A uniquely with MEM. This method is applicable to the extraction of the spectral function from the temporal correlator obtained by lattice QCD. Temporal correlator from lattice QCD K A B Desired spectral function →can be extracted !!

23. MEM results for χc1 channel Lattice setup: Wilson quark action with β=7.0 (at-1=20.2GeV) as/at=4.0, lattice size 203×46 (L=0.78fm, T=1.62Tc) ω=6GeV PBC ω=3.5GeV ① No compact bound state of χc1 (～3.51GeV) is observed. ② In the high energy region, there emerges a sharp peak around 6GeV. → χc1 already dissolves at T=1.62Tc

24. APBC Comparison between PBC and APBC Almost no difference between PBC and APBC →The peak around 6GeV is considered as a compact bound state. This peak may be considered to be a lattice artifact of Wilson fermion. The bound state of doubler(s) ? (pointed out by other groups)

25. MEM results of J/ψ PBC APBC Comp. ω=3GeV No BCD (a) Spectral function on PBC (b) SPF on APBC (c) Comparison between PBC and APBC

26. MEM results of ηC PBC APBC Comp. ω=3GeV No BCD Peak around 3GeV + No Boundary Condition dep. → Survival of J/ψ and ηc above Tc Different from the P-wave channel

27. Summary and Conclusion ・We have investigated J/ψ and ηc above Tc using lattice QCD. ・We have used the O(a) improved Wilson action for quarks. ・For the accurate measurement, we have used anisotropic lattice QCD. ・Changing the spatial boundary condition, we have examined whether J/ψ and ηc above Tc are compact states or scattering states of c and c. ・We have observed almost no boundary condition dependence of cc state above Tc. This suggests that J/ψ and ηc survive above Tc(～２Tc）. (・The level inversion of J/ψ and ηc may occur.)

28. Summary and Conclusions ・ We have also investigated in χc1 channel above Tc using lattice QCD, because the dissociation temperature of χc1 may differ from those of J/ψ and ηc. ・Unfortunately, we cannot extract a low-lying state (due to the difficulty of optimization of operators). → MEM on PBC and APBC ・We extract the spectral function in χc1 channel with maximally entropy method (MEM). No peak structure corresponding to χc1 is observed at T～1.6Tc. (Consistent with other work) ・The spectral functions in J/ψ and ηc channel has the peaks corresponding to J/ψ and ηc and those are independent of BC. → Compact state (・There may be a compact bound state in high energy region (doubler(s)).)

29. Perspectives ・Analysis of P-wave meson with effective mass ・Further analysis of Maximum Entropy Method (MEM) + Spatial boundary condition dependence (By K. Tsumura (Kyoto Univ.)) ・Other charmonium and charmed mesons, D mesons… ( D meson…If D becomes lighter, J/ψ→DD channel open. The width of charmonium possibly change. ) →Ongoing ・Mechanism of the formation of the bound state above Tc