1. Charmonium prospects from anisotropic lattice study 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.) - Using spatial boundary condition -

2. Introduction ・ 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, … Study of charmonium at high temperatures (Review)

3. Boundary condition dependence Lattice studies suggest J/ψ may survive even above T c. 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 If a state is Compact charmonium cc scattering state No boundary condition dependence Boundary condition dependence due to the relative momentum of cc We impose a periodic boundary condition or an anti-periodic boundary condition on c and c quark, respectively for x,y,z direction. 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 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: Calculation of hadron masses at high temperatures are difficult. At finite temperature, the temporal lattice points N t becomes the smaller as the temperature becomes the higher. space Imaginary time anisotropic lattice

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

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

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

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. Periodic Boundary Condition (PBC) Anti-periodic Boundary Condition (APBC) ( 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. ☆ After zero momentum projection, the total momentum of the system vanishes. However, c and c can have non-vanishing momentum, respectively.

10. S-wave case J/ψ(J P =1 - ), m J/ψ ＝ 3100MeV η c (J P =0 - ), m J/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. APBC for cc scattering state c c PBC for cc scattering state c,c

12. 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.) Spatial boundary condition cc scattering state A compact J/ψ

13. Estimation of ground state overlap t In the plateau region, we assume that the lowest state dominates. We fit the correlator in this region by the following cosh type function form: Then is considered as an index of the ground state overlap. : correlator We define (If this value is near 1, then the ground state overlap is large.) Optimal extension of the operator correlator fit range 1 Ground state overlap R=(0.2-0.3)fm is optimal. Therefore, we show the effective mass at R=0.2fm in the following.

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

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

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

17. Behavior of η c mass …Almost no boundary condition dependence of the energy. η c is a compact state above T c ( ～ 2T c ). Compact η c ⇒ Scattering state ⇒

18. Crossing of the level of J/ψ and η c Level crossing??? J/ψ ηcηc

19. Example of a scattering state (NK) raised up Penta quark Θ + (uudds)? NK scattering state? ⇒ Not a penta quark, but an NK scattering state!! N. Ishii et al. Phys.Rev.D71(2004)034001 ⇒ Hybrid Boundary Condition The energy of the observed state on standard boundary condition is raised up on hybrid boundary condition. Due to the small enegry difference between Θ+ and NK, it is difficult to distinguish the two states.

20. P-wave case χ c1 (J P =1 + ), m χ c1 =3510MeV

21. Calculation in χ c1 channel (J P =1 + )  χ c1 … 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 χ c1 would be lower than that of J/ψ and η c.

22. Threshold of P-wave state  In the P-wave case: PBC: (BC x,BC y,BC z )=(P,P,P) APBC: (BC x,BC y,BC z )=(A,A,A) → Lowest quanta: (n x,n y,n z )=(0,0,1) → Lowest quanta: (0,0,0) Threshold is lower than that in PBC case Hybrid boundary condition (HBC): (BC x,BC y,BC z )=(P,P,A) BC is different in the direction → Lowest quanta: (0,0,0) The lowest-threshold Largestbetween PBC and HBC The highest-threshold

23. Difficulty with the optimization of the operator Gaussian type and spherical extension of the operator may notbe suitable for P-wave state. → We examined the extension radius ρ=(0-0.5)fm Point-source, Point-sink Extended, Point Extended, Extended

24. Threshold of P-wave state between PBC and HBC HBC2: (BC x,BC y,BC z )=(A,A,P)

25. Effective mass in χ c1 channel ρ= 0fm (Point-source, Point-sink) ： PBC ： APBC

26. Effective mass in χ c1 channel Extended-source, Point-sink (ρ=0.2fm) HBC2: (BC x,BC y,BC z )=(A,A,P) PBC APBC HBC HBC2 No plateau region even at T=1.1T c ! between PBC and HBC T=1.11T c

27. Effective mass in χ c1 channel Extended-source, Extended-sink (ρ=0.2fm) In all the case, we cannot extract the χ c1 state. (No plateau region!!) No clear boundary-condition dependence. Until now, this method does not work

28. Analysis ofχ c1 channel with maximally entropy method (K. Tsumura (Kyoto Univ.))  Maximally entropy method (MEM) A method to solve an inverse problem: B = K A Obtained imageMapping function which “dirty” the information Information we want to know B KA This method is applicable to the extraction of the spectral function from the temporal correlator obtained by lattice QCD. Temporal correlator from lattice QCD Desired spectral function →can be extracted !! … We can obtain B from A uniquely with MEM. [M. Asakawa, Y. Nakahara and T. Hatsuda, Prog. In Part. And Nucl. Phys 46 (2001) 459.]

29. MEM results for χ c1 channel PBC Lattice setup: Wilson quark action with β=7.0 (a t -1 =20.2GeV) a s /a t =4.0, lattice size 20 3 ×46 (L=0.78fm, T=1.62T c ) ① No compact bound state of χ c1 ( ～ 3.51GeV) is observed. ② In the high energy region, there emerges a sharp peak around 6GeV. → χ c1 already dissolves at T=1.62Tc ω=6GeV ω=3.5GeV

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

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

32. MEM results of η C PBCAPBC Comp. ω=3GeV Peak around 3GeV + No Boundary Condition dep. → Survival of J/ψ and η c above T c Different from the P-wave channel No BCD

33. MEM results of χ c1 (a) Spectral function on PBC (b) SPF on APBC (c) Comparison between PBC and APBC PBCAPBC Comp.

34. Summary and Conclusion ・ We have investigated J/ψ and η c above T c 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 η c above T c are compact states or scattering states of c and c. ・ We have observed almost no boundary condition dependence of cc state above T c. This suggests that J/ψ and η c survive above T c ( ～２ T c ）. ( ・ The level inversion of J/ψ and η c may occur.)

35. Summary and Conclusions ・ We have also investigated in χ c1 channel above T c using lattice QCD, because the dissociation temperature of χ c1 may differ from those of J/ψ and η c. ・ 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 χ c1 channel with maximally entropy method (MEM). No peak structure corresponding to χ c1 is observed at T ～ 1.6T c. (Consistent with other work) ・ The spectral functions in J/ψ and η c channel has the peaks corresponding to J/ψ and η c and those are independent of BC. → Compact state ( ・ There may be a compact bound state in high energy region (doubler(s)).)

36. 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 T c