1. Komaba seminarT.Umeda (Tsukuba)1 A study of charmonium dissociation temperatures using a finite volume technique in the lattice QCD T. Umeda and H. Ohno for the WHOT-QCD Collaboration Komaba seminar, Tokyo Univ., 14 Nov. 2007

2. Komaba seminarT.Umeda (Tsukuba)2 Contents Introduction - J/ψ suppression scenarios - Spectral functions in Lattice QCD - Comments on MEM Charmonium in a finite box - Spectral function in a finite box - Finite volume technique - Variational analysis - Test with free quarks Numerical results Summary & outlook total 30 pages Preliminary results !!

3. Komaba seminarT.Umeda (Tsukuba)3 Experiments SPS : CERN ( – 2005) Super Proton Synchrotron RHIC: BNL (2000 – ) Relativistic Heavy Ion Collider LHC : CERN (2009 - ) Large Hadron Collider from the Phenix group web-site

4. Komaba seminarT.Umeda (Tsukuba)4 J/ψ suppression in Exp. Phys.Lett.B477(2000)28 NA50 Collaboration QM2006 PHENIX Collaboration Energy density 0.5-1.5GeV/fm 3 = 1.0Tc 10 GeV/fm 3 = 1.5Tc 30 GeV/fm 3 = 2.0Tc

5. Komaba seminarT.Umeda (Tsukuba)5 Charmonium spectral function Quenched QCD - T. Umeda et al., EPJC39S1, 9, (2005). - S. Datta et al., PRD69, 094507, (2004). - T. Hatsuda & M. Asakawa, PRL92, 012001, (2004). - A. Jakovac et al., PRD75, 014506 (2007). Full QCD - G. Aarts et al., arXiv:0705.2198. Maximum entropy method C(t)  ρ(ω)

6. Komaba seminarT.Umeda (Tsukuba)6 Charmonium spectral function Quenched QCD - T. Umeda et al., EPJC39S1, 9, (2005). - S. Datta et al., PRD69, 094507, (2004). - T. Hatsuda & M. Asakawa, PRL92, 012001, (2004). - A. Jakovac et al., PRD75, 014506 (2007). Full QCD - G. Aarts et al., arXiv:0705.2198. All studies indicate survival of J/ψ state above T c (1.5T c ?) Maximum entropy method C(t)  ρ(ω)

7. Komaba seminarT.Umeda (Tsukuba)7 Sequential J/ψsuppression Particle Data Group (2006) PsV S Av E705 Collaboration, PRL70, 383, (1993). 10% 30% 60% Dissociation temperatures of J/ψ and ψ’& χ c are important for QGP phenomenology. for the total yield of J/ψ

8. Komaba seminarT.Umeda (Tsukuba)8 χ c states dissolve ? A.Jakovac et al., PRD75,014506 )2007. the results of SPFs for P-states are not so reliable. e.g. large default model dep. the drastic change just above Tc is reliable results. S. Datta et al., PRD69, 094507 (2004). A.Jakovac et al., PRD75, 014506 (2007). G.Aarts et al., arXiv:0705.2198 charmonium spectral func. for P-wave states  χ c states

9. Komaba seminarT.Umeda (Tsukuba)9 Constant mode exp(-m q t) x exp(-m q t) = exp(-2m q t) exp(-m q t) x exp(-m q (L t -t)) = exp(-m q L t ) m q is quark mass or single quark energy L t = temporal extent Imaginary time formalism L t = 1/Temp. gauge field : periodic b.c. quark field : anti-periodic b.c. confined phase (T<T c ) m q = ∞ the constant mode disappears deconfined phase (T>T c ) m q = finite the constant mode appears

10. Komaba seminarT.Umeda (Tsukuba)10 Effective mass (local mass) Definition of effective mass In the (anti) periodic b.c. when

11. Komaba seminarT.Umeda (Tsukuba)11 χ c states dissolve ? Small changes even in P-states ( apart from the const. mode ) Midpoint subtracted correlator

12. Komaba seminarT.Umeda (Tsukuba)12 Comments on MEM MEM is a kind of constrained χ 2 fitting # of fit parameters ≤ # of data points Prior knowledge of SPFs is needed as a default model function, m(w) The default model function plays crucial roles in MEM MEM maximizes L : Likelihood function (χ 2 term ) default model func. α: parameter ( to be integrated out )

13. Komaba seminarT.Umeda (Tsukuba)13 diagonal matrix basis in singular space fit parameters fit form for spectral function : ρ(ω) Singular Value Decomposition (SVD) Fit form in MEM

14. Komaba seminarT.Umeda (Tsukuba)14 Fit form in MEM

15. Komaba seminarT.Umeda (Tsukuba)15 Goal of our study Dissociation temperatures for J/ψ,χ c,ψ(2S),… from Lattice QCD calculations Comments on our Lattice QCD calculations  Thermal equilibrium system  Charmonium with zero total momentum  Quenched approximation - Tc 〜 260MeV ( 〜 190MeV in full QCD) - 1st order transition (cross over? in full QCD) - Static free energy vs T/Tc - 〜 10% deviation in hadron spectra  Homogeneous in finite volume

16. Komaba seminarT.Umeda (Tsukuba)16 Spectral functions in finite volume infinite volume case In a finite box of (La) 3, momenta are discretized for the periodic BC finite volume case

17. Komaba seminarT.Umeda (Tsukuba)17 bound state or scattering state ? (1)volume dependence of the discrete spectra bound states  small volume dependence scattering state  volume dependence from discretized momenta (2) wave function bound states  localized shape scattering states  broad shape (3) boundary condition dependence bound states  insensitive to B.C. scattering states  sensitive to B.C.

18. Komaba seminarT.Umeda (Tsukuba)18 finite volume techniques Spectrum on different (spatial) boundary conditions Periodic B.C.  Anti-periodic B.C.  H. Iida et al., PRD74, 074502 (2006). E bound (PBC) ≃ E bound (APBC) E scatt (PBC) < E scatt (APBC)

19. Komaba seminarT.Umeda (Tsukuba)19 Variational analysis We prepare N operators for charmonia and calculate the correlator matrix

20. Komaba seminarT.Umeda (Tsukuba)20 Numerical results Test with free quarks - Wilson quarks in 20 3 x 32 lattice - bare quark mass is tuned as 2 * am q ≃ m charmonium /8GeV Quenched QCD results

21. Komaba seminarT.Umeda (Tsukuba)21 Midpoint subtraction An analysis to avoid the constant mode Midpoint subtracted correlator Free quarks

22. Komaba seminarT.Umeda (Tsukuba)22 Test with free quarks S-wave states P-wave states S-wave : PBC APBC P-wave : PBC APBC 4 3 21 43 12

23. Komaba seminarT.Umeda (Tsukuba)23 Lattice QCD results Lattice setup Quenched approximation ( no dynamical quark effect ) Anisotropic lattices lattice size : 20 3 x N t lattice spacing : 1/a s = 2.03(1) GeV, anisotropy : a s /a t = 4 Quark mass charm quark (tuned with J/ψ mass) t x,y,z

24. Komaba seminarT.Umeda (Tsukuba)24 At zero temperature (our lattice results) M PS = 3033(19) MeV M V = 3107(19) MeV (exp. results from PDG06) M ηc = 2980 MeV M J/ψ = 3097 MeV M χc0 = 3415 MeV M χc1 = 3511 MeV In our lattice N t ⋍ 28 at T c t = 1 – 14 is available near T c Spatially extended (smeared) op. is discussed later

25. Komaba seminarT.Umeda (Tsukuba)25 Results for PS channel

26. Komaba seminarT.Umeda (Tsukuba)26 Results for V channel

27. Komaba seminarT.Umeda (Tsukuba)27 Results for S channel

28. Komaba seminarT.Umeda (Tsukuba)28 Results for Av channel

29. Komaba seminarT.Umeda (Tsukuba)29 Summary We investigated T dis of charmonia from Lattice QCD using the finite volume technique - boundary condition dependence - variational analysis Our approach can control the system with free quarks In quenched QCD case, we found no boundary condition dependence and no mass shift (except for 2.3Tc)  Our results may suggest both S & P states survive upto 2Tc (?)

30. Komaba seminarT.Umeda (Tsukuba)30 Outlook more efficient smearing func. more configurations & temperatures up to 1000confs. at each temp. Nt=32,26,20,(16),12,(8) volume dependence L=20, (16)  wave function (Bethe-Salpeter amplitude)

31. Komaba seminarT.Umeda (Tsukuba)31 Wave functions at T>0

32. Komaba seminarT.Umeda (Tsukuba)32 Wave functions at T>0

33. Komaba seminarT.Umeda (Tsukuba)33 Wave functions at T>0

34. Komaba seminarT.Umeda (Tsukuba)34 Wave functions at T>0

35. Komaba seminarT.Umeda (Tsukuba)35 Wave functions at T>0

36. Komaba seminarT.Umeda (Tsukuba)36 Wave functions at T>0