1. CATHIE-INT 09T.Umeda (Hiroshima Univ.)1 Quarkonium correlators on the lattice T. Umeda (Hiroshima Univ.) H. Ohno, K. Kanaya (Univ. of Tsukuba) for WHOT-QCD Collaboration Joint CATHIE-INT mini-program INT, Seattle, USA, June 16th 2009 /20

2. CATHIE-INT 09T.Umeda (Hiroshima Univ.)2 Contents of this talk Introduction Introduction -- Quark Gluon Plasma & J/ψ suppression -- Quark Gluon Plasma & J/ψ suppression -- Lattice studies on J/ψ suppression -- Lattice studies on J/ψ suppression Our approach to study charmonium dissociation Our approach to study charmonium dissociation Charmonium wave functions at T>0 Charmonium wave functions at T>0 Discussion & Summary Discussion & Summary from the Phenix group web-site

3. CATHIE-INT 09T.Umeda (Hiroshima Univ.)3 J/ψ suppression as a signal of QGP Lattice QCD calculations: Spectral function by MEM: T.Umeda et al.(’02), S.Datta et al.(’04), Spectral function by MEM: T.Umeda et al.(’02), S.Datta et al.(’04), Asakawa&Hatsuda(’04), A.Jakovac et al.(’07), G.Aatz et al.(’06) Asakawa&Hatsuda(’04), A.Jakovac et al.(’07), G.Aatz et al.(’06) Wave func.: T.Umeda et al.(’00) Wave func.: T.Umeda et al.(’00) B. C. dep.: H.Iida et al. (’06) B. C. dep.: H.Iida et al. (’06)  all calculations conclude that J/ψ survives till 1.5T c or higher  all calculations conclude that J/ψ survives till 1.5T c or higher Confined phase: linear raising potential linear raising potential  bound state of c - c  bound state of c - c De-confined phase: Debye screening Debye screening  scattering state of c - c  scattering state of c - c T.Hashimoto et al.(’86), Matsui&Satz(’86)

4. CATHIE-INT 09T.Umeda (Hiroshima Univ.)4 E705 Collab.(’93) 10% 30% 60% It is important to study dissociation temperatures for not only J/ψ but also ψ(2S), χ c ’s for not only J/ψ but also ψ(2S), χ c ’s AA collisions J/ψ ψ’ψ’ χcχc Sequential J/ψ suppression scenario J /ψ (1S) : J PC = 1 – – M=3097MeV (Vector) ψ (2S) : J PC = 1 – – M=3686MeV (Vector) ψ (2S) : J PC = 1 – – M=3686MeV (Vector) χ c0 (1P) : J PC = 0 ++ M=3415MeV (Scalar) χ c1 (1P) : J PC = 1 ++ M=3511MeV (AxialVector) PDG(’06)

5. CATHIE-INT 09T.Umeda (Hiroshima Univ.)5 Hot QCD on the lattice Finite T Field Theory on the lattice 4dim. Euclidean lattice 4dim. Euclidean lattice gauge field U μ (x)  periodic B.C. gauge field U μ (x)  periodic B.C. quark field q(x)  anti-periodic B.C. quark field q(x)  anti-periodic B.C. Temperature T=1/(N t a) Temperature T=1/(N t a) Lattice QCD enables us to perform nonperturbative calculations of QCD Path integral by Monte Carlo integration QCD action on a lattice

6. CATHIE-INT 09T.Umeda (Hiroshima Univ.)6 Spectral function on the lattice Thermal hadron (charmonium) correlation functions Michael E. Peskin, Perseus books (1995) Spectral function discrete spectra  bound states charmonium states continuum spectra  2-particle states scattering states melted charmonium

7. CATHIE-INT 09T.Umeda (Hiroshima Univ.)7 infinite volume casefinite volume case In a finite volume (e.g. Lattice simulations), discrete spectra does not always indicate bound states ! discrete spectra does not always indicate bound states ! Spectral functions in a finite volume Momenta are discretized in finite (V=L 3 ) volume p i /a = 2n i π/L (n i =0,±1,±2,…) for Periodic boundary condition localized wave func. broad wave func. Shape of wave functions may be good signature to find out the charmonium melting. to find out the charmonium melting.

8. CATHIE-INT 09T.Umeda (Hiroshima Univ.)8 Bound state or scattering state ? r φ (r) scattering state r φ (r) bound state r φ (r) bound state r φ (r) scattering state Φ(r): wave function r : c - c distance r : c - c distance lowest state lowest state next lowest state next lowest state examples for examples for · S-wave · S-wave · Periodic B.C · Periodic B.C localized wave function small vol. dependence broad wave function vol. dependence

9. CATHIE-INT 09T.Umeda (Hiroshima Univ.)9 Temp. dependence of ( Bethe-Salpeter ) “Wave function” Wave functions at finite temperature Remarks on wave function of quark-antiquark gauge variant  Coulomb gauge fixing large or small components of quark/antiqaurk  wave func. for large components

10. CATHIE-INT 09T.Umeda (Hiroshima Univ.)10 finite volume case Technique to calculate wave function at T>0 In order to study a few lowest states, the variational analysis is one of the most reliable methods ! the variational analysis is one of the most reliable methods ! N x N correlation matrix : C(t) It is difficult to extract higher states from lattice correlators (at T>0) from lattice correlators (at T>0) even if we use MEM !! even if we use MEM !! It is important to investigate a few lowest states (at T>0) a few lowest states (at T>0) Constant mode can be separated by the Midpoint subtraction by the Midpoint subtraction T. Umeda (2007) T. Umeda (2007)

11. CATHIE-INT 09T.Umeda (Hiroshima Univ.)11 Lattice setup Quenched approximation ( no dynamical quark effect ) Quenched approximation ( no dynamical quark effect ) Anisotropic lattices Anisotropic lattices lattice spacing : a s = 0.0970(5) fm lattice spacing : a s = 0.0970(5) fm anisotropy : a s /a t = 4 anisotropy : a s /a t = 4 r s =1 to suppress doubler effects r s =1 to suppress doubler effects Variational analysis with 4 x 4 correlation matrix Variational analysis with 4 x 4 correlation matrix t x,y,z

12. CATHIE-INT 09T.Umeda (Hiroshima Univ.)12 Wave functions in free quark case Test with free quarks ( L s /a=20, ma=0.17 ) in case of S-wave channels in case of S-wave channels Free quarks make trivial waves with an allowed momentum in a box an allowed momentum in a box The wave function is constructed with The wave function is constructed with eigen functions of 6 x 6 correlators eigen functions of 6 x 6 correlators 6 types of Gaussian smeared operators 6 types of Gaussian smeared operators φ(x) = exp(-A|x| 2 ), φ(x) = exp(-A|x| 2 ), A = 0.02, 0.05, 0.1, 0.15, 0.2, 0.25 A = 0.02, 0.05, 0.1, 0.15, 0.2, 0.25 Our method well reproduces Our method well reproduces the known result ( ! ) the known result ( ! )

13. CATHIE-INT 09T.Umeda (Hiroshima Univ.)13 Charmonium wave functions at finite temperatures Small temperature dependence in each channels Clear signals of bound states even at T=2.3Tc ( ! ) (2fm) 3 may be small for P-wave states. (2fm) 3 may be small for P-wave states.

14. CATHIE-INT 09T.Umeda (Hiroshima Univ.)14

15. CATHIE-INT 09T.Umeda (Hiroshima Univ.)15 J/ψ(1S) ψ(2S) Volume dependence at T=2.3Tc Clear signals of bound states even at T=2.3Tc ( ! ) Large volume is necessary for P-wave states. Large volume is necessary for P-wave states. χ c1 (1P) χ c1 (2P)

16. CATHIE-INT 09T.Umeda (Hiroshima Univ.)16 Discussion We found localized wave functions up to 2.3Tc for S- & P- wave channels. up to 2.3Tc for S- & P- wave channels. (1) Does variational analysis does work well ? wave function for lowest/next-lowest state wave function for lowest/next-lowest state + contributions from higher states  contaminations + contributions from higher states  contaminations our results suggest our results suggest there are no/small broad wave functions there are no/small broad wave functions even in the higher states (!) even in the higher states (!) (2) Unbound state with localized wave function ? anyway, anyway, tight wave function is incompatible tight wave function is incompatible with the J/ψ suppression (!) with the J/ψ suppression (!) (3) Small changes in wave functions contradict some potential model results contradict some potential model results

17. CATHIE-INT 09T.Umeda (Hiroshima Univ.)17 The idea has been originally applied for the charmonium study in H. Iida et al., Phys. Rev. D74 (2006) 074502. M BCBC’ bound M BCBC’ scattering r φ (r) bound 0 r φ (r) scattering 0 The wave functions are localized, their energies are insensitive to B.C. The momenta depends on BC, the scattering state energies are sensitive to B.C. Boundary condition dependence

18. CATHIE-INT 09T.Umeda (Hiroshima Univ.)18 Variational analysis in free quark case Test with free quarks ( L s /a=20, ma=0.17 ) S wave P wave PBC APBC MBC Mass diff. between the lowest masses in each BC PBC : b=( 1, 1, 1) PBC : b=( 1, 1, 1) APBC : b=(-1,-1,-1) MBC : b=(-1, 1, 1) MBC : b=(-1, 1, 1) an expected diff. in V=(2fm) 3 (free quark case) ~ 200MeV ~ 200MeV

19. CATHIE-INT 09T.Umeda (Hiroshima Univ.)19 No significant differences in the different B.C. Analysis is difficult at higher temperature ( 2T c ～ ) PBC : b=( 1, 1, 1) PBC : b=( 1, 1, 1) APBC : b=(-1,-1,-1) MBC : b=(-1, 1, 1) MBC : b=(-1, 1, 1) Temperature dependence of charmonium spectra an expected gap in V=(2fm) 3 (free quark case) ~ 200MeV ~ 200MeV η c (2S) χ c0 (2P) Ψ(2S) χ c1 (2P) η c (1S)J/Ψ(1S) χ c0 (1P) χ c1 (1P) PBC APBC PBC APBC PBC MBC PBC MBC

20. CATHIE-INT 09T.Umeda (Hiroshima Univ.)20 Summary and future plan We investigated T dis of charmonia from Lattice QCD without Bayesian (MEM) analysis using... without Bayesian (MEM) analysis using... - Bethe-Salpeter “wave function” - Bethe-Salpeter “wave function” - Volume dependence of the “wave function” - Volume dependence of the “wave function” - Boundary condition dependence - Boundary condition dependence  The result may affect the scenario of J/ψ suppression. Future plan Possible scenarios for the experimental J/ψ suppression Higher Temp. calculations ( T/Tc=3~5 ) Higher Temp. calculations ( T/Tc=3~5 ) Full QCD calculations ( Nf=2+1 Wilson is now in progress ) Full QCD calculations ( Nf=2+1 Wilson is now in progress ) No evidence for unbound c-c quarks up to T = 2.3 Tc -