Lattice QCD results on heavy quark potentials and bound states in the quark gluon plasma

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1 Lattice QCD results on heavy quark potentials and bound states in the quark gluon plasma Olaf Kaczmarek Felix Zantow Universität Bielefeld April 5, 26 Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p./27

2 Machines for QCD - Experiments RHIC@BNL LHC@CERN STAR detector = only indirect signals from collision region theoretical input needed Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.2/27

3 Machines for QCD - lattice theory QCDOC apenext Installation in Bielefeld 999/2 44 Gflops APEmille 25/26 5 Tflops apenext Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.3/27

4 Charmonium suppression Measured / Expected J/ψ / DY Pb-Pb S-U p-a observed charmonium yields at SPS nuclear absorption length L using Bjorken formula (τ = f m) + lattice EOS: ε = de T/dη τ A T ψ, / DY Pb-Pb S-U p-a L (fm) L [ f m] ε [GeV/ f m 3 ] T/T c suppression patterns in a narrow temperature regime detailed (lattice) equation of state needed detailed knowledge on heavy quark bound states needed What are the processes of charmonium formation/suppression? Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.4/27

5 Charmonium suppression ψ S(J/ ) observed charmonium yields at SPS nuclear absorption length L using Bjorken formula (τ = f m) + lattice EOS: ε = de T/dη τ A T.25 In In, SPS Pb Pb, SPS Au Au, RHIC, y <.35 Au Au, RHIC, y =[.2,2.2] ε (GeV/fm 3) L [ f m] ε [GeV/ f m 3 ] T/T c [F. Karsch,D. Kharzeev,H. Satz,hep-ph/52239] suppression patterns in a narrow temperature regime detailed (lattice) equation of state needed detailed knowledge on heavy quark bound states needed What are the processes of charmonium formation/suppression? Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.4/27

6 Motivation - Heavy quark bound states above deconfinement Quarkonium suppression as a probe for thermal properties of hot and dense matter [Matsui and Satz] - heavy quark potential gets screened - screening radius related to parton density r D g n/t - at high T screening radius smaller than size of a quarkonium state Typical length scales of heavy quark bound states: /Λ QCD fm - screening has to be strong enough to modify short distance behaviour - detailed analysis of heavy quark potentials - temperature and r dependence - screening properties above deconfinement - What is the correct effective potential at finite temperature? F (r,t) = U (r,t) T S (r,t) Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.5/27

7 Heavy quark bound states above deconfinement Strong interactions in the deconfined phase T> T c Possibility of heavy quark bound states? Charmonium (χ c,j/ψ) as thermometer above T c Suppression patterns of charmonium/bottomonium = Potential models heavy quark potential (T =) V (r) = 4 3 α(r) r + σ r heavy quark free energies (T > T c ) F (r,t) 4 3 α(r,t) r e m(t)r heavy quark internal energies (T ) F (r,t) = U (r,t) T S (r,t) = Charmonium correlation functions/spectral functions Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.6/27

8 The lattice set-up Polyakov loop correlation function and free energy: L. McLerran, B. Svetitsky (98) Z Q Q Z (T) Z (T) Z D A... L(x) L (y) exp ( Z /T Z dt ) d 3 xl [A,...] F Q Q (r,t) T Q Q =, 8, av Lattice data used in our analysis: N f = : ,8,6-lattices ( Symanzik ) O. Kaczmarek, F. Karsch, P. Petreczky, F.Z. (22, 24) N f = 2: lattices ( Symanzik, p4-stagg. ) m π /m ρ.7 (m/t =.4) O. Kaczmarek, F.Z. (25), O. Kaczmarek et al. (23) N f = 3: lattices m π /m ρ.4 P. Petreczky, K. Petrov (24) Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.7/27

9 The lattice set-up Polyakov loop correlation function and free energy: L. McLerran, B. Svetitsky (98) Z Q Q Z (T) Z (T) Z D A... L(x) L (y) exp ( Z /T Z dt ) d 3 xl [A,...] F Q Q (r,t) T Q Q =, 8, av a /3 V =N σ a ln /T =N τ a O. Philipsen (22) O. Jahn, O. Philipsen (24) ( ln TrL(x) TrL ) (y) ( ln TrL(x)L ) (y) ( 9 8 TrL(x) TrL (y) 8 TrL(x)L (y) GF GF ) = F qq(r,t) T = F (r,t) T = F 8(r,T) T Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.7/27

10 Heavy quark free energy F [MeV] 5-5 r [fm].76t c.8t c.9t c.96t c.t c.2t c.7t c.23t c.5t c.98t c 4.T c Renormalization of F(r, T) by matching with T = potential e F (r,t)/t = ( Z r (g 2 ) ) 2N τ Tr (L x L y) Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.8/27

11 Heavy quark free energy F [MeV] 5-5 r [fm].76t c.8t c.9t c.96t c.t c.2t c.7t c.23t c.5t c.98t c 4.T c F(r,T)/σ /2 T/T c Renormalization of F(r, T) by matching with T = potential e F (r,t)/t = ( Z r (g 2 ) ) 2N τ Tr (L x L y) T -independent r / σ F(r,T) g 2 (r)/r r σ / Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.8/27

12 Heavy quark free energy F [MeV] String breaking T < T c F(r σ,t) < 5-5 r [fm].76t c.8t c.9t c.96t c.t c.2t c.7t c.23t c.5t c.98t c 4.T c F [MeV] N f = N f =2 N f =3 T -independent r / σ F(r,T) g 2 (r)/r T/T c 4 Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.8/27

13 Heavy quark free energy F [MeV] String breaking T < T c F(r σ,t) < 5-5 r [fm].76t c.8t c.9t c.96t c.t c.2t c.7t c.23t c.5t c.98t c 4.T c high-t physics rt ; screening µ(t) g(t)t F(,T) T F [MeV] N f = N f =2 N f =3 T -independent r / σ F(r,T) g 2 (r)/r T/T c 4 Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.8/27

14 Heavy quark free energy F [MeV] String breaking T < T c F(r σ,t) < 5-5 r [fm].76t c.8t c.9t c.96t c.t c.2t c.7t c.23t c.5t c.98t c 4.T c high-t physics rt ; screening µ(t) g(t)t F(,T) T T -independent r / σ F(r,T) g 2 (r)/r Complex r and T dependence Small vs. large distance behavior Running coupling vs. screening Where do temperature effects set in? Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.8/27

15 Temperature depending running coupling 2.5 α qq (r,t) Free energy in perturbation theory: T/T c r [fm] F (r,t) V(r) 4 α(r) 3 r F (r,t) 4 α(t) e m D(T)r 3 r for rλ QCD for rt QCD running coupling in the qq-scheme α qq (r,t) = 3 4 r2 df (r,t) dr Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.9/27

16 Temperature depending running coupling 2.5 α qq (r,t) Free energy in perturbation theory: T/T c r [fm] F (r,t) V(r) 4 α(r) 3 r F (r,t) 4 α(t) e m D(T)r 3 r for rλ QCD for rt QCD running coupling in the qq-scheme α qq (r,t) = 3 4 r2 df (r,t) dr non-perturbative confining part for r>.4 fm α qq (r) 3/4r 2 σ present below and just above T c temperature effects set in at smaller r with increasing T maximum due to screening Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.9/27

17 Temperature depending running coupling 2.5 α qq (r,t) Free energy in perturbation theory: T/T c r [fm] F (r,t) V(r) 4 α(r) 3 r F (r,t) 4 α(t) e m D(T)r 3 r for rλ QCD for rt QCD running coupling in the qq-scheme α qq (r,t) = 3 4 r2 df (r,t) dr non-perturbative confining part for r>.4 fm α qq (r) 3/4r 2 σ present below and just above T c temperature effects set in at smaller r with increasing T maximum due to screening = At which distance do T -effects set in? = definition of the screening radius/mass = definition of the T -dependent coupling Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.9/27

18 Heavy quark bound states above T c?.5 r [fm] r med r D bound states above deconfinement? first estimate: ψ mean charge radii of charmonium states compared to screening radius: V(r med ) F (T) χ c.5 J/ψ thermal modifications on ψ and χ c already at T c J/ψ may survive above deconfinement 2 3 T/T c 4 r n N Z d 3 r r n F (r,t) r D [fm] <r> r med r entropy 2/m D (a) r (b) 2r cloud (c). High T: Using a screened Coulomb form r 2/m D T = : Using parameterization of V(r) at T = : r T= fm T/T c Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p./27

19 Heavy quark bound states above T c?.5 r [fm] r med r D bound states above deconfinement? first estimate:.5 ψ χ c J/ψ mean charge radii of charmonium states compared to screening radius: V(r med ) F (T) thermal modifications on ψ and χ c already at T c J/ψ may survive above deconfinement 2 3 T/T c 4 Better estimates: Better estimates: effective potentials in Schrödinger Equation Potential models, effective potential V e f f (r,t) But: Free energies vs. internal energies F(r,T) = U(r,T) TS(r,T) direct calculation using correlation functions Maximum entropy method spectral function Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p./27

20 Free energy vs. Entropy at large separations 5 F [MeV] N f = N f =2 N f =3 Free energies not only determined by potential energy F = U T S 5 Entropy contributions play a role at finite T S = F T 2 3 T/T c 4 Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p./27

21 Free energy vs. Entropy at large separations 4 U [MeV] High temperatures: 3 2 U = T 2 F /T T F (T) 4 3 m D(T)α(T) O (g 3 T) TS (T) 4 3 m D(T)α(T) U (T) 4m D (T)α(T) β(g) g O (g 5 T) 2 3 T/T c 4 (a) (b) (c) r 2r cloud 4 3 TS [MeV] N f = N f =2 N f =3 The large distance behavior of the finite temperature energies is related to screening rather than to the temperature dependence of masses of corresponding heavy-light mesons! 2 S = F T 2 3 T/T c Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p./27

22 r-dependence of internal energies TS [MeV] (b) 5 F (r,t) = U (r,t) TS (r,t) S (r,t) = F (r,t) T U (r,t) = T 2 F (r,t)/t T Entropy contributions vanish in the limit r F (r,t) = U (r,t) V (r).5.5 r [fm] important at intermediate/large distances (a) r (b) 2r cloud (c) The large distance behavior of the finite temperature energies is related to screening rather than to the temperature dependence of masses of corresponding heavy-light mesons! Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.2/27

23 r-dependence of internal energies U [MeV] r [fm].88t c.93t c.98t c U [MeV] 5 F (r,t) = U (r,t) TS (r,t) S (r,t) = F (r,t) T U (r,t) = T 2 F (r,t)/t T Entropy contributions vanish in the limit r F (r,t) = U (r,t) V (r) important at intermediate/large distances = Implications on heavy quark bound states? 5 r [fm].9t c.3t c.9t c.29t c.43t c.57t c.89t c = What is the correct V e f f (r,t)? Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.2/27

24 Heavy quark bound states V eff (r,t) [MeV] V( ) E J/ψ (T=) r [fm] steeper slope of V e f f (r,t) = U (r,t) = J/ψ stronger bound using V e f f = U (r,t) = dissociation at higher temperatures compared to V e f f (r,t) = F (r,t) Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.3/27

25 Estimates on bound states from Schrödinger equation Schrödinger equation for heavy quarks: [ 2m f + ] 2 +V e f f (r,t) Φ f i = E f i m (T)Φ f i f, f = charm, bottom T -dependent color singlet heavy quark potential mimics in-medium modifications of q q interaction reduction to 2-particle interaction clearly too simple, in particular close to T c recent analysis: using V e f f = F : S.Digal, P.Petreczky, H.Satz, Phys. Lett. B54 (2)57 using V e f f = V : C.-Y. Wong, hep-ph/482 using V e f f = V : W.M. Alberico, A. Beraudo, A. De Pace, A. Molinari, hep-ph/5784 state J/ψ χ c ψ ϒ χ b ϒ χ b ϒ E i s[gev] r [ f m] T d /T c T d /T c T d /T c Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.4/27

26 Correlation functions of hadronic states Direct lattice calculations on bound states Thermal correlation functions: G H (τ,r,t) = J H (τ,r)j H (,) with J H = q(τ,r)γ H q(τ,r) Example using light quarks, scalar (δ) and pseudo-scalar (π): G(τ,T)/T 3 pseudo-scalar scalar G(τ,T)/T 3 pseudo-scalar scalar.6 T c.5 T c τ T τ T splitting below T c due to chiral and axial U() symmetry breaking symmetry restauration at high temperatures Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.5/27

27 Spectral functions of charmonium states Where do thermal effects on charmonium states set in? G(τ)/G recon (τ) χ c, χ c,.75t c AX+. SC.2T c.5t c τ[fm] G(τ)/G recon (τ) η c J/ψ.9T c PS+.2 VC.5T c 2.25T c 3T c τ[fm] strong effects already near T c no effect in η c up to.5t c modifications on J/ψ only of order % at.5t c bound states above T c possible? = reconstruction of spectral functions Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.6/27

28 Spectral functions of charmonium states spectral representation of correlators = in-medium properties of hadrons spectral representation of Euclidean correlation functions: G β H (τ, r) = Z Z dω d 3 p (2π) 3 σ i p r cosh(ω(τ /2T)) H(ω, p,t) e sinh(ω/2t) calculate the most probable spectral function Bayesian data analysis, e.g. Maximum Entropy Method (MEM) Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.6/27

29 Spectral functions of charmonium states J/ψ spectral function T =.78Tc T =.38Tc.8 σ(ω)/ω 2 T =.62Tc.6.9T c.5t c T =.87Tc T = 2.33Tc.4.2 ω[gev] T c T c.8 σ(ω)/ω ω [GeV] M. Asakawa, T. Hatsuda, Phys.Rev.Lett.92 (24) 2.2 ω[gev] S. Datta et al., Phys.Rev.D69 (24) 9457 Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.6/27

30 Spectral functions of charmonium states J/ψ spectral function T =.78Tc T =.38Tc.8 σ(ω)/ω 2 T =.62Tc.6.9T c.5t c T =.87Tc T = 2.33Tc.4 J/ψ dissociates for.6t c < T<.9T c rather abrupt disappearance of J/ψ.2 ω[gev] T c T c.8 σ(ω)/ω J/ψ gradually disappears for T>.5T c strength reduced by 25% at T = 2.25T c.5 ω [GeV] M. Asakawa, T. Hatsuda, Phys.Rev.Lett.92 (24) 2.2 ω[gev] S. Datta et al., Phys.Rev.D69 (24) 9457 Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.6/27

31 Heavy quark bound states above deconfinement - Diquarks [OK, K. Hübner, O. Vogt] Coloured bound states speculated just above T c (SQGP) [E.V. Shuryak,I. Zahed (24/5)] - Is the interaction strong enough to support diquarks? - Potential models for coloured bound states - Diquark free and internal energies Colour antitriplet (anti-symmetric) or color sextet (symmetric) state: 3 3 = 3 6. Cqq(R,T) 3 = 3 2 Tr L()Tr L(R) Tr L()L(R) 2 Cqq(R,T) 6 = 3 4 Tr L()Tr L(R) + Tr L()L(R) 4 F 3,6 qq (R,T) = T lnc 3,6 qq (R,T) Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.7/27

32 Z(3) symmetry and volume scaling < Tr L >. N τ =4. T/T c Polyakov loop and diquark correlation functions not Z(3) symmetric - In the confined phase all non Z(3) symmetric observables vanish - Z(3) rotation such that Re L > - At β = the Polyakov loop scales like N σ L = V x Tr L x V. - We assume this scaling behaviour for L and C 3,6 qq (r,t) for T < T c ( Cqq(r,T) 3 N σ ) 3ν(r,T) = V ν(r,t). Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.8/27

33 Z(3) symmetry and volume scaling ln C 3 qq ν RT < TrL > ln(/n 3 σ ) Polyakov loop and diquark correlation functions not Z(3) symmetric - In the confined phase all non Z(3) symmetric observables vanish - Z(3) rotation such that Re L > - At β = the Polyakov loop scales like RT ln < TrL >.45 T/T c L = V x Tr L x V. - We assume this scaling behaviour for L and C 3,6 qq (r,t) for T < T c ( Cqq(r,T) 3 N σ ) 3ν(r,T) = V ν(r,t). Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.8/27

34 Z(3) symmetry and volume scaling C 3 qq - Below T c the diquark free energies diverge RT /V c< TrL >.55.e+ 5.e-5.e-4.5e-4 2.e-4 2.5e-4 3.e-4 3.5e-4 - Polyakov loop and diquark correlation functions not Z(3) symmetric - In the confined phase all non Z(3) symmetric observables vanish - Z(3) rotation such that Re L > - At β = the Polyakov loop scales like - Above T c the Z(3) symmetry is spontanously broken - In full QCD the Z(3) symmetry is explicitly broken at all T - All observables are finite in the thermodynamic limit L = V x Tr L x V. - We assume this scaling behaviour for L and C 3,6 qq (r,t) for T < T c ( Cqq(r,T) 3 N σ ) 3ν(r,T) = V ν(r,t). Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.8/27

35 Diquark free energies - qq vs. q q in the deconfined phase. normalized free energy -.5 F i (r,t) := F i (r,t) F i (r,t) F q q (R,T)/T 2 F 3 qq (R,T)/T RT T/T c is a renormalization invariant quantity. perturbative relation between diquark and quar antiquark free energies F 3 qq(r,t) 2 F q q(r,t) good approximation above T c. Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.9/27

36 Diquark free energies - qq vs. q q in the deconfined phase. normalized free energy -.5 F i (r,t) := F i (r,t) F i (r,t) F q q (R,T)/T 2 F 3 qq (R,T)/T RT T/T c is a renormalization invariant quantity. perturbative relation between diquark and quar antiquark free energies F 3 qq(r,t) 2 F q q(r,t) good approximation above T c. temperature dependence for all separations = entropy contributions play a role for all r F q q (R,T)/σ/2 F 3 qq (R,T)/σ/2 Rσ /2 T/T c same asymptotic value for qq and q q = quarks in both systems are screened independently by the medium Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.9/27

37 3-quark free energies [ K. Hubner, OK, O. Vogt., hep-lat/483 ] Different color channels: = F(R,T)/σ (/2) exp ( F qqq/t ) = 6 27Tr L Tr L 2 Tr L 3 9Tr L Tr (L 2 L 3 ) 9Tr L 2 Tr (L L 3 ) 9Tr L 3 Tr (L L 2 ) + 3Tr (L L 2 L 3 )+3Tr (L L 3 L 2 ) T c exp ( F 8 qqq/t ) = 27Tr L Tr L 2 Tr L 3 + 9Tr L Tr (L 2 L 3 ) Tr L 3 Tr (L L 2 ) 3Tr (L L 3 L 2 ) -6 Rσ (/2) Tqqq= average singlet octet decuplet ( ) exp Fqqq/T 8 = 27Tr L Tr L 2 Tr L 3 + 9Tr L 3 Tr (L L 2 ) 24 9Tr L Tr (L 2 L 3 ) 3Tr (L L 2 L 3 ) exp ( F qqq/t ) = 27Tr L Tr L 2 Tr L 3 + 9Tr L Tr (L 2 L 3 ) 6 + 9Tr L 2 Tr (L L 3 )+9Tr L 3 Tr (L L 2 ) + 3Tr (L L 2 L 3 )+3Tr (L L 3 L 2 ) Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.2/27

38 3-quark free energies [ K. Hubner, OK, O. Vogt., hep-lat/483 ] F qqq (R,T)/σ/2 3(F 3 qq (R,T)-F q (T))/σ/2 2.5 F(R,T)/σ (/2) Rσ /2 T/T c Renormalization of 3-quark Polyakov loop correlation functions T c Tqqq= -6 average singlet Rσ (/2) octet decuplet F qqq (r,t) = ( Z R (g 2 ) ) 3N τ f (L x,l y,l z ) - Renormalization of n-point functions with the same Z R (g 2 ) - Comparison with qq free energies F qqq (r) F qq (r,t) 3F q (T) 3 2 F q q(r,t) with F q (T) = /2F q q (r ) qq - Baryonic interaction dominated by two-particle interactions Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.2/27

39 3-quark free energies [ K. Hubner, OK, O. Vogt., hep-lat/483 ] 2.5 F qqq (R,T)/σ/2 3(F 3 qq (R,T)-F q (T))/σ/ Quenched Rσ /2 T/T c F qqq (R,T)/σ/2 - Renormalization of 3-quark Polyakov loop correlation functions Full QCD 3(F 3 qq (R,T)-F q (T))/σ/2 Rσ /2 T/T c F qqq (r,t) = ( Z R (g 2 ) ) 3N τ f (L x,l y,l z ) - Renormalization of n-point functions with the same Z R (g 2 ) - Comparison with qq free energies F qqq (r) F qq (r,t) 3F q (T) 3 2 F q q(r,t) with F q (T) = /2F q q (r ) qq - Baryonic interaction dominated by two-particle interactions Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.2/27

40 qq and qqq entropy and internal energies The r-dependence of static qq, qq and qqq interactions are related by Casimir factors The relations for entropy and internal energy follow directly (F q (T) = /2F q q (r )): F 3 qq(r,t) 2 F qq(r,t)+f q (T) S 3 qq(r,t) 2 S qq(r,t)+s q (T) U 3 qq(r,t) 2 U qq(r,t)+u q (T) and for the baryonic system: F qqq(r,t) 3 2 F qq(r,t) S qqq(r,t) 3 2 S qq(r,t) U qqq(r,t) 3 2 U qq(r,t) Potential models based on this relations possible Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.2/27

41 SQGP - Strongly Coupled QGP??? J. Liao, E.V. Shuryak, hep-ph/5835 Bound states of quasi-particles above T c? (here m = 8 MeV)? T c E b 2 4 Is the interaction in the different channels strong enough to support bound states above T c? 6 meson bound states diquark bound states baryon bound states 3 gluon bound states T T c channel rep. charge factor no. of states gg 9/4 9 s gg 8 9/8 9 s 6 qg+ qg 3 9/8 3 c 6 s 2 N f qg+ qg 6 3/8 6 c 6 s 2 N f qq 8 s N 2 f qq+ q q 3 /2 4 s 3 c 2 N 2 f Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.22/27

42 SQGP - Strongly Coupled QGP??? J. Liao, E.V. Shuryak, hep-ph/5835 Bound states of quasi-particles above T c? (here m = 8 MeV)? T c Is the interaction in the different channels strong enough to support bound states above T c? E b meson bound states diquark bound states baryon bound states 3 gluon bound states Binding energy E b of the order of T already at T c! Coloured bound states and Polymer Chains are speculative T T c channel rep. charge factor no. of states (a) T<Tc (b) T=Tc gg 9/4 9 s gg 8 9/8 9 s 6 qg+ qg 3 9/8 3 c 6 s 2 N f qg+ qg 6 3/8 6 c 6 s 2 N f qq 8 s N 2 f qq+ q q 3 /2 4 s 3 c 2 N 2 f (c) (d) T > 3Tc Tc<T<.5Tc Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.22/27

43 What are the constituents of matter at high temperature? [S.Ejiri,F.Karsch,K.Redlich Recent discussion of quark number and charge fluctuations in the QGP Specific ratios of cummulants of (net) quark number and charge can characterize the particular degrees of freedom which are relevant in the low and high temperature phase 5 5 q R 4,2 HRG SB d x 2 V T 3 d x 4 V T 3 quark number R q 4,2 = dq 4 d q 2 2 lnz (µ x /T) 2 4 lnz (µ x /T) 4 qq T/T µu =µ d = µu =µ d = Q R 4,2 HRG charge R Q 4,2 = dq 4 d Q 2 SB qq T/T Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.23/27

44 What are the constituents of matter at high temperature? [S.Ejiri,F.Karsch,K.Redlich Recent discussion of quark number and charge fluctuations in the QGP Specific ratios of cummulants of (net) quark number and charge can characterize the particular degrees of freedom which are relevant in the low and high temperature phase 5 5 q R 4,2 HRG SB d x 2 V T 3 d x 4 V T 3 quark number R q 4,2 = dq 4 d q 2 2 lnz (µ x /T) 2 4 lnz (µ x /T) 4 qq T/T µu =µ d = µu =µ d = Q R 4,2 HRG charge R Q 4,2 = dq 4 d Q 2 SB qq T/T Already for T >.5T c quark number and charge are predominantly carried by states with quantum numbers of quarks See also V. Koch, A. Majumder, J. Randrup (25) Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.23/27

45 Renormalized Polyakov loop ( L ren = exp ) F(r =,T) 2T Defined by long distance behaviour of F(r,T). Renormalized by renormalization of F(r, T) at small distances. F [MeV].76T c.8t c 5.9T c.96t c.t c.2t c.7t c.23t c.5t c.98t c r [fm] 4.T c L ren = ( Z R (g 2 ) ) N t L lattice 5 F [MeV] N f = N f =2 N f = T/T c 4 Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.24/27

46 Renormalized Polyakov loop L ren ( L ren = exp ) F(r =,T) 2T F [MeV].76T c.8t c 5.9T c.96t c.t c.2t c.7t c.23t c.5t c.98t c r [fm] 4.T c F-QCD: N τ =4 SU(3): N τ =4 N τ =8 5 F [MeV] N f = N f =2 N f =3 5 T/T c T/T c 4 Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.24/27

47 Renormalized Polyakov loop ( L ren = exp ) F(r =,T) 2T L ren Quenched QCD: L ren = for T < T c. Finite gap at T c.5 2F-QCD: N τ =4 SU(3): N τ =4 N τ =8 Full QCD: L ren finite for all T. Strong increase near T c. T/T c Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.24/27

48 Renormalized Polyakov loop ( L ren = exp ) F(r =,T) 2T L ren Quenched QCD: L ren = for T < T c. Finite gap at T c.5 2F-QCD: N τ =4 SU(3): N τ =4 N τ =8 Full QCD: L ren finite for all T. Strong increase near T c. Renormalized T/T c Polyakov loop well4defined in quenched and full QCD - non-zero for finite quark mass - strong increase near T c Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.24/27

49 Renormalized Polyakov loop ( L ren = exp ) F(r =,T) 2T.5.5 L ren N τ =4 N τ =8 perturbation theory: L ren (T) approaches high temperature limit from above L ren (T) m D (T) α(t) T T/T c Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.24/27

50 Polyakov loops in higher representations [S. Gupta,K. Hübner,OK] use character property of direct product rep: χ p q (g) = χ p (g)χ q (g) Z(3) symmetry: L D e itφ L D triality (Z(3) charge): t p q mod 3 adjoint link variable [ U D=8] i j := 2 Tr [ U D=3 λ i (U D=3 ) λ j ] D (p,q) t C 2 (r) d D = C D /C F L D (x) 3 (,) 4/3 L 3 6 (2,) 2 /3 5/2 L3 2 L 3 8 (,) 3 9/4 L 3 2 (3,) 6 9/2 L 3 L 6 L 8 5 (2,) 6/3 4 L3 L 6 L 3 5 (4,) 28/3 7 L 3 L L 5 24 (3,) 2 25/3 25/4 L3 L L 6 27 (2,2) 8 6 L 6 2 L 8 Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.25/27

51 Polyakov loops in higher representations [S. Gupta,K. Hübner,OK] perturbation theory: F D (r,t) = C D α s (r) r for rλ QCD renormalization of the Polykov loop: L ren D = ( Z D (g 2 ) ) N τ d D L bare D, i.e. the renormalization constants are related by Casimir [G.Bali, Phys.Rev.D62 (2) 453] 3 2 F sing 8 (r,t)/σ / rσ /2 T/T c d 8 V 3 (T=) Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.25/27

52 Polyakov loops in higher representations [S. Gupta,K. Hübner,OK] perturbation theory: F D (r,t) = C D α s (r) r for rλ QCD renormalization of the Polykov loop: L ren D = ( Z D (g 2 ) ) N τ d D L bare D, i.e. the renormalization constants are related by Casimir [G.Bali, Phys.Rev.D62 (2) 453].6 Z D (g 2 ).5.4 r3 r6 r Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.25/27 g 2

53 Polyakov loops in higher representations [S. Gupta,K. Hübner,OK] Does Casimir scaling hold beyond leading order? Singlet free energies of static quarks in representation D=3,8: F sing D (r,t) T ( = ln TrL ) D (x)l D (y) GF with L D made up of U D=3 and [ U D=8] i j := 2 Tr [ U D=3 λ i (U D=3 ) λ j ].2 F sing D (r,t)/c 2 (D)/T D=8 D=3 rt c T/T c Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.25/27

54 Polyakov loops in higher representations [S. Gupta,K. Hübner,OK] Does Casimir scaling hold beyond leading order? Yes! Even the Polyakov loops are related by Casimir L d L r d D, with d D = C 2 (D)/C 2 (3). This relation holds for bare and renormalized Polykov loops:.6.5 L D bare D=3 D=6 D=8 D= D=5. T/T c Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.25/27

55 Polyakov loops in higher representations [S. Gupta,K. Hübner,OK] Does Casimir scaling hold beyond leading order? Yes! Even the Polyakov loops are related by Casimir L d L r d D, with d D = C 2 (D)/C 2 (3). This relation holds for bare and renormalized Polykov loops:.6 L ren L 3 L 6 L 8 L L 5.2 T/T c Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.25/27

56 Conclusions - Heavy quark free and internal energies Complex r and T dependence Running coupling shows remnants of confinenement above T c Entropy contributions play a role at finite T Separation of entropy and internal energy contributions possible Potential energy larger than free energy - Charmonium in the quark gluon plasma First estimates from potential models Higher dissociation temperature using V (directly produced) J/ψ exist well above T c Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.26/27

57 Outlook - Heavy quark free and internal energies Smaller quark masses Smaller lattice spacing to analyze small r dependence Better understanding of the different contributions - Charmonium in the quark gluon plasma What is the correct V e f f in potential models Full QCD calculations of correlation/spectral functions Relevant processes for charmonium production/dissociation? Exploration of Hadron Structure and Spectroscopy, Seattle, April, 26 p.27/27