Heavy quark free energies, screening and the renormalized Polyakov loop

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1 Heavy quark free energies, screening and the renormalized Polyakov loop Olaf Kaczmarek Felix Zantow Universität Bielefeld October 6, 26 VI Workshop III, Rathen, October, 26 p./9

2 Charmonium suppression ψ S(J/ ) observed charmonium yields at SPS+RHIC 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? VI Workshop III, Rathen, October, 26 p.2/9

3 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) VI Workshop III, Rathen, October, 26 p.3/9

4 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 VI Workshop III, Rathen, October, 26 p.4/9

5 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. Zantow (22, 24) N f = 2: lattices ( Symanzik, p4-stagg. ) m π /m ρ.7 (m/t =.4) O. Kaczmarek, F. Zantow (25), O. Kaczmarek et al. (23) N f = 3: lattices m π /m ρ.4 P. Petreczky, K. Petrov (24) VI Workshop III, Rathen, October, 26 p.5/9

6 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 VI Workshop III, Rathen, October, 26 p.5/9

7 Heavy quark free energy F [MeV] 5-5 (a) r [fm].8t c.9t c.96t 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) VI Workshop III, Rathen, October, 26 p.6/9

8 Heavy quark free energy F [MeV] 5-5 (a) r [fm].8t c.9t c.96t c.2t c.7t c.23t c.5t c.98t c 4.T c F [MeV] 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 -2 r [fm] VI Workshop III, Rathen, October, 26 p.6/9

9 Heavy quark free energy F [MeV] (a) String breaking T < T c F(r σ,t) < 5-5 r [fm].8t c.9t c.96t 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 VI Workshop III, Rathen, October, 26 p.6/9

10 Heavy quark free energy F [MeV] (a) String breaking T < T c F(r σ,t) < 5-5 r [fm].8t c.9t c.96t 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 VI Workshop III, Rathen, October, 26 p.6/9

11 Heavy quark free energy F [MeV] (a) String breaking T < T c F(r σ,t) < 5-5 r [fm].8t c.9t c.96t 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? VI Workshop III, Rathen, October, 26 p.6/9

12 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 VI Workshop III, Rathen, October, 26 p.7/9

13 Temperature depending running coupling 2.5 α qq (r,t) Free energy in perturbation theory: T/T c 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..5 r [fm] non-perturbative confining part for r>.4 fm α qq (r,t) = 3 4 r2 df (r,t) dr α qq (r) 3/4r 2 σ present below and just above T c remnants of confinement at T> T c temperature effects set in at smaller r with increasing T maximum due to screening VI Workshop III, Rathen, October, 26 p.7/9

14 Temperature depending running coupling 2.5 α qq (r,t) Free energy in perturbation theory: T/T c 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..5 r [fm] non-perturbative confining part for r>.4 fm α qq (r,t) = 3 4 r2 df (r,t) dr α qq (r) 3/4r 2 σ present below and just above T c remnants of confinement at T> 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 VI Workshop III, Rathen, October, 26 p.7/9

15 Temperature depending running coupling ~ define α qq (T) by maximum of α qq (r,t): α qq (T) 2-flavor QCD ւ quenched α qq (T) α qq (r max,t) perturbative behaviour at high T : ( ) µt g 2 2-loop (T) = 2β ln + β ( ( )) µt ln 2ln, Λ MS β Λ MS T/T c 2.5 Using T c /Λ MS =.77(2) we find µ=.4(2)π VI Workshop III, Rathen, October, 26 p.8/9

16 Temperature depending running coupling ~ define α qq (T) by maximum of α qq (r,t): α qq (T) 2-flavor QCD ւ quenched α qq (T) α qq (r max,t) perturbative behaviour at high T : ( ) µt g 2 2-loop (T) = 2β ln + β ( ( )) µt ln 2ln, Λ MS β Λ MS T/T c 2.5 Using T c /Λ MS =.77(2) we find µ=.4(2)π non-perturbative large values near T c not a large Coulombic coupling remnants of confinement at T> T c string breaking and screening difficult to separate slope at high T well described by perturbation theory VI Workshop III, Rathen, October, 26 p.8/9

17 Temperature depending running coupling ~ define α qq (T) by maximum of α qq (r,t): α qq (T) 2-flavor QCD ւ quenched α qq (T) α qq (r max,t) perturbative behaviour at high T : ( ) µt g 2 2-loop (T) = 2β ln + β ( ( )) µt ln 2ln, Λ MS β Λ MS T/T c 2.5 Using T c /Λ MS =.77(2) we find µ=.4(2)π non-perturbative large values near T c not a large Coulombic coupling remnants of confinement at T> T c string breaking and screening difficult to separate = At which distance do T -effects set in? = calculation of the screening mass/radius slope at high T well described by perturbation theory VI Workshop III, Rathen, October, 26 p.8/9

18 Screening mass - perturbative vs. non-perturbative effects m D /T N f = N f =2 Screening masses obtained from fits to: F (r,t) F (r =,T) = 4α(T) e m D(T)r 3r at large distances rt > T/T c leading order perturbation theory: m D (T) T ( = A + N f 6 ) /2 g(t) VI Workshop III, Rathen, October, 26 p.9/9

19 Screening mass - perturbative vs. non-perturbative effects 4 3 m D /T A Nf = =.52(2) A Nf =2 =.42(2) N f = N f =2 Screening masses obtained from fits to: F (r,t) F (r =,T) = 4α(T) e m D(T)r 3r 2 m D /T 3 A Nf = =.39(2) 2 T/T c at large distances rt > leading order perturbation theory: m D (T) T ( = A + N f 6 ) /2 g(t) perturbative limit reached very slowly T/T c T depencence qualitatively described by perturbation theory But A = non-perturbative effects A in the (very) high temperature limit Difference between N f =,2 disappears when converting to physical units VI Workshop III, Rathen, October, 26 p.9/9

20 Screening mass - density dependence [M.Döring et al.] leading order perturbation theory: m D (T,µ q ) = g(t) + N f T 6 + N ( f µq 2π 2 T ) 2 Taylor expansion: m D (T) = m (T)+m 2 (T) ( µq T ) 2 +O (µ 4 q ) m /T 2 m 2 /T /2 m av 2 /T T/T c T/T c m 2 (T) agrees with perturbation theory for T>.5T c non-perturbative effects dominated by gluonic sector VI Workshop III, Rathen, October, 26 p./9

21 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 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 med : V(r med ) F (r,t) r D [fm] <r> r med r entropy 2/m D (a) r (b) 2r cloud (c).. T/T c VI Workshop III, Rathen, October, 26 p./9

22 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 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 VI Workshop III, Rathen, October, 26 p./9

23 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] T d /T c T d /T c T d /T c VI Workshop III, Rathen, October, 26 p.2/9

24 Renormalized Polyakov loop Using short distance behaviour of free energies Renormalization of F(r, T) at short distances e F(r,T)/T = ( Z r (g 2 ) ) 2N τ Tr (L x L y) Renormalization of the Polyakov loop 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 =3 L ren defined by long distance behaviour of F(r,T) ( L ren = exp ) F(r =,T) 2T T/T c 4 VI Workshop III, Rathen, October, 26 p.3/9

25 Renormalized Polyakov loop F [MeV].5 L ren 2F-QCD: N τ =4 SU(3): N τ =4 N τ =8.76T 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 [MeV] N f = N f =2 N f =3 T/T c ( L ren = exp ) F(r =,T) 2T T/T c 4 VI Workshop III, Rathen, October, 26 p.3/9

26 Renormalized Polyakov loop.5 L ren.5 N τ =4 N τ =8 High temperature limit, L ren =, reached from above as expected from PT Clearly non-perturbative effects below 5T c Renormalized T/T c Polyakov loop well defined 25 in quenched and full QCD - non-zero for finite quark mass - strong increase near T c ( L ren = exp ) F(r =,T) 2T VI Workshop III, Rathen, October, 26 p.3/9

27 Renormalization constants Renormalization constants obtained from heavy quark free energies.4 Z R (g 2 ).3.2. g 2 4 =6/β The renormalization constants depend on the bare coupling, i.e. Z R (g 2 ) N τ ( ) Z R (g 2 ) exp g 2 (N 2 )/NQ (2) + g 4 Q (4) +O (g 6 ) with Q (2) =.597(3) consistent with lattice perturbation theory (Heller + Karsch, 985) VI Workshop III, Rathen, October, 26 p.4/9

28 Direct renormalization of the Polyakov loop [S. Gupta,K. Hübner,OK] Instead of renormalizing heavy quark free energies Use Polyakov loops obtained at different N τ Assume no volume dependence (T > T c ) The renormalization constants only depend on coupling, i.e. Z(g 2 ) L R D (T) L D (g 2,N τ ) g fixed N τ, N τ,2 T T start VI Workshop III, Rathen, October, 26 p.5/9

29 Direct renormalization of the Polyakov loop [S. Gupta,K. Hübner,OK] Instead of renormalizing heavy quark free energies Use Polyakov loops obtained at different N τ Assume no volume dependence (T > T c ) The renormalization constants only depend on coupling, i.e. Z(g 2 ) L R D (T) g fixed.2 L D (g 2,N τ ) N τ,. N τ,2.8.6 L 3 R N τ =4 N τ =6 N τ =8.4.2 T T start. T/T c VI Workshop III, Rathen, October, 26 p.5/9

30 Direct renormalization of the Polyakov loop [S. Gupta,K. Hübner,OK] Instead of renormalizing heavy quark free energies Use Polyakov loops obtained at different N τ Assume no volume dependence (T > T c ) The renormalization constants only depend on coupling, i.e. Z(g 2 ) L ren direct renormalization plc renormalization Z R (g 2 ) direct renormalization plc renormalization.2 T/T c Both renormalization procedures are equivalent g 2 solely based on gauge-invariant quantities VI Workshop III, Rathen, October, 26 p.5/9

31 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 VI Workshop III, Rathen, October, 26 p.6/9

32 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 free energies: e F D (r,t)/t = ( Z r (g 2 ) ) 2d D N τ Tr (L D x L D y ) i.e. the renormalization constants are related by Casimir [G.Bali, Phys.Rev.D62 (2) 453] 3. F,R Q Q,8 (r,t)/σ/ rσ /2 T/T c VI Workshop III, Rathen, October, 26 p.6/9

33 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 ]. F,R Q Q,D (r,t)/c(d)/σ/ D=8 D=3 rσ /2 T/T c VI Workshop III, Rathen, October, 26 p.6/9

34 Polyakov loops in higher representations [S. Gupta,K. Hübner,OK] Does Casimir scaling, L /C F 3 L /C D D, hold beyond two-loop order? 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] Z R D (g2 ) D=3 D=6 D=8 D= D=5 D sing =3 D sing = g 2 VI Workshop III, Rathen, October, 26 p.6/9

35 Polyakov loops in higher representations [S. Gupta,K. Hübner,OK] Does Casimir scaling, L /C F 3 L /C D D, hold beyond two-loop order? 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] Casimir scaling equivalent for bare or renormalized Polyakov loops.5 <L D > /d D D=3.3 D=6 D=8 D=.25 D=5 D=5 D=24.2 D=27.5 T/T c VI Workshop III, Rathen, October, 26 p.6/9

36 Polyakov loops in higher representations [S. Gupta,K. Hübner,OK] Does Casimir scaling, L /C F 3 L /C D D, hold beyond two-loop order? 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] Casimir scaling equivalent for bare or renormalized Polyakov loops.6 L R D D=3 D=6 D=8 D= D=5 D sing =3 D sing = T/T c VI Workshop III, Rathen, October, 26 p.6/9

37 String breaking for adjoint Polyakov loops [S. Gupta,K. Hübner,OK] string breaking expected for representations with triality t = non-zero for L R 8 even below T c.5.4 L R D D=3 D= T/T c VI Workshop III, Rathen, October, 26 p.7/9

38 String breaking for adjoint Polyakov loops [S. Gupta,K. Hübner,OK] string breaking expected for representations with triality t = non-zero for L R 8 even below T c finite values of F 8 (r) at large distances [GeV] F R Q Q,8 d 8 F,R Q Q,3 F,R Q Q,8. r string (F 8 ) r string (V 8 ) r [fm] VI Workshop III, Rathen, October, 26 p.7/9

39 String breaking for adjoint Polyakov loops [S. Gupta,K. Hübner,OK] string breaking expected for representations with triality t = non-zero for L R 8 even below T c finite values of F 8 (r) at large distances might be related to binding energy of gluelumps 2.8 F [GeV] r string [fm]..8 V 8 F 8 T/T c VI Workshop III, Rathen, October, 26 p.7/9

40 Polyakov loops in higher representations [S. Gupta,K. Hübner,OK] 2-flavour QCD:.3 <L D > /d D D=3. D=6 D=8 D=.5 D= T/T c VI Workshop III, Rathen, October, 26 p.8/9

41 Polyakov loops in higher representations [S. Gupta,K. Hübner,OK] 2-flavour QCD:.6.2 L R D D=3 D=6 D=8 D= D= Freedom to set the scale:. T/T. c V T= (r) V T= (r)+c Be carefull to extract T c by slope of L R Susceptibility not renormalized in this way L R D L R D exp(c/2t) Z R D Z R D exp(ca(g 2 )/2) VI Workshop III, Rathen, October, 26 p.8/9

42 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 Non-perturbative effects in m D up to high T Non-perturbative effects dominated by gluonic sector - Renormalized Polyakov loop Two consistent renormalization procedures Renormalization constant depend on the bare coupling Applicable for higher representations Casimir scaling good approximation at high T - 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 VI Workshop III, Rathen, October, 26 p.9/9