Heavy quark free energies, screening and the renormalized Polyakov loop

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

Charmonium suppression..75.5 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) 2 3 4 5 L [ f m] ε [GeV/ f m 3 ] T/T c 4. 8 3.6 4.25 [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

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

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

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 = : 32 3 4,8,6-lattices ( Symanzik ) O. Kaczmarek, F. Karsch, P. Petreczky, F. Zantow (22, 24) N f = 2: 6 3 4-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: 6 3 4-lattices m π /m ρ.4 P. Petreczky, K. Petrov (24) VI Workshop III, Rathen, October, 26 p.5/9

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

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.5.5 2 2.5 3 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

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 6 4.5.5 2 2.5 3 2 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]..2.3.4.5.6.7 VI Workshop III, Rathen, October, 26 p.6/9

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 5.5.5 2 2.5 3 F [MeV] N f = N f =2 N f =3 T -independent r / σ F(r,T) g 2 (r)/r 5 2 3 T/T c 4 VI Workshop III, Rathen, October, 26 p.6/9

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 5.5.5 2 2.5 3 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 5 2 3 T/T c 4 VI Workshop III, Rathen, October, 26 p.6/9

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.5.5 2 2.5 3 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

Temperature depending running coupling 2.5 α qq (r,t) Free energy in perturbation theory: 2.5.5 T/T c.8.9.96.2.7.23.5.98 4...5 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

Temperature depending running coupling 2.5 α qq (r,t) Free energy in perturbation theory: 2.5.5 T/T c.8.9.96.2.7.23.5.98 4. 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

Temperature depending running coupling 2.5 2.5.5 ~ 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.5.5 2 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

Temperature depending running coupling 2.5 2.5.5 ~ 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.5.5 2 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

Screening mass - perturbative vs. non-perturbative effects 4 3 2 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.5 2 2.5 3 3.5 4 leading order perturbation theory: m D (T) T ( = A + N f 6 ) /2 g(t) VI Workshop III, Rathen, October, 26 p.9/9

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.5 2 2.5 3 3.5 4 at large distances rt > leading order perturbation theory: m D (T) T ( = A + N f 6 ) /2 g(t) perturbative limit reached very slowly 5 5 2 25 T/T c T depencence qualitatively described by perturbation theory But A.4 -.5 = 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

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 ) 4 3.5 m /T 2 m 2 /T /2 m av 2 /T 3.5 2.5 2.5.5.5 T/T c.5 2 2.5 3 3.5 4 T/T c.5 2 2.5 3 3.5 4 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

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 5 5 2 25 VI Workshop III, Rathen, October, 26 p./9

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

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].64.2.5..67.54.3.2 T d /T c..74.-.2 2.3.3..83.74 T d /T c 2..2.2 5..95.65 - - T d /T c.75-.95.3-.5.-. 4.4-4.7.5-.6.4-.5.2.2 VI Workshop III, Rathen, October, 26 p.2/9

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 -5.5.5 2 2.5 3 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 5 2 3 T/T c 4 VI Workshop III, Rathen, October, 26 p.3/9

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 -5.5.5 2 2.5 3 5 F [MeV] N f = N f =2 N f =3 T/T c.5.5 2 2.5 3 3.5 4 ( L ren = exp ) F(r =,T) 2T 5 2 3 T/T c 4 VI Workshop III, Rathen, October, 26 p.3/9

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 5 5 2 - 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

Renormalization constants Renormalization constants obtained from heavy quark free energies.4 Z R (g 2 ).3.2. g 2 4 =6/β 8.6.8.2.4.6 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

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 2 3 4 5 6 7 8 9 2 VI Workshop III, Rathen, October, 26 p.5/9

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 ).2.8.6.4 L ren direct renormalization plc renormalization.6.5.4.3 Z R (g 2 ) direct renormalization plc renormalization.2 T/T c 2 4 6 8 2.2 Both renormalization procedures are equivalent..8.9..2.3.4.5 g 2 solely based on gauge-invariant quantities VI Workshop III, Rathen, October, 26 p.5/9

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

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/2 2... -. -2. -3. -4. -5. rσ /2 T/T c.97.959.977.3.3.49.682 2.995.5..5 2. 2.5 VI Workshop III, Rathen, October, 26 p.6/9

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/2.5. -.5 -. -.5 D=8 D=3 rσ /2 T/T c.97.959.977.3.3.49.682 2.995.2.4.6.8..2.4.6.8 2. VI Workshop III, Rathen, October, 26 p.6/9

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].6.5.4.3 Z R D (g2 ) D=3 D=6 D=8 D= D=5 D sing =3 D sing =8.2..8.9..2.3.4.5 g 2 VI Workshop III, Rathen, October, 26 p.6/9

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.4.35.3.2. D=3.3 D=6 D=8 D=.25 D=5 D=5 D=24.2 D=27.5 T/T c..5 2. 2.5 3. 2 3 4 5 6 7 8 9 2 3 VI Workshop III, Rathen, October, 26 p.6/9

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.4.2..8.6.4 D=3 D=6 D=8 D= D=5 D sing =3 D sing =8.2.8.4.2. T/T..5 2. 2.5 3. c 2 3 4 5 6 7 8 9 2 VI Workshop III, Rathen, October, 26 p.6/9

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=8.3.2.25.2.5...9.95. T/T c..9.92.94.96.98..2.4.6 VI Workshop III, Rathen, October, 26 p.7/9

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 3.5 3. [GeV] 2.5 2..5..5 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] -.5.5..5 2. 2.5 VI Workshop III, Rathen, October, 26 p.7/9

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] 2.4 2..2 r string [fm]..8 V 8 F 8 T/T c.9.92.94.96.98. VI Workshop III, Rathen, October, 26 p.7/9

Polyakov loops in higher representations [S. Gupta,K. Hübner,OK] 2-flavour QCD:.3 <L D > /d D.2.5.. D=3. D=6 D=8 D=.5 D=5..8..2 T/T c 2 3 4 VI Workshop III, Rathen, October, 26 p.8/9

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=5.8.4.4.2 Freedom to set the scale:. T/T. c.8..2 2 3 4 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

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