Heavy quark free energies and screening from lattice QCD Olaf Kaczmarek Universität Bielefeld February 9, 29 RBC-Bielefeld collaboration O. Kaczmarek, PoS CPOD7 (27) 43 RBC-Bielefeld, Phys.Rev.D77 (28) 45 O. Kaczmarek, F. Zantow, Phys.Rev.D7 (25) 45 O. Kaczmarek, F. Zantow, hep-lat/569 Excited QCD 29, Zakopane, February 9, 29 p./7
Heavy quark bound states above deconfinement Strong interactions in the deconfined phase T> T c Possibility of heavy quark bound states? Suppression patterns of charmonium/bottomonium Charmonium (χ c,j/ψ) as thermometer above T c = 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 Excited QCD 29, Zakopane, February 9, 29 p.2/7
Zero temperature potential - n f =2+ 8 6 4 r V qq - (r) 2+ flavor QCD highly improved p4-staggered almost realistic quark masses.6.55.5.45.4.35 2-2 r/r 3.382 3.43 3.5 3.57 3.63 3.69 3.76 3.82 3.92 V string.5.5 2 2.5 3 ( ) renormalization: V T= (r) = log (Z ren (β)) 2 W(r,τ) W(r,τ+) Z ren (β) β.3 3. 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4. 4. r 2 dv qq(r) dr r 2 dv qq(r) dr =.65 r=r =. r=r (r =.469(7) fm) m π 22 MeV, physical m s Large distance behaviour consistent with string model prediction:.5.4.3.2. V(r) = π 2r used for renormalization cut-off effects are small.5..5.2.25.3 r /r r σ /2 + σ r, for large r a [fm] a/r....2.3.4.5.6.7 Excited QCD 29, Zakopane, February 9, 29 p.3/7
Zero temperature potential - n f =2+ 8 6 4 r V qq - (r) 2+ flavor QCD highly improved p4-staggered almost realistic quark masses 2-2 r/r 3.382 3.43 3.5 3.57 3.63 3.69 3.76 3.82 3.92 V string.5.5 2 2.5 3.5 m π 22 MeV, physical m s Large distance behaviour consistent with string model prediction: V(r) = π 2r used for renormalization r V(r) + σ r, for large r Short distance behaviour deviations at small r enhancement of the running coupling fit: V(r) =.392(6)/r+ σ r r-dependent running coupling α(r).5 -.5 - -.5-2 -2.5-3 -3.5-4 r/r V string Cornell Cornell+/r 2..2.3.4.5.6.7.8.9 Excited QCD 29, Zakopane, February 9, 29 p.3/7
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,...] log() = 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 ) HB-OR O. Kaczmarek, F. Karsch, P. Petreczky, F. Zantow (22, 24) N f = 2: 6 3 4-lattices ( Symanzik, p4-stagg. ) hybrid-r 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 ( stagg., Asqtad ) hybrid-r m π /m ρ.4 P. Petreczky, K. Petrov (24) N f = 2+: 24 4 6-lattices ( Symanzik, p4fat3 ) RHMC m π 22 MeV, phys. m s O. Kaczmarek (27), RBC-Bielefeld (28) Excited QCD 29, Zakopane, February 9, 29 p.4/7
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,...] log() = 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 Excited QCD 29, Zakopane, February 9, 29 p.4/7
Heavy quark free energy - 2+-flavors F(r,T) [MeV] Renormalization of F(r, T) 5-5 - T[MeV] 75 86 97 2 23 25 226 24 259 28 326 365 424 459 548 r [fm].5.5 2 2.5 using Z ren (g 2 ) obtained at T= e F (r,t)/t = ( Z r (g 2 ) ) 2N τ Tr (L x L y) alternative renormalization procedures all equivalent! (O. Kaczmarek et al.,plb543(22)4, S. Gupta et al., Phys.Rev.D77 (28) 3453) Excited QCD 29, Zakopane, February 9, 29 p.5/7
Heavy quark free energy - 2+-flavors F(r,T) [MeV] Renormalization of F(r, T) 5-5 - T[MeV] 75 86 97 2 23 25 226 24 259 28 326 365 424 459 548 r [fm] 4 xxx xxx xxx.5.5 2 2.5 T -independent r / σ F(r,T) g 2 (r)/r 2-2 -4-6 -8 F(r,T) [MeV] using Z ren (g 2 ) obtained at T= e F (r,t)/t = ( Z r (g 2 ) ) 2N τ Tr (L x L y) alternative renormalization procedures all equivalent! (O. Kaczmarek et al.,plb543(22)4, S. Gupta et al., Phys.Rev.D77 (28) 3453) - r [fm] -2.5..5.2.25.3.35.4.45.5 Excited QCD 29, Zakopane, February 9, 29 p.5/7
Heavy quark free energy - 2+-flavors 5-5 - F(r,T) [MeV] T[MeV] 75 86 97 2 23 25 226 24 259 28 326 365 424 459 548 r [fm] 2 xxx xxx xxx.5.5 2 2.5 T -independent r / σ F(r,T) g 2 (r)/r 8 6 4 2 F [MeV] String breaking T < T c F(r σ,t) < high-t physics rt ; screening µ(t) g(t)t F(,T) T SU(3): N τ =4 SU(3): N τ =8 2F-QCD: N τ =4 2+F-QCD: N τ =4 2+F-QCD: N τ =6-2 -4 2 3 4 5 Excited QCD 29, Zakopane, February 9, 29 p.5/7
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 L ren = ( Z R (g 2 ) ) N t L lattice L ren defined by long distance behaviour of F(r,T) ( L ren = exp ) F(r =,T) 2T 5-5 - 2 8 6 4 2-2 -4 F(r,T) [MeV].5.5 2 2.5 F [MeV] SU(3): N τ =4 SU(3): N τ =8 2F-QCD: N τ =4 2+F-QCD: N τ =4 2+F-QCD: N τ =6 T[MeV] 75 86 97 2 23 25 226 24 259 28 326 365 424 459 548 r [fm] 2 3 4 5 Excited QCD 29, Zakopane, February 9, 29 p.6/7
Renormalized Polyakov loop L ren..8.6.3.4.5.6.7.8.9. L ren Tr.5 SU(3): N τ =4 SU(3): N τ =8 2F-QCD: N τ =4 2+F-QCD: N τ =4 2+F-QCD: N τ =6.4.2 N τ =4 6 8 T [MeV]. 5 2 25 3 35 4 45.5.5 2 2.5 3 3.5 4 2 8 6 4 F [MeV] SU(3): N τ =4 SU(3): N τ =8 2F-QCD: N τ =4 2+F-QCD: N τ =4 2+F-QCD: N τ =6 ( L ren = exp ) F(r =,T) 2T 2-2 -4 2 3 4 5 Excited QCD 29, Zakopane, February 9, 29 p.6/7
Renormalized Polyakov loop SU(3) pure gauge results.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 5 5 2 25 ( L ren = exp ) F(r =,T) 2T Excited QCD 29, Zakopane, February 9, 29 p.6/7
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) Excited QCD 29, Zakopane, February 9, 29 p.7/7
Temperature depending running coupling.8.6.4.2.8.6.4 α eff (r,t) T[MeV] 56 75 8 86 97 2 23 25 226 24 259 28 326 365 424 459 548 Free energy in perturbation theory: 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.2 r[fm].5..2.3.4.5.6.7 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 = At which distance do T -effects set in? remnants of confinement at T> T c temperature effects set in at smaller r with increasing T maximum due to screening = definition of the screening radius/mass = definition of the T -dependent coupling Excited QCD 29, Zakopane, February 9, 29 p.8/7
Temperature depending running coupling.2 α max (T) nf= nf=2 nf=2+ Nt=6 nf=2+ Nt=4 define α qq (T) by maximum of α qq (r,t):.8 α qq (T) α qq (r max,t).6.4.2 perturbative behaviour at high T : ( ) µt g 2 2-loop (T) = 2β ln + β ( ( )) µt ln 2ln, Λ MS β Λ MS.5 2 2.5 3 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 Excited QCD 29, Zakopane, February 9, 29 p.9/7
Screening mass - perturbative vs. non-perturbative effects 5 4 m D /T N f =2 N f = Screening masses obtained from fits to: F (r,t) F (r =,T) = 4α(T) e m D(T)r 3r 3 2 at large distances rt > leading order perturbation theory:.5 2 2.5 3 3.5 4 m D (T) T ( = A + N f 6 ) /2 g(t) Excited QCD 29, Zakopane, February 9, 29 p./7
Screening mass - perturbative vs. non-perturbative effects 5 4 m D /T N f =2+ N f =2 N f = Screening masses obtained from fits to: F (r,t) F (r =,T) = 4α(T) e m D(T)r 3r 3 2 4 3 A Nf =2+ =.66(2) A Nf =2 =.42(2) A Nf = =.52(2).5 2 2.5 3 3.5 4 N f =2+ N f =2 N f = α α max at large distances rt > leading order perturbation theory: m D (T) T ( = A + N f 6 ) /2 g(t) perturbative limit reached very slowly 2 2 3 4 Excited QCD 29, Zakopane, February 9, 29 p./7
Screening mass - perturbative vs. non-perturbative effects 5 4 m D /T N f =2+ N f =2 N f = Screening masses obtained from fits to: F (r,t) F (r =,T) = 4α(T) e m D(T)r 3r 3 2 A Nf =2+ =.66(2) A Nf =2 =.42(2) A Nf = =.52(2).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 T depencence qualitatively described by perturbation theory But A.4 -.5 = non-perturbative effects A in the (very) high temperature limit 3 2 m D /T A Nf = =.39(2) 5 5 2 25 Excited QCD 29, Zakopane, February 9, 29 p./7
Screening mass - density dependence (N f = 2) [M.Doring et al., Eur.Phys.J. C46 (26) 79] leading order perturbation theory: m D (T,µ q ) = g(t) + N f T 6 + N ( f µq 2π 2 T Taylor expansion: m D (T) = m (T)+m 2 (T) ( µq T ) 2 ) 2 +O (µ 4 q ) 4 3.5 m /T 2 m 2 /T 3.5 2.5 2.5.5.5.5 2 2.5 3 3.5 4.5 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 Excited QCD 29, Zakopane, February 9, 29 p./7
Diquark free energies - Screening and string breaking [M.Doring et al., Phys.Rev.D75 (27) 5454].5 N QQ =.87 =.9 =.96 Net quark number induced by a qq-pair: N (c) QQ (r,t) = N q L (c) QQ N q = QQ (r,t), L (c) QQ (r,t) -.5 -.5..5 rt.5.5 2 N QQ =.2 =.2 =.7 =.6 =.5 where N q is the quark number operator, N q = ( ) ] D( ˆm,µ) [D 2 Tr ( ˆm,). µ µ= Net quark number induced by a single static quark source, N Q (T) = N q Tr P( ) N q Q =. Tr P( ) -.5 -. -.5 -.2 rt -.25.5.5 2 Excited QCD 29, Zakopane, February 9, 29 p.2/7
Diquark free energies - Screening and string breaking [M.Doring et al., Phys.Rev.D75 (27) 5454].5 N QQ =.87 =.9 =.96 Net quark number induced by a qq-pair: N (c) QQ (r,t) = N q L (c) QQ N q = QQ (r,t), L (c) QQ (r,t) -.5 -.5 rt.5.5 2 where N q is the quark number operator, N q = ( ) ] D( ˆm,µ) [D 2 Tr ( ˆm,). µ µ= Net quark number induced by a single static quark source, N Q (T) = N q Tr P( ) N q Q =. Tr P( ) -.5 - N QQ (r ) N QQ ( ) -.5-2.5.5 2 Diquark is neutralized by quarks or antiquarks from the vacuum to be color neutral overall lim N QQ(r,T) = T, r < r c, 2, r > r c Excited QCD 29, Zakopane, February 9, 29 p.2/7
Diquark free energies - Screening and string breaking [M.Doring et al., Phys.Rev.D75 (27) 5454].5 N QQ =.87 =.9 =.96 Net quark number induced by a qq-pair: N (c) QQ (r,t) = N q L (c) QQ N q = QQ (r,t), L (c) QQ (r,t) -.5 -.5 rt.5.5 2 where N q is the quark number operator, N q = ( ) ] D( ˆm,µ) [D 2 Tr ( ˆm,). µ µ= Net quark number induced by a single static quark source, N Q (T) = N q Tr P( ) N q Q =. Tr P( ) -.5 - N QQ (r ) N QQ ( ) -.5-2.5.5 2 Diquark is neutralized by quarks or antiquarks from the vacuum to be color neutral overall lim N QQ(r,T) = T, r < r c, 2, r > r c Excited QCD 29, Zakopane, February 9, 29 p.2/7
Heavy quark bound states above T c?.5 r [fm] r med 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 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).. 5 5 2 25 Excited QCD 29, Zakopane, February 9, 29 p.3/7
Heavy quark bound states above T c?.5 r [fm] r med 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 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 Excited QCD 29, Zakopane, February 9, 29 p.3/7
Free energy vs. Entropy at large separations 2 8 6 4 2-2 F [MeV] SU(3): N τ =4 SU(3): N τ =8 2F-QCD: N τ =4 2+F-QCD: N τ =4 2+F-QCD: N τ =6 Free energies not only determined by potential energy F = U T S Entropy contributions play a role at finite T S = F T -4 2 3 4 5 Excited QCD 29, Zakopane, February 9, 29 p.4/7
Free energy vs. Entropy at large separations 5 U [MeV] 45 F [MeV] High temperatures: 4 35 3 25 2 5 5-5 (a) U = T 2 F /T T.5.5 2 2.5 3 r (b) 2r cloud (c) 4 35 3 25 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) TS [MeV] F [MeV] The large distance behavior of the finite temperature energies is rather related to screening than to t he temperature dependence of masses of corresponding heavylight mesons! 2 5 5-5 S = F T.5.5 2 2.5 3 Excited QCD 29, Zakopane, February 9, 29 p.4/7
Free energy vs. Entropy at large separations 5 U [MeV] 45 F [MeV] High temperatures: 4 35 3 25 2 5 5-5 (a) U = T 2 F /T T.5.5 2 2.5 3 r (b) 2r cloud (c) 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 4 3 TS [MeV] O (g 5 T) N f = N f =2 N f =3 N f =2+ The large distance behavior of the finite temperature energies is rather related to screening than to t he temperature dependence of masses of corresponding heavylight mesons! 2 S = F T 2 3 Excited QCD 29, Zakopane, February 9, 29 p.4/7
r-dependence of internal energies (N f = 2+) 2 5 5-5 TS(r,T) [MeV] T[MeV] 9 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) - 9 8 7 6 5 4 3 2 r[fm].2.4.6.8.2 TS(r,T) [MeV] T[MeV] 233 249 345 394 important at intermediate/large distances - r[fm].2.4.6.8 Excited QCD 29, Zakopane, February 9, 29 p.5/7
r-dependence of internal energies (N f = 2+) 8 7 6 5 4 3 2 U(r,T) [MeV] T[MeV] 66 9 99 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) - -2 5 r[fm].2.4.6.8.2 U(r,T) [MeV] important at intermediate/large distances 5-5 - -5-2 T[MeV] 233 249 345 394 r[fm].2.4.6.8 = Implications on heavy quark bound states? = What is the correct V e f f (r,t)? Excited QCD 29, Zakopane, February 9, 29 p.5/7
Heavy quark bound states from potential models V eff (r,t) [MeV] V( ) E J/ψ (T=) 5.5.5 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) Excited QCD 29, Zakopane, February 9, 29 p.6/7
Conclusions - Heavy quark free energies, internal energies and entropy Results for almost physical quark masses, n f = 2+ 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 - Bound states in the quark gluon plasma Estimates from potential models? What is the correct potential for such models? Higher dissociation temperature using V (directly produced) J/ψ may exist well above T c Full QCD calculations of correlation/spectral functions needed What are relevant processes for charmonium? Excited QCD 29, Zakopane, February 9, 29 p.7/7