Hot and Dense QCD on the lattice Frithjof Karsch, BNL Introduction: T , gT , g 2 T ,... screening and the running coupling Bulk thermodynamics T c and the equation of state in (2+1)-flavor QCD with an almost realistic quark mass spectrum Thermodynamics at non-zero baryon number density hadronic fluctuations isentropic equation of state Conclusions F . Karsch, apeNEXT, Florence 2007 – p.1/32
Critical behavior in hot and dense matter: QCD phase diagram crossover vs. phase transition T deconfined, quark-gluon plasma χ -symmetric 170 ~ MeV hadron gas confined, χ -SB color superconductor µ o µ few times nuclear matter density F . Karsch, apeNEXT, Florence 2007 – p.2/32
Critical behavior in hot and dense matter: QCD phase diagram continuous/rapid continuous transition for (crossover) transition small chemical potential and small quark masses T deconfined, quark-gluon plasma χ -symmetric 170 ~ MeV hadron gas confined, χ -SB color superconductor µ o µ few times nuclear matter density F . Karsch, apeNEXT, Florence 2007 – p.2/32
Critical behavior in hot and dense matter: QCD phase diagram continuous/rapid continuous transition for (crossover) transition small chemical potential and small quark masses T deconfined, quark-gluon plasma χ -symmetric 170 ~ MeV 2nd order phase transition; chiral critical Ising universality class point T c ( µ ) under investigation hadron gas confined, location of CCP uncertain: χ -SB color volume and quark mass dependence superconductor µ o µ few times nuclear matter density F . Karsch, apeNEXT, Florence 2007 – p.2/32
Critical behavior in hot and dense matter: QCD phase diagram continuous/rapid continuous transition for (crossover) transition small chemical potential and small quark masses T deconfined, quark-gluon plasma χ -symmetric 170 ~ MeV 2nd order phase transition; chiral critical Ising universality class point T c ( µ ) under investigation hadron gas confined, location of CCP uncertain: χ -SB color volume and quark mass dependence superconductor µ o µ improving accuracy on T c , ǫ c , ǫ ( p ) and few times nuclear matter density the phase boundary is mandatory to make contact to HIC phenomenology F . Karsch, apeNEXT, Florence 2007 – p.2/32
Non-perturbative QGP Perturbation theory provides a hierachy of length scales T ≫ gT ≫ g 2 T ... ⇒ guiding principle for effective theories, resummation, dimensional reduction... Early lattice results show that g 2 ( T ) > 1 even at T ∼ 5 T c G. Boyd et al, NP B469 (1996) 419: SU(3) thermodynamics.. ...one has to conclude that the temperature dependent running coupling has to be large, g 2 ( T ) ≃ 2 even at T ≃ 5 T c the Debye screening mass is large close to T c the spatial string tension does not vanish above T c √ σ s � = 0 ⇒ the QGP is ”non-perturbative” up to very high T F . Karsch, apeNEXT, Florence 2007 – p.3/32
Screening of heavy quark free energies – remnant of confinement above T c – pure gauge: O.Kaczmarek, FK, P . Petreczky, F. Zantow, PRD70 (2005) 074505 2-flavor QCD: O.Kaczmarek, F. Zantow, Phys. Rev. D71 (2005) 114510 singlet free energy F 1 [MeV] 1000 0.76T c T ≃ T c : screening for 0.81T c r> ∼ 0 . 5 fm 0.90T c 500 0.96T c F 1 ( r, T ) ∼ α ( T ) 1.00T c e − µ ( T ) r 1.02T c r 1.07T c 0 1.23T c + const. 1.50T c 1.98T c 4.01T c r [fm] -500 0 0.5 1 1.5 2 2.5 3 qq ( r, T = 0) = − 4 α ( r, T = 0) F 1 ( r, T ) follows linear rise of V ¯ + σr 3 r for T < ∼ 1 . 5 T c , r< ∼ 0 . 3 fm F . Karsch, apeNEXT, Florence 2007 – p.4/32
��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� Singlet free energy and asymptotic freedom pure gauge: O.Kaczmarek, FK, P . Petreczky, F. Zantow, PRD70 (2005) 074505 2-flavor QCD: O.Kaczmarek, F. Zantow, Phys. Rev. D71 (2005) 114510 singlet free energy defines a running coupling: α eff = 3 r 2 0.6 d F 1 ( r, T ) T/T c α qq (r,T) T=0 4 d r 0.5 1.05 (in Coulomb gauge) 1.10 1.20 0.4 1.30 1.50 1.60 0.3 3.00 6.00 9.00 0.2 12.0 0.1 r [fm] 0 0.01 0.1 F . Karsch, apeNEXT, Florence 2007 – p.5/32
��� ��� ��� ��� ��� ��� ��� ��� Singlet free energy and asymptotic freedom pure gauge: O.Kaczmarek, FK, P . Petreczky, F. Zantow, PRD70 (2005) 074505 2-flavor QCD: O.Kaczmarek, F. Zantow, Phys. Rev. D71 (2005) 114510 singlet free energy defines a running coupling: α eff = 3 r 2 0.6 d F 1 ( r, T ) T/T c α qq (r,T) T=0 4 d r 0.5 1.05 (in Coulomb gauge) 1.10 1.20 0.4 1.30 1.50 1.60 0.3 large distance: constant 3.00 �� �� α ≡ π/ 16 �� �� 6.00 �� �� Coulomb term (string model) 9.00 0.2 12.0 short distance: running coupling α ( r ) from ( T = 0) , 3-loop ��� ��� ��� ��� ��� ��� 0.1 ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� (S. Necco, R. Sommer, r [fm] Nucl. Phys. B622 (2002) 328) 0 0.01 0.1 T-dependence starts in non-perturbative short distance physics ⇔ vacuum physics regime for T < ∼ 3 T c F . Karsch, apeNEXT, Florence 2007 – p.5/32
Singlet free energy and asymptotic freedom pure gauge: O.Kaczmarek, FK, P . Petreczky, F. Zantow, PRD70 (2005) 074505 2-flavor QCD: O.Kaczmarek, F. Zantow, Phys. Rev. D71 (2005) 114510 singlet free energy defines a running coupling: α eff = 3 r 2 0.6 d F 1 ( r, T ) T/T c α qq (r,T) T=0 4 d r 0.5 1.05 rise due to (in Coulomb gauge) 1.10 confinement 1.20 �� �� α eff ∼ σr 2 �� �� �� �� 0.4 1.30 1.50 1.60 0.3 large distance: constant 3.00 �� �� α ≡ π/ 16 �� �� 6.00 �� �� Coulomb term (string model) 9.00 0.2 12.0 short distance: running coupling α ( r ) from ( T = 0) , 3-loop ��� ��� ��� ��� ��� ��� 0.1 ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� (S. Necco, R. Sommer, r [fm] Nucl. Phys. B622 (2002) 328) 0 0.01 0.1 T-dependence starts in non-perturbative short distance physics ⇔ vacuum physics regime for T < ∼ 3 T c F . Karsch, apeNEXT, Florence 2007 – p.5/32
Non-perturbative Debye screening � 1 + n f leading order perturbation theory: m D = g ( T ) T 6 T c < T < ∼ 10 T c : non-perturbative effects are well represented by an ”A-factor”: m D ≡ Ag ( T ) T, A ≃ 1 . 5 perturbative limit is reached very slowly 100 10 10 10 10 1 0.5 4.0 (logarithms at work!!) T / Λ MS SU(3) m / gT m D /T N f =0 4 3.0 N f =2 m D /T=Ag(T) 3 A=1.42(2) 2.0 2 β G =144 32 3 1.0 β G =72 24 3 1 β G =40 24 3 g ( T ) ≃ 1 . 5 ⇔ α ( T ) ≃ 0 . 18 β G =24 24 3 β G =12 24 3 x T/T c 0 0.0 −3 −2 −1 10 10 10 1 1.5 2 2.5 3 3.5 4 O.Kaczmarek,F .Zantow, PRD 71 (2005) 114510 K.Kajantie et al, PRL 79 (1997) 3130 F . Karsch, apeNEXT, Florence 2007 – p.6/32
The spatial string tension Non-perturbative, vanishes in high-T perturbation theory: R x ,R y →∞ ln W ( R x , R y ) √ σ s = − lim R x R y √ σ s c M : 3-d SU(3), LGT g 2 ( T ) T = c M f M ( g ( T )) , c M = 0 . 553(1) g M ≡ g 2 f M : dim. red. pert. th. 1.2 g 2 ( T ) ≃ 2 ⇔ α ( T ) ≃ 0 . 16 T/ σ 1/2 dimensional reduction works for T> 4-d SU(3) and QCD ∼ 2 T c 1 - c M (almost) flavor independent - g 2 ( T ) shows 2-loop running 0.8 c = 0 . 566(13) [SU(3)] c = 0 . 594(39) [QCD] 0.6 N f =0 T/T c G. Boyd et al. NP B469 (1996) 419 N f =2+1 0.4 RBC-Bielefeld, preliminary 1 2 4 F . Karsch, apeNEXT, Florence 2007 – p.7/32
µ = 0 : Equation of State and T c 280 ε SB /T4 16.0 T c [MeV] strong deviations from ε = 3p 15% 260 14.0 dev. 12.0 ε /T 4 240 3p /T 4 10.0 220 8.0 n f =2, p4 200 6.0 n f =3, p4 3 flavor, N τ =4, p4 staggered n f =2, std 4.0 180 m π =770 MeV 2.0 m PS [MeV] 160 T/T c 0.0 0 500 1000 1500 2000 2500 3000 3500 1.0 1.5 2.0 2.5 3.0 3.5 4.0 QCD EoS transition temperature strong deviations from ideal T c = (173 ± 8 ± sys ) MeV gas behavior ( ǫ = 3 p ) for weak quark mass and flavor T c ≤ T ∼ 3 T c and even dependence at high T improved staggered fermions but still on rather coarse lattices: N τ = 4 , i.e. a − 1 ≃ 0 . 8 GeV with moderately light quarks FK, E. Laermann, A. Peikert, Nucl. Phys. B605 (2001) 579 F . Karsch, apeNEXT, Florence 2007 – p.8/32
EoS and T c Goal: QCD thermodynamics with realistic quark masses and controlled extrapolation to the continuum limit T c , EoS, µ q > 0 , ... use an improved staggered fermion action that removes O ( a 2 ) errors in bulk thermodynamic quantities and reduces flavor symmetry breaking inherent to the staggered formulation RBC-Bielefeld choice: p4-action + 3-link smearing (p4fat3) MILC: Naik-action + (3,5,7)-link smearing (asqtad); Wuppertal: standard staggered + exponentiated 3-link smearing (stout) F . Karsch, apeNEXT, Florence 2007 – p.9/32
Recommend
More recommend
