Thermodynamics of the Polyakov-Quark-Meson Model Bernd-Jochen Schaefer 1, Jan Pawlowski and Jochen Wambach 3 1 KFU Graz, Austria U Heidelberg, Germany 3 TU and GSI Darmstadt, Germany Virtual Institute "Dense Hadronic Matter and QCD Phase Transitions 15 th Oct. - 17 th Oct, 006 Rathen (Sächsische Schweiz), Germany B.-J. Schaefer (KFU Graz) 1 / 16
Outline 1 Motivation Polyakov-Quark-Meson Model 3 Results 4 Summary & Outlook B.-J. Schaefer (KFU Graz) / 16
QCD Phase Diagram QCD: two phase transitions 1 restoration of chiral symmetry SU L+R(N f ) SU L(N f ) SU R(N f ) order parameter: qq de/confinement in heavy-quark limit: Z(N c)-symmetry order parameter: φ (Polyakov loop) dynamical quarks: Z(N c)-symmetry explicitly broken combined in effective description: Polyakov quark-meson or PNJL models B.-J. Schaefer (KFU Graz) 3 / 16
Polyakov-quark-meson model quark-meson model: L qm = q[iγ µ µ +g(σ+i τ πγ 5 )]q + 1 ( µσ µ σ + µ π µ π) + V (σ, π) mesonic potential: V (σ, π) = λ 4 (σ + π v ) cσ B.-J. Schaefer (KFU Graz) 4 / 16
Polyakov-quark-meson model quark-meson model: L qm = q[iγ µ µ +g(σ+i τ πγ 5 )]q + 1 ( µσ µ σ + µ π µ π) + V (σ, π) background gauge field: µ D µ = µ ia µ with A µ = δ µ 0 g sa 0 a λa L pol = qγ 0 A 0 q U(φ, φ) B.-J. Schaefer (KFU Graz) 4 / 16
Polyakov-quark-meson model quark-meson model: L qm = q[iγ µ µ +g(σ+i τ πγ 5 )]q + 1 ( µσ µ σ + µ π µ π) + V (σ, π) background gauge field: µ D µ = µ ia µ with A µ = δ µ 0 g sa 0 a λa L pol = qγ 0 A 0 q U(φ, φ) Polyakov loop ( φ( x) = 1 Tr c P exp i N c β ) dτa 0 ( x, τ) 0 β B.-J. Schaefer (KFU Graz) 4 / 16
Polyakov-quark-meson model quark-meson model: L qm = q[iγ µ µ +g(σ+i τ πγ 5 )]q + 1 ( µσ µ σ + µ π µ π) + V (σ, π) background gauge field: µ D µ = µ ia µ with A µ = δ µ 0 g sa 0 a λa L pol = qγ 0 A 0 q U(φ, φ) Polyakov loop potential: U(φ, φ) T 4 = b (T, T 0 ) φ φ b 3 6 ( φ 3 + φ 3) + b 4 ( φ φ) 16 B.-J. Schaefer (KFU Graz) 4 / 16
Polyakov-quark-meson model quark-meson model: L qm = q[iγ µ µ +g(σ+i τ πγ 5 )]q + 1 ( µσ µ σ + µ π µ π) + V (σ, π) background gauge field: µ D µ = µ ia µ with A µ = δ µ 0 g sa 0 a λa L pol = qγ 0 A 0 q U(φ, φ) Polyakov loop potential: U(φ, φ) T 4 = b (T, T 0 ) φ φ b 3 6 ( φ 3 + φ 3) + b 4 ( φ φ) 16 Polyakov-quark-meson model: L PQM = L qm + L pol B.-J. Schaefer (KFU Graz) 4 / 16
Polyakov-quark-meson model quark-meson model: L qm = q[iγ µ µ +g(σ+i τ πγ 5 )]q + 1 ( µσ µ σ + µ π µ π) + V (σ, π) background gauge field: µ D µ = µ ia µ with A µ = δ µ 0 g sa 0 a λa L pol = qγ 0 A 0 q U(φ, φ) Polyakov loop potential: U(φ, φ) T 4 = b (T, T 0 ) φ φ b 3 6 ( φ 3 + φ 3) + b 4 ( φ φ) 16 parameters b i i =,..., 4 fixed to pure gauge QCD B.-J. Schaefer (KFU Graz) 4 / 16
Polyakov-quark-meson model quark-meson model: L qm = q[iγ µ µ +g(σ+i τ πγ 5 )]q + 1 ( µσ µ σ + µ π µ π) + V (σ, π) background gauge field: µ D µ = µ ia µ with A µ = δ µ 0 g sa 0 a λa L pol = qγ 0 A 0 q U(φ, φ) Polyakov loop potential: U(φ, φ) T 4 = b (T, T 0 ) φ φ b 3 6 parameter b (T, T 0 ) = b (α(t, T 0 )) ( φ 3 + φ 3) + b 4 ( φ φ) 16 B.-J. Schaefer (KFU Graz) 4 / 16
Polyakov-quark-meson model quark-meson model: L qm = q[iγ µ µ +g(σ+i τ πγ 5 )]q + 1 ( µσ µ σ + µ π µ π) + V (σ, π) background gauge field: µ D µ = µ ia µ with A µ = δ µ 0 g sa 0 a λa L pol = qγ 0 A 0 q U(φ, φ) Polyakov loop potential: U(φ, φ) T 4 = b (T, T 0 ) φ φ b 3 6 ( φ 3 + φ 3) + b 4 ( φ φ) 16 T 0 = T 0 (N f ): N f 0 1 + 1 3 T 0 [MeV] 70 40 08 187 178 B.-J. Schaefer (KFU Graz) 4 / 16
Polyakov-quark-meson model Polyakov loop potential: U(φ, φ) T 4 = b (T, T 0 (N f )) φ φ b 3 6 ( φ 3 + φ 3) + b 4 ( φ φ) 16 finite µ φ φ φ φ 1 (φ φ + (φ φ) ) or 1 ( φ + φ ) B.-J. Schaefer (KFU Graz) 5 / 16
Polyakov-quark-meson model Polyakov loop potential: U(φ, φ) T 4 = b (T, T 0 (N f ; µ)) ( φ + 4 φ ) b 3 6 (φ3 + φ 3 )+ b 4 ( φ + 16 φ ) B.-J. Schaefer (KFU Graz) 5 / 16
Polyakov-quark-meson model Polyakov loop potential: U(φ, φ) T 4 = b (T, T 0 (N f ; µ)) ( φ + 4 φ ) b 3 6 (φ3 + φ 3 )+ b 4 ( φ + 16 φ ) this potential has a minimum and no saddle point determinant of Hessian: 3000 500 000 1500 1000 500 positive minimum 0 50 100 150 00 50 300 B.-J. Schaefer (KFU Graz) 5 / 16
Polyakov-quark-meson model Polyakov loop potential: U(φ, φ) T 4 = b (T, T 0 (N f ; µ)) ( φ + 4 φ ) b 3 6 (φ3 + φ 3 )+ b 4 ( φ + 16 φ ) mean-field grand canonical potential: with fermi contribution: d 3 p { Ω qq = N f T (π) 3 Ω(T, µ) = U(φ, φ) + V renorm ( σ, 0) + Ω qq (T, µ) ln [1+3(φ + φe ] (Ep µ)/t )e (Ep µ)/t +e 3(Ep µ)/t + ln [1+3( φ ]} + φe (Ep+µ)/T )e (Ep+µ)/T 3(Ep+µ)/T +e E p = p p + (g σ ) B.-J. Schaefer (KFU Graz) 5 / 16
Polyakov-quark-meson model Polyakov loop potential: U(φ, φ) T 4 = b (T, T 0 (N f ; µ)) ( φ + 4 φ ) b 3 6 (φ3 + φ 3 )+ b 4 ( φ + 16 φ ) equations of motion: Ω σ = 0 Ω φ = 0 Ω φ = 0 B.-J. Schaefer (KFU Graz) 5 / 16
Outline 1 Motivation Polyakov-Quark-Meson Model 3 Results 4 Summary & Outlook B.-J. Schaefer (KFU Graz) 6 / 16
Order parameters: µ = 0 1.4 1. 1 <qq> <phi> 0.8 0.6 0.4 0. 0 10 140 160 180 00 0 40 T [MeV] B.-J. Schaefer (KFU Graz) 7 / 16
Order parameters: µ = 0 0.1 0.08 d<qq>/dt d<phi>/dt 0.06 0.04 0.0 0 10 140 160 180 00 0 40 T [MeV] B.-J. Schaefer (KFU Graz) 8 / 16
Order parameters: finite µ 1.6 1.4 1. 1 0.8 0.6 0.4 0. 0 <qq> <phi> <phibar> 0 50 100 150 00 50 300 350 T [MeV] µ=0 µ=00 µ=70 B.-J. Schaefer (KFU Graz) 9 / 16
Phase diagram 00 150 1st order crossover CEP T [MeV] 100 50 0 0 50 100 150 00 50 300 350 µ [MeV] B.-J. Schaefer (KFU Graz) 10 / 16
Phase diagram 00 150 1st order crossover CEP T [MeV] 100 50 0 0 50 100 150 00 50 300 350 µ [MeV] B.-J. Schaefer (KFU Graz) 10 / 16
Pressure perturbative pressure of QCD with N f massless quarks [ p T 4 = (N c 1) π 7π 45 + N f 60 + 1 ( µ ) 1 ( µ ) ] 4 + T 4π T µ = 0: 1 N f =. 0.8 p/p SB 0.6 0.4 0. 0 0 0.5 1 1.5 T/T c B.-J. Schaefer (KFU Graz) 11 / 16
Scaled pressure difference p p = p(t, µ) p(t, µ = 0) ; T c (µ = 0) 184 MeV p/t 4 0.8 0.7 0.6 0.5 0.4 0.3 0. 0.1 0 µ =0. T c µ =0.4 T c µ =0.6 T c µ =0.8 T c 0 0.5 1 1.5.5 T/T c B.-J. Schaefer (KFU Graz) 1 / 16
Quark number density n q Ω(T, µ) n q = µ ; T c (µ = 0) 184 MeV n q /T 3 1. 1 0.8 0.6 0.4 0. µ = 0. T c µ = 0.4 T c µ = 0.6 T c µ = 0.8 T c 0 0 0.5 1 1.5 T/T c B.-J. Schaefer (KFU Graz) 13 / 16
Outline 1 Motivation Polyakov-Quark-Meson Model 3 Results 4 Summary & Outlook B.-J. Schaefer (KFU Graz) 14 / 16
Summary parameter in Polyakov loop potential: T 0 pure gauge: T 0 70 MeV T 0 10 MeV for N f = T 0 (N f, µ) quark-meson model is renormalizable no cutoff parameter mean-field approximation encouraging Outlook include quark-meson dynamics with RG include glue dynamics with RG full QCD (step by step) B.-J. Schaefer (KFU Graz) 15 / 16
Phase diagrams: MF versus RG 00 150 1st order crossover CEP T [MeV] 100 50 0 0 50 100 150 00 50 300 350 µ [MeV] B.-J. Schaefer (KFU Graz) 16 / 16