Thermodynamics of the Polyakov-Quark-Meson Model

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1 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 th Oct, 006 Rathen (Sächsische Schweiz), Germany B.-J. Schaefer (KFU Graz) 1 / 16

2 Outline 1 Motivation Polyakov-Quark-Meson Model 3 Results 4 Summary & Outlook B.-J. Schaefer (KFU Graz) / 16

3 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

4 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

5 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

6 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

7 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

8 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

9 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

10 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

11 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 T 0 [MeV] B.-J. Schaefer (KFU Graz) 4 / 16

12 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

13 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

14 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: positive minimum B.-J. Schaefer (KFU Graz) 5 / 16

15 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

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

17 Outline 1 Motivation Polyakov-Quark-Meson Model 3 Results 4 Summary & Outlook B.-J. Schaefer (KFU Graz) 6 / 16

18 Order parameters: µ = <qq> <phi> T [MeV] B.-J. Schaefer (KFU Graz) 7 / 16

19 Order parameters: µ = d<qq>/dt d<phi>/dt T [MeV] B.-J. Schaefer (KFU Graz) 8 / 16

20 Order parameters: finite µ <qq> <phi> <phibar> T [MeV] µ=0 µ=00 µ=70 B.-J. Schaefer (KFU Graz) 9 / 16

21 Phase diagram st order crossover CEP T [MeV] µ [MeV] B.-J. Schaefer (KFU Graz) 10 / 16

22 Phase diagram st order crossover CEP T [MeV] µ [MeV] B.-J. Schaefer (KFU Graz) 10 / 16

23 Pressure perturbative pressure of QCD with N f massless quarks [ p T 4 = (N c 1) π 7π 45 + N f ( µ ) 1 ( µ ) ] 4 + T 4π T µ = 0: 1 N f =. 0.8 p/p SB T/T c B.-J. Schaefer (KFU Graz) 11 / 16

24 Scaled pressure difference p p = p(t, µ) p(t, µ = 0) ; T c (µ = 0) 184 MeV p/t µ =0. T c µ =0.4 T c µ =0.6 T c µ =0.8 T c T/T c B.-J. Schaefer (KFU Graz) 1 / 16

25 Quark number density n q Ω(T, µ) n q = µ ; T c (µ = 0) 184 MeV n q /T µ = 0. T c µ = 0.4 T c µ = 0.6 T c µ = 0.8 T c T/T c B.-J. Schaefer (KFU Graz) 13 / 16

26 Outline 1 Motivation Polyakov-Quark-Meson Model 3 Results 4 Summary & Outlook B.-J. Schaefer (KFU Graz) 14 / 16

27 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

28 Phase diagrams: MF versus RG st order crossover CEP T [MeV] µ [MeV] B.-J. Schaefer (KFU Graz) 16 / 16