The Phase Structure of the Polyakov Quark-Meson Model beyond Mean Field Tina Katharina Herbst In Collaboration with B.-J. Schaefer and J.M. Pawlowski arxiv: 18.81 [hep-ph] 5th International Conference on the Exact Renormalization Group September 12-19, 21 Corfu, Greece Recipient of a DOC-fFORTE-fellowship of the Austrian Academy of Sciences at the Institute of Physics.
QCD Phase Structure Temperature early universe LHC quark gluon plasma RHIC SPS <ψψ> crossover FAIR/NICA <ψψ> = / vacuum hadronic fluid n B= n B> AGS SIS nuclear matter µ quark matter crossover superfluid/superconducting 2SC phases? <ψψ> = / CFL neutron star cores (approx.) order parameters j = symmetric qq broken Chiral Symmetry Z Nc m q chiral condensate qq Center Symmetry m q Polyakov loop Φ = l( x) β l( x) = 1 N c Tr cp exp{i Z β dτa 4( x, τ)} relation to confinement: j = Φ Φ e βfq confined deconfined
QCD Phase Structure Temperature early universe LHC quark gluon plasma RHIC SPS <ψψ> crossover FAIR/NICA <ψψ> = / vacuum hadronic fluid n B= n B> AGS SIS nuclear matter µ quark matter crossover superfluid/superconducting 2SC phases? <ψψ> = / CFL neutron star cores (approx.) order parameters j = symmetric qq broken Other Phases? Chiral Symmetry Z Nc m q chiral condensate qq Center Symmetry m q Polyakov loop Φ = l( x) β l( x) = 1 N c Tr cp exp{i Z β dτa 4( x, τ)} relation to confinement: j = Φ Φ e βfq confined deconfined
Polyakov-Quark-Meson Model Lagrangian L PQM = q [ i /D h(σ + iγ 5 τ π) ] q + 1 2 ( µφ) 2 U(σ, π) U(Φ, Φ) φ = (σ, π)... O(4)-representation of the meson field (N f = 2) D/ (Φ) = γ µ µ i gγ A (Φ) g... gauge coupling h... Yukawa coupling Meson Potential U(σ, π) = λ 4 (σ2 + π 2 v 2 ) 2 cσ
Polyakov Loop Potential Polynomial Ansatz [C. Ratti, M.A. Thaler, W. Weise, Phys.Rev. D73, 1419 (26)] U(Φ, Φ) T 4 = b 2(T ) Φ Φ b 3 2 6 (Φ3 + Φ 3 ) + b 4 4 (Φ Φ) 2 coefficients fitted to lattice data (pure glue): b 2 (T ) = a + a 1 T T «+ a 2 T T «2 «3 T + a 3 a a 1 a 2 a 3 b 3 b 4 6.75 1.95 2.625 7.44.75 7.5 27 MeV (pure glue)? T = 28 MeV? something else? T
T (µ) - One Motivation: Experiment experimental information on the QCD phase diagram: chemical freezeout points not raw data, but interpretation using Statistical Model increase of entropy (red band) and density suggests position of phase transition PNJL computation with T = 2 MeV inconsistent (green band) polynomial ansatz for T (µ) greater overlap (blue band) [in these plots: N c = N f = 3] [K. Fukushima, arxiv:16.2596]
T (µ) - Another Motivation: Theory [B.-J. Schaefer, J.M. Pawlowski, J. Wambach, Phys.Rev. D76, 7423 (27)] FRG flow for QCD: Talks by Jan Pawlowski and Lisa Haas t Γ k [φ] = 1 2 + 1 2 dynamical quarks modify the gluon contribution: gluons ghosts quarks mesons Polyakov Loop potential: from pure YM contribution t Γ k [φ] = 1 2 T T (N f, µ) = T τ e 1/(αb(N f,µ))
Functional Renormalization Group (FRG) Flow Equation [C. Wetterich, 1993] tγ k [ϕ] = 1 j ff 2 Tr tr k,b Γ (2,) 1 [ϕ] + R k,b k Polyakov Quark-Meson Truncation Z Γ k = j d 4 x q (D/ + µγ + ih(σ + iγ 5 τ π)) q + 1 ff 2 ( µφ)2 + Ω k [σ, π, Φ, Φ] at initial scale Λ: Ω Λ [σ, π, Φ, Φ] = U(Φ, Φ) + U(σ, π) + Ω Λ [σ, π, Φ, Φ] t Γ k [φ] = 1 2 + 1 2
Functional Renormalization Group (FRG) Flow Equation [C. Wetterich, 1993] tγ k [ϕ] = 1 j ff 2 Tr tr k,b Γ (2,) 1 [ϕ] + R k,b k Polyakov Quark-Meson Truncation Z Γ k = j d 4 x q (D/ + µγ + ih(σ + iγ 5 τ π)) q + 1 ff 2 ( µφ)2 + Ω k [σ, π, Φ, Φ] at initial scale Λ: tγk[φ] = 1 2 + 1 2 Ω Λ [σ, π, Φ, Φ] = U(Φ, Φ) + U(σ, π) + Ω Λ [σ, π, Φ, Φ] k Ω k Λ = N cn f k 4 3π 2 E q [ 1 Nq (Φ, Φ) N q (Φ, Φ) ] [J. Braun, K. Schwenzer, H.J. Pirner, Phys.Rev. D7, 8516 (24)] [V. Skokov, B. Stokic, B. Friman, Phys.Rev. C82, 1526 (21)]
PQM Flow Equation k Ω k (T, µ) = [V. Skokov, B. Stokic, B. Friman, Phys.Rev. C82, 1526 (21)] k 4 [ ( ) 3 Eπ 12π 2 coth + 1 ( ) Eσ coth E π 2T E σ 2T 2ν q E q { 1 Nq (T, µ; Φ, Φ) N q (T, µ; Φ, Φ) } ] N q(t, µ; Φ, Φ) = 1 + 2 Φe (Eq µ)/t + Φe 2(Eq µ)/t 1 + 3 Φe (Eq µ)/t + 3Φe 2(Eq µ)/t + e 3(Eq µ)/t N q(t, µ; Φ, Φ) N q(t, µ; Φ, Φ) E π = qk 2 + 2Ω k, Eσ = q k 2 + 2Ω k + 4σ2 Ω k, νq = 2NcN f
Phase Structure T = 28 MeV const. 2 15 1 5 5 1 15 2 25 3 35 [TKH, J.M. Pawlowski, B.-J. Schaefer, arxiv:18.81]
Phase Structure T = 28 MeV const. 2 15 1 5 5 1 15 2 25 3 35 [TKH, J.M. Pawlowski, B.-J. Schaefer and M. Wagner, Work in Progress]
Phase Structure T (µ), T () = 28 MeV 2 15 1 5 5 1 15 2 25 3 35 [TKH, J.M. Pawlowski, B.-J. Schaefer, arxiv:18.81]
Normalized Pressure [TKH, J.M. Pawlowski, B.-J. Schaefer, arxiv:18.81] 2 2 15 15 1 5 5 1 15 2 25 3 35 1 5 5 1 15 2 25 3 35 1.2 1.3 µ = MeV µ = 15 MeV µ = 29 MeV 1.2 1.3 µ = MeV µ = 15 MeV µ = 29 MeV p/p SB.8.6 25 5 p/p SB.8.6 25 5.4.4.2.2 5 1 15 2 25 3 35 5 1 15 2 25 3 35
Normalized Pressure [TKH, J.M. Pawlowski, B.-J. Schaefer, arxiv:18.81] [B.-J. Schaefer, J.M. Pawlowski, J. Wambach, Phys.Rev. D76, 7423 (27)] 2 15 1 5 5 1 15 2 25 3 35 1.8 µ = MeV µ = 15 MeV µ = 29 MeV p/p SB.6.4.2 5 1 15 2 25 3 35
Normalized Entropy Density [TKH, J.M. Pawlowski, B.-J. Schaefer, arxiv:18.81] 2 2 15 15 1 5 5 1 15 2 25 3 35 1 5 5 1 15 2 25 3 35 1.8 µ= MeV µ=15 MeV µ=29 MeV 1.8 s/s SB.6.4 s/s SB.6.4.2 5 1 15 2 25 3 35.2 µ= MeV µ=15 MeV µ=29 MeV 5 1 15 2 25 3 35
Quark Number Density [TKH, J.M. Pawlowski, B.-J. Schaefer, arxiv:18.81] 2 2 15 15 1 5 5 1 15 2 25 3 35 1 5 5 1 15 2 25 3 35 2 15 µ=15 MeV µ=29 MeV 2 15 µ=15 MeV µ=29 MeV n q /T 3 1 n q /T 3 1 5 5 5 1 15 2 25 3 35 5 1 15 2 25 3 35
Vicinity of the [TKH, J.M. Pawlowski, B.-J. Schaefer, arxiv:18.81] 2 2 15 15 1 5 5 1 15 2 25 3 35 1 5 5 1 15 2 25 3 35.8.7.6 T<T T=T T>T.8.7.6 T<T T=T T>T.5.5 n q /µ 3.4.3 n q /µ 3.4.3.2.2.1.1 288 29 292 294 296 298 288 29 292 294 296 298
Vicinity of the [TKH, J.M. Pawlowski, B.-J. Schaefer, arxiv:18.81] 2 2 15 15 1 5 5 1 15 2 25 3 35 1 5 5 1 15 2 25 3 35 14 12 1 T<T T=T T>T 14 12 1 T<T T=T T>T χ q /µ 2 8 6 χ q /µ 2 8 6 4 4 2 2 288 29 292 294 296 298 288 29 292 294 296 298
Summary PQM model beyond Mean Field quark-meson fluctuations included within FRG approach Important feature: back-reaction to gluonic sector T T (N f, µ) Modifications of the Phase Structure fluctuations push downwards T (µ): chiral and deconfinement transitions coincide no quarkyonic phase Thermodynamics agree well with lattice studies at µ = 2 15 1 5 5 1 15 2 25 3 35