With the FRG towards the QCD Phase diagram

With the FRG towards the QCD Phase diagram Bernd-Jochen Schaefer University of Graz, Austria RG Approach from Ultra Cold Atoms to the Hot QGP 22 nd Aug - 9 th Sept, 211 Helmholtz Alliance Extremes of Density and Temperature: Cosmic Matter in the Laboratory (EMMI) in collaboration with Yukawa Institute for Theoretical Physics (YITP) Kyoto, Japan With the FRG towards the QCD Phase Diagram B.-J. Schaefer (KFU Graz) 1/35

QCD two phase transitions: 1 restoration of chiral symmetry SU L+R(N f ) SU L(N f ) SU R(N f ) order parameter: { > symmetry broken, T < Tc qq = symmetric phase, T > T c 2 de/confinement order parameter: Polyakov loop variable { = confined phase, T < Tc Φ > deconfined phase, T > T c ( β ) Φ = tr cp exp i dτa (τ, x) /N c alternative: dressed Polyakov loop (dual condensate) QCD Phase Transitions Temperature early universe LHC quark gluon plasma RHIC SPS <ψψ> crossover FAIR/JINR <ψψ> = / vacuum hadronic fluid n B= n B> relates chiral and deconfinement transition spectral properties of Dirac operator effective models: AGS SIS nuclear matter µ quark matter crossover superfluid/superconducting phases? 2SC <ψψ> = / CFL neutron star cores At densities/temperatures of interest only model calculations available 1 Quark-meson model or other models e.g. NJL 2 Polyakov quark-meson model or PNJL models With the FRG towards the QCD Phase Diagram B.-J. Schaefer (KFU Graz) 2/35

The conjectured QCD Phase Diagram Temperature early universe LHC crossover vacuum RHIC SPS <ψψ> = / hadronic fluid n = n > B AGS SIS nuclear matter µ quark gluon plasma FAIR/JINR B <ψψ> quark matter crossover superfluid/superconducting 2SC phases? <ψψ> = / CFL neutron star cores At densities/temperatures of interest only model calculations available non-perturbative functional methods (FunMethods) here: FRG complementary to lattice Open issues: related to chiral & deconfinement transition existence of CEP? Its location? additional CEPs? How many? coincidence of both transitions at µ =? quarkyonic phase at µ >? chiral CEP/ deconfinement CEP? finite volume effects? lattice comparison so far only MFA results effects of fluctuations? size of crit. region no sign problem µ > chiral symmetry/fermions (small masses/chiral limit)... With the FRG towards the QCD Phase Diagram B.-J. Schaefer (KFU Graz) 3/35

The conjectured QCD Phase Diagram Temperature early universe LHC crossover vacuum RHIC SPS <ψψ> = / hadronic fluid n = n > B AGS SIS nuclear matter µ quark gluon plasma FAIR/JINR B <ψψ> quark matter crossover superfluid/superconducting 2SC phases? <ψψ> = / CFL neutron star cores At densities/temperatures of interest only model calculations available non-perturbative functional methods (FunMethods) here: FRG complementary to lattice Open issues: related to chiral & deconfinement transition existence of CEP? Its location? additional CEPs? How many? coincidence of both transitions at µ =? quarkyonic phase at µ >? chiral CEP/ deconfinement CEP? finite volume effects? lattice comparison so far only MFA results effects of fluctuations? size of crit. region lattice versus FRG N c = 2 Polyakov-quark-meson-diquark (PQMD) model [Strodthoff, BJS, von Smekal; in prep. 11] With the FRG towards the QCD Phase Diagram B.-J. Schaefer (KFU Graz) 3/35

Outline Three-Flavor Chiral Quark-Meson Model...with Polyakov loop dynamics Taylor expansions and generalized susceptibilities Functional Renormalization Group Finite volume effects With the FRG towards the QCD Phase Diagram B.-J. Schaefer (KFU Graz) 4/35

N f = 3 Quark-Meson (QM) model Model Lagrangian: L qm = L quark + L meson Quark part with Yukawa coupling h: L quark = q(i / h λa 2 (σa + iγ 5π a))q Meson part: scalar σ a and pseudoscalar π a nonet meson fields: M = 8 a= λ a (σa + iπa) 2 L meson = tr[ µm µ M] m 2 tr[m M] λ 1 (tr[m M]) 2 λ 2 tr[(m M) 2 ]+c[det(m) + det(m )] +tr[h(m + M )] explicit symmetry breaking matrix: H = a λ a 2 ha U(1) A symmetry breaking implemented by t Hooft interaction talk by Mario Mitter With the FRG towards the QCD Phase Diagram B.-J. Schaefer (KFU Graz) 5/35

Phase diagram for N f = 2 + 1 (µ µ q = µ s ) Model parameter fitted to (pseudo)scalar meson spectrum: one parameter precarious: f (6) particle (i.e. sigma) broad resonance PDG: mass=(4... 12) MeV we use fit values for m σ between (4... 12) MeV existence of CEP depends on m σ! Example: m σ = 6 MeV (lower lines), 8 and 9 MeV (here mean-field approximation) with U(1) A [BJS, M. Wagner 9] 25 2 crossover 1st order CEP T [MeV] 15 1 5 5 1 15 2 25 3 35 4 µ [MeV] With the FRG towards the QCD Phase Diagram B.-J. Schaefer (KFU Graz) 6/35

Chiral critical surface (m σ = 8 MeV) standard scenario for m σ = 8 MeV (as expected) here: t Hooft coupling µ-independent (might change if µ-dependence is considered) with U(1) A without U(1) A physical point physical point µ [MeV] µ [MeV] 4 3 2 1 4 3 2 1 6 5 4 3 m K [MeV] 2 1 15 175 75 1125 5 m π [MeV] 25 6 5 4 3 m K [MeV] 2 1 15 175 75 1125 5 m π [MeV] 25 [BJS, M. Wagner, 9] non-standard scenario in PNJL with (unrealistic) large vector int. bending of surface With the FRG towards the QCD Phase Diagram B.-J. Schaefer (KFU Graz) 7/35

Polyakov-quark-meson (PQM) model Lagrangian L PQM = L qm + L pol with L pol = qγ A q U(φ, φ) 1. polynomial Polyakov loop potential: Polyakov 1978, Meisinger 1996, Pisarski 2 U(φ, φ) T 4 = b 2(T, T ) 2 φ φ b ( 3 φ 3 + φ 3) + b 4 ( 2 φ φ) 6 16 b 2 (T, T ) = a + a 1 (T /T) + a 2 (T /T) 2 + a 3 (T /T) 3 2. logarithmic potential: Rößner et al. 27 U log T 4 = 1 2 a(t) φφ [ + b(t) ln 1 6 φφ + 4 (φ 3 + φ 3) 3 ( ) ] 2 φφ a(t) = a + a 1 (T /T) + a 2 (T /T) 2 and b(t) = b 3 (T /T) 3 3. Fukushima Fukushima 28 { U Fuku = bt 54e a/t φ φ [ + ln 1 6 φφ + 4 (φ 3 + φ 3) 3 ( ) ]} 2 φφ a controls deconfinement b strength of mixing chiral & deconfinement With the FRG towards the QCD Phase Diagram B.-J. Schaefer (KFU Graz) 9/35

Polyakov-quark-meson (PQM) model Lagrangian L PQM = L qm + L pol with L pol = qγ A q U(φ, φ) 1. polynomial Polyakov loop potential: Polyakov 1978, Meisinger 1996, Pisarski 2 U(φ, φ) T 4 = b 2(T, T ) 2 φ φ b ( 3 φ 3 + φ 3) + b 4 ( 2 φ φ) 6 16 b 2 (T, T ) = a + a 1 (T /T) + a 2 (T /T) 2 + a 3 (T /T) 3 back reaction of the matter sector to the YM sector: N f and µ-modifications in presence of dynamical quarks: T = T (N f, µ, m q) BJS, Pawlowski, Wambach; 27 N f 1 2 2 + 1 3 T [MeV] 27 24 28 187 178 matter back reaction to the YM sector important at µ for µ : φ > φ is expected since φ is related to free energy gain of antiquarks in medium with more quarks antiquarks are more easily screened. With the FRG towards the QCD Phase Diagram B.-J. Schaefer (KFU Graz) 9/35

QCD Thermodynamics N f = 2 + 1 [BJS, M. Wagner, J. Wambach 1] (P)QM models (three different Polyakov loop potentials) versus QCD lattice simulations p/p SB 1.8.6 QM.4 PQM log PQM pol.2 PQM Fuku p4 asqtad.5 1 1.5 2 2.5 3 T/T χ solid lines: PQM with lattice masses (HotQCD) m π 22, m K 53 MeV dashed lines: (P)QM with realistic masses data taken from: [Bazavov et al. 9] Stefan-Boltzmann limit: psb T 4 = 2(N 2 c 1) π2 9 + Nf Nc 7π 2 18 With the FRG towards the QCD Phase Diagram B.-J. Schaefer (KFU Graz) 1/35

N f = 2 + 1 (P)QM phase diagrams Summary of QM and PQM models in mean field approximation T [MeV] chiral transition and deconfinement coincide at small densities 25 2 15 1 CEP χ crossover 5 χ first Φ,Φ order crossover 5 1 15 2 25 3 35 4 µ [MeV] for PQM model (upper lines) for QM model (lower lines) [BJS, M. Wagner; in prep. 11] N f = 2: [BJS, Pawlowski, Wambach; 27] With the FRG towards the QCD Phase Diagram B.-J. Schaefer (KFU Graz) 11/35

N f = 2 + 1 (P)QM phase diagrams Summary of QM and PQM models in mean field approximation T [MeV] chiral transition and deconfinement coincide at small densities 25 2 15 1 CEP χ crossover 5 χ first Φ,Φ order crossover 5 1 15 2 25 3 35 4 µ [MeV] for PQM model (upper lines) with matter back reaction in Polyakov loop potential shrinking of possible quarkyonic phase for QM model (lower lines) [BJS, M. Wagner; in prep. 11] N f = 2: [BJS, Pawlowski, Wambach; 27] With the FRG towards the QCD Phase Diagram B.-J. Schaefer (KFU Graz) 11/35

Critical region contour plot of size of the critical region around CEP defined via fixed ratio of susceptibilities: R = χ q/χ free q compressed with Polyakov loop QM model (2+1) PQM model (2+1) (T-T c ) [MeV] 2 1-1 -2 R=2 R=3 R=5-6 -4-2 2 4 (µ-µ c ) [MeV] (T-T c ) [MeV] 15 1 5-5 -1-15 -2 R=2 R=3 R=5-4 -3-2 -1 1 2 3 4 (µ-µ c ) [MeV] [BJS, M. Wagner; in preparation 11] With the FRG towards the QCD Phase Diagram B.-J. Schaefer (KFU Graz) 12/35

at nonzero density Finite(I) density extrapolations 12 N f = 2 + 1 ure Taylor expansion: p(t, µ) ( ) µ n = c n(t) with c n(t) = 1 n ( p(t, µ)/t 4) d, µ s ) = c u,d,s µu i µd j µs k i,j,k T 4 T T T n! (µ/t) n i,j,k n= µ= fixed temperature convergence radii: convergence-radius has to T method for be determined nonperturbatively limited by first-order line? small µ/t k µu,d,s= 19MeV hadron gas confined, χ-broken quark-gluon plasma deconfined, χ-symmetric color superconductor ρ 2n = c 2 1/(2n 2) c 2n nuclear matter density µ B [C. Schmidt 9] With the FRG towards the QCD Phase Diagram B.-J. Schaefer (KFU Graz) 14/35

Finite density extrapolations N f = 2 + 1 Taylor expansion: p(t, µ) ( ) µ n = c n(t) with c n(t) = 1 n ( p(t, µ)/t 4) Non-zero density QCD T 4 by the Taylor expansion method Chuan Miao T n! (µ/t) n and Christian Schmidt n= µ=.6.5.4 c 2 u SB.1.8 c 4 u n f =2+1, m " =22 MeV n f =2, m " =77 MeV filled: N! = 4 open: N! = 6.6.3 n f =2+1, m " =22 MeV SB.2 n f =2, m " =77 MeV.4 filled: N! = 4 SB -.1.1 open: N! = 6.2 T[MeV] T[MeV] -.2 T[MeV] 15 2 25 3 35 4 45 15 2 25 3 35 4 45 15 2 25 3 35 4 45.3.2.1 c 6 u n f =2+1, m " =22 MeV n f =2, m " =77 MeV [Miao et al. 8] Figure 1: Taylor coefficients of the pressure in term of the up-quark chemical potential. Results are obtained with the p4fat3 action on N! = 4 (full) and N! = 6 (open symbols) lattices. We compare preliminary results of (2+1)-flavor a pion mass of m " 22 MeV to previous results of 2-flavor simulations with a corresponding pion mass of m p 77 [2]. 1. Introduction With the FRG towards the QCD Phase Diagram B.-J. Schaefer (KFU Graz) 14/35

New method: based Finite on algorithmic densitydifferentiation extrapolations N f [M. = Wagner, 2 + A. 1Walther, BJS, CPC 1] Taylor coefficients c n numerically known to high order, e.g. n = 22 4 2-2 -4-6 5 c 8 3 2-5 1-1 -1 c 6-15 c 1-2 -3 5e+7 1e+6-5 c 12 c 14-1e+6 c 16-5e+7 3e+9 2e+9 1e+9-1e+9-2e+9-3e+9-4e+9 c 18,98 1 1,2 c 2 5e+1-5e+1,98 1 1,2 T/T χ c 22,98 1 1,2 2e+12-2e+12-4e+12 With the FRG towards the QCD Phase Diagram B.-J. Schaefer (KFU Graz) 15/35

Taylor coefficients for N f = 2 + 1 PQM model 4 2-2 -4-6 5-5 3e+9 2e+9 1e+9-1e+9-2e+9-3e+9-4e+9 c 8 3 2-5 1-1 -1 c 6-15 c 1-2 -3 5e+7 1e+6 c 12 c 18,98 1 1,2 c 14 c 2-1e+6 5e+1-5e+1,98 1 1,2 T/T χ c 16 c 22,98 1 1,2-5e+7 2e+12-2e+12-4e+12 this technique applied to PQM model investigation of convergence properties of Taylor series properties of c n oscillating increasing amplitude no numerical noise small outside transition region number of roots increasing 26th order [F. Karsch, BJS, M. Wagner, J. Wambach; arxiv:19.5211] Can we locate the QCD critical endpoint with the Taylor expansion? With the FRG towards the QCD Phase Diagram B.-J. Schaefer (KFU Graz) 16/35

Findings: Taylor expansions simply Taylor expansion: slow convergence high orders needed disadvantage for lattice simulations Taylor applicable within convergence radius also for µ/t > 1 r 2n = c 2n 1/2 c, r χ 2n = (2n + 2)(2n + 1) 1/2 2n+2 (2n + 3)(2n + 4) r 2n+2 T/T χ 1.8.6.4.2 n=8 n=16 n=24.5 1 1.5 2 µ/t χ µ/t 1.5 1.4 1.3 1.2 1.1 1.9.8 Padé [N/N] r n p [N/N] p [N+2/N] p [N+2/N-2] r χ n χ [N/N] χ [N+2/N] 4 6 8 1 12 14 16 18 2 22 24 n max [F.Karsch, BJS, M.Wagner, J.Wambach; arxiv:19.5211] With the FRG towards the QCD Phase Diagram B.-J. Schaefer (KFU Graz) 17/35

Findings: Taylor expansions simply Taylor expansion: slow convergence high orders needed disadvantage for lattice simulations Taylor applicable within convergence radius also for µ/t > 1 but 1st order transition not resolvable expansion around µ = 1 21 8 25 σ x [MeV] 6 4 24th order PQM T c,n [MeV] 2 195 2 19 9 95 1 15 11 T [MeV] 185 6 8 1 12 14 16 18 2 n [F.Karsch, BJS, M.Wagner, J.Wambach; arxiv:19.5211] With the FRG towards the QCD Phase Diagram B.-J. Schaefer (KFU Graz) 17/35

Generalized Susceptibilities Can we use these coefficients to locate CEP experimentally? Is there a memory effect that the system (HIC) has passed through the QCD phase transition? Probing phase diagram with fluctuations of e.g. net baryon number Example: Kurtosis R B 4,2 = 1 c 4 probe of deconfinement? 9 c 2 It measures quark content of particles carrying baryon number B in HRG model R 4,2 = 1 (always positive) c 4 / c 2 6 4 2-2 -4-6 -8..1.2.3.4 196 198 2 22 24 26 28 T [MeV] R 4,2 2 1-1 1-1 2 23 26 29 1 2 3 4 5 6 µ [MeV] T [MeV] [BJS, M.Wagner; in prep. 11] With the FRG towards the QCD Phase Diagram B.-J. Schaefer (KFU Graz) 18/35

Generalized Susceptibilities Fluctuations of higher moments (more sensitive to criticality) exhibit strong variation from HRG model turn negative [Karsch, Redlich; 11] see also [Friman, Karsch, Redlich, Skokov; 11] see talk by V. Skokov higher moments: R q n,m = c n/c m regions where R n,m are negative along crossover line in the phase diagram T [MeV] 21 25 2 PQM with T (µ) 195 R 4,2 19 R 6,2 185 R 8,2 R 1,2 18 R 12,2 175 crossover CEP 17 2 4 6 8 1 12 14 16 18 µ [MeV] T [MeV] 16 15 14 QM model 13 R 4,2 12 R 6,2 11 R 8,2 R 1,2 1 R 12,2 9 crossover CEP 8 5 1 15 2 µ [MeV] [BJS, M.Wagner; in prep. 11] With the FRG towards the QCD Phase Diagram B.-J. Schaefer (KFU Graz) 19/35

Functional RG Approach Γ k [φ] scale dependent effective action ; t = ln(k/λ) ; R k regulators FRG (average effective action) [Wetterich 93] tγ k [φ] = 1 2 Tr tr k ( 1 Γ (2) k + R k ) ; Γ (2) k = δ2 Γ k δφδφ k k Γ k [φ] 1 2 With the FRG towards the QCD Phase Diagram B.-J. Schaefer (KFU Graz) 21/35

T (N f, µ) modification full dynamical QCD FRG flow: fluctuations of gluon, ghost, quark and meson (via hadronization) fluctuations [Braun, Haas, Marhauser, Pawlowski; 9] t Γ k [φ] = 1 2 + 1 2 in presence of dynamical quarks gluon propagator modified: pure Yang Mills flow + these modifications pure Yang Mills flow replaced by effective Polyakov loop potential: (fit to YM thermodynamics) T Λ QCD : T T (N f, µ, m q) [BJS, Pawlowski, Wambach; 27] With the FRG towards the QCD Phase Diagram B.-J. Schaefer (KFU Graz) 22/35

Functional Renormalization Group tγ k [φ] = 1 2 Tr tr k ( 1 Γ (2) k + R k ) ; Γ (2) k = δ2 Γ k δφδφ [Wetterich 93] tγ k[φ] = 1 2 + 1 2 Γ k[φ] scale dependent effective action ; t = ln(k/λ) ; R k regulators PQM truncation N f = 2 [Herbst, Pawlowski, BJS; 211] Γ k = see also [Skokov, Friman, Redlich; 21] { d 4 x ψ (D/ + µγ + ih(σ + iγ 5 τ π)) ψ + 1 2 ( µσ)2 + 1 } 2 ( µ π)2 + Ω k [σ, π, Φ, Φ] Initial action at UV scale Λ: Ω Λ [σ, π, Φ, Φ] = U(σ, π) + U(Φ, Φ) + Ω Λ [σ, π, Φ, Φ] U(σ, π) = λ 4 (σ2 + π 2 v 2 ) 2 cσ With the FRG towards the QCD Phase Diagram B.-J. Schaefer (KFU Graz) 23/35

Phase diagram T = 28 MeV 2 T [MeV] 15 1 5 χ crossover Φ crossover Φ crossover CEP χ first order 5 1 15 2 25 3 35 µ [MeV] [Herbst, Pawlowski, BJS; 211] With the FRG towards the QCD Phase Diagram B.-J. Schaefer (KFU Graz) 24/35

Phase diagram T (µ), T () = 28 MeV 2 15 T [MeV] 1 5 χ crossover Φ crossover Φ crossover CEP χ first order 5 1 15 2 25 3 35 µ [MeV] [Herbst, Pawlowski, BJS; 211] With the FRG towards the QCD Phase Diagram B.-J. Schaefer (KFU Graz) 25/35

Phase diagram T (µ), T () = 28 MeV 2 15 T [MeV] 1 5 χ crossover Φ crossover Φ crossover CEP χ first order 5 1 15 2 25 3 35 µ [MeV] CEP unlikely for small µ B /T baryons [Herbst, Pawlowski, BJS; 211] With the FRG towards the QCD Phase Diagram B.-J. Schaefer (KFU Graz) 26/35

Phase diagram (DSE - HTL) 2 T [MeV] 15 1 5 dressed Polyakov-loop dressed conjugate P.-loop chiral crossover chiral CEP chiral first order region 1 2 3 µ [MeV] [Ch. S. Fischer, J. Luecker, J. A. Mueller; 211] With the FRG towards the QCD Phase Diagram B.-J. Schaefer (KFU Graz) 27/35

Critical region similar conclusion if fluctuations are included fluctuations via Functional Renormalization Group comparison: N f = 2 QM model Mean Field RG analysis (T-T c )/T c 1.2 1.8.6.4.2 -.2 -.4 -.6 Mean Field RG 1st order -.8 -.6 -.4 -.2.2.4.6 (µ-µ c )/µ c [BJS, Wambach 6] With the FRG towards the QCD Phase Diagram B.-J. Schaefer (KFU Graz) 28/35

Finite volume effects Finite Euclidean Volume: L 3 1/T d 3 p ( ) 2π 3 L n 1 Z n 2 Z n 3 Z with cf. talk by Bertram Klein p 2 4π2 ( ) L 2 n 2 1 + n2 2 + n2 3 grand potential (N f = 2 quark-meson truncation) [A Tripolt, J Braun, B Klein, BJS; 211] [ tω k = B p(k, L) k5 12π 2 2N f N c E q ( Eσ + 1 E σ coth 2T { tanh ( Eq µ 2T ( )] Eπ ) + 3 E π coth ) + tanh 2T ( )} Eq + µ with E σ,π,q = k 2 + m 2 σ,π,q, m 2 σ = 2Ω k + 4σ2 Ω k, m2 π = 2Ω k, m2 q = h 2 σ 2 2T mode counting functions B p(k, L) : periodic B ap(k, L) : antiperiodic With the FRG towards the QCD Phase Diagram B.-J. Schaefer (KFU Graz) 3/35

Finite volume effects mode counting functions (periodic and antiperiodic (lower) boundary) B ap(k, L) = B p(k, L) = 6π2 (kl) 3 n 1,n 2,n 3 Z 6π 2 (kl) 3 Θ ( n 1,n 2,n 3 Z k 2 4π2 L 2 Θ (k 2 4π2 ( ) ) L 2 n 2 1 + n2 2 + n2 3 ( ( n 1 + 1 ) 2 ( + n 2 + 1 ) 2 ( + n 3 + 1 ) )) 2 2 2 2 3 periodic L = 5 fm 3 antiperiodic L = 5 fm 2 2 1 1 2 4 6 8 1 k [MeV] 2 4 6 8 1 k [MeV] With the FRG towards the QCD Phase Diagram B.-J. Schaefer (KFU Graz) 31/35

Finite volume effects mode counting functions (periodic and antiperiodic (lower) boundary) B ap(k, L) = B p(k, L) = 6π2 (kl) 3 n 1,n 2,n 3 Z 6π 2 (kl) 3 Θ ( n 1,n 2,n 3 Z k 2 4π2 L 2 Θ (k 2 4π2 ( ) ) L 2 n 2 1 + n2 2 + n2 3 ( ( n 1 + 1 ) 2 ( + n 2 + 1 ) 2 ( + n 3 + 1 ) )) 2 2 2 2 3 periodic L = 2 fm 3 antiperiodic L = 2 fm 2.5 2 2 1.5 1 1.5 2 4 6 8 1 k [MeV] 2 4 6 8 1 k [MeV] With the FRG towards the QCD Phase Diagram B.-J. Schaefer (KFU Graz) 31/35

Finite volume effects [A. Tripolt, J. Braun, B. Klein, BJS, in preparation 11] preliminary results for periodic BC L With the FRG towards the QCD Phase Diagram B.-J. Schaefer (KFU Graz) 32/35

Finite volume effects [A. Tripolt, J. Braun, B. Klein, BJS, in preparation 11] preliminary results for periodic BC With the FRG towards the QCD Phase Diagram B.-J. Schaefer (KFU Graz) 32/35

Finite volume effects [A. Tripolt, J. Braun, B. Klein, BJS, in preparation 11] preliminary results for periodic BC movement of the CEP (L... 2 fm) With the FRG towards the QCD Phase Diagram B.-J. Schaefer (KFU Graz) 32/35

Finite volume effects [A. Tripolt, J. Braun, B. Klein, BJS, in preparation 11] preliminary results for antiperiodic BC L With the FRG towards the QCD Phase Diagram B.-J. Schaefer (KFU Graz) 33/35

Finite volume effects [A. Tripolt, J. Braun, B. Klein, BJS, in preparation 11] preliminary results for antiperiodic BC With the FRG towards the QCD Phase Diagram B.-J. Schaefer (KFU Graz) 33/35

Finite volume effects [A. Tripolt, J. Braun, B. Klein, BJS, in preparation 11] preliminary results for antiperiodic BC movement of the CEP (L... 2 fm) With the FRG towards the QCD Phase Diagram B.-J. Schaefer (KFU Graz) 33/35

Finite volume effects [A. Tripolt, J. Braun, B. Klein, BJS, in preparation 11] curvature κ relative change ( ) T χ(l, µ) T χ(l, ) = 1 κ(l) µ 2 +... πt χ(l, ) κ(l) = κ(l) κ( ) κ( ).5 -.5 -.1 κ (periodic) κ (antiperiodic) 2.5 3 3.5 4 4.5 5 L [fm] With the FRG towards the QCD Phase Diagram B.-J. Schaefer (KFU Graz) 34/35

Summary N f = 2 and N f = 2 + 1 chiral (Polyakov)-quark-meson model study Mean-field approximation and FRG fluctuations are important functional approaches (such as the presented FRG) are suitable and controllable tools to investigate the QCD phase diagram and its phase boundaries Findings: matter back-reaction to YM sector: T T (N f, µ) FRG with PQM truncation: Chiral & deconfinement transition coincide for N f = 2 with T (µ)-corrections similar conclusion for N f = 2 + 1 size of the critical region around CEP smaller Finite volume effects Higher moments Outlook: T [MeV] 2 15 1 χ crossover Φ crossover 5 Φ crossover CEP χ first order 5 1 15 2 25 3 35 µ [MeV] FRG methods suitable test lattice predictions (such as finite volume or N c = 2,...) include glue dynamics with FRG towards full QCD With the FRG towards the QCD Phase Diagram B.-J. Schaefer (KFU Graz) 35/35