On the role of fluctuations in (2+1)-flavor QCD Bernd-Jochen Schaefer Austria Germany Germany September 21 st , 2017 September 20-22, 2017 Zalakaros
Conjectured QC 3 D phase diagram expected experimental • CEP: existence/location/number trajectory early universe quark − gluon plasma LHC RHIC • relation between chiral & deconfinement? SPS < ψψ > ∼ 0 chiral ⇔ ! deconfinement CEP? [Braun, Janot, Herbst 12/14] Temperature crossover FAIR/JINR • Quarkyonic phase: coincidence of both transitions AGS quark matter at μ =0 & μ >0? SIS < ψψ > = 0 / • inhomogeneous phases? ➜ more favored? crossover hadronic fluid [Carignano, BJS, Buballa 14] superfluid/superconducting • axial anomaly restoration around chiral transition? phases ? n > 0 n = 0 2SC < ψψ > = 0 / B B CFL [Mitter, BJS 14] vacuum nuclear matter • finite volume effects? ➜ lattice comparison/ neutron star cores µ influence boundary conditions Experiment: • role of fluctuations? so far mostly Mean-Field results ➜ effects of fluctuations important [Rennecke, BJS 16] examples: size of crit reg. around CEP • What are good experimental signatures? ➜ higher moments more sensitive to criticality deviation from HRG model 21.09.2017 | B.-J. Schaefer | Giessen University | 2
Conjectured QC 3 D phase diagram expected experimental trajectory early universe Theory: quark − gluon plasma LHC RHIC but simulations restricted to small μ ➜ Lattice: SPS < ψψ > ∼ 0 Temperature crossover FAIR/JINR ➜ Functional QFT methods: FRG,DSE, nPI AGS quark matter SIS < ψψ > = 0 / ➜ Models: effective theories parameter dependency crossover hadronic fluid superfluid/superconducting phases ? Experiment: (non-equilibrium? ➜ most likely thermal equilibrium ) n > 0 n = 0 2SC < ψψ > = 0 / B B CFL ➜ in a finite box (HBT radii: freeze-out vol. ~ 2000-3000 fm 3 ) vacuum nuclear matter neutron star cores µ √ s (UrQMD ( ): system vol. ~ 50 - 250 fm 3 ) Experiment: Theoretical aim: deeper understanding & more realistic HIC description ➜ existence of critical end point(s)? 21.09.2017 | B.-J. Schaefer | Giessen University | 3
Agenda complementary to • Motivation: QCD phase diagram ➜ long term project • Role of Fluctuations: from mean-field approximations to FRG • Instabilities of quark-meson model truncations 21.09.2017 | B.-J. Schaefer | Giessen University | 4
Chiral transition ������������������������������ → ∞ ���� �� ���������������� Taylor expansion: � µ �������������������� � ��������������� ∞ p ( T, µ ) � n � with = c n ( T ) T 4 T n =0 How can we probe a transition? � µ ∂ n p ( T, µ ) /T 4 � n � c n ( T ) = 1 � � ��������������������� ∂ n p ( X ) � n ! ∂ ( µ/T ) n T ���� X = T, µ, . . . � µ =0 ∂ X n freeze-out close to chiral crossover line � ���������������������� c n ≡ ∂ n p ( T, µ ) ∂ ( µ/T ) n . . . ����������������������������� [HotQCD, QM 2012] 27.05.2015 | B.-J. Schaefer | Giessen University | 5
Functional Renormalization Group = δ 2 Γ k Γ (2) Γ k [ φ ] scale dependent e ff ective action t = ln( k/ Λ ) R k ���������� � k δφδφ FRG (average e ff ective action) Regulator [Wetterich 1993] � ���������� Γ k � Example: Leading order derivative expansion arbitrary potential ⇥� 5 )] q + 1 2( ⌃ µ ⇤ ) 2 + 1 � ⇥ ) 2 + V k ( ⇧ 2 ) d 4 x ¯ q [ i � µ ⌃ µ − g ( ⇤ + i ⌥ Γ k = 2( ⌃ µ ⌥ ⌅⌥ V k = Λ ( ⌅ 2 ) = � 4 ( ⇤ 2 + ⇧ ⇥ 2 − v 2 ) 2 − c ⇤ 21.09.2017 | B.-J. Schaefer | Giessen University | 6
Solution techniques Taylor expansion around some point Discretize potential on 1dim-grid � N max a k, 2 n X n ! ( φ 2 − φ 2 0 ) n V k ( φ ) = n =1 input: initial condition at high UV cutoff (usually a symmetric potential) here: fix parametrization at high scales (UV) such to reproduce vacuum physics (IR) φ k → 0 ≡ f π ∼ 93 MeV 27.05.2015 | B.-J. Schaefer | Giessen University | 7
Example: grid technique phase transition: First order Scale evolution UV of meson potential: V_k [a.u.] 0 10 20 30 40 50 φ [MeV] evolution phase transition: Second Order alternative solution techniques: • polynomial (static and co-moving) • bilocal expansion technique V_k [a.u.] • pseudo-spectral IR 0 10 20 30 40 50 60 70 80 90 φ [MeV] 27.05.2015 | B.-J. Schaefer | Giessen University | 8
FRG and QCD � full dynamical QCD FRG flow: [Pawlowski et al. 2009/12] fluctuations of gluon , ghost , quark and (via hadronization) meson in presence of dynamical quarks : gluon propagator is modified pure Yang Mills flow + matter back-coupling 27.05.2015 | B.-J. Schaefer | Giessen University | 9
FRG: quark-meson truncation first step: flow for quark-meson model truncation: neglect YM contributions and bosonic fluctuations without bosonic fluctuations: MFA 27.05.2015 | B.-J. Schaefer | Giessen University | 10
Phase diagram N f =2 QM m π = 138 MeV chiral limit [BJS, J Wambach 2005] 27.05.2015 | B.-J. Schaefer | Giessen University | 11
FRG and QCD � Polyakov-loop improved quark-meson flow: [Herbst, Pawlowski, BJS 2007 2013] fluctuations of Polyakov-loop , quark and meson → U Pol ( Φ ) Yang-Mills flow is replaced by → U Pol ( Φ ) effective Polyakov-loop potential fitted to lattice Yang-Mills thermodynamics 27.05.2015 | B.-J. Schaefer | Giessen University | 12
FRG: Quark-Meson with Polyakov ( T 0 ( µ ) = const) with back reaction ( T 0 ( µ )) without back reaction 200 200 m π =138 MeV 150 150 T [MeV] T [MeV] m π =138 MeV 100 100 χ crossover χ crossover Φ crossover Φ crossover — Φ crossover — Φ crossover 50 50 χ 1st order χ 1st order σ (T=0)/2 σ (T=0)/2 0 0 0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350 µ [MeV] µ [MeV] [Herbst, Pawlowski, BJS 2010,2013] 27.05.2015 | B.-J. Schaefer | Giessen University | 13
Critical Endpoint Exact location of CEP not ( yet ) accessible with lattice, FRG & DSE 200 µ B /T = 4 , 5 µ B /T=2 µ B /T=3 150 T [MeV] 100 Lattice: curvature range κ =0.0066-0.0180 DSE: chiral crossover DSE: critical end point 50 DSE: chiral first order DSE: deconfinement crossover 0 0 50 100 150 200 µ q [MeV] [Herbst, Pawlowski, BJS 2011] [Fischer, Luecker, Welzbacher 2014] so far: we can exclude CEP for small densities: μ B / Τ <2 Higher densities: dynamical baryons needed! 21.09.2017 | B.-J. Schaefer | Giessen University | 14
(2+1)-flavor effective action [F. Rennecke, BJS 2017] k . 2 π T c ≈ 1 GeV • effective action for scales below • relevant current quarks: u,d,s m u ≈ m d < m s ⇒ N f = 2 + 1 • dominant 4-quark channel with chiral U(N)xU(N) symmetry � 2 + h� � 2 i � λ S − P q T a q q i γ 5 T a q ¯ ¯ • (Pseudo)scalar meson nonet via bosonization κ − + iK − 0 1 � 0 + i η l + i π 0 � 1 σ l + a 0 0 + i π − a − √ 2 1 κ 0 + iK 0 a + 1 � 0 + i η l − i π 0 � Σ = T a ( σ a + i π a ) = 0 + i π + σ l − a 0 B C √ √ 2 2 @ A κ + + iK + κ 0 + i ¯ K 0 1 ¯ 2 ( σ s + i η s ) √ • (Pseudo)scalar mixing angels mass eigenstates flavor eigenstates 27.05.2015 | B.-J. Schaefer | Giessen University | 15
(2+1)-flavor quark-meson model [F. Rennecke, BJS 2017] • gluons fully integrated out: effective action Z n o + ˜ Z Σ ,k ∂ µ Σ · ∂ µ Σ † � � � � Γ k = ¯ γ µ ∂ µ + γ 0 µ q + ¯ qh k · Σ 5 q + tr U k ( Σ ) q Z q,k x q T = ( l, l, s ) Σ 5 = T a ( σ a + i γ 5 π a ) ˜ • effective potential U k = U k ( ρ 1 , ˜ ρ 2 ) − j l σ l − j s σ s − c k ξ ξ = det( Σ + Σ † ) explicit chiral symmetry breaking: U(3) x U(3) sym. potential anomalous U(1)A breaking finite light & strange current quark masses (two chiral invariants) via ’t Hooft determinant ρ i = tr( Σ · Σ † ) i p 2 + m 2 → Z k p 2 + m 2 • wave function renormalizations • Yukawa couplings Z l,k 0 0 h l,k h l,k h ls,k Z l,k Z q,k = 0 0 h l,k h l,k h ls,k 0 0 Z s,k h k = h sl,k h sl,k h s,k Z φ ,k 0 0 Z Σ ,k = 0 Z φ ,k 0 Z φ ,k 0 0 27.05.2015 | B.-J. Schaefer | Giessen University | 16
Chiral Phase Diagram [F. Rennecke, BJS 2017] • different truncations: Z n o + ˜ Z Σ ,k ∂ µ Σ · ∂ µ Σ † � � � � Γ k = ¯ γ µ ∂ µ + γ 0 µ q + ¯ qh k · Σ 5 q + tr U k ( Σ ) q Z q,k x truncation running couplings ( T CEP , µ CEP ) [MeV] ¯ U k , ¯ h l,k , ¯ ˜ LPA 0 +Y h s,k , Z l,k , Z s,k , Z φ ,k (61,235) ‣ ¯ U k , ¯ ˜ h l,k , ¯ LPA+Y (46,255) h s,k ¯ ˜ LPA (44,265) U k 21.09.2017 | B.-J. Schaefer | Giessen University | 17
Chiral Phase Diagram [F. Rennecke, BJS 2017] • Critical Endpoint for different truncations: truncation running couplings ( T CEP , µ CEP ) [MeV] ¯ U k , ¯ h l,k , ¯ ˜ LPA 0 +Y h s,k , Z l,k , Z s,k , Z φ ,k (61,235) ¯ U k , ¯ ˜ h l,k , ¯ LPA+Y (46,255) h s,k ¯ ˜ LPA (44,265) U k 21.09.2017 | B.-J. Schaefer | Giessen University | 18
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