Aspects of Two- and Three-Flavor Chiral Phase Transitions

Transcription

1 Aspects of Two- and Three-Flavor Chiral Phase Transitions Mario Karl-Franzens-Universität Graz Institut für Physik Fachbereich Theoretische Physik Kyoto, September 6, 211

2 Table of Contents 1 Motivation 2 The N f -Flavor Quark-Meson (QM) Model Functional Renormalization Group Flavor QM Model U(2) SU(2) QM Model

3 Conjecture for the 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/NICA B <ψψ> quark matter crossover superfluid/superconducting 2SC phases? <ψψ> = / CFL neutron star cores (non-)standard scenario (philipsen, de forcrand last decade) weakening of crossover for small µ > (endrodi et al. 211) influence of anomaly? (pisarski, wiczek 1984, chen et al. 29)

4 How To Spoil Symmetries explicit symmetry breaking: symmetry only approximate anomalies: violation by quantum terms via regularization/renormalization procedure spontaneously broken symmetry: ground state/vacuum less symmetric than Lagrangian

5 How To Spoil Symmetries explicit symmetry breaking: symmetry only approximate anomalies: violation by quantum terms via regularization/renormalization procedure spontaneously broken symmetry: ground state/vacuum less symmetric than Lagrangian Chiral Symmetry U B (1) SU L (N f ) SU R (N f ) U A (1) SU L (N f ) SU R (N f ) U A (1) expl. broken by quark-masses U A (1) broken by axial anomaly SU L (N f ) SU R (N f ) spontaneously broken to SU R+L (N f )

6 The Quark-Meson Model (QM-Model) The N f -Flavor Quark-Meson (QM) Model Functional Renormalization Group Mesons bound states of quark and antiquark: treated as separate dofs in QM model representation of χ sym coupled to quarks: Yukawa theory of quarks and mesons [ L M = tr µ Σ µ Σ ] +U({ρ i i N},ξ) h b σ b Σ = t b (σ b +iπ b ), t b generators of U(N f ) ( ρ i = tr[ ΣΣ ) ] i, ξ = det(σ)+det (Σ ) L QM = L m + q( µ γ µ iht b (σ b +iγ 5 π b ))q

7 Order Parameter in the QM-Model The N f -Flavor Quark-Meson (QM) Model Functional Renormalization Group The Chiral Condensate as Order Parameter qq > : SU R (N f ) SU L (N f ) SU R+L (N f ) N f = 2 : SO(4) SO(3)

8 Order Parameter in the QM-Model The N f -Flavor Quark-Meson (QM) Model Functional Renormalization Group The Chiral Condensate as Order Parameter qq > : SU R (N f ) SU L (N f ) SU R+L (N f ) N f = 2 : SO(4) SO(3) usual 2-flavor qm-model only ρ 1 = tr [ ΣΣ ] O(4)-representation: (σ, π) (without σ 1,2,3,π ) σ cl σ qq rotations in R 4 rotations around σ

9 Order Parameter in the QM-Model The N f -Flavor Quark-Meson (QM) Model Functional Renormalization Group The Chiral Condensate as Order Parameter qq > : SU R (N f ) SU L (N f ) SU R+L (N f ) N f = 2 : SO(4) SO(3) usual 2-flavor qm-model only ρ 1 = tr [ ΣΣ ] O(4)-representation: (σ, π) (without σ 1,2,3,π ) σ cl σ qq rotations in R 4 rotations around σ In General one quark-condensate for every flavor

10 The N f -Flavor Quark-Meson (QM) Model Functional Renormalization Group Wetterich RG for Effective Potential (FDE PDE) Derivative Expansion ( Low Momentum Expansion) Σ(x) Σ only scale-dependency in mesonic potential U k three dimensional version of optimized ( Litim ) regulator flow for U k with Γ k=λ [Σ] = S[Σ,,]: k U k = k4 12π 2 [ 2N 2 ( ) f 1 Eb coth E b 2T b=1 N f 2N c 1 E f f=1 { ( ) ( )} ] Ef +µ f Ef µ f tanh +tanh 2T 2T

11 Solution Techniques (PDE ODE) Grid Method The N f -Flavor Quark-Meson (QM) Model Functional Renormalization Group discretize domain of effective potential: D R n {x i } coupled ODE for {U k (x i )} non-local information available chiral limit boundary of domain Taylor Method Taylor expand potential (around minimum) coupled ODE for expansion coefficients (couplings) numerically cheaper critical exponents interpretation in terms of couplings chiral limit first order transitions

12 2+1 Flavor Quark Meson Model with FRG condensates rotated to non-strange/strange basis (σ,σ 8 ) (σ x,σ y ) potential as function U k (ρ 1,ρ 2,ξ(ρ 1,ρ 2 )) with and without coupling of ξ: c k c different realizations depending on σ x,y explicit symmetry breaking in light and strange direction h x < h y mesonic spectrum: σ,a,κ,f,η,π,k,η degenerate chemical potential µ x = µ y Taylor expansion through O(ρ 3 1,2 )

13 Mesonic Masses With Anomaly: FRG vs. Mean-Field (MF) fix initial potential U Λ in vacuum... masses [MeV] 12 π σ η a masses [MeV] κ K f η solid line: FRG, dashed line: MF MF: Schaefer, Wagner 29 FRG: MM, Schaefer in preparation 211

14 Mesonic Masses Without Anomaly: FRG vs. MF... to experimental values (physical point) masses [MeV] π σ a masses [MeV] κ K f η solid line: FRG, dashed line: MF MF: Schaefer, Wagner 29 FRG: MM, Schaefer in preparation 211

15 Condensates With and Without Anomaly: FRG vs. MF fluctuations wash out chiral crossover in light sector effect on strange sector mitigated 12 1 σ x σ y 12 1 σ x σ y 8 8 σ [MeV] 6 4 σ [MeV] solid line: FRG, dashed line: MF MF: Schaefer, Wagner 29 FRG: MM, Schaefer in preparation 211

16 Critical Endpoint (CEP) With Anomaly there exists a critical endpoint... µ c = 196 MF µ c = 35 FRG 12 1 π σ a η 12 1 π σ a η masses [MeV] masses [MeV] MF: Schaefer, Wagner 29 FRG: MM, Schaefer in preparation 211

17 CEP Without Anomaly... driven towards higher µ/lower T by fluctuations µ c = 157 MF µ c = 262 FRG masses [MeV] π 2 π 1 σ 1 σ a a masses [MeV] MF: Schaefer, Wagner 29 FRG: MM, Schaefer in preparation 211

18 Condensates at CEP: FRG vs. MF with anomaly without anomaly 12 1 σ x σ y 12 1 σ x σ y 8 8 σ [MeV] 6 4 σ [MeV] solid line: FRG, dashed line: MF MF: Schaefer, Wagner 29 FRG: MM, Schaefer in preparation 211

19 Light Condensate in Phase Diagram with anomaly without anomaly σ 6 x µ [MeV] σ 6 x µ [MeV] 2 25 MF: Schaefer, Wagner 29 FRG: MM, Schaefer in preparation 211

20 Gradient of Light Condensate in Phase Diagram with anomaly without anomaly dσ x dσ x µ [MeV] µ [MeV] MM, Schaefer in preparation 211

21 Critical Band Motivation T max,µ : maximizes σ x(t,µ) σx(t,µ) I: σx(tmax,µ,µ).9 σx(t,µ) σx(tmax,,µ) II: σx(tmax,,µ).9 III: the same as II with T max, T max,µc crit. band I crit. band II crit. band III MF I µ [MeV] 5 crit. band I crit. band II crit. band III MF I µ [MeV] with anomaly without anomaly MF: Schaefer, Wagner 29 FRG: MM, Schaefer in preparation 211

22 First Order Line Motivation quark number susceptibility with anomaly without anomaly χ q /µ T = χ q /µ µ [MeV] µ [MeV] MM, Schaefer in preparation 211

23 Influence of Anomaly Motivation with anomaly different strange quark masses influence of anomaly m qs = 47 m qs = 426 m qs = µ [MeV] 5 m qs = 41 m qs = 47, c = µ [MeV] MM, Schaefer in preparation 211

24 Sharpness of Crossover with anomaly without anomaly max T ( dσ x ) [MeV] µ [MeV] max T ( dσ x ) [MeV] µ [MeV] MM, Schaefer in preparation 211

25 Influence of σ-mass Motivation with anomaly - all other masses fixed no weakening of crossover without anomaly? m σ = 558 m σ = µ [MeV] max T ( dσ x ) [MeV] m σ = 558 m σ = µ [MeV] MM, Schaefer in preparation 211

26 Comparison to Two Flavors I comparison to 2 flavor quark-meson model masses [MeV] π σ O(4)-model condensates [MeV] σ x σ y O(4)-model MM, Schaefer in preparation 211

27 Comparison to Two Flavors II comparison to 2 flavor quark-meson model U(3)xSU(3) O(4)-model µ [MeV] max T ( dσ ) [MeV] U(3)xSU(3) O(4)-model µ [MeV] MM, Schaefer in preparation 211

28 Effect of Taylor-Expansion Order and Cutoff Scale Λ with anomaly O(ρ 3 ) O(φ 6 ) O(ρ 2 ) O(φ 4 ) µ [MeV] 5 Λ = 1 Λ = µ [MeV] MM, Schaefer in preparation 211

29 3 Flavor 2 Flavor Motivation U(2) SU(2)

30 Taylor-RG vs. Grid-RG vs. MF fixed initial anomaly coupling such that m η (k = ) = 98 MeV solid line: Taylor-RG crosses: grid-rg dashed: MF masses [MeV] π σ a η σ [MeV] σ 1*σ [MM,Schaefer,Strodthoff,von Smekal in preparation 211]

31 Sensitivity to Variation of h,3 solid line: h = h 3 = h phys short dashed: h = h phys, h 3 =.1 h phys long dashed: h = h 3 =.1 h phys masses [MeV] π σ a,η condensates [MeV] σ 1*σ [MM,Schaefer,Strodthoff,von Smekal in preparation 211]

32 U A (1) Violating Couplings gauge theories with fermions: not U V (1) U A (1)

33 U A (1) Violating Couplings gauge theories with fermions: not U V (1) U A (1) a ij (k) = ( ρ1 ) i ( ξ ) j U k (ρ 1,ξ) min no U A (1) violating couplings in initial action solid: with quarks, dashed: without quarks σ 2 couplings at minimum [MeV 2 ] k [MeV] a 1 a 1 σ 4 couplings at minimum a 2 a 11-6 a k [MeV] [MM,Schaefer,Strodthoff,von Smekal in preparation 211]

34 Restoration of Mesonic Mass Spectrum Degeneracy no U A (1) violating couplings at k = Λ h = h phys, h 3 =.1 h phys masses [MeV] π σ a a,η condensates [MeV] σ 1*σ [MM,Schaefer,Strodthoff,von Smekal in preparation 211]

35 Partial Chiral Limit Motivation no U A (1) violating couplings at k = Λ h = h 3 = h phys additional (exact!) symmetry above chiral T c masses [MeV] π σ a a,η condensates [MeV] σ σ [MM,Schaefer,Strodthoff,von Smekal in preparation 211]

36 Couplings in Partial Chiral Limit additional symmetry above T c is σ σ 3 (Z 2 ) calculated order parameter critical exponent: β =.366 (.32.33) σ 2 couplings at minimum [MeV 2 ] σ 4 couplings at minimum a 1-4 a a 2 a 11 a 2 [MM,Schaefer,Strodthoff,von Smekal in preparation 211]

37 Taylor vs. Grid Revisited σ 2 couplings at origin [MeV 2 ] a 1 a 1 σ 2 couplings at minimum [MeV 2 ] a 1 a 1 [MM,Schaefer,Strodthoff,von Smekal in preparation 211]

38 2+1 flavor Quark-Meson Model weakening of crossover for small µ with CEP anomaly seems to weaken transition possibly no back-bending of first order line U(2) SU(2) Quark-Meson Model partial restoration of mass U A (1) at chiral T c spontanous generation of anomalous mass by quarks second order phase transition in partial chiral limit

39 2+1 flavor Quark-Meson Model weakening of crossover for small µ with CEP anomaly seems to weaken transition possibly no back-bending of first order line U(2) SU(2) Quark-Meson Model partial restoration of mass U A (1) at chiral T c spontanous generation of anomalous mass by quarks second order phase transition in partial chiral limit 2+1 flavor grid technique for Columbia plot scale dependency of anomalous coupling U(2) SU(2): µ > include ρ 2 first order transition? compare next talk

40 U(2) SU(2) Quark Meson Model with FRG condensates (σ,σ 3 ): σ 3 quark-mass splitting potential as function U k (ρ 1,ξ): scale dependent anomaly explicit symmetry breakings h and h 3 (quark splitting) mesonic spectrum: σ,a,η,π grid and Taylor technique Realizations σ = σ 3 = (m π, m η,a1/2 ): σ >, σ 3 = (m π =, m η,a1/2 ): σ σ 3 (m π =, m η,a1/2 = ): σ = σ 3 (m π =, m η,a1/2 = ):