QCD Phases with Functional Methods

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1 QCD Phases with Mario PhD-Advisors: Bernd-Jochen Schaefer Reinhard Alkofer Karl-Franzens-Universität Graz Institut für Physik Fachbereich Theoretische Physik Rab, September 2010 QCD Phases with

2 Table of Contents QCD Phases with

3 Strong Interaction Quantum Chromodynamics (QCD) fundamental theory of the strong interaction degrees of freedom: quarks, gluons non Abelian gauge theory running coupling: asymptotic freedom confinement QCD phase diagram - nonperturbative methods QCD Phases with

4 Conjecture for the QCD Phase Diagram Temperature early universe LHC crossover vacuum RHIC SPS <ψψ> = / 0 hadronic fluid n = 0 n > 0 B AGS SIS nuclear matter µ quark gluon plasma FAIR/NICA B <ψψ> 0 quark matter crossover superfluid/superconducting 2SC phases? <ψψ> = / 0 CFL neutron star cores [The CBM Physics Book, 2010] QCD Phases with

5 A Theorist s Playground 2nd order O(4)? N f = 2 2nd order Z(2) 1st Pure Gauge m tric s phys. point N f = 3 crossover N f = 1 [de Forcrand, Philipsen, 2009] m s 0 0 1st 2nd order Z(2) m, m u d QCD Phases with

6 Chiral Critical Surface and Triple Line - Standard Scenario [de Forcrand, Philipsen, 2009] QCD Phases with

7 χ Crit. Surface and Triple Line - Non-Standard Scenario [de Forcrand, Philipsen, 2009] QCD Phases with

8 Approaches to the Phase Diagram Methods perturbation theory lattice QCD functional approaches Dyson-Schwinger equations renormalization group effective theories QCD Phases with

9 Order Parameters Accesible with chiral symmetry (chiral limit) quark condensate ψψ scalar dressing function of quark propagator B( p, ω) QCD Phases with

10 Order Parameters Accesible with chiral symmetry (chiral limit) quark condensate ψψ scalar dressing function of quark propagator B( p, ω) center symmetry (pure Yang-Mills) Polyakov loop QCD Phases with

11 Order Parameters Accesible with chiral symmetry (chiral limit) quark condensate ψψ scalar dressing function of quark propagator B( p, ω) center symmetry (pure Yang-Mills) Polyakov loop dual condensate ( dressed Polyakov loop ) [Bilgici, et al., 2008] Σ 1 = 2π dφ 0 2π e iφ ψψ φ ψ ( x, T) 1 = e iφ ψ( x,0) QCD Phases with

12 Order Parameters Accesible with chiral symmetry (chiral limit) quark condensate ψψ scalar dressing function of quark propagator B( p, ω) center symmetry (pure Yang-Mills) Polyakov loop dual condensate ( dressed Polyakov loop ) [Bilgici, et al., 2008] Σ 1 = 2π dφ 0 2π e iφ ψψ φ ψ ( x, T) 1 = e iφ ψ( x,0) dual scalar dressing function [Fischer, Mueller, 2009] Σ B = 2π dφ 0 2π e iφ B( 0,ω 0 (φ)) QCD Phases with

13 Effective Average Action [Wetterich, 1993] effective action Γ = (logz) effective average action Γ k = (logz k ) S k take scale derivative, i.e. integrate infinitesimal momentum shell RG flow k Γ k Functional RG flow equation for Γ k UV IR (micro macro) initial value Γ Λ? QCD Phases with

14 Exact Renormalization Group Flow Equation for Effective Average Action [Wetterich 1993] { ( ) } k Γ k = 1 2 Tr Γ (2) 1 k k +R k R k k Γ k = 1 2 example for R k : QCD Phases with

15 (2+1) Flavor Quark-Meson Truncation Mesonic SU(3) SU(3)-Symmetric Theory { [ ] Γ k=λ [Φ] = d 4 x Tr ( µ Φ)( µ Φ ) } +U k=λ (Φ) Φ = T j (σ j +iπ j ), T j : generators of U(3) QCD Phases with

16 (2+1) Flavor Quark-Meson Truncation Mesonic SU(3) SU(3)-Symmetric Theory { [ ] Γ k=λ [Φ] = d 4 x Tr ( µ Φ)( µ Φ ) } +U k=λ (Φ) Φ = T j (σ j +iπ j ), T j : generators of U(3) Yukawa-Coupling to Three Quark Flavors [Schaefer, Wagner 2009] Γ k=λ [Φ, q,q] = Γ k=λ [Φ] d 4 x { q ( ) } / +igφ 5 q Φ 5 = T j (σ j +iγ 5 π j ) QCD Phases with

17 (2+1) Flavor Quark-Meson Truncation Mesonic SU(3) SU(3)-Symmetric Theory { [ ] Γ k=λ [Φ] = d 4 x Tr ( µ Φ)( µ Φ ) } +U k=λ (Φ) Φ = T j (σ j +iπ j ), T j : generators of U(3) Yukawa-Coupling to Three Quark Flavors [Schaefer, Wagner 2009] Γ k=λ [Φ, q,q] = Γ k=λ [Φ] d 4 x { q ( ) } / +igφ 5 q Φ 5 = T j (σ j +iγ 5 π j ) PDE for grand potential solve by Taylor expansion in chiral invariants QCD Phases with

18 Features coupled system of ordinary non-linear differential equations Taylor expansion: around one (global) minimum flow of minimum first order phase transitions not accessible expansion order in Φ? chiral anomaly? QCD Phases with

19 Features coupled system of ordinary non-linear differential equations Taylor expansion: around one (global) minimum flow of minimum first order phase transitions not accessible expansion order in Φ? chiral anomaly? initial values fixed in vacuum predictions for non-vanishing T, µ vary explicit symmetry breaking term to explore Columbia plot tric m s m s 0 0 2nd order O(4)? phys. point 1st N f = 2 2nd order Z(2) m u, md 2nd order Z(2) N f = 3 crossover QCD Phases with 1st Pure Gauge N f = 1

20 Masses and Condensates at Physical Point for T 0 Λ = 1500MeV expansion through O(Φ 6 ) optimized regulator m π = 138MeV m σ = 600MeV m K = 496MeV f π = 93MeV f K = 113MeV m 2 η + m2 η = 1.22e6MeV 2 meson masses [MeV] T [MeV] π σ a 0 η K κ condensates [MeV] σ ud σ s T [MeV] QCD Phases with

21 Towards the Chiral Limit condensates [MeV] σ ud σ s 0.5h ud, σ ud 0.5h ud, σ s 0.2h ud, σ ud 0.2h ud, σ s T [MeV] QCD Phases with

22 DSEs - Basic Concepts based on generalization of Gauss law in Gribov region Ω i.e. 0 = Ω DA δ δa e S[A,...]+ A,J +... infinite set of equations for n-point functions QCD Phases with

23 DSEs - Basic Concepts based on generalization of Gauss law in Gribov region Ω i.e. 0 = Ω DA δ δa e S[A,...]+ A,J +... infinite set of equations for n-point functions DSE for Quark Propagator ( D 1 Q ) ij(p) = Z 2 ( D 1 Q,0 ) ij(p) +Z 1F C Nc g 2 (2π) 4 s R 4 γ ) µν ) µ il ( Dγ (p s) ( DQ (s) Γ ν kj(s,p) lk QCD Phases with

24 Quark Propagator Structure of the Quark Propagator T = 0: ( D 1 Q )(p) γµ p µ A(p)+B(p) T 0: ( D 1 Q )(p) γ4 p 4 C(p)+γ i p i A(p)+B(p)+γ i p i γ 4 p 4 D(p) QCD Phases with

25 Quark Propagator Structure of the Quark Propagator T = 0: ( D 1 Q )(p) γµ p µ A(p)+B(p) T 0: ( D 1 Q )(p) γ4 p 4 C(p)+γ i p i A(p)+B(p)+γ i p i γ 4 p 4 D(p) DSE set of coupled equations for dressing functions (A, B,...) gluon propagator, e.g. from lattice data presence quark gluon vertex: truncations/modeling necesarry depending on model: regularization/renormalization QCD Phases with

26 Representative Solutions in Vacuum FM model for quark gluon vertex Fischer/Mueller model 0.6 Dressing Function B/A [GeV] Ln(p 2 ) [GeV] QCD Phases with

27 objective: QCD phase diagram functional renormalization group (FRG): investigate the phase-boundaries Dyson-Schwinger equations (DSEs): on T-axis with QCD QCD Phases with