SUNY Stony Brook August 16, Wolfram Weise. with. Thomas Hell Simon Rössner Claudia Ratti

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1 SUNY Stony Brook August 16, 27 PHASES of QCD POLYAKOV LOOP and QUASIPARTICLES Wolfram Weise with Thomas Hell Simon Rössner Claudia Ratti C. Ratti, M. Thaler, W. Weise: Phys. Rev. D 73 (26) 1419 C. Ratti, S. Rößner, M. Thaler, W. Weise: Eur. Phys. J. C 49 (27) 213 S. Rößner, C. Ratti, W. Weise: Phys. Rev. D 75 (27) 347 C. Ratti, S. Rößner, W. Weise: Phys. Lett. B 649 (27) 57

2 QCD PHASE DIAGRAM T [GeV] N f = 2 (q = u, d).2 T c quark gluon phase temperature hadron phase qq nuclear matter critical point CSC phase qq 1 GeV baryon chemical potential µ B

3 1. SYMMETRIES and SYMMETRY BREAKING PATTERNS SU(2) R SU(2) L CHIRAL SYMMETRY SU(3) R SU(3) L s quark mass u,d quark mass Z(3) SYMMETRY Center of SU(N c = 3) gauge group E. Laermann, O. Philipsen: Ann. Rev. Nucl. Part. Sci. 53 (23) 163

4 1.1 Z(3) SYMMETRY Consider PURE GAUGE ( pure glue ) QCD Thermodynamics: periodic boundary condition on gauge fields A µ ( x, τ + β) = A µ ( x, τ) (β = 1/T ) Gauge transformations: g( x, τ + β) = z g( x, τ) g A µ = g(a µ + µ )g [ z = exp i 2π n ] N c (n = 1, 2, 3,...) Polyakov loop: Φ( x ) = 1 N c T r [ exp ( i 1 T dτ A 4 ( x, τ) )] Φ z Φ = e 2πi n/3 Φ

5 DECONFINEMENT and spontaneous breaking of Z(3) SYMMETRY Polyakov loop: Φ( x ) = 1 N c T r Free energy of static color sources: [ exp ( i 1 T dτ A 4 ( x, τ) F (T ) = lim r F Q Q(r = x y, T ) = T ln Φ 2 Order parameter for confinement / deconfinement: )] U(Φ) T < T c confinement: Z(3) symmetry intact U(Φ) T > T c deconfinement: Z(3) symmetry spontaneously broken Φ = Φ Φ

6 QCD with (almost) MASSLESS u- and d-quarks (N = 2) spin 1.2 CHIRAL SYMMETRY momentum spin left - handed right - handed SU(2) L SU(2) R f ψ = (u, d) pseudoscalar - isovector π a ψ γ 5 t a ψ T SPONTANEOUS SYMMETRY BREAKING (Nambu - Goldstone) LOW T HIGH T U eff U eff σ ψ ψ scalar - isoscalar σ < φ > CHIRAL (QUARK) CONDENSATE qq qq = σ < φ >

7 Spontaneously Broken CHIRAL SYMMETRY NAMBU - GOLDSTONE BOSON: PION ORDER PARAMETER: PION DECAY CONSTANT A a µ() π b (p) = iδ ab p µ f π π µ Axial current f π = 92.4 MeV ν SYMMETRY BREAKING SCALE Λ χ = 4π f π 1 GeV MASS GAP PCAC: m 2 π f 2 π = m q ψψ + O(m 2 q) Gell-Mann - Oakes - Renner Relation

8 Tests of Chiral Symmetry Breaking Scenario: Lattice QCD.8 (am! ) 2.6 m! "676 MeV perfect consistency with leading-order Gell-Mann - Oakes - Renner relation M. Lüscher 294 Proc. Lattice 25 physical point amq m 2 π = m u + m d f 2 π + qq + O(m 2 q) O(m 2 q log m q ) confirmation of standard (Nambu-Goldstone) spontaneous chiral symmetry breaking with Pions as Goldstone Bosons and large quark condensate: qq (.23 GeV) fm 3 fm 3

9 2. Introducing the PNJL MODEL POLYAKOV LOOP dynamics Confinement Synthesis of and NAMBU & JONA-LASINIO model Chiral Symmetry... very much like Ginsburg - Landau approach: identify order parameters as collective degrees of freedom which drive dynamics and thermodynamics

10 2.1 Data base: QCD Thermodynamics on the Lattice (e.g. : F. Karsch, J. Phys. G31 (25) 633) Equation of State of pure gauge (purely gluonic) QCD Energy density Entropy density Pressure T c 27 MeV lattice: G. Boyd et al. Nucl. Phys. B 469 (1996) 419 Extrapolations of full-qcd thermodynamics to non-zero quark chemical potentials nq T n q T 3 Μ.6 T c Μ.4 T c Μ.2 T c T T c C. Ratti, M. Thaler, W. W. : Phys. Rev. D 73 (26) Quark number density lattice: C.R. Allton et al. Phys. Rev. D 68 (23) 1457

11 2.2 Sketch of the PNJL MODEL Action : S(ψ, ψ, φ) = β=1/t dτ V d 3 x [ ψ τ ψ + H(ψ, ψ, φ) ] T V U(φ, T) VT Fermionic Hamiltonian density (NJL) : Four-Fermion interaction H = iψ ( α + γ 4 m φ) ψ + V(ψ, ψ ) quark glue V = L int quark Temporal background gauge field Φ = 1 N c Tr [ exp ( i 1/T )] dτ A 4 1 Tr exp(iφ/t) 3 φ = φ 3 λ 3 + φ 8 λ 8 Polyakov loop SU(3) Effective potential : U(Φ) confinement T < T c U(Φ) T > T c deconfinement

12 2.3 Basics of the NJL MODEL Y. Nambu, G. Jona-Lasinio: Phys. Rev. 122 (1961) applications to HADRON PHYSICS: U. Vogl, W. W. : Prog. Part. Nucl. Phys. 27 (1991) 195 T. Hatsuda, T. Kunihiro: Phys. Reports 247 (1994) many others QUARK COLOR CURRENT: J a µ(x) = ψ(x)γ µ λa 2 ψ(x) Assume: short correlation range for color transport between quarks l c <.2 fm G c g 2 l 2 c quark glue quark L int = G c J a µ(x) J µ a(x) (chiral invariant) LOCAL SU(N c) GLOBAL SU(N c ) gauge symmetry of symmetry of QCD NJL model

13 Fierz transform of Color - Current-Current Interaction (N f = 2 flavors): QUARK-ANTIQUARK channels vector + axial vector L q q = G 2 [( ψψ) 2 + ( ψiγ 5 τψ) 2 ] color octet terms H DIQUARK channels L qq = ( ψiγ 5 τ 2 λ A C ψ T )(ψ T Ciγ 5 τ 2 λ A ψ) Self-consistent MEAN FIELD approximation: GAP equation: M = m o G ψψ quark CHIRAL CONDENSATE: ψψ = 2iN f N c d 4 p M θ(λ 2 p 2 ) (2π) 4 p 2 M 2 + iɛ G = 4 3 H 1 GeV 1 m 5 MeV Λ.65 GeV SPONTANEOUS CHIRAL SYMMETRY BREAKING GOLDSTONE BOSONS (pions)...but: no confinement!

14 π the MESON sector Bethe-Salpeter Equation in (colour singlet) QUARK-ANTIQUARK channels: antiquark quark = + T K K T SU(3) NJL model including Axial U(1) breaking t Hooft interaction (INSTANTONS) PSEUDOSCALAR MESON SPECTRUM Gell-Mann - Oakes -Renner Relation satisfied: mass 1. S ].8 [GeV] 1..8 U(1) A U(1) A breaking breaking η m 2 π = m u + m d f 2 π qq + O(m 2 q).6.6 η η f 2 π = 4iN c d 4 p M 2 θ(λ 2 p 2 ) (2π) 4 (p 2 M 2 + iɛ) U(3) x U(3).2 L R SU(3) x SU(3) L R K m s = MeV MeV s π S. Klimt, M. Lutz, U. Vogl, W. W. : Nucl. Phys. A 516 (199) 429 SU(3) m u L = m SU(3) d = m s = R π, K, η, η 8 π, K, η 8 m u,d 5 MeV m = m = 5 MeV u d

15 2.3 Polyakov Loop (Thermal Wilson Line) Order parameter of spontaneously broken Z(N c ) center symmetry of SU(N c ) pure gauge theory ( DECONFINEMENT ) [ ( Φ = 1 )] 1/T Tr exp i dτ A 4 1 Tr exp(iφ/t) N c 3 φ = φ 3 λ 3 + φ 8 λ 8 { SU(3) Effective potential : (R. Pisarsky (2); K. Fukushima (24)) U(Φ, T) = 1 2 a(t) Φ Φ b(t) ln[1 6 Φ Φ + 4(Φ 3 + Φ 3 ) 3(Φ Φ) 2 ] U(Φ) T < T c confinement: Z(3) symmetry intact Φ = U(Φ) T > T c Φ deconfinement: Z(3) symmetry spontaneously broken Φ

16 Polyakov Loop EFFECTIVE POTENTIAL and Z(3) SYMMETRY Φ z Φ = e 2πi n/3 Φ (n = 1, 2, 3,...) U(Φ, T) = 1 2 a(t) Φ Φ b(t) ln[1 6 Φ Φ + 4(Φ 3 + Φ 3 ) 3(Φ Φ) 2 ] 5 U(Φ) Φ Φ 8 S. Rößner

17 Comparison with PURE GLUE Lattice Thermodynamics Minimization of U(Φ(T), T) = p(t) fit a(t), b(t) energy density, entropy density, pressure ε T 4 3 s 4 T 3 3 p T 4 U T Effective potential.5 T 1. T.75 T 1.25 T 2. T T T c O. Kaczmarek et al. Phys. Lett. B 543 (22) S. Rößner, C. Ratti, W. W. Phys. Rev. D 75 (27) 347 first order phase transition T c (pure gauge) T 27 MeV

18 2.4 Action : S S(ψ, ψ, φ) = Bosonisation: Thermodynamics of the PNJL Model β=1/t dτ V d 3 x [ ψ τ ψ + H(ψ, ψ, φ) ] V T U(φ, T) TV Goldstone Bosons of spontaneously broken Chiral Symmetry π a ψ iγ 5 τ a ψ Scalar Field σ ψψ Chiral Condensate σ ψψ Diquark Field ψ T Γψ Cooper Pair Condensate ψ T Γψ Thermodynamical Potential: Ω = T V ln Z Z = Dϕ exp[ S(T, V, µ)] ( τ τ µ)

19 Thermodynamics of the PNJL Model Thermodynamical Potential after Matsubara sums: Ω = U(Φ, T ) + σ2 2G + 2H { [ ] 2N d 3 p f (2π) 3 T ln expressions with ε( p ) = p + m : Quasiparticle branches : E 1,2 = ε( p ) ± µ b, E 3,4 = (ε( p ) + µ r ) ± i φ 3, E 5,6 = (ε( p ) µ r ) ± i φ 3, ε( p ) = p 2 + m 2 m = m σ = m G ψψ µ b = µ + 2i φ 8 3, µ r = µ i φ 8 3. Minimize Thermodynamical Potential = E j ε ± µ is the difference of the quasip fermions (quarks), where the lower sign ap Re Ω ϕ = (ϕ = σ,, φ 3, φ 8 ) chiral condensate j diquark [ 1 + e E j/t ] + const Mean Field Equations: Polyakov loop generators

20 3. Results PART 1: DECONFINEMENT and CHIRAL SYMMETRY RESTORATION Ψ Ψ Ψ Ψ T T T c CHIRAL and DECONFINEMENT transitions almost coincide! T T GeV (µ = ) PNJL T Ψ Ψ T T GeV PNJL: S. Rößner, C. Ratti, W. W. : Phys. Rev. D 75 (27) 347 Lattice results: G. Boyd et al., Phys. Lett. B 349 (1995)

21 Entanglement of DECONFINEMENT and CHIRAL SYMMETRY RESTORATION: Chiral Limit chiral condensate 1. Polyakov loop.8 PNJL Ψ Ψ Ψ Ψ.6.4 NJL Pure Gauge T GeV 2nd order chiral transition Thomas Hell 1st order deconfinement transition

22 POLYAKOV LOOP at zero chemical potential PNJL prediction in comparison with 2-flavour Lattice QCD Thermodynamics S. Rößner, C. Ratti, W. W. Phys. Rev. D 75 (27) 347 Lattice data: O. Kaczmarek, F. Zantow: Phys. Rev. D 71 (25) 5458 first order deconfinement transition (pure gauge) cross-over (with quarks)

23 PNJL : Comparisons with N = 2 Lattice Thermodynamics f PRESSURE and ENERGY DENSITY at zero chemical potential p = Ω(T, µ = ) ε = T p(t, µ = ) T C. Ratti, M. Thaler, W. W.: Phys. Rev. D 73 (26) 1419 p(t, µ = ) p psb PNJL NJL pressure (confinement) (no conf.) N t 6 N t 4 8 (ε-3p)/t 4 N t = interaction measure T T c T/T c Lattice data: F. Karsch, F. Laermann, A. Peikert; Nucl. Phys. B 65 (22) 579

24 Results PART I1: Non-zero QUARK (contd.) CHEMICAL POTENTIAL Quark number density: n q (T, µ) = Ω(T, µ) µ.8.6 C. Ratti, M. Thaler, W.W. PRD 73 (26) Μ.6 T c Μ T c nq T Μ.4 T c Μ.2 T c nq T Μ.8 T c T T c T T c Lattice data: Allton et al. Phys. Rev. D 68 (23)

25 Non-zero QUARK CHEMICAL POTENTIAL (contd.) Role of CONFINEMENT (POLYAKOV loop dynamics) 1.8 NJL classic (no confinement) quark number density nq T PNJL (incl. confinement) Μ 12 MeV T T c C. Ratti, M. Thaler, W.W. PRD 73 (26)

26 Non-zero QUARK CHEMICAL POTENTIAL (contd.) Taylor expansion of pressure: p(t, µ) = T 4 n ( µ n c n (T) T) c 2 c 4 c 6 S. Rößner, C. Ratti, W. W. Phys. Rev. D 75 (27) 347 Lattice: C.R. Allton et al. Phys. Rev. D 71 (25) 5458

27 4. PHASE DIAGRAM Issues: Critical point Diquark (superconducting) phase S. Rößner, C. Ratti, W. W.: Phys. Rev. D 75 (27) 347 hadronic phase qq quark gluon phase diquark phase qq... for comparison: critical temperature from Lattice QCD T c 22 MeV ( 2 flavors ) T c = 192 ± 11 MeV ( 2+1 flavors ) M. Cheng et al., Phys. Rev. D74 (26) 5457 O. Kaczmarek, F. Zantow: Phys. Rev. D 71 (25) 5458

28 PHASE DIAGRAM (contd.) Effect of Polyakov Loop Effect of Diquarks T [GeV] T [GeV].1 PNJL without diquarks PNJL without diquarks µ [GeV] µ [GeV] Location of critical point depends sensitively on active degrees of freedom involved S. Rößner, C. Ratti, W. W.: Phys. Rev. D 75 (27) 347

29 PHASE DIAGRAM (contd.).22 T.2 [GeV].18 m = 5 MeV Critical Point and Quark Mass.16 m = m = 5.5 MeV µ [GeV] T c [GeV].16 chiral limit Location of critical point depends sensitively on quark mass.14 m q in MeV.3.4 µ c [GeV]

30 PHASE DIAGRAM (contd.) Status: Lattice QCD and chemical freezeout phenomenology T (MeV) Z. Fodor, S. Katz, JHEP44, Z. Fodor, S. Katz: JHEP 44 (24) 5 P. Braun-Munzinger et al. (26)

31 5. Summary QUASIPARTICLE approach (PNJL model) encoding CHIRAL SYMMETRY and CONFINEMENT in terms of coupled order parameter fields (CHIRAL CONDENSATE, POLYAKOV LOOP) surprisingly successful in comparison with QCD THERMODYNAMICS on the Lattice (T 2 T c ) further developments: PNJL for flavors (including DIQUARKS) Establish contacts with high temperature limit (Transverse Gluons, Hard Thermal Loops, Running Coupling) Extensions beyond MEAN FIELD

32 6. Outlook : PNJL with Running Coupling Running Nambu-Coupling III Α.2 α s (Q) Particle Data Group [1] Lattice (Sternbeck et al., 25 [8]) Q GeV 3 Definition of the Q Running [GeV] Nambu-Coupling, S.Rößner, C. Ratti, W. Weise Thermodynamics of an NJL-Model with R ψψ = N cn f 4π 4 G(Q 2 ) = 2π α(q 2 ) dq for Q Γ Q 2 d 3 M(Q 2 ) Q Q 2 + M(Q 2 ) g(q 2 ) for Q 2 < Γ 2 smooth function α s (Q 2 ) gq continues a δ bg Q 2 p µ heat bath p k ν p G(Q 2 ) = 2π 3 in the IR M(Q 2 ) = G(Q 2 ) ψψ at large Q: k quark condensate Running Nambu-Coupling G GeV Q GeV Q [GeV] 4 Introduction of a three momentum cutoff Ω Γ in order to Q G(Q) [GeV 2 ] Thomas Hell NJL with Simon Rößner contacts with PQCD: replace NJL cutoff by sliding Q-scale see also: Dyson-Schwinger approach

33 NJL with running coupling: dynamical (constituent) quark mass onstituent Quark Mass: Comparison to the Latt ANJL Constituent Quark Mass compared to Lattice Data.3 NJL.25 M(Q) M GeV [GeV] M(Q 2 ) = G(Q 2 ) ψψ Lattice (quenched).5 Quark Self-Energy at Finite Temper Q GeV Q [GeV] Quark Self-Energy Diagram at Finite Temperatu perfect description of pion properties and current algebra k in progress: implementation of Polyakov loop and a δ b thermodynamics incl. thermal self-energies α p µ heat bathν p p k β