QCD Green s Functions, Confinement and Phenomenology ECT*, Trento, 1 September 29 SYMMETRY BREAKING PATTERNS in QCD: CHIRAL and DECONFINEMENT Transitions Wolfram Weise Modelling the PHASES of QCD in contact with Dyson - Schwinger and Lattice QCD results (the non-local PNJL Model) Entanglement of CHIRAL and DECONFINEMENT transitions Role of the AXIAL U(1) ANOMALY PHASE DIAGRAM of QCD at finite BARYON DENSITY, NUCLEAR MATTER, CRITICAL POINT, and all that...
TWO SYMMETRIES that govern LOW-ENERGY QCD pure glue SU(2) R SU(2) L CHIRAL SYMMETRY s quark mass ms physical point? Z(3) SYMMETRY Center of gauge group SU(N c = 3) SU(3) R SU(3) L exact for massless quarks u,d quark mass Columbia plot m u,d exact for infinitely heavy quarks
LATTICE QCD THERMODYNAMICS: CHIRAL and DECONFINEMENT TRANSITIONS 1..8.6.4.2 spontaneously broken Chiral Symmetry.3.4.5.6.7 ψψ T l,s ψψ T= Tr Chiral Condensate p4fat3: N τ =4 6 8 T [MeV]. 1 15 2 25 3 crossover transitions no critical temperature in strict sense chiral and deconfinement transitions seem to coincide 1..8.6.4.2 Lattice QCD (2+1 flavours) almost physical quark masses.3.4.5.6.7.8.9 1. LΦ ren M. Cheng et al. Bielefeld/BNL-Riken/Columbia Phys. Rev. D77 (28) 14511 F. Karsch et al. arxiv:711.661 [hep-lat] spontaneously broken Z(3) Symmetry Tr Polyakov Loop N τ =4 6 8 T [MeV]. 1 15 2 25 3 35 4 45
POLYAKOV LOOP dynamics Confinement Synthesis Pisarsky (2) Fukushima (24) NAMBU & JONA-LASINIO model Ratti, Thaler, W.W. (25) PNJL MODEL Spontaneous Chiral Symmetry Breaking Nambu, Jona-Lasinio (1961) Action : S(ψ, ψ, φ) = β=1/t dτ V Fermion (quark) effective Hamiltonian V d 3 x [ ψ τ ψ + H(ψ, ψ, φ) ] T VT Polyakov loop effective potential U(φ, T) identify dominant collective degrees of freedom ( order parameters) quarks as quasiparticles with dynamically generated masses
Sketch of (non-local) PNJL MODEL Action : S(ψ, ψ, φ) = β=1/t dτ V d 3 x [ ψ τ ψ + H(ψ, ψ, φ) ] T V VT U(φ, T) Fermionic Hamiltonian density (NJL) : H = iψ ( α + γ 4 m φ) ψ + V(ψ, ψ ) chiral invariant Non-local fermion interaction Temporal background gauge field Φ = 1 N c Tr [ exp ( i 1/T φ = φ 3 λ 3 + φ 8 λ 8 )] dτ A 4 1 Tr exp(iφ/t) 3 SU(3) Polyakov loop Effective potential : U(Φ) confinement T < T c U(Φ) T > T c deconfinement
Polyakov Loop Effective Potential from PURE GLUE Lattice Thermodynamics Minimization of U(Φ(T), T) = p(t) [ 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 ] 5 energy density, entropy density, pressure 1.5 Polyakov loop effective potential 4 3 2 1 ε T 4 3 s 4 T 3 3 p T 4 1 2 3 4 5 TT c lattice results: O. Kaczmarek et al. PLB 543 (22) 41 first order phase transition U T 4 T 1..5..5 1..5 T.75 T 1. T 2. T 1.25 T 1.5..2.4.6.8 1. Φ S. Rößner, C. Ratti, W. W. PRD 75 (27) 347 T c (pure gauge) T 27 MeV
Non-Local PNJL Model: GAP EQUATION momentum dependent, dynamical quark mass M(p) = m + 8 N c G d 4 d 4 q M(q) C(p q) (2π) 4 q 2 + M 2 (q) p G C(p q) q.4 old (local) NJL correlation length d 1 3 fm (typical instanton size) Mp GeV.3.2.1 instanton model non-local PNJL dynamical quark dynamical mass quark mass lattice QCD P. Bowman et al. (23). coupling strength G 1 fm consistent with self-energy from Dyson-Schwinger calculations (Landau gauge).5 1. 1.5 2. 2.5 p GeV T. Hell, S. Rößner, M. Cristoforetti, W. W. Phys. Rev. D79 (29) 1422, and preprint iσ(p) = C.D. Roberts, S.M. Schmidt, et al. + many others
THREE - FLAVOUR non-local PNJL MODEL includes: G chiral U(3) R U(3) L invariant plus q q Kobayashi-Maskawa- t Hooft interaction K[det ψ(1 γ 5 )ψ + det ψ(1 + γ 5 )ψ] u u d d K s s breaks axial U(1) A input: G 1 fm 2 K (.63 fm) 5 m u, d m s 3. MeV 7 MeV T. Hell, S. Rößner, M. Cristoforetti, W. W. : Phys. Rev. D79 (29) 1422 and preprint
THREE - FLAVOUR non-local PNJL MODEL (contd.) Chiral low-energy theorems and Current Algebra relations o.k. e.g.: Gell-Mann, Oakes, Renner relation m 2 π f 2 π = m q ψψ + O(m 2 q) output: ūu = dd ss (.34 GeV) 3 (.323 GeV) 3 m π m K m η m η f π f K 139 MeV 495 MeV 547 MeV 964 MeV 92.8 MeV 11.1 MeV η η mixing angle θ η 29º T. Hell, S. Rößner, M. Cristoforetti, W. W. : preprint (29)
PNJL THERMODYNAMICS Construct grand-canonical partition function and minimize thermodynamic potential Relevant piece of thermodynamic potential involving quark quasiparticles: Ω T d 3 p [ ] (2π) 3 ln 1 + 3 Φ e (Ep µ)/t + 3 Φ e 2(Ep µ)/t + e 3(E p µ)/t + d 3 p [ ] (2π) 3 ln 1 + 3 Φ e (Ep+µ)/T + 3 Φ e 2(Ep+µ)/T + e 3(E p+µ)/t in the hadronic phase with Φ : single quarks and diquarks (i.e. color non-singlets) suppressed
Entanglement of CONFINEMENT and SPONTANEOUS CHIRAL SYMMETRY BREAKING Thermodynamics of the PNJL model ψψ T 1. 1. chiral ψψ Φ condensate.8.8 N f = 2 chiral limit m q = σ/σ.6.4.2 2nd order Polyakov loop 1st order.6.4.2 Φ..1.2.3.4.5 T [GeV] S. Rössner, C. Ratti, W. W. Phys. Rev. D 75 (27) 347 T. Hell, S. Rössner, M. Cristoforetti, W. W. Phys. Rev. D 79 (29) 1422
Entanglement of CONFINEMENT and SPONTANEOUS CHIRAL SYMMETRY BREAKING N f = 2 + 1 Thermodynamics of the PNJL model in comparison with Lattice QCD (with almost physical quark masses) chiral condensate qqqq 1..8.6.4.2. ūu / ūu Φ ss / ss.5 1. 1.5 2. 2.5 TT c GeV 1..8.6.4.2 Polyakov loop PNJL: T. Hell, M. Cristoforetti, S. Rössner, W. W. (29) Lattice: M. Cheng et al. (Bielefeld/BNL/Columbia) Phys. Rev. D77 (28) 14511
pressure PT 4 Beyond Mean Field: Mesonic Excitations contribution of mesonic quark-antiquark modes to pressure 4 3 2 1 MFΠ, K, Σ Π, Σ K inmedium MFΠ, Σ Π inmedium MF p [ ] [ mesons quarks.1.2.3 T GeV P meson (T ) = M M=π,σ = K,... + glue d M 2 T m Z N f = 2 + 1 qk p q + pk [ T. Hell, S. Rössner, M. Cristoforetti, W. W. (29) d 3 p (2π) 3 ln [1 GΠ M(ν m, p )]
Energy Density and Interaction Measure E Ε T 4 T 4 15 1 5 energy density mesons..5 1. 1.5 2. TT c T/T c quarks + glue Lattice: M. Cheng et al. (Bielefeld/BNL/Columbia) Phys. Rev. D77 (28) 14511 PNJL: T. Hell, M. Cristoforetti, S. Rössner, W. W. (29) 8 6 4 2 3-flavour non-local PNJL model versus lattice QCD trace of energy-momentum tensor E 3P T 4.5 1. 1.5 2. 2.5 T/T c TT interaction measure
Non-zero QUARK CHEMICAL POTENTIAL Taylor expansion of pressure: Pp(T, µ) = T 4 n c n (T) ( µ T) n c 2 c 4 c 6 S. Rößner, C. Ratti, W. W. Phys. Rev. D 75 (27) 347 N f = 2 Lattice: C.R. Allton et al. Phys. Rev. D 71 (25) 5458
Flavour non-diagonal Susceptibilities Example: -.5 c ud 2 -.1 -.15 -.2. pionic LT=4 fluctuations LT=4.6 LT=6.3 LT=8 LT=1 LT=13.6 lattice data Preliminary c ud 2 2 P µ u µ d Polyakov loop........... N f = 2.4.6.8 1 1.2 T/T T/T c Monte Carlo simulations with PNJL model M. Cristoforetti, T. Hell, B. Klein, W.W. (29) Sensitivity to effects beyond mean field (e.g. pionic fluctuations)
PHASE DIAGRAM and CRITICAL POINT N f = 2 + 1 Σ u.4 GeV.2. profile of scalar field / chiral order parameter from non-local 3-flavour PNJL model..1.2 T GeV.3.1.2 Μ GeV.3 T. Hell, S. Rössner, M. Cristoforetti, W. W. preprint (29) temperature.4. non-strange quark chemical potential
PHASE DIAGRAM (contd.) from non-local 3-flavour PNJL model.2 chiral crossover deconfinement T GeV.15.1 hadronic quarkyonic? L. McLerran, R. Pisarsky.5. 1 st order chiral Polyakov loop.5.1.15.2.25.3 Μ GeV warning: too low transition chemical potential at T = T. Hell, S. Rössner, M. Cristoforetti, W. W. preprint (29) wrong degrees of freedom at low temperature, non-zero baryon density!
PHASE DIAGRAM (contd.) 2"-3"&.$%&" Issues: Crossover Critical Point Diquarks and CSC Phase K. Fukushima (9) 2-nd 1-st 2-nd?,+"-(#./'1$")$(./''! The location of the critica depend on m Existence and location of critical point(s) crucially dependent on AXIAL U(1) ANOMALY Yamamoto, Hatsuda, Baym (27) K. Fukushima (28) T MeV 2 16 12 8 4 Critical point: Role of axial anomaly u u K 1.4 K K 1.2 K d d K s s N f = 2 + 1 24 28 32 Μ MeV K K K.9 K K.8 K K.7 K K.6 K K.55 K K.525 K K.5 K N. Bratovich, T. Hell, S. Rößner, W.W. (29) Location (and existence) of critical point(s) depends sensitively on no. of flavours, quark masses, axial U(1) A anomaly, etc....
crossover T.2.1 [GeV] Critical point Vacuum Hadron gas QGP PHASE DIAGRAM Nuclear matter Quark matter phases (contd.) CFL 1. µ baryon [GeV] Question: first order chiral transition boundary all the way down to T =? Nuclear liquid-gas phase diagram from Chiral Thermodynamics 945678 symmetric gas &' gas :.; )*+,+-./12+3, (N = Z) critical point nuclear matter "( "' (?6=,*23@*.-,+23A'B(' gas :.; liquid!"#!"$!"%!&'!&&!&#!"45678 liquid /+<=+> symmetric (N = Z) nuclear matter
From NUCLEI to COMPRESSED BARYONIC MATTER Framework: Effective Field Theory implementing the Chiral Symmetry Breaking pattern of Low-Energy QCD: In-medium chiral perturbation theory Active degrees of freedom in the hadronic phase: pions, nucleons, delta-isobars Compute Free Energy Density (3-loop order) N. Kaiser, S. Fritsch, W. W. (22-24)
NUCLEAR THERMODYNAMICS NUCLEAR CHIRAL (PION) DYNAMICS BINDING & SATURATION: Yukawa + Van der Waals + Pauli N π π N V(r) e 2m πr + N N contact terms N, r 6 P(m π r) + 3-body forces P [MeV/fm 3 ] 4 3 2 1 nuclear matter: equation of state pressure 3-loop in-medium ChEFT T = 25 MeV 2 T=25MeV T=2MeV T=15MeV T=1MeV T=5MeV -1.5.1.15.2 ρ [fm -3 ] 15 1 5 T = T=MeV Liquid - Gas Transition at Critical Temperature T = 15 MeV c (empirical: T = 16-18 MeV) c baryon density S. Fritsch, N. Kaiser, W. W. : Nucl. Phys. A 75 (25) 259
PHASE DIAGRAM of NUCLEAR MATTER 8/34567 "$ "# "' "% "& $ # ' % & 9-: gas critical point >5<+)12/?)-,+*12/@/&AB& gas 9-:!"#!"$!%&!%%!"34567 µ b [MeV] baryon chemical potential ()*+*,-./1*2+ liquid.*;<*= Pion-nucleon dynamics (incl. delta isobars) Short distance: NN contact terms Three-body forces symmetric (N = Z) nuclear matter 7+289:6 '% '$ '# '! & % $ #! gas ()* S. Fiorilla, N. Kaiser, W. W. (29) In-medium chiral effective field theory (3-loop in the free energy density) S. Fritsch, N. Kaiser, W. W. : NPA 75 (25) 259 ()* ()*+,+-./.1 -./.1!!"!#!"!$!"!%!"!&!"'!"'#!"'$!"'%!"234,5 6 ;9<=>?+3=)@<.>?+A+!"B! liquid
CHIRAL CONDENSATE at finite DENSITY (T = ) sigma term m q M N m q in-medium chiral effective field theory N π π N qq ρ qq = 1 ρ f 2 π [ σn m 2 π ( 1 3 p2 F 1 M 2 N ) +... + m 2 π ( )] Eint (p F ) A (free) Fermi gas of nucleons nuclear interactions (dependence on pion mass)
CHIRAL CONDENSATE: DENSITY DEPENDENCE In-medium Chiral Effective Field Theory (NLO 3-loop) constrained by realistic nuclear equation of state N. Kaiser, Ph. de Homont, W. W. Phys. Rev. C 77 (28) 2524 Symmetric Nuclear Matter condensate ratio 1.8.6.4.2... ψψ (ρ) ψψ (ρ = ). chiral limit m π ρ.5.1.15.2.25.3 ρ [fmρ [fm 3-3 ] chiral limit m π T = chiral in-medium dynamics m π =.14 GeV leading order leading order (Fermi gas) Substantial change of symmetry breaking scenario between chiral limit m q = and physical quark mass m q 5 MeV Nuclear Physics would be very different in the chiral limit!
Summary and Conclusions PNJL-Modelling of Phases based on QCD Symmetries SU(N f ) L SU(N f ) R and Z(3) Entanglement of CHIRAL and DECONFINEMENT crossover transitions in QCD transition temperatures (at zero chemical potential) coincide in PNJL models and on the Lattice PHASE DIAGRAM at low T, large BARYON DENSITY, CRITICAL POINT, and all that... role of axial U(1) anomaly constraints from realistic nuclear EoS thanks to: Nino Bratovic Marco Cristoforetti Salvatore Fiorilla Thomas Hell Bertram Klein Norbert Kaiser Claudia Ratti Simon Rössner