The interplay of flavour- and Polyakov-loop- degrees of freedom

Transcription

1 The interplay of flavour- and Polyakov-loopdegrees of freedom A PNJL model analysis Simon Rößner, Nino Bratović, Thomas Hell and Wolfram Weise Physik Department Technische Universität München Thursday, August 1 th, 2008 The QCD Critical Point Institute for Nuclear Theory, Seattle

2 Outline Connections between colour and flavour (N f = 2 thermodynamics) Flavour blind dofs couple to up- and down- quark densities Up- and down- quark densities couple to flavour blind dofs Up- and down- quarks communicate via an intermediary: Polyakov loop dofs Quantitative investigation of induced flavour mixing: NJL-model + Polyakov-loop model PNJL model A perturbative approach 1 to investigate: The Polyakov loop Φ and its conjugate Φ Non-vanishing up- and down-quark susceptibilies χ uu and χ ud Conclusion & Outlook 1 R., Hell, Ratti, Weise arxiv: [hep-ph], [RHRW07]

3 QCD thermodynamics Symmetry breaking patterns of QCD at finite T Chiral symmetry 1 Explicit breaking m q > 0 2 Dynamic breaking at low T 3 Order parameter: Chiral condensate qq Quarks are coloured objects Dynamic quark masses Confinement-deconfinement 1 Explicit breaking m q < 2 Dynamic breaking at high T 3 Order parameter: Polyakov loop Φ and Φ Colour screening by vacuum fluctuations of quarks Colour confinement Chiral symmetry breaking + Z(3) symmetry breaking are closely linked Joint crossover transition

4 Modelling colour and flavour dynamics The Polyakov loop extended Nambu and Jona-Lasinio model (PNJL model) NJL-model + Polyakov loop model = PNJL model Joint crossover of Φ and qq Ψ Ψ Ψ Ψ NJL chiral limit PNJL pure gauge nd order 1 st order T T c R., Hell, Ratti, Weise arxiv: [hep-ph] [RHRW07] PNJL: Joint effects of quarks and Polyakov loop Confinement (colour) affecting quark densities

5 Modelling colour and flavour dynamics The Polyakov loop extended Nambu and Jona-Lasinio model (PNJL model) NJL-model + Polyakov loop model = PNJL model PNJL: Ψ Ψ Ψ Ψ 0 Joint crossover of Φ and qq NJL chiral limit Quark densities n q T 3 Μ.55 T PNJL c 1 pure gauge Joint effects of quarks and Polyakov loop Confinement (colour) affecting quark densities NJL nd order 0. 1 st order 0.2 PNJL SB limit lattice T T c T T c R., Hell, Ratti, Weise arxiv: [hep-ph] [RHRW07] Ratti, Thaler, Weise PRD 73 (2006) [RTW06]

6 Diagrammatic view to quark number densities n qx = Ω µ x = ω n d 3 [ ] p S 1 (2π) 3 tr S = µ x Ω : thermodynamic potential Ω = d 3 p ω n tr ln [ βs 1] (2π) 3 : γ 0 τ x, where τ x is a matrix in flavour space Quark number susceptibilities χ ux n u n x n u n x = = γ 0 τ u = γ 0 τ x No explicit isospin breaking (thoughout this presentation)

7 Introduction of a perturbative interaction δu Ω = Ω MF + δu(ζ) (Ω MF indep. of ζ) Propagators remain unchanged Use the same quasiparticles as in MF Quark density operators induce new Feynman rules: = S 1 ζ = [ 2 δu ζ ζ ] 1 Corrections to the susceptibilities: δχ ux = ζ couples to n qx ζ χ ζ charge ζ conjugation [ 1 2 δu ζ ζ] 0 ζ-susceptibility χ ζ [ 2 δu ζ ζ] 1 a δχ ud n qx -susceptibility a The mean field contribution of χ ud vanishes due to flavour symmetry.

8 Polyakov loop degrees of freedom Polyakov loop Φ( x) is the trace of a time-like { Wilson-line Φ( x) = 1 } β tr c L( x) L( x) = P exp i dτ A a N ( x) t a c Order parameter for de-confinement Φ = 0 confined Φ 0 deconfined Define Polyakov loop fields with good charge conjugation parity: Φ + = 1 2 Φ + Φ Φ = 1 2 Φ Φ QCD toy model (Ginzburg-Landau-type) S QCD eff = S QCD eff, 0 }{{} indep. of Φ Treat S QCD eff, 0 in mean field + δu(φ ) 0 Treat δu pertubatively χ ud = χ MF ud + 2 S eff µ u Φ [ 2 ] 1 δu 2 S eff Φ Φ µ d Φ < 0

9 Part 1: NJL model Free quarks L NJL = ψ ( /p m 0 ) ψ g ( ψγ µ λ a ψ ) ( ψγ µ λ a ψ ) Integrated out gluons Local SU(3) c Local color current interaction Chiral symmetry QCD NJL Global SU(3) c No confinement Spontaneous chiral symmetry breaking Ω NJL = σ2 + N 2 2G T d 3 p 2 (2π) 3 Tr log S 1 (ω n, p) T ω n Hartree-Fock approximation using Fierz-transformations Bosonization in channels with large -quark coupling σ = G ψψ M = m 0 σ N = G ψiγ 5 τ 1 ψ ( ) S 1 /p M + γ0 (µ + µ = I ) iγ 5 N iγ 5 N /p M + γ 0 (µ µ I ) No explicit isospin breaking terms (Zhang, Liu [ZL07]) N = 0

10 Part 2: Polyakov loop model Model for SU(3) c -gauge theory Confinement 1 st -order Spontaneous breakdown of Z(3)-center sym. Order parameter for de-confinement Polyakov loop Polyakov loop Φ( x) is the trace of a time-like Wilson-line { Φ( x) = 1 } β tr c L( x) L( x) = P exp i dτ A a N ( x) t a c Φ = 0 confined Φ 0 deconfined 0 Ginzburg-Landau effective potential U = U(Φ, Φ, T) { } Simplified loop: Φ = 1 N c Tr exp i Aa ta T a { 3, 8 } Integrate out all dofs that do not change order parameters DΦ DΦ e U(Φ, Φ,T) = DA e S eff(φ(a),φ (A),T)

11 Polyakov loop model adjusted to lattice QCD data Ansatz for the Polyakov loop potential (K. Fukushima [Fuk0]) U(Φ,Φ, T) T = 1 2 b 2 (T) Φ Φ + 1 b (T) log [J(Φ,Φ )] ( J(Φ,Φ ) = 1 6Φ Φ + Φ 3 + Φ 3) 3(Φ Φ) 2 ( ) 3 ( ) ( T0 T0 T0 b (T) = b b 2 (T) = a 0 + a 1 + a 2 T T T Temperature dependent coupling strength b 2 = b 2 (T) ) 2 G. Boyd et. al. [B + 96], O. Kaczmarek et. al. [KKPZ02, KZ05] Φ = Φ at N f = 0: U(Φ,Φ, T) only fixed in 1 2 (Φ + Φ) Stiffness of U(Φ,Φ, T) in 1 2 (Φ Φ) is free to be adjusted

12 Polyakov loop extended NJL (PNJL) Substitute the Matsubara frequencies ω n by ω n + A Formal substitution µ µ ia after Matsubara summation Ω 0 = Ω NJL µ µ ia + U(Φ,Φ, T) Defining mean field as 0 th perturbative order Fermion sign problem: µ µ ia Identification in 0 th order: Mean field: Ω MF = Re Ω 0 Ω 0 = T V S E C p (T) = Ω MF(T) + Ω MF(T=0) Maximization of e S E /T ( quenched mean field) Re Ω 0 σ = Re Ω 0 = Re Ω 0 A (3) = Re Ω 0 A (8) = 0 Constraints: Ω MF! R ΦMF = Φ MF...

13 Polyakov loop extended NJL (PNJL) Substitute the Matsubara frequencies ω n by ω n + A Formal substitution µ µ ia after Matsubara summation Ω 0 = Ω NJL µ µ ia + U(Φ,Φ, T) Joint crossover of Φ and qq T c in MeV Ψ Ψ Ψ Ψ NJL chiral limit 2 nd order PNJL pure gauge 1 st order Kaczmarek et al. [KZ05] (N f = 2) Cheng et al. [C + 06] (N f = 2+1) PNJL (N f = 2) 215 T T c R., Hell, Ratti, Weise arxiv: [hep-ph], [RHRW07]

14 Polyakov loop extended NJL (PNJL) Substitute the Matsubara frequencies ω n by ω n + A Formal substitution µ µ ia after Matsubara summation Ω 0 = Ω NJL µ µ ia + U(Φ,Φ, T) Joint crossover of Φ and qq Ψ Ψ Ψ Ψ NJL chiral limit 2 nd order PNJL pure gauge 1 st order T T c R., Hell, Ratti, Weise arxiv: [hep-ph], [RHRW07] PNJL-Confinement Confinement at T < T c : Polyakov loop Φ 1 Free quarks suppressed Statisical confinement: Active (color-neutral) quasi-particles: m = 3 M M N

15 Corrections to the PNJL mean field solutions Unquenching the PNJL model Goal: Release constraints on the mean fields Integrate out the approximate order parameters (mean fields) Subtile cancellation of imaginary parts is guaranteed How? Perturbative expansion about the (constraint) MF solution Taylor expansion of the action with respect to the fields S = V T Ω 0 = V 1 T k! ω k ξ k with ξ = θ θ 0 k The vector arrow p Set of all fields θ = (σ, N, A (3), A(8) )T θ 0 is the new minimum after SSB Separate free and perturbative parts: Free part: k = 0, 1, 2 Interactions: k 3 Note: Im[ω 1 ] 0 for the (former) gauge field A (8)

16 Expectation values of the Polyakov loop Φ and Φ In mean field MF + corrections Φ MF = Φ MF No split of Φ and Φ Φ R and Φ R Φ Φ at µ 0 Fluctuation effects beyond mean field produce Φ Φ R., Hell, Ratti, Weise arxiv: [hep-ph], [RHRW07]

17 Susceptibilities: c2 uu, cud 2, cuu, cud beyond mean field c2 uu c2 ud in MF c n (T) = 1 n (Ω/T ) n! (µ/t) n cn(t) I = 1 n (Ω/T ) µ=µi =0 n! (µ I /T) 2 (µ/t) (n 2) cn uu = 1 ( cn + c I n) cn ud = 1 ( cn c I n) Isovector moments in agreement with lattice data as well Lattice data: Allton et al. [A + 05], Bielefeld-Swansea coll. µ=µi =0

18 Susceptibilities: c2 uu, cud 2, cuu, cud beyond mean field c uu c2 ud in MF c n (T) = 1 n (Ω/T ) n! (µ/t) n cn(t) I = 1 n (Ω/T ) µ=µi =0 n! (µ I /T) 2 (µ/t) (n 2) cn uu = 1 ( cn + c I n) cn ud = 1 ( cn c I n) Isovector moments in agreement with lattice data as well Lattice data: Allton et al. [A + 05], Bielefeld-Swansea coll. µ=µi =0

19 Susceptibilities: c2 uu, cud 2, cuu, cud beyond mean field c uu c2 ud in MF c n (T) = 1 n (Ω/T ) n! (µ/t) n cn(t) I = 1 n (Ω/T ) µ=µi =0 n! (µ I /T) 2 (µ/t) (n 2) cn uu = 1 ( cn + c I n) cn ud = 1 ( cn c I n) Isovector moments in agreement with lattice data as well Lattice data: Allton et al. [A + 05], Bielefeld-Swansea coll. µ=µi =0

20 Susceptibilities: c2 uu, cud 2, cuu, cud beyond mean field c uu c2 ud beyond MF c n (T) = 1 n (Ω/T ) n! (µ/t) n cn(t) I = 1 n (Ω/T ) µ=µi =0 n! (µ I /T) 2 (µ/t) (n 2) cn uu = 1 ( cn + c I n) cn ud = 1 ( cn c I n) Isovector moments in agreement with lattice data as well Lattice data: Allton et al. [A + 05], Bielefeld-Swansea coll. µ=µi =0

21 Susceptibilities: c2 uu, cud 2, cuu, cud beyond mean field c uu c ud c n (T) = 1 n (Ω/T ) n! (µ/t) n cn(t) I = 1 n (Ω/T ) µ=µi =0 n! (µ I /T) 2 (µ/t) (n 2) cn uu = 1 ( cn + c I n) cn ud = 1 ( cn c I n) Isovector moments in agreement with lattice data as well Lattice data: Allton et al. [A + 05], Bielefeld-Swansea coll. µ=µi =0

22 Adjusted Polyakov loop effective potentials Connection to the Polyakov loop degrees of freedom Polyakov loop effective potential adjusted at N f = 0 δu(φ,φ, T) T Φ Φ = Φ+2 Φ 2 Φ+2 k Φ 2 Thermodynamics unchanged by varying k c ud 2 with modified Φ -potential Φ at µ > 0 Φ -dof c ud 2 < 0

23 c ud 2 controlled by Φ = 1 2 Φ Φ Φ around µ = 0 c ud 2 normalized to Φ -variation Lattice: Döring [PhD Thesis] Universal for varying k Variations of Φ are responsable for c ud 2 < 0.

24 Susceptibilities χ ud beyond mean field χ ud /χ uu in MF χ ux (T, µ) T 2 ( µ = 2c2 ux + 12cux T Fluctuation effects: χ ud 0 ) 2 ( + 30c ux µ ) 6 + with x { u, d } T Lattice data: Allton et al. [A + 05], Bielefeld-Swansea coll.

25 Susceptibilities χ ud beyond mean field χ ud /χ uu beyond MF χ ux (T, µ) T 2 ( µ = 2c2 ux + 12cux T Fluctuation effects: χ ud 0 ) 2 ( + 30c ux µ ) 6 + with x { u, d } T Lattice data: Allton et al. [A + 05], Bielefeld-Swansea coll.

26 Conclusion PNJL: Chiral symmetry breaking Confinement Entanglement of chiral and deconfinement crossover Perturbative approach used to investigate Polyakov loop: Φ Φ at µ 0 Isovector susceptibilities Variations of Φ are responsable for c2 ud < 0 Stiffness of Polyakov loop effective potential directly governs c2 ud c2 ud contains information about Polyakov loop effective potential Outlook flavors Extract Polyakov loop effective potential using c ud 2 from the lattice

27 The end. Thank you for your attention We thank the BMBF, GSI, INFN, by the DFG excellence cluster Origin and Structure of the Universe and the Elitenetzwerk Bayern for supporting this work.

28 Bibliography I C. R. Allton et al., Thermodynamics of two flavor qcd to sixth order in quark chemical potential, Phys. Rev. D71 (2005), G. Boyd et al., Thermodynamics of su(3) lattice gauge theory, Nucl. Phys. B69 (1996), 19. M. Cheng et al., The transition temperature in QCD, Phys. Rev. D7 (2006), Kenji Fukushima, Chiral effective model with the polyakov loop, Phys. Lett. B591 (200), O. Kaczmarek, F. Karsch, P. Petreczky, and F. Zantow, Heavy quark anti-quark free energy and the renormalized polyakov loop, Phys. Lett. B53 (2002), 1 7. Olaf Kaczmarek and Felix Zantow, Static quark anti-quark interactions in zero and finite temperature qcd. i: Heavy quark free energies, running coupling and quarkonium binding, Phys. Rev. D71 (2005),

29 Bibliography II S. Roessner, T. Hell, C. Ratti, and W. Weise, The chiral and deconfinement crossover transitions: PNJL model beyond mean field. Claudia Ratti, Michael A. Thaler, and Wolfram Weise, Phases of QCD: Lattice thermodynamics and a field theoretical model, Phys. Rev. D73 (2006), Zhao Zhang and Yu-Xin Liu, Coupling of pion condensate, chiral condensate and polyakov loop in an extended njl model, Phys. Rev. C75 (2007),