The Polyakov loop and the Hadron Resonance Gas Model

Issues The Polyakov loop and the Hadron Resonance Gas Model 1, E. Ruiz Arriola 2 and L.L. Salcedo 2 1 Grup de Física Teòrica and IFAE, Departament de Física, Universitat Autònoma de Barcelona, Spain 2 Departamento de Física Atómica, Molecular y Nuclear, Universidad de Granada, Spain. 1 Supported by the Juan de la Cierva Program of the Spanish MICINN. QCD 2012: 16 th International QCD Conference July 5, 2012, Montpellier, France Some References: E. Megías et al. PRD74 (2006) 065005, 114014, AIP Conf. Proc. 892 (2007), arxiv:1204.2424[hep-ph].

Issues Issues 1 QCD at finite temperature 2 3

Issues Issues 1 QCD at finite temperature 2 3

Issues Issues 1 QCD at finite temperature 2 3

Symmetries in full QCD The chiral-deconfinement cross over is a unique prediction of lattice QCD Order parameter of chiral symmetry breaking (m q = 0) Quark condensate SU(N f ) SU(N f ) SU V (N f ) qq 0 T < T c, qq = 0 T > T c. Order parameter of deconfinement (m q = ) Polyakov loop: Center symmetry Z(N c ) broken L T = 1 N c tr c e ia 0/T = 0 T < T c, L T = 1 N c tr c e ia 0/T = 1 T > T c, L T = e Fq/T, F q Free energy of a heavy quark. In the real world m q is finite but inflexion points nearly coincide d 2 dt 2 L T = 0, For about the same T c = 155(10) MeV. d 2 dt 2 qq T = 0.

Chiral Perturbation Theory Suppose the non-vanishing of chiral condensate. It cannot describe the QCD phase transition. Chiral Quark Models Dynamics of QCD at low energies (low temperatures). Mean field approximation. Minimal coupling of Polyakov loop (analogy with chemical potential). Ogilvie and Meissinger PLB (1995) K. Fukushima, PLB591, 277 (2004). W. Weise et al. PRD73, 014019 (2006), N. Scoccola, D. G. Dumm (2008), S.K. Ghosh et al. PRD73, 114007 (2006), Quantum and local Polyakov loop. E.Megías, E.Ruiz Arriola and L.L.Salcedo, PRD74: 065005 (2006).

Issues QCD at finite temperature 1 QCD at finite temperature 2 3

Old Treatment of Chiral Quark Models at Finite T [E. Ruiz Arriola, Chr. Christov, K. Goeke, PLB 1989, APP 1990] The standard rule (to pass from T = 0 to T 0) is: dk0 2π F( k, k 0 ) it The chiral condensate at one loop is: qq = 4MT Tr c ω n n= F( k, iω n ). d 3 k (2π) 3 1 ω 2 n + k 2 + M 2, ω n = 2πT(n+ 1 2 ). After application of the Poisson s summation formula: qq Low T qq N c 2 n=1 ( 1) n ( 2MT nπ ) 3/2 e nm/t.

Interpretation: Consider the fermionic propagator at low T S( x, T) = d 4 k (2π) 4 e ik x k/ M e M/T, (exponential suppression of a single quark). qq can be written in terms of Boltzmann factors with M n = nm: qq = qq +O q e M/T + O qq e 2M/T +... Problem 1: When temperature rises, every 1,2,3,... quark state is generated. Problem 2: In Chiral Perturbation Theory: qq ChPT = qq (1 T 2 8fπ 2 T 4 384f 4 π Finite temperature corrections are N c suppressed. ) +, fπ 2 N c.

Issues QCD at finite temperature 1 QCD at finite temperature 2 3

Constituent Quark model: L QC = q D q, D = / + V/ f + A/ f +MU γ 5 + ˆm 0. Consider the minimal coupling of the gluons in the model: V f µ V f µ + gv c µ, V c µ = δ µ0 V c 0. Covariant derivative expansion (E. Megías et al. PLB563(2003), PRD69(2004), Oswald and Dyakonov PRD (2004) ). L(x) = n tr[ f n (Ω(x)) O n (x) ], Ω( x, x 0 ) = P e i x 0 +β x 0 dx 0 V c 0 ( x,x 0 ). Ω enters in: ˆω n = 2πT(n+1/2+ ˆν), Ω = e i2πˆν.

The rule to pass from T = 0 to T 0 is: S(x; x) ( Ω( x)) n S( x, x0 + nβ; x, x 0 ). n= The quark condensate writes: qq = n 1 N c tr c ( Ω) n q(nβ)q(0). ( Ω) n

Peierls-Yoccoz projection on color singlets [E. Megías, E. Ruiz Arriola, L.L. Salcedo, PRD74 (2006) 065005, 114014] We introduce a colour source (Polyakov loop). We obtain the projection onto the color neutral states by integrating over the A 0 field. Quenched approximation: Group integration in SU(N c ). tr c ( Ω) n SU(N c) DΩ tr c( Ω) n = { N c, n = 0 1, n = ±N c 0, otherwise There is only contribution from n = 0,±N c. ( ) 3/2 qq Low T MT qq +4 e NcM/T. 2πN c The N c suppression is consistent with ChPT.

Beyond the Quenched approximation: Z = DUDΩ e ΓG[Ω] e ΓQ[U,Ω]. For any observable: O = 1 Z DUDΩ e Γ G [Ω] e ΓQ[U,Ω] O. Analytically Expand the exponents and compute correlation functions of Polyakov loops: DΩ tr c Ω( x) tr c Ω 1 ( y) = e σ x y /T. L 1 tr c Ω N c qq qq Low T O(e M/T ), Low T 1+O(e 2M/T )+O(e NcM/T ). Taking into account the quark binding effects: O q = O q + m π O mπ 1 N c e mπ/t + B O B e M B/T +...

Numerically Consider the Polyakov gauge. 1 0.8 <qq>/<qq> 0 NJL Model Standard K. Fukushima Integrated Fuk. L8 Integrated Fuk. <L> 1 0.8 <qq>/<qq> 0 NJL Model <L> Order Parameters 0.6 0.4 Order Parameters 0.6 0.4 K. Fukushima VT Integrated 0.2 0.2 L adj Lattice 0 0 0 50 100 150 200 250 300 350 400 450 T ( MeV) 0 50 100 150 200 250 300 350 400 450 T ( MeV) Polyakov cooling : The condensate does not change at low temperatures.

Issues QCD at finite temperature 1 QCD at finite temperature 2 3

Multiquark states: Create/Annihilate a quark at point x and momentum p Ω(x)e E P/T, Ω(x) + e E P/T. At low temperatures quark Boltzmann factor small e Ep/T < 1. The action becomes small d 3 xd 3 p [ S q [Ω] = 2N f trc (2π) 3 Ω(x)+tr c Ω + (x) ] e Ep/T +..., ( Z = DΩ e S[Ω] = DΩ 1 S[Ω]+ 1 ) 2 S[Ω]2 +....

Polyakov loop in the quark model For the Polyakov loop 1 d 3 xd 3 p L(T) = tr c Ω = 2N 1 f N c (2π) 3 e Ep/T tr c Ω( x 0 ) tr c Ω 1 ( x) +... N c A single q Hamiltonian = 2N f d 3 xd 3 p (2π) 3 e H( x, p)/t +... H( x, p) = p 2 + M 2 + V q (r). Quantization Hψ α = α ψ α. : L(T) 1 g q e q/t + g q,q e q,q /T +.... 2N c q q,q Heavy-light system heavy quark + light dynamical quarks q, q Hadrons with a heavy quark h q Meson,hqq Baryon,...

Issues QCD at finite temperature 1 QCD at finite temperature 2 3

Polyakov loop in the HRG model (arxiv:1204.2424) L(T) 1 g h,α e h,α/t, 2N c h,α α = lim m h (M h,α m h ). h, α Heavy-light system Mesons and baryons with one heavy quark. We use single charm states from PDG (a): Mesons D 0, D +, D s +, D 0, D +, Ds +. Baryons Σ 0 c,σ + c,σ ++ c,ξ + c,ξ 0 c,ω 0 c,λ + c. 10 8 d T log L dt 6 4 2 d b c a 0 150 160 170 180 190 200 T MeV (a) These lowest-lying hadrons underestimate the lattice data for L(T) by a factor 3.

Polyakov loop in the HRG model (arxiv:1204.2424) We need to take into account more states. Two possibilities: Relativized quark model (RQM), Isgur et al. 85. (b) and (c). States with one c-quark: 117 for mesons and 1470 for baryons. States with one b-quark: 87 for mesons and 1740 for baryons. Add missing states with a s-quark by using Equal Spacing Rule: Ξ c Σ c m s, Ω c Ξ c m s. Cut-off: = M m c < 1500 MeV. 10 8 d T log L dt 6 4 2 d b c a 0 150 160 170 180 190 200 T MeV (b) and (c) still bellow L Lattice (T) in the range T < 175 MeV.

Polyakov loop in the HRG model (arxiv:1204.2424) MIT Bag Model, R.Jaffe et al. 74. (d). i = n iω i Z + 4π R 3 R3 B + i Bag energy. m i dynamical quark mass. m i, m u = m d = 0, m s = 109 MeV, m c = 1390 MeV. B = (166MeV) 4, Z = 0.5. Cut-off: = M m c < 5500 MeV. 10 8 d T log L dt 6 4 2 d b c a 0 150 160 170 180 190 200 T MeV (d) Good description of L Lattice (T) in regime 145 MeV < T < 175 MeV.

Conclusions: Chiral quark models coupled to Polyakov loops are useful as a first estimate of QCD features. They reproduce qualitatively the cross-over observed in QCD. Chiral quark models at finite temperature have much better properties when the Polyakov loop (colour source) is projected a la Peierls-Yoccoz onto singlet colour states Quantum and local Polyakov loop. With simple modifications the transition to the hadronic degrees of freedom becomes possible. Polyakov loops in fundamental and higher representations allow to deduce multiquark states, gluelumps, etc, containing one or several heavy quark states. This goes beyond the models and opens up the possibility of a Polyakov loop spectroscopy.