We investigate the chiral phase transition of the strongly interacting matter
at nonzero temperature and baryon chemical potential $\mu_B$ within an extended
(2+1) flavor Polyakov constituent quark-meson model which incorporates the
effect of the vector and axial vector mesons. The parameters of the model are
determined by comparing masses and tree-level decay widths with experimental
values. We examine the restoration of the chiral symmetry by monitoring the
temperature evolution of condensates. We study the $T-\rho_B$ phase diagram of
the model and find that a critical end point exists, although at very low
density.Comment: 5 pages, 2 figures, Presented at CPOD 2016, Wrocla

Kovács, P.

Wolf, Gy.

English

arXiv

Gy. Wolf

P. Kovács

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Acta Physica Polonica B Proceedings Supplement

The Phase Diagram in the Vector Meson Extended Linear Sigma Model

Wolf, György

Kovács, Péter

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arXiv:1711.10879v1  [hep-ph]  29 Nov 2017The phase diagram in the vector meson extended linearsigma model∗Gy. Wolf and P. KovácsWigner Research Centre for Physics, Hungarian Academy of Sciences, H-1525Budapest, POB 49, HungaryWe investigate the chiral phase transition of the strongly interactingmatter at nonzero temperature and baryon chemical potential µB withinan extended (2+1) flavor Polyakov constituent quark-meson model whichincorporates the effect of the vector and axial vector mesons. The parame-ters of the model are determined by comparing masses and tree-level decaywidths with experimental values. We examine the restoration of the chiralsymmetry by monitoring the temperature evolution of condensates. Westudy the T-ρB phase diagram of the model and find that a critical endpoint exists, although at very low density.PACS numbers: 12.39.Fe, 11.30.Rd, 11.30.Qc, 14.40.Be1. IntroductionWe investigate the thermodynamics of the strongly interacting mat-ter at high temperature and/or density with the (2 + 1) flavor Polyakovquark meson model in which, beyond the vector and axial vector mesonsincluded alongside the scalar and pseudoscalar ones. We take into account,as fermionic degrees of freedom, the constituent quarks propagating on aconstant gluon background in the temporal direction, which naturally leadsin a mean-field treatment to the appearance of the Polyakov loop. We in-corporate the vacuum fluctuations of the fermions in the grand potential.We compare our results at µB = 0 with lattice data and then move to finiteµB . The details of the model can be found in [1, 2].2. Formulation of the ModelWe work with the Lagrangian used in [2]. Compared to the usual chiralLagrangian, we introduce additional kinetic and Yukawa coupling terms for∗ CPOD 2016, Wroclaw(1)2 PhaseTrans printed on October 13, 2018the constituent quarks Ψ = (qu, qd, qs)T . Another important modification isthe presence of the gluon field in the covariant derivative of the quark field.In the mean-field approximation, this will give rise in the grand potentialto the appearance of the Polyakov loop, which mimics some properties ofthe quark confinement. We use the logarithmic Polyakov loop potential [3].This parametrization does not include the backreaction of the dynamicalquarks on the gauge sector and therefore the influence of the quarks onthe deconfinement phase transition. In the present study we shall use theimproved Polyakov loop potential proposed in [4].There are altogether 16 unknown parameters, 15 parameters found inthe Lagrangian and the renormalization scale. In the parametrization pro-cedure we use alongside 29 vacuum quantities, and also the pseudocriticaltemperature Tc at µB = 0 [2]. The value of the masses and decay con-stants are compared with the corresponding experimental value taken fromthe PDG [5]. The Tc pseudocritical temperature, which should be around150 MeV, can reach very high values (≥ 350 MeV) in case of some solutions.Thus we chose to include the physical value of Tc in the parametrizationwith a 10% error to throw away solutions with high Tc, and unrealisticthermodynamics.2.1. The Grand PotentialTo study the thermodynamics of a symmetric quark matter (µu = µd =µs ≡ µq = µB/3), we shall use the grand potential Ω(T, µq) obtained fromthe partition function of a three-dimensional spatially uniform system of vol-ume V in thermal equilibrium at temperature T = 1/β. Since the Polyakovloop is treated at mean-field level, there is no integration over the gluonsand in this case the Polyakov loop potential is simply added to the grandpotential.In the present case the vacuum and thermal fluctuations for the fermionsare taken into account, while the mesonic vacuum fluctuations are neglectedand the effects of the lightest mesonic thermal fluctuations (π, K, fL0 ) areincluded only in the pressure and the thermodynamical quantities derivedfrom it. In this approximation the grand potential readsΩH(T, µq) = U(〈M〉) + U(〈Φ〉, 〈Φ̄〉) + Ω(0)q̄q (T, µq), (1)where U(〈M〉) is the tree-level meson potential, U(〈Φ〉, 〈Φ̄〉) is the Polyakovloop potential and Ω(0)q̄q is the contribution of the fermions.PhaseTrans printed on October 13, 2018 32.2. The curvature meson massesThe squared masses of the scalar and pseudoscalar mesons are calculatedfrom the elements of the corresponding curvature matrix:m2i,ab =∂2Ω(T, µq)∂ϕi,a∂ϕi,b∣∣∣∣min= m2i,ab +∆m2i,ab + δm2i,ab, (2)where the three terms on the right-hand side are as follows: m2i,ab is thetree-level mass matrix, and ∆m2i,ab and δm2i,ab are the contributions of thefermionic vacuum and thermal fluctuations, respectively.The field equations, which determine the dependence on T and µB = 3µqof the chiral condensates φN and φS and the Polyakov loop variables Φ andΦ̄, are obtained by extremizing the grand potential,∂ΩH∂φN=∂ΩH∂φS=∂ΩH∂Φ=∂ΩH∂Φ̄= 0. (3)To calculate various thermodynamical quantities we add the contributionof a meson b ∈ {π,K, fL0 } to the pressure with the formula∆pb(T ) = −nbT∫d3p(2π)3ln(1− e−βEb(p)), (4)where Eb(p) =√p2 +m2b , with mb being the meson mass, and nb is themeson multiplicity (nπ = 3, nK = 4, and nfL0= 1).3. ResultsOn the left hand side of Fig. 1 we study at µB = 0 the temperaturevariation of the nonstrange and strange chiral condensates and Polyakovloop expectation value. We see that the chiral condensates stay close totheir vacuum values up to some quite high value of the temperature oforder 100 MeV.On the right hand side of Fig. 1 we see that the constituent quarks andthe Polyakov loop potential give the major part of the contribution to thepressure around and beyond Tc and that at small temperature the pressureis pion dominated.Some thermodynamical quantities derived from the pressure, like thescale interaction measure ∆, the square of the speed of sound c2s, and theequation of state parameter p/ǫ (pressure over energy density), are shownin [2] to be described very well by the model.4 PhaseTrans printed on October 13, 2018 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0  0.05  0.1  0.15  0.2  0.25  0.3  0.35  0.4  0.45 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1[GeV]T [GeV] 1 2 3 4 5-0.5  0  0.5  1  1.5  2  2.5  3SBp/T4tπ, K, f0L includedπ, K includedπ includedno mesonic contributionlatticeFig. 1. At µB = 0 temperature dependence of condensates from the improvedPolyakov loop potential (left) and the pressure (right) as function of reduced tem-perature (T/Tc − 1). Lattice data are taken from [6]. Here Tgluec= 182 MeV.3.1. µB − T phase diagram and the critical endpointWe turn now to the study of the chiral phase transition at finite quarkchemical potentials: µu = µd = µs = µq = 1/3µB . The lattice result of thecurvature at µB = 0 is very well reproduced [2]. With the best set of pa-rameters, the location of the CEP in our model is given by (µCEPB , TCEPc ) =(885, 52.7) MeV when the improved Polyakov loop potential is used withT gluec = 210 MeV. The phase diagram is similar to that obtained in [7], withcomparable values of the CEP’s coordinates. The large value of µCEPB we ob-tained is typical of a linear sigma model without (axial)vector mesons in thecase when the vacuum fluctuation of the fermions is included. See, e.g., [8]where the value (µCEPB , TCEPc ) = (849, 81) MeV was reported. Without theinclusion of the fermionic vacuum fluctuations, as is the case of Refs. [9, 10],the value of µCEPB is smaller and TCEPc higher, compared to the case whenthey are properly taken into account.On Fig. 2 we show the phase diagram on the T, ρB plane. The dashedline shows the first order transition line whereas the solid curve is wherethe transition is cross over. The critical point is at rather low density, note,however, it is not the usual µu = µd = µq = 1/3µB , µs = 0 case.4. ConclusionsWe have studied at finite temperature and baryonic densities the ther-modynamical properties of the Polyakov loop extended quark meson modelcontaining also vector and axial vector mesons. These latter ingredientsmanifest themselves in a nontrivial way in the vacuum parametrizationof the model through their tree-level masses and decay widths. The χ2-minimization procedure was applied to fix the parameters. We have studiedPhaseTrans printed on October 13, 2018 5 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0  0.05  0.1  0.15  0.2  0.25  0.3T [GeV]ρΒ  [fm-3]cross overfirst orderFig. 2. The phase diagram obtained by using the improved Polyakov loop potential.The dashed line shows the first order phase transition usinf Maxwell construction.the thermodynamics of the model, by using an improved Polyakov loop po-tential, and obtained fairly good agreement with lattice data. We found thata CEP of the crossover transition line exists in the T − ρB phase diagramat rather low values of ρB .AcknowledgmentsThe authors were supported by the Hungarian Research Fund (OTKA)under Contract No. K109462 and by the HIC for FAIR Guest Funds of theGoethe University Frankfurt.REFERENCES[1] D. Parganlija, P. Kovács, Gy. Wolf, F. Giacosa and D. H. Rischke, Phys. Rev.D 87, 014011 (2013).[2] P. Kovács, Zs. Szép and Gy. Wolf, Phys. Rev. D 93, 114014 (2016).[3] K. Fukushima, Phys. Lett. B 591, 277 (2004).[4] L. M. Haas, R. Stiele, J. Braun, J. M. Pawlowski and J. Schaffner-Bielich,Phys. Rev. D 87, 076004 (2013).[5] J. Beringer et al. [PDG], Phys. Rev. D 86, 010001 (2012).[6] S. Borsányi, G. Endrődi, Z. Fodor, A. Jakovác, S. D. Katz, S. Krieg, C. Rattiand K. K. Szabó, JHEP 1011, 077 (2010).[7] R. Stiele and J. Schaffner-Bielich, Phys. Rev. D 93, 094014 (2016).[8] S. Chatterjee and K. A. Mohan, Phys. Rev. D 85, 074018 (2012).[9] B. J. Schaefer and M. Wagner, Phys. Rev. D 79, 014018 (2009).[10] H. Mao, J. Jin and M. Huang, J. Phys. G 37, 035001 (2010).

The phase diagram in the vector meson extended linear sigma model

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The phase diagram in the vector meson extended linear sigma model

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