Meson properties and phase diagrams in a SU(3) nlPNJL model with lQCD-inspired form factors

We study the features of a nonlocal SU(3) Polyakov-Nambu-Jona-Lasinio model that includes wave function renormalization. Model parameters are determined from vacuum phenomenology considering lattice QCD-inspired nonlocal form factors. Within this framework we analyze the properties of light scalar and pseudoscalar mesons at finite temperature and chemical potential determining characteristics of deconfinement and chiral restoration transitions.


I. INTRODUCTION
The strong interaction among quarks depends on their color charge. When quarks are placed in a medium, this charge is screened due to density and temperature effects [1]. If either of these increase beyond a certain critical value, the interactions between quarks no longer confine them inside hadrons. This is usually referred to as the deconfinement phase transition. In addition, another transition takes place when the realization of chiral symmetry shifts from a Nambu-Goldstone to a Wigner-Weyl phase. Based on lattice QCD (lQCD) evidence at zero chemical potential [2] one expects these two phase transitions to occur at approximately the same temperature.
At finite density, in principle, they could arise at different critical temperatures, leading to a quarkyonic phase, in which the chiral symmetry is restored while quarks and gluons remain confined.
Although QCD is a first principle theory of hadron interactions, it has the drawback of being a theory where the low energy regime is not available using standard perturbative methods. This problem can be addressed from first principles through lattice calculations [3][4][5][6][7]. However, this approach has difficulties when dealing with small current quark masses and/or finite chemical potential. Thus, some of the present knowledge about the * carlomagno@fisica.unlp.edu.ar behavior of strongly interacting matter arises from the study of effective models, which offer the possibility to get predictions of the transition features at regions that are not accessible through lattice techniques.
Here we will concentrate on one particular class of effective theories, viz. the nonlocal Polyakov−Nambu−Jona-Lasinio (nlPNJL) models (see [8] and references therein), in which quarks interact through covariant nonlocal chirally symmetric four and six point couplings in a background color field and gluon self-interactions are effectively introduced by a Polyakov loop effective potential. These approaches, which can be considered as an improvement over the (local) PNJL model, offer a common framework to study both the chiral restoration and deconfinement transitions. In fact, the nonlocal character of the interactions leads to a momentum dependence in the quark propagator that can be made consistent [9] with lattice results.
Some previous works have addressed the study of meson properties and/or phase transitions using nlPNJL models with Gaussian nonlocal form factors, for specific Polyakov potentials [10]. These functional forms can be improved, since it is possible to choose model parameters and momentum dependences for the form factors so as to fit the quark propagators obtained in lattice QCD.
The aim of this work is to extend the above references to finite chemical potential with lQCD-inspired form fac- Here, ψ(x) is the N f = 3 fermion triplet ψ = (u d s) T , andm = diag(m u , m d , m s ) is the current quark mass matrix. We will work in the isospin symmetry limit, assuming m u = m d . The fermion currents are given by where f (z) and g(z) are covariant form factors responsible for the nonlocal character of the interactions, and λ a , a = 0, ..., 8, are the standard Gell-Mann matrices, plus λ 0 = 2/3 1 3×3 . The relative weight of the interaction driven by j r (x), responsible for the quark wave function renormalization (WFR), is controlled by the parameter κ.
The interaction between fermions and color gauge fields G a µ takes place through the covariant derivative in the fermion kinetic term, D µ ≡ ∂ µ − ıA µ , where A µ = g G a µ λ a /2. In this effective model we will assume that fermions move on a static and constant background gauge field φ. The traced Polyakov loop (PL) Φ, which in the infinite quark mass limit can be taken as the order parameter for confinement [12,13], is given by The effective gauge field self-interactions in Eq. (1) where T τ = 1.77 GeV, α 0 = 0.304 and b(µ) = 1.508 − 32/π (µ/T τ ) 2 . This dependence is motivated by the calculation of hard dense loop and hard thermal loop contributions to the effective charge [16].
A possible Ansatz for the PL potential is given by a logarithmic form based on the Haar measure of the SU (3) color group, namely [17] U log (Φ, T ) where Another widely used potential is that given by a polynomial function based on a Ginzburg-Landau ansatz [18,19]: Numerical values for parameters a i and b i in these potentials can be found in Refs. [17][18][19].

Mean Field Approximation
To determine the QCD phase diagram in the T − µ plane we consider the thermodynamic potential per unit volume at mean field level (MF). We proceed by using the standard Matsubara formalism. Following the same procedure as in Refs. [8,20,21], we perform a standard bosonization of the fermionic theory, Eq. (1), introducing scalar fields σ a (x), ζ(x) and pseudoscalar fields π a (x), with a = 0, ..., 8. We obtain where Here we have defined p 2 nc = [(2n + 1)πT + φ c − ı µ] 2 + p 2 , f p = p 2 + m 2 f . The sums over color and flavor indices run over c = r, g, b and f = u, d, s, respectively, and the color background fields are φ r = −φ g = φ, φ b = 0. The term Ω free is the regularized expression for the thermodynamical potential of a free fermion gas, while Ω 0 is just a constant that fixes the value of the thermodynamical potential at T = µ = 0.
The functions M f (p) and Z(p) correspond to momentum-dependent effective masses and WFR of the quark propagators. In terms of the model parameters and form factors, these are given by whereσ f andζ, are the vacuum expectation values of the scalar fields introduced to bosonize the fermionic theory.
We use the stationary phase approximation, where the path integrals over the corresponding auxiliary fields S f and R are replaced by the arguments evaluated at the minimizing valuesS f andR. The procedure is similar to that carried out in Ref. [20], where more details can be found. From the minimization of this regularized thermodynamic potential it is possible to obtain a set of coupled gap equations that determineσ f ,ζ and φ at a given temperature T and chemical potential µ To characterize the chiral and deconfinement phase transitions it is necessary to define the corresponding order parameters. It is well known that the chiral quark condensates qq are appropriate order parameters for the restoration of the chiral symmetry. Their expression can be obtained by varying the MF partition function with respect to the current quark masses. In general, these quantities are divergent, and can be regularized by subtracting the free quark contributions. Therefore, it is usual to define a subtracted chiral condensate, normalized to its value at T = 0, as Regarding the description of the deconfinement tran-sition, a crucial role is played by the center symmetry Z(N ) of the pure Yang-Mills theory. As stated, we will take as the corresponding order parameter the trace of the Polyakov line, given by [11] If Φ = 0, Z(N ) symmetry is manifest, and this situation indicates confinement. Above the critical temperature one has Φ = 0, therefore, the symmetry is broken, which corresponds to the deconfined phase. For the light quark sector, Φ turns out to be an approximate order parameter for the deconfinement transition in the same way that the chiral quark condensate is an approximate order parameter for the chiral symmetry restoration outside the chiral limit.

Observables beyond mean field
The study of meson properties at finite temperature has to be carried out beyond mean field. The quadratic contribution (in powers of mesonic fluctuations) to the thermodynamical potential is given by where φ M correspond to the meson fields in the SU (3) charge basis. Here M labels the scalar and pseudoscalar mesons in the lowest mass realization, plus the ζ field, and q k = ( q, ν k ), where ν k = 2kπT , are bosonic Matsubara frequencies.
Meson masses are then given by the equations [8,10] The mass values determined by these equations at q = For the pseudoscalar meson sector one can also evaluate mixing angles and weak decay constants. The latter are given by the matrix elements of the axial currents between the vacuum and the physical meson states. Since the I = 0 states get mixed, it is necessary to introduce mixing angles θ η and θ η to diagonalize this coupled sector.
Calculation details, together with the definitions of above quantities at zero temperature, can be found in Ref. [8]. Our aim is to extend here those results to a finite temperature system.

Model parameters and form factors
The model includes five free parameters, namely the current quark masses m u,s and the coupling constants G, H and κ. In addition, one has to specify the form factors f (z) and g(z) in the nonlocal fermion currents of Eq. (2). Here, we will consider for the form factors a momentum dependence based on lQCD results for the quark propagators. Therefore, following the analysis of Ref. [22], we parametrize the effective mass M f (p) as where where f z (p) = 1 It is found that lQCD results favor a relatively low value for the exponent, hence we take here β = 5/2, which is the smallest exponent compatible with the ultraviolet convergence of the loop integrals. The coefficients α m and α z can be expressed in terms of the mean field values σ u andζ (see Eq. (8)). From Eqs. (8), Eq. (12) and Eq (14) one can relate the functions f (p) and g(p) to f m (p) and f z (p).
Given the form factor functions, it is possible to set the model parameters to reproduce the observed meson phenomenology. To the above mentioned free parameters (m u , m s , G, H and κ) one has to add the cutoffs Λ 0 and Λ 1 , introduced in the form factors. Through a fit to lQCD results quoted in Ref. [24] for the functions f m (p) and Z(p), we obtain The curves corresponding to the functions f m (p) and Z(p), together with N f = 2 + 1 lattice data are shown in  Table I.   As in precedent analyses [8,9,25],  Table I dif-fers from the one used in Ref. [8]. As it was explained by the authors in Ref. [26], the numerical evaluation of loop  Figure 1: Fit to lattice data from Ref. [24] for the functions fm(p) and Z(p), Eqs. (13) and (14).
integrals has to be treated with some care given the func-  [8].

Finite temperature phenomenology
In previous works [8,27] we have analyzed the thermal behavior of thermodynamic quantities such as entropy, energy density and interaction measure in this kind of models. Here, we will describe the temperature dependence of meson masses, mixing angles and decay constants, which has not been previously addressed in SU (3) nonlocal models with WFR and/or lQCD-inspired form factors.
In addition, in Ref. [8] we have studied the mentioned thermal properties for Gaussian form factors, which guarantee a fast ultraviolet convergence of the loop integrals. However, this kind of exponential momentum dependence provides unfavorable predictions in comparison with lQCD estimations and results coming from the previously introduced lQCD-inspired form factors. This same improvement is also appreciable in the temperature dependence of meson masses, mixing angles and decay constants presented in this section (see [10] for an analysis with Gaussian form factors).
In Fig. 2 we show the behavior of spatial masses of mesons σ (thin line) and π (thick line) as functions of the temperature, for the logarithmic (upper panel) and polynomial (lower panel) effective potentials given by Eqs. (6) and (7), respectively. Around the critical temperature, it is possible to distinguish a stronger steepness in the curves for the logarithmic potential. In addition, the higher the temperature, the larger is the splitting be-  by the thermal energy. In the case of the η meson and its chiral partner f 0 , and similarly for K and K * 0 , the degeneracy is achieved at larger temperatures than in the case of the other mesons (see Fig. 4). This a consequence of the strange quark content, which becomes larger compared to the content of other flavors as the temperature increases.   it is seen (as in Ref. [10]) that above T c , θ 0 and θ 8 tend to a common value, the so-called "ideal" mixing angle Ref. [14] within a Polyakov-quark-meson model.
In heavy ion collisions it is believed that before the occurrence of the kinetic freeze out a mixed phase of quarks and hadrons could exist [32]. As discussed above, a µdependent T 0 leads to a QCD phase diagram without such a mixed phase. Therefore, we concentrate mainly on the case of a constant T 0 , where for large densities and for a certain temperature range, where the chiral symmetry is restored, the trace of the Polyakov loop still indicates confinement.
As stated, for the deconfinement and chiral symmetry restoration transitions we take as order parameters the traced Polyakov loop Φ and the subtracted chiral condensate qq sub , respectively. The associated susceptibilities χ Φ and χ q are given by the derivatives The associated critical temperatures T χ and T Φ are defined by the position of the peaks in the chiral susceptibilities in the region where the transition occurs as a smooth crossover.
When the chiral restoration occurs as a first order phase transition, the PL susceptibility present a divergent behavior at the chiral critical temperature even when the order parameter Φ remains close to zero.
Therefore we need another definition for the deconfinement critical temperatures in this region of the phase diagram. Here, we employ the same prescription as in Ref [33], namely, we define the critical temperature requiring that Φ takes a given value. We choose a range be-  However, for chemical potentials larger than µ CEP these transitions begin to separate. This can be seen in to µ = 100 MeV, it is seen that the chiral and deconfinement transitions proceed as smooth crossovers occurring at the same critical temperature. When the chemical potential becomes larger than µ CEP (see Table III  referred to as a quarkyonic phase [35][36][37].
We quote in Fig. 8   Within this framework we have obtained a parametrization that reproduces lattice QCD results for the momentum dependence of the effective quark mass and WFR, and at the same time leads to an acceptable phenomenological pattern for particle masses and decay constants in both scalar and pseudoscalar meson sectors.
In our calculations we have included the contributions from branch cuts in the momentum complex plane that arise from the lattice inspired nonlocal form factors.
As a second step, we have analyzed the temperature dependence of several meson properties, like meson masses, decay constants and mixing angles. As expected, it is found that meson masses get increased beyond the chiral critical temperature, becoming degenerated with their chiral partners. The temperatures at which this happens depend on the strange quark content of the cor-responding mesons.
Meson masses and weak decay constants remain approximately constant up to the critical chiral temperature. In addition, light hadrons with strange degrees of freedom present a decay constant with a less steep decrease. Regarding the properties of the mixing angles, they tend to converge to the so-called "ideal" mixing, which indicates that the U(1) A anomaly tends to vanish as the temperature increases.
Finally, we study the characteristics of deconfinement and chiral restoration transitions at finite temperature and chemical potential. As expected, at zero µ, the model shows a crossover phase transition, corresponding to the restoration of the SU(2) chiral symmetry. The transition temperature is found to be T c ∼ 165 MeV, in very good agreement with lattice results. In addition, one finds a deconfinement phase transition, which occurs at the same critical temperature. On the other hand, at zero temperature chiral restoration takes place via a first order transition at a critical density µ χ ∼ 290 MeV, in agreement with estimations coming from compact objects.
For chemical potentials larger than µ CEP , the critical temperatures for the restoration of the chiral symmetry and deconfinement transition begin to separate. The region between them denotes a phase where the chiral symmetry is restored but quarks remains confined, known as quarkyonic phase. This splitting is strongly dependent of the parameter T 0 entering in the PL potential. If we consider for this parameter an explicit dependence with µ, both transitions are always simultaneous, and therefore there is no such mixed phase, in contradiction with some results from heavy-ion collisions.