Deconfinement and chiral restoration in nonlocal PNJL models at zero and imaginary chemical potential

We study the deconfinement and chiral restoration transitions in the context of non-local PNJL models, considering the impact of the presence of dynamical quarks on the scale parameter appearing in the Polyakov potential. We show that the corresponding critical temperatures are naturally entangled for both zero and imaginary chemical potential, in good agreement with lattice QCD results. We also analyze the Roberge Weiss transition, which is found to be first order at the associated endpoint.

provide a satisfactory description of hadron properties at zero temperature and density. Moreover, it has been found that, under certain conditions, it is possible to derive the main features of nonlocal PNJL models starting directly from QCD [21]. Related Polyakov-Dyson-Schwinger equation models have also been recently considered [22].
The aim of the present work is to analyze the relation between the deconfinement and chiral restoration transitions at both zero and imaginary chemical potential µ in the context of non-local chiral quark models. One of the problems of the standard (local) PNJL model is that once the PL potential is adjusted to reproduce the pure gauge lattice QCD results, it is found [8] that the critical temperature for the chiral and deconfinement transitions at vanishing chemical potential, T c ≈ 220 MeV, is somewhat too high in comparison with the presently most accepted lattice result, namely T c = 173 (8) MeV for two light flavors [4]. A solution to this difficulty follows from the observation made in the context of the Polyakov-Quark-Meson model [23], where it is claimed that in the presence of dynamical quarks one should decrease the parameter T 0 which sets the scale of the PL potential. However, in contradiction to lattice results, in the PNJL model this sort of rescaling leads to a rather noticeable splitting between the deconfinement and chiral restoration temperatures. This splitting can be avoided only after the inclusion of extra eight-quark interactions [24], or by assuming that the quark-quark coupling constant is some ad-hoc function of the Polyakov loop [25]. Here we show that in the case of the non-local SU(2) PNJL model the critical temperature can be made naturally compatible with lattice QCD estimates, without spoiling the entanglement between deconfinement and chiral restoration transition temperatures even for imaginary chemical potential. It should be stressed that the extension to imaginary chemical potential deserves significant theoretical interest, since lattice calculations [26][27][28] become free of the sign problem and the corresponding results can be compared with effective model predictions. Moreover, the behavior in the region of imaginary chemical potential is expected to have implications on the QCD phase diagram at finite real values of µ. Lattice calculations, as well as analyses based on the exact renormalization group equations [29], suggest a close relation between the deconfinement and chiral restoration transitions for imaginary chemical potentials.
Thus, we extend our study of these transitions to the region of imaginary µ, where we also analyze the characteristics of the so-called Roberge-Weiss (RW) transition [30], in particular, at the RW endpoint.
Let us briefly describe the model under consideration, namely a non-local SU(2) chiral quark theory that includes couplings to a background color gauge field. The Euclidean effective action is given by [31] where ψ is the N f = 2 fermion doublet ψ ≡ (u, d) T ,m = m q 1 1 2×2 is the current quark mass matrix in the isospin limit, and D µ ≡ ∂ µ − iA µ is a covariant derivative, A µ being color gauge fields. The nonlocal currents j a (x), j P (x) are given by where Γ a = (1 1, iγ 5 τ ), and the functions F(z) and G(z) are non-local form factors that characterize the interactions. Notice that even if we take for convenience the same coupling parameter G S for both interaction terms, the relative strength between them is controlled by the mass parameter κ p .
To proceed we perform a standard bosonization of the theory, introducing bosonic fields σ 1,2 (x) and π a (x), and integrating out the quark fields. We will work within the mean field approximation (MFA), in which these bosonic fields are replaced by their vacuum expectation values σ 1,2 and π a = 0. Since we are interested in studying the characteristics of the chiral phase transition, we extend the bosonized effective action to finite temperature T and chemical potential µ. This can be done by using the Matsubara formalism. Concerning the gauge fields A µ , we assume that quarks move on a constant background field φ = A 4 = iA 0 = ig δ µ0 G µ a λ a /2, where G µ a are the SU(3) color gauge fields. Then the traced Polyakov loop, which is taken as order parameter of confinement, is given by Φ = 1 3 Tr exp(iφ/T ). We will work in the so-called Polyakov gauge, in which the matrix φ is given a diagonal representation φ = φ 3 λ 3 + φ 8 λ 8 . This leaves only two independent variables, φ 3 and φ 8 . The mean field traced Polyakov loop reads then Within this framework the mean field thermodynamical potential Ω MFA at finite temperature and arbitrary (in general, complex) chemical potential is given by Here, M (p) and Z(p) are given by where g(p) and f (p) are Fourier transforms of G(z) and F(z). We have also defined where the quantities φ c are given by the relation To proceed we need to specify the explicit form of the Polyakov loop effective potential U (Φ, Φ * , T ). Following Ref. [9] we take where the coefficients are parameterized as The values of a i and b 3 are fitted [9] to lattice QCD results, which in absence of dynamical quarks lead to a deconfinement temperature T 0 ≈ 270 MeV. However, as mentioned above, it has been argued [23] that in the presence of light dynamical quarks this value has to be modified accordingly, e.g. T 0 ≃ 208 MeV for N f = 2 and T 0 ≃ 180 MeV for N f = 3. Effects of this change in T 0 will be discussed below. In addition, it is seen that Ω MFA turns out to be divergent, thus it has to be regularized. Here we use the same prescription as e.g. in Ref. [32], namely where Ω free is obtained from Eq. (4) by setting σ 1 = σ 2 = 0, and Ω free reg is the regularized expression for the quark thermodynamical potential in the absence of fermion interactions: Re with ǫ p = p 2 + m 2 q . The last term in Eq. (9) is just a constant fixed by the condition that Ω MFA reg vanishes at T = µ = 0.
The mean field values σ 1,2 and φ 3,8 can be obtained from a set of four coupled "gap" equations that follow from the minimization of the regularized thermodynamical potential, Once the mean field values are obtained, the behavior of other relevant quantities as functions of the temperature and chemical potential can be determined. We concentrate in particular in the chiral quark condensate qq = ∂Ω MFA reg /∂m q , which together with the modulus of the Polyakov loop |Φ| will be taken as order parameters of the chiral restoration and deconfinement transitions, respectively. For simplicity, the associated susceptibilities will be defined as χ cond = d qq /dT and In order to fully specify the model under consideration we have to fix the model parameters as well as the form factors g(q) and f (q) that characterize the non-local interactions. Here we consider three parameter sets A, B and C, which have been introduced in Ref. [16]. Set A corresponds to the relatively simple case in which there is no wave function renormalization (WFR) of the quark propagator, i.e. f (p) = 0, Z(p) = 1, and g(p) has a Gaussian behavior, g(p) = exp −p 2 /Λ 2 0 . In set B we consider a more general case that includes quark WFR, taking also an exponential shape for the corresponding form factor, f (p) = exp −p 2 /Λ 2 1 . Finally, in set C we take Lorentzian-like form factors, chosen in such a way that one can well reproduce the momentum dependence of mass and WFR functions obtained in lattice calculations. The parameter values for sets A, B and C, together with the corresponding predictions for several meson properties, can be found in Ref. [16].
Let us analyze the deconfinement and chiral transitions at vanishing chemical potential in the framework of the model presented above. Taking T 0 as a parameter, and solving numerically Eqs. (11) for sets A, B and C, it is found that both the deconfinement and chiral restoration temperatures are coincident in a wide range of values of T 0 . This is illustrated in Fig. 1, where we show the behavior of the relevant order parameters and the corresponding susceptibilities for the lattice inspired parameterization set C. We consider three characteristic values T 0 = 270, 208 and 180 MeV, corresponding to the presence of 0, 2 and 3 dynamical fermions, respectively [23]. It is clear that for this set both transitions are crossover-like, and they occur at basically the same critical temperature, as it is indicated by the peaks of the corresponding susceptibilities. One might notice that as long as T 0 decreases the chiral susceptibility tends to become asymmetric around T c , being somewhat broader on the high temperature side. Though this could be considered as an indication that for smaller values of T 0 chiral symmetry is restored at a slightly higher temperature, even at T 0 = 180 MeV the splitting between the main peak and what might be considered as a second broad peak is less than 10 MeV. In addition, it is worth to point out that although the coincidence of the deconfinement and chiral restoration critical temperatures holds for all three parameter sets A, B and C, the character of the transitions may be different from one another. This is shown in  [33][34][35][36][37] favor a second order transition in the chiral limit, there are also claims for a first order transition [38,39]. Given that in the context of the present non-local model the parameterizations that include WFR appear to be more realistic, the second order scenario turns out to be preferred.
We consider now the situation at nonzero imaginary chemical potential. As it is well known, Roberge and Weiss found [30] that the thermodynamical potential of QCD in presence of an imaginary chemical potential µ = i θ T is a periodic function of θ with period 2π/3. This means that QCD is invariant under a combination of a Z 3 transformation of the quark and gauge fields and a shift θ → θ + 2 k π/3, with integer k, in the chemical potential. Recently, it has been shown that this so-called extended Z 3 transformation is also a symmetry of the local Polyakov-Nambu-Jona-Lasinio model [40]. Indeed, in the context of this model the thermodynamical potential is invariant under the transformations The RW periodicity is a remnant of the Z 3 symmetry in the pure gauge limit. In QCD with dynamical quarks, if the temperature is larger than a certain value T RW it can be seen that three According to lattice calculations, it appears that for two light flavors the RW endpoint is first order for realistically small values of the current quark mass. Following these considerations it is important to check whether the thermodynamical potential of the non-local PNJL models studied in this work does respect the extended Z 3 symmetry. In fact, it is easy to show that this is the case. The last two terms in Eq. (4) are obviously invariant under the transformations in Eq. (12), whereas to check the invariance of the first term it is convenient to write these transformations in the equivalent way Thus it can be easily proven that any sum of the form c=r,g,b F [(ρ c n, p ) 2 ], where F is an arbitrary function, turns out to be invariant under the extended Z 3 transformations. The invariance of the terms introduced in the regularization procedure [c.f. Eq.(9)] can be shown in the same way.
Having checked that our non-local PNJL models possess the extended Z 3 invariance we turn now to the results of the numerical analysis of the behavior of the different order parameters as functions of T and θ using the value T 0 = 208 MeV corresponding to two light flavors. We first keep T fixed, verifying that the expected periodicity of the different thermodynamical quantities as functions of θ is indeed satisfied. Moreover, for T > T RW we find the mentioned RW first order phase transition at θ = π/3, which is signalled by a discontinuity in the phase of the Polyakov loop field. The values obtained for T RW are 191 MeV, 188 MeV and 191 MeV for sets A, B and C, respectively, in good agreement with the lattice QCD estimate T RW = 185(9) [28]. Concentrating on the sector 0 ≤ θ ≤ π/3 we observe that for values of the temperature T c (θ = 0) ≤ T ≤ T RW the order parameters for both deconfinement and chiral symmetry show signals of a phase transition at a given value of θ. This is clearly seen in Fig. 3 where we plot the behavior of order parameters and susceptibilities as functions of T taking now θ fixed at two representative values θ = π/6 and π/3.
The plots correspond to parameter set C. We note that while for θ = π/6 both deconfinement and chiral restoration are crossover-like, they are first order for θ = π/3. As in the case of θ = 0 (see the curves corresponding to T 0 = 208 MeV in Fig. 1), for both values of θ the deconfinement and chiral restoration transitions occur at the same temperature, given by the peaks of the susceptibilities or the positions of the discontinuities. Although one might argue that there is a certain tendency of the chiral susceptibility to decay more slowly or even, in the case of θ = π/3, to display a very broad peak on the high temperature side, one can hardly conclude that both transitions apart from each other more than about 20 MeV, even for θ = π/3. The situation is quite similar for parameter set B (which also includes WFR), whereas for set A one gets just first order transitions for all values of θ in the range of temperatures considered. The dependence of the critical temperature T c as a function of θ is shown in Fig. 4 for our three parameter sets. For comparison, we also show the corresponding lattice results given in Ref. [28], which include an error of about 10% due to the uncertainty in the lattice determination of T c (θ = 0). As already mentioned, while for set A both the deconfinement and chiral restoration transitions are always first order, in the case of sets B and C there is a critical value θ ∼ 0.7 × π/3 below which the transitions become crossover-like.
Thus, we find that for all three parameterizations the corresponding transition lines are first order when they meet the RW endpoint. This implies that the RW endpoint is a triple point, the RW transition being also first order there. The character of the RW transition at the RW endpoint is clearly seen in Fig. 5, where we plot the behavior of the phase of the extended Polyakov loop Ψ at θ = π/3 as a function of T (the figure corresponds to set C). The extended Polyakov loop, defined by Ψ = exp(iθ)Φ, is by construction invariant under the transformations in Eq. (12), and its phase ψ can be taken as order parameter of the RW transition [40].
In summary, we have considered the impact of the feedback of the dynamical quarks on the parameter T 0 appearing in the Polyakov potential, as proposed in Refs. [23]. This has been done here in the context of non-local PNJL models, considering three different types of non-local form factors. We have studied the deconfinement and chiral restoration transitions, determining the corresponding critical temperatures and the character of the transitions. The results are found to be in agreement with those obtained in lattice QCD, showing a natural entanglement between both critical temperatures for zero and imaginary chemical potential. We have also analyzed the Roberge Weiss transition, which is found to be first order at the RW endpoint for all three parameter sets.