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We analyze the two flavor version of the Nambu-Jona-Lasinio model with a repulsive vector coupling (G_{V}), at finite temperature and quark chemical potential, in the strong scalar coupling (G_{s}) regime. Considering G_{V} = 0, we review how finite N_{c} effects are introduced by means of the Optimized Perturbation Theory (OPT) which adds a term to the thermodynamical potential. This 1/ N_{c} suppressed term is similar to the contribution obtained at the large-N_{c} limit when G_{V} ≠ 0. Then, scanning over the quark current mass values, we compare these two different model approximations showing that both predict the appearance of two critical points when chiral symmetry is weakly broken. By mapping the first order transition region in the chemical potential-current mass plane, we show that, for low chemical potential values, the first order region shrinks as μ increases but the behavior gets reversed at higher values leading to the back-bending of the critical line. This result, which could help to conciliate some lattice results with model predictions, shows the important role played by finite N_{c} corrections which are neglected in the majority of the works devoted to the determination of the QCD phase diagram. Recently the OPT, with G_{V} = 0, and the large-N_{c} approximation, with G_{V} ≠ 0, were compared at zero temperature and finite density for one quark flavor only. The present work extends this comparison to finite temperatures, and two quark flavors, supporting the result that the OPT finite N_{}

Although most of the results obtained up to now seem to support the quantum chromodynamics (QCD) critical point (CP), an interesting observation against its existence comes from the numerical simulations of QCD at imaginary chemical potential by de Forcrand and Philipsen [1-3] which shows that the region of quark masses (m_{c}) where the transition is presumably of the first order (for quark masses smaller than the physical ones), tends to shrink for small positive values of the chemical potential as shown in the upper panel of

tion lines where one of them represents the usual line which starts at zero temperature and chemical potential of the order of the constituent quark mass while the other is an unusual line which starts at zero chemical potential and high temperature [6,7].

In [_{c} with an explicit repulsive vector interaction, with coupling G_{V}, in order to produce a back-bending that would conciliate the lattice results obtained by de Forcrand and Philipsen with most model predictions. It is well known that within the NJL this type of interaction weakens the first order transition line [_{c} application of [_{V }r_{q}^{2} (here ρ_{q} represents the quark number density) to the pressure and as a result the size of the first order covers a smaller range of temperatures as compared to the G_{V} = 0 case.

At the same time, the value of the coexistence chemical potential for a given temperature occurs at a higher value when G_{V} ≠ 0 and, as a consequence, the critical end point happens at smaller temperatures to be higher chemical potentials than in the case of vanishing G_{V}. Although such a vector term is known to be important at high densities in theories such as the Walecka model for nuclear matter, its consideration is more delicate within a non renormalizable model such as the NJL where usually the integrals are regulated by a momentum cut-off, Λ. Within this model, G_{S} and Λ are usually fixed to reproduce the pion mass, the pion decay constant

and the quark condensate

which yields Λ ~ 560 - 670 MeV, G_{S }Λ^{2} ~ 2 - 3.2 and m_{c} ~ 5 - 5.6 MeV (see [

However, fixing G_{V} poses and additional problem since this quantity should be fixed using the ρ meson mass which, in general, happens to be higher than the maximum energy scale set by Λ. Then, G_{V} is usually considered to be a free parameter whose estimated value ranges between 0.25 G_{S} and 0.5G_{S} [11,12].

Alternatively, when going beyond the large-N_{c} (or mean field) level one may induce quantum (loop) corrections which mimic the physical effects caused by a classical (tree) term such as G_{V}. This is precisely what has been observed in an application of the nonperturbative Optimized Perturbation Theory (OPT) method to the two flavor NJL model with vanishing G_{V} [_{c} with an explicit repulsive vector channel. The reason is that the OPT two loop contributions add a term like to the pressure.

The relationship between the OPT, at G_{V} = 0, and the large-N_{c} approximation, at G_{V} ≠ 0, has been recently investigated in great detail in the framework of the abelian NJL at finite densities and zero temperature in [_{c} plane, the OPT has also been previously employed with success in [_{S} in order to obtain a T − μ phase diagram dominated by first order chiral transitions only.

Then, the OPT with its term was used at different quark mass values showing that, in this case, two critical points emerge at low m_{c} due to the weakening of the first order line at intermediate μ values leading to the back-bending behavior observed in [

Note that a repulsive vector type of coupling was not explicitly considered in the two flavor LSM application performed by Bowman and Kapusta which, on the other hand, was carried out beyond the mean field level through the consideration of thermal fluctuations.

In the present work, we extend the comparison between the OPT (at G_{V} = 0) and the large-N_{c} approximation (at G_{V} ≠ 0) to the non abelian NJL model at finite temperature and density in the strong coupling and small quark mass regime showing that, as expected, both methods agree from the qualitative point of view leading to a back-bending which would be completely missed by a standard large-N_{c} evaluation.

Our results also emphasize the importance played by terms which are easily taken into account by the OPT so that this method may be viewed as a robust alternative to investigate nonperturbative effects related to the chiral transition of strongly interacting matter. The work is organized as follows. In the next section, we perform a large-N_{c} application to the two flavor NJL version in the strong coupling regime for the G_{V} ≠ 0 case. In Section 3 we review the OPT results, at G_{V} = 0, which were originally obtained in [

The standard version of the two flavor Nambu-JonaLasinio model lagrangian density L with a repulsive vector channel reads [10,15]

where y (a sum over flavors and color degrees of freedom is implicit) represents a flavor isodoublet (u and d type of quarks) N_{c}-plet quark fields while are isospin Pauli matrices.

As emphasized in [_{S }Λ^{2} = 2.44, and m_{c} = 5.6 MeV so that, with these inputs, one obtains f_{π} = 93 MeV, m_{π} = 135 MeV, and M = 400 MeV at T = 0, and μ = 0 [

However, as will be shown in the next subsection, in order to simulate the back-bending behavior in the present model we will keep Λ = 587.9 MeV considering while varying the current quark mass from m_{c} = 0 to m_{c} = m_{phys}= 5.6 MeV. As discussed in [_{π }, m_{π} and. On the other hand the quark effective mass, which is directly proportional to G_{S}, assumes very high values (around 800 MeV > Λ). However, this is not a problem for our present purpose of simulating the back-bending in a qualitative way (see [

The large-N_{c} (or MFA) evaluation of the thermodynamical potential within this model is standard and yields [

where the dressed “free gas” term is given by

the scalar density is given by

and the quark number density is given by

The integrals appearing in the above equations are defined by

and

where Here, the divergent contributions corresponding to the first term on the right hand side of Equations (6) and (7) are regulated by Λ. The effective quark mass, M, and the effective chemical potential, µ, are obtained from solving the following coupled self consistent equations

and

Let us now review the main effects of the repulsive vector interaction in the phase diagram by considering the standard parametrization. _{S} Λ^{2} = 2.44, Λ = 587.9 MeV and m_{c} = 0 (chiral limit) at G_{V }/G_{S} = 0, 0.1, and 0.4. As expected the G_{V} term has little effect at low chemical potential values where the second order chiral transition dominates since m_{c} = 0.

One also notices that, as G_{V} increases, the first order chiral transition line weakens so that the tricritical point occurs at smaller temperatures. It is also clear that for a given temperature the coexistence chemical potential takes place at higher values with increasing G_{V}. This scenario is also observed away from the chiral limit except that now the second order transition is replaced by a cross over region and the tricritical point turns into a critical end point. This type of phase diagram where a CP naturally appears at the physical quark mass point is predicted by most model approximations. Now, in order to simulate the lattice results by de Forcrand and Philipsen we first need to obtain a first order phase transition at μ = 0 for low m_{c} values so that at vanishing densities our results would be consistent with the lattice results furnished by the Columbia plot [

Within the NJL this is easily achieved by increasing G_{S} [_{V} = 0 and m_{c} = 0, the whole phase diagram is dominated by a first order phase transition. _{V} and m_{c}. Let us first analyze the case G_{V} = 0 by noticing that if one increases m_{c} the first order line weakens and turns into a cross over at μ = 0. _{c}

plane showing that, for G_{V} = 0, the first order line recedes from the temperature axis as m_{c} increases without reproducing the back-bending scenario which would conciliate the de Forcrand and Philipsen results with model predictions. Now, if we turn on the vector interaction, still at m_{c} = 0, the weakening of the first order transition happens in a different way so that two segments of first order chiral transitions appear. One of them is the usual one which starts at T = 0 while the other is an unusual first order line which starts at μ = 0 as _{S} and a finite G_{V} we have managed to induce the appearance of two tricritical points at vanishing m_{c} which, as will be shown in Section 3, leads to the back-bending scenario.

The basic idea of the OPT method is to deform the original lagrangian density by adding a quadratic term like to the original lagrangian density as well as by multiplying all coupling constants by δ [^{1}.

Therefore, the fermionic propagator is dressed by η which may also be viewed as an infrared regulator in the case of massless theories. After a physical quantity, such as the thermodynamical potential (Ω), is evaluated to the k-order and δ set to the unity only a residual η dependence remains. Then, optimal nonperturbative results can be obtained by requiring that Ω(k) (η) be evaluated where it is less sensitive to variations of the arbitrary mass parameter. This requirement translates into the criterion known as the Principle of Minimal Sensitivity (PMS) [

In general, the solution to this equation implies in self consistent relations generating a nonperturbative coupling dependence. In most cases nonperturbative corrections appear already at the first nontrivial order while the large-N_{c} (or MFA) results can be recovered at any time simply by considering N_{c} → ∞. Finally, note that the OPT has the same spirit as the Hartree and the Hartree-Fock approximation in which one also adds and subtracts a mass term. However, within these two traditional approximations the topology of the dressing is fixed from the start: direct (tadpole) terms for Hartree and direct plus exchange terms for Hartree-Fock. On the other hand, within the OPT, the dressed mass term acquires characteristics which change order by order progressively incorporating direct, exchange, vertex corrections, etc, effects. The differences between these three different methods have been recently discussed in [_{V} = 0 one follows the prescription used in [

Then, the order-δ thermodynamical potential can be written as (see [

where now all the integrals defining the quantities ω_{FG}, ρ_{s} , and ρ_{q} are redefined as. Then, for each pair of values the optimum mass parameter, , can be obtained by solving the PMS equation given by [

Note that when N_{c} → ∞ the PMS optimization procedure sets = −2G_{S}ρ_{s} exactly reproducing the large-N_{c} result (for the standard NJL model) with no ρ_{q} dependence. By comparing the OPT thermodynamical potential given by Equation (13) with its large-N_{c} counterpart given by Equation (2) one notices that the OPT induces a finite N_{c} correction of the form while, as discussed in [_{V} term gives a net contribution of the form G_{V} ρ_{q}^{2}. Then, one could expect that the two different model approximations given by the OPT (at G_{V} = 0) and the large-N_{c} approximation (at G_{V} ≠ 0) lead to the same qualitative picture of the phase diagram. A recent comparison performed with one flavor at vanishing temperature in [

Here, we are now in position to extend that comparison to the more realistic two flavor case at finite temperatures.

Let us now compare the results furnished by the two approximations for the NJL model at high G_{S}. _{c} result for G_{V }= 0 which predicts a first order transition taking place in the whole plane. At the same time the OPT with its contribution and the large-N_{c} at G_{V} ≠ 0 weaken that line at intermediate chemical potential values so that two critical points appear in this case of small current mass as expected from the discussion related to _{c} values towards the physical one (m_{phys}) one can map the T − μ diagram into the μ − m_{c} plane as shown in

ing of the critical line so that the CP will be recovered at m_{c} = m_{phys} even if initially (at low values of μ) the line bends in such a way which is reminiscent of the “exotic” scenario displayed by the right panel of _{c} have been discussed in great detail in [_{c} → m_{phys} the unusual first order line disappears and only the usual “liquid-gas” type of first order line survives in accordance with most model predictions.

We have considered the two flavor NJL model in the strong scalar coupling regime (G_{S}Λ^{2} @ 4) in order to compare two distinct model approximations. The first is the traditional large-N_{c} approximation which was applied by explicitly considering a finite repulsive vector interaction (proportional to G_{V}) which was introduced at the classical (tree) level. The second is the alternative OPT method which was applied to the standard version of the NJL model.

Our first step towards the simulation of the backbending behavior was to tune G_{S} at G_{V} = 0 so that the large-N_{c} approximation predicts that the first order transition line, which usually starts at T = 0, will touch the T axis at μ = 0 for very small m_{c} values.

For mass values closer to the physical ones, this approximation recovers the expected cross over behavior at small μ with the appearance of a single critical point at intermediate chemical potentials. Next, we have shown that by considering finite values for G_{V} at this single first order transition line splits into two lines in the T − μ plane. One of them is similar to the usual “liquid-gas” line which starts at T = 0 and ends at intermediate temperature values. The other one, which has a more “chiral” behavior according to the analysis of [_{c} approaches m_{phys} from below. In this way, we were able to induce the back-bending behavior for two flavors in a manner analogous to the one adopted in [_{q} represents the fermionic density. This term is similar to the net −G_{V }ρ_{q}^{2} contribution considered at large-N_{c} [_{s}, weakening the first order line at intermediate values of μ and enhancing the appearance of two critical points in the T − μ plane for m_{c} values which are smaller than the physical ones. Finally by scanning the values of m_{c}, we have mapped the T − μ phase diagram into the μ − m_{c} plane observing that, for strong couplings, the large-N_{c} approximation, at finite G_{V}_{ }, and the OPT, at vanishing G_{V} , predict that the first order transition region shrinks for low values of μ as observed in the lattice simulations of [_{V }ρ_{q}^{2} (large-N_{c}) and (OPT) change the first order phase transition region into a cross over region. Finally, at higher chemical potentials, the first order transition region reappears and then expands as μ is increased. So, our results suggest that even if an initial shrinkage of the first order region is confirmed by lattice simulations, it does not necessary rule out the existence of the CP which is expected to occur at intermediate chemical potentials for physical quark masses. In this case, a back-bending will be observed on the μ − m_{c} plane outlining the importance of a repulsive vector contribution in agreement with [_{c} case or by radiatively generating it by going beyond the mean field level. Within the NJL model, the advantage of the second procedure, which can easily be implemented within the OPT, is that it does not require the fixing of G_{V} which is a drawback of the first procedure. The results obtained in the present application support those obtained in [

This work was partially supported by CAPES, CNPq and FAPESC (Fundação de Amparo à Pesquisa e Inovação do Estado de Santa Catarina).