Cold isospin asymmetric baryonic rich matter in nonlocal NJL-like models

We study the features of low energy strong interactions for a system at zero temperature and finite baryon and isospin chemical potentials, in the framework of a Nambu--Jona-Lasinio-like model that includes nonlocal four-point interactions. We analyze the phase transitions corresponding to chiral symmetry restoration and pion condensation, comparing our results with those obtained from local NJL-like models and lattice QCD calculations.


I. INTRODUCTION
Over the past few decades, there has been a significant amount of research focused on the study of quark and hadronic matter under conditions of finite temperature T and baryon chemical potential µ B .At high temperatures and low densities, it is well known that quantum chromodynamics (QCD) predicts the formation of a quark-gluon plasma (QGP) [1], in which quark and gluons are expected to be weakly coupled.In this limit, strong interactions can be described through perturbative calculations based on expansions in powers of the QCD coupling constant.In the region of intermediate temperatures one can rely on lattice QCD (LQCD) calculations, which indicate that at vanishing chemical potential the transition from the hadronic phase to the QGP occurs in the form of a smooth crossover [2].
On the other corner of the µ B − T phase diagram, at sufficiently high densities and low temperatures, one expects to find a "color-flavor locked" phase, in which the existence of strongly correlated quark pairs is predicted [3].At moderate densities, however, the situation is much more uncertain.The main reason for this is that first-principle nonperturbative QCD calculations at nonzero µ B are hardly accessible by Monte Carlo simulations, due to the presence of a complex fermion determinant in the corresponding partition function (the so-called "sign problem") [4].In this region most theoretical analyses of the phase structure rely on the predictions from effective models for strong interactions.
In addition to T and µ B , the system may show an imbalance in the isospin charge, which can be characterized by an isospin chemical potential µ I .This situation, which can be applicable e.g. to the study of the physics of heavy ion collisions and the structure of stellar objects, is worth to be considered in order to get more insight into the properties of strongly interacting matter.In general, the QCD phase diagram in the T − µ B − µ I thermodynamic space is expected to show a rich structure that can be addressed both from LQCD techniques and effective approaches to strong interactions [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22].In the case of systems at µ B = 0 and finite µ I , LQCD calculations are not affected by the sign problem [23]; thus, the corresponding phase diagram in the T − µ I plane has been studied in several works that use different lattice techniques [20,[24][25][26][27][28].In particular, one important feature confirmed by these calculations is that at µ I ≃ m π one finds the onset of a Bose-Einstein pion condensation phase [5,29], which could enable the existence of pion stars [30].
In the case of nonzero µ B and µ I , lattice analyses are not free from the sign problem and require some extrapolations.Hence, it is remarkably important to get definite predictions from effective models.In this work we study the properties of quark matter under finite µ B and µ I conditions considering quark models in which the fermions interact through covariant nonlocal four-point couplings [31].These models models can be viewed as improved versions of the standard (local) Nambu−Jona-Lasinio (NJL) scheme [32,33].In fact, nonlocal interactions naturally emerge within most approaches to low energy QCD, leading to a momentum dependence in quark propagators that can be successfully reconciled with LQCD results [34].Moreover, these so-called "nonlocal NJL" (nlNJL) models do not exhibit some of the drawbacks observed in the local approach.For example, through the usage of well-behaved nonlocal form factors it is possible to regularize ultraviolet loop integrals while preserving anomalies [35] and ensuring proper charge quantization.In addition, the absence of sharp cutoffs implies that model predictions are more stable against changes in the input parameters [36].Within this framework, in a previous work [37] we have studied the phase diagram in the µ I − T plane for µ B = 0, finding a good agreement with LQCD calculations.In the present article our aim is to extend this research considering a system at zero temperature and finite µ I and µ B .We study the condensate formation and the corresponding phase transitions, comparing our findings with those obtained from the local NJL model [38][39][40][41].
In addition, within the nlNJL model we analyze the behavior of the speed of sound c s as a function of the isospin chemical potential.Recent LQCD calculations [28] have found that c 2 s reaches a maximum at intermediate values of µ I (µ I ∼ 2m π ), and then decreases slowly towards the limit predicted by duality for 4D conformal field theories [42].This result has been discussed in the framework of several effective models, see Refs.[43][44][45][46][47].
This article is organized as follows.In Sec.II we present the general formalism to describe two-flavor nlNJL models at zero temperature and finite baryon and isospin chemical potential, including theoretical expressions for chiral and pion condensates and for the speed of sound.In Sec.III we discuss our numerical results for condensates and phase transitions, for both vanishing and nonvanishing µ B .Finally, in Sec.IV we summarize our results and present our main conclusions.

II. THEORETICAL FORMALISM
We consider a two-flavor quark model that includes nonlocal scalar and pseudoscalar quark-antiquark currents.The Euclidean action reads [37] where ψ = (ψ u ψ d ) T stands for the u, d quark field doublet, and m = diag(m u , m d ) is the current quark mass matrix.For simplicity, we assume that the current quark masses m u and m d are equal and we denote them generically by m c .The nonlocal currents j a (x) in Eq. ( 1) are given by where we have defined Γ a = (1 1, iγ 5 ⃗ τ ), τ i being Pauli matrices that act on flavor space.
The function G(z) is a form factor responsible for the nonlocal character of the four-point interactions.The action for the standard (local) two-flavor quark version of the NJL model is recovered by taking G(z) = δ (4) (z).
To study strongly interacting matter in a system at nonzero chemical potential we introduce the partition function Z = D ψ Dψ exp[−S E ].As stated, we are interested in studying isospin asymmetric matter; this can be effectively implemented by introducing quark chemical potentials µ u and µ d that in general are different from each other.Thus, we consider the effective action in Eq. ( 1) and perform the replacement In fact, it is convenient to write the quark chemical potentials in terms of average and isospin chemical potentials denoted by µ (= µ B /3) and µ I , respectively.One has In addition, owing to the nonlocal character of the interactions, to obtain the appropriate conserved currents one has to complement the replacement in Eq. ( 3) with a modification of the nonlocal currents in Eq. ( 2).This procedure is similar to the one used e.g. in Refs.[37,48,49].
We proceed now by carrying out a standard bosonization of the effective theory, introducing bosonic degrees of freedom σ and π i , i = 1, 2, 3, and integrating out the fermionic fields.Then we consider a mean field approximation (MFA) in which the bosonic fields are replaced by their vacuum expectation values (VEVs) σ and πi .As is well known, in the chi- We consider the above described general situation in which both σ and ∆ can be nonvanishing.The mean field grand canonical thermodynamic potential is found to be given by where Here we have defined p f ν ≡ (⃗ p, p 4 + iµ f ), with f = u, d, and p = (p u + p d )/2.The function is the Fourier transform of the form factor G(z) in Eq. ( 2).
As usual in this type of model, it is seen that Ω MFA turns out to be divergent and has to be regularized.We adopt here a prescription similar as the one considered e.g. in Refs.[37,51], in which one subtracts the thermodynamic potential obtained for σ = ∆ = 0 and adds it in a regularized form.In this way, the regularized expression Ω MFA,reg is given by where with The regularized form of the free piece, after subtraction of divergent terms, reads where ϵ f = ⃗ p 2 + m 2 f .The mean field values σ and ∆ can now be obtained from a set of two coupled "gap equations" that follow from the minimization of the regularized thermodynamic potential, Quark-antiquark and pion condensates are also relevant quantities, since they can be taken as order parameters of the spontaneous symmetry breaking transitions.As usual, we consider the scalar condensate Σ = Σ u + Σ d , where Σ f = ⟨ ψf ψ f ⟩; the latter can be obtained by differentiating the thermodynamic potential with respect to the current up and down current quark masses, namely For µ I ̸ = 0 one can also have nonvanishing pseudoscalar condensates.According to our choice πi = δ i1 ∆, we define the charged pion condensate Π = ⟨ ψiγ 5 τ 1 ψ⟩.The analytical expression for this condensate can be obtained by taking the derivative of the thermodynamic potential with respect to an auxiliary parameter added to ρ(p) in Eq. ( 5), and then set to zero after the calculation [37].
To study the phase transitions, we also introduce the susceptibilities associated to the above defined order parameters [52].They are given by Finally, as mentioned in the Introduction, it is interesting to consider the speed of sound c s .At zero temperature, one has where the energy density ε is given by with n I = − ∂Ω MFA,reg /∂µ I , n B = − ∂Ω MFA,reg /∂µ B .

III. NUMERICAL RESULTS
To fully define our model it is necessary to specify the form factor entering the nonlocal fermion current given by Eq. ( 2).In this work we consider an exponential momentum dependence for the form factor (in momentum space), This form, which is widely used, guarantees a fast ultraviolet convergence of quark loop integrals.Notice that the energy scale Λ, which acts as an effective momentum cutoff, has to be taken as an additional parameter of the model.Other functional forms, as e.g.
Lorentzian form factors with integer or fractional momentum dependences, have also been considered in the literature [49,53].In any case, it is seen that the form factor choice does not have in general major impact in the qualitative predictions for most relevant thermodynamic quantities [54].This leads to m c = 5.67 MeV, Λ = 752 MeV and GΛ 2 = 20.67 [48].In Ref. [37] this parametrization has been used to study the features of this type of model for a system at

Order parameters and phase transitions
In Fig. 1 we show the behavior of the above introduced Σ and Π condensates at µ B = 0 and finite µ I .Although these results have been previously presented in Ref. [37], we find it convenient to include a brief review for the sake comparison with the case of nonzero µ B .
For µ I < m π one finds the usual low energy situation in which chiral symmetry is spontaneously broken, which is reflected in a large value Σ = Σ 0 for the quark-antiquark condensate, while the pion condensate vanishes (Π = Π 0 = 0).Then, it can be analytically shown that at µ I = m π the model predicts the onset of a phase in which one has pion condensation.For µ I > m π , as seen in Fig. 1, the chiral condensate decreases monotonically and the charged pion condensate gets strongly increased.Thus, one has a second order phase transition in which the isospin symmetry U(1) I 3 gets spontaneously broken, whereas one finds a smooth partial restoration of the U(1) I 3 A symmetry when reaching large values of µ I .It can be seen that the results from local and nonlocal versions of the NJL model are similar to each other, and they are found to be in good qualitative agreement with lattice QCD calculations (also shown in the figure) [30].In addition, as discussed in Ref. [37], for this range of values of µ I the results are consistent with the relation which can be obtained from lowest-order chiral perturbation theory [56].Lattice results from Ref. [30] are included for comparison.

Speed of sound
As mentioned above, the speed of sound c s has been studied within various effective models.For the case of systems at nonzero isospin chemical potential, recent LQCD calculations [22,28] have found that the curve of c 2 s as a function of µ I shows a maximum for µ I /m π ∼ 2. This maximum is shown to be well above the limit c 2 s = 1/3, which is obtained for QCD at large temperature on the basis of gauge/gravity duality for 4D conformal field theories [42].It is worth noticing that a similar behavior has been obtained in the framework of two-color QCD [45,57,58] and quarkyonic models for dense quark matter [59][60][61][62].
Our numerical results are shown in Fig. 2, where we also include for comparison the results arising from the local NJL model and those obtained from LQCD in Refs.[22,28].
To provide an estimate of the dependence on model parameterizations, we have considered the results for the nonlocal NJL model for parameters corresponding to quark-antiquark condensates lying within the range from −(260 MeV) 3 to −(230 MeV) 3 (dark gray band), and similarly for the local NJL model, taking a range from −(250 MeV) 3 to −(240 MeV) 3   (light gray band).It can be seen that the results obtained within the nonlocal model do not show a strong dependence on the parameterization.Moreover, they reproduce with good qualitative agreement the behavior observed by the most recent LQCD analysis -see Ref. [28]-, where a large range of values of µ I is covered.On the other hand, in agreement with the results in Ref. [47], it is seen that for the local NJL model c 2 s does not show a clear peak at intermediate values of µ I .In fact, in Ref. [47] it is found that such a peak can be obtained once the coupling constants are allowed to have an explicit dependence on µ I .

B. Phase transitions for finite baryon chemical potential
Let us consider a more general situation in which both the quark chemical potential µ = µ B /3 and the isospin chemical potential are nonzero.To describe the picture obtained in the (µ, µ I ) thermodynamic space we take some representative values of µ and study how the order parameters Σ and Π evolve with µ I .This is shown in Fig. 3. Left and right panels correspond to the results from nonlocal and local NJL models, respectively.
For low values of µ the situation is similar to the one described in the previous section for µ = 0.In the left and right upper panels of Fig. 3 we show the behavior of Σ and Π for µ = 100 MeV; one finds at µ I /m π = 1 the onset of a pion condensation phase (a second order phase transition), while chiral symmetry gets smoothly restored when µ I is increased.
Notice that, in the case of the nonlocal model, for large values of µ I there is a significant deviation from the chiral relation in Eq. ( 16); this deviation is not observed for the local model (at least, for values of µ I up to 5 m π ).   3 .LQCD data from Refs.[22,28] are also included for comparison.
To get a better understanding of the transitions let us also show contour plots that describe the behavior of the mean field thermodynamic potential Ω MFA,reg as a function of the VEVs σ and ∆ for particular values of µ I .In Fig. 4 we consider the case µ = 100 MeV, Left and right panels correspond to the nonlocal and local models, respectively.
Next, in Fig. 5  which there is no pion condensation (Π = 0); it corresponds to the dashed lines in the central panels of Fig. 3.The onset of this metastable solution occurs at some critical isospin chemical potential that we denote by µ I ; for the chosen parameterizations, is seen that µ (sp) I (µ = 200 MeV) ≃ 2 m π and 4 m π for the nonlocal and local models, respectively.We notice that a similar picture has been obtained in Ref. [41] for the case of a three-flavor NJL model.However, in that article the saddle point is interpreted as maximum, since only the dependence of the thermodynamical potential with ∆ is analyzed.Then, if µ I /m π is further increased, at some critical value µ I,c a first order phase transition occurs: as illustrated in the lower panels of Fig. 6, the minima for which one has ∆ = 0 are the ones that become energetically favored; thus, the system jumps into a phase in which there is no pion condensation and the U (1) I 3 symmetry gets restored.The behavior of the order parameters for µ = 280 MeV is shown in the lower panels of Fig. 3.In the case of the nonlocal model, it is seen that at the first order transition the value of the quark-antiquark condensate Σ shows also a jump that implies an approximate restoration of chiral symmetry (in the case of the local model, the value of Σ is already very low when the transition is reached).
For even larger values of µ the region in which there is a stable nonvanishing pion condensate gets subsequently reduced, until one reaches a triple point in which three phases coexist.
The full phase diagrams in the µ − µ I plane for both nonlocal (upper panel) and local (lower panel) models are shown in Fig. 7. Solid and dashed lines denote first and second order phase transitions, respectively, while dotted lines denote the spinodals [51] -boundaries of the region in which energetically unfavored solutions exist as metastable states.
The phase diagrams show a region of normal matter (NM) in which one has Π = 0 and the chiral symmetry is spontaneously broken, a quark-gluon plasma phase in which Π = 0  and local NJL models.Solid (dashed) lines correspond to first (second) order phase transitions.
NM, πC and QGP stand for normal hadronic matter, pion condensation and quark-gluon plasma, respectively.
is increased: at µ ∼ 350 MeV one has a transition from the πC phase to the NM phase, followed by a transition to the QGP phase.In fact, these features of the phase diagram for the local NJL model are in agreement with the results obtained in Refs.[38,41] for twoand three-flavor (local) NJL models.
In Fig. 7 we also show the spinodals, represented by blue dotted lines.As stated, these lines indicate the critical isospin chemical potentials µ (sp) I at which metastable solutions are found to appear.For the local NJL model we find spinodals at both sides of the first order transition line -delimiting a band in the phase diagram where metastable solutions exist-, while in the case of the nonlocal model no upper spinodal is found.In fact, for large values of µ one can always found a metastable solution in which quarks have a relatively large effective dynamical mass.This is a well-known feature of nonlocal NJL models; it is related to the fact that -depending on the model parameterization-the quark propagators may not have purely real poles in Minkowski space [63].
If one goes further to larger values of µ I , the nlNJL and NJL approaches show significant differences.For low values of µ, the order parameter Π increases monotonically for the nlNJL model, while for the local model (as shown in Ref. [39]) it starts to decrease and goes to zero at some point beyond µ I ∼ 10 m π .We understand that these different behaviors are artifacts that arise from the regularization of the (nonrenormalizable) models, which become hardly trustable in that limit.Therefore we present our results for more conservative values of µ I , up to a few m π .

IV. SUMMARY AND CONCLUSIONS
The phase diagram of strongly interacting matter has been examined in two-flavor NJLtype models, considering zero temperature and nonzero baryon and isospin chemical potentials.Specifically, we have investigated the transitions related to the order parameters Σ and Π, which characterize the spontaneous breakdowns of chiral and isospin symmetries, in models with local and non-local four-quark interactions.We have also studied the behavior of the speed of sound as a function of the isospin chemical potential.symmetry.On the other hand, by increasing µ one arrives at a first order transition to a quark-gluon plasma phase in which there is no pion condensation and chiral symmetry is approximately restored.In the nonlocal NJL approach it is seen that the first and second order transition meet at a triple point located at µ I = m π , µ ≃ 300 MeV.In the case of the local model, in agreement with previous works, we find that the second order transition line has a critical endpoint where it becomes of first order, and then, for µ I > m π , it smoothly merges the first order chiral restoration line.We have also studied metastable phases, identifying saddle points for the thermodynamic potential as a function of the order parameters; in general, it is seen that metastable phases cover a larger region in the case of the nonlocal model.Concerning the speed of sound c s , we find that for the nonlocal model the behavior of c 2 s with µ I for vanishing baryon chemical potential shows a maximum at µ I ∼ 2 m π , improving the qualitative agreement with lattice QCD calculations in comparison with the results obtained for the local NJL approach.It would be interesting to extend these studies to systems at finite temperature, with the aim of determining the behavior of the triple point and the phase transition lines (the case µ = 0 has been already considered in Ref. [37]).In addition, it would be worthwhile to consider the case of neutral matter conditions, to be applied to the composition of compact objects like neutron or pion stars.
ral limit (m c = 0), for µ I = 0 the action is invariant under global U(1) B ⊗ SU(2) I ⊗ SU(2) IA transformations.The group U(1) B is associated to baryon number conservation, while the chiral group SU(2) I ⊗SU(2) IA corresponds to the symmetries under isospin and axial-isospin transformations.Now, in the presence of a nonzero isospin chemical potential, the full symmetry group is explicitly broken down to the U(1) I 3 ⊗ U(1) I 3 A subgroup.If σ develops a nonzero VEV, the U(1) I 3 A symmetry gets spontaneously broken.Moreover, while even for finite current quark masses one has π3 = 0[50], for µ I ̸ = 0 it can happen that π 1 and π 2 develop nonvanishing VEVs, leading to a spontaneous breakdown of the remaining U(1) I 3 symmetry.Since the action is still invariant under U(1) I 3 transformations, without loss of generality one can choose πi = δ i1 ∆.
Given the form factor shape, the model parameters m c , G and Λ can be fixed by requiring that the model can reproduce the phenomenological values of some selected physical quantities.Here we take as inputs the empirical values of the pion mass, m π = 138 MeV, and the pion weak decay constant, f π = 92.4MeV, together with phenomenologically reasonable values of the quark-antiquark condensates at µ = µ I = 0, viz.Σ u = Σ d = −(240 MeV)3 .
finite temperature and isospin chemical potential, getting a good agreement with LQCD results for the T − µ I phase diagram.As mentioned in the Introduction, one of the aims of this work is to confront the results obtained within the nonlocal model with those obtained in the framework of the standard, local version of the NJL model.For the latter we use the parametrization m c = 5.83 MeV, Λ 0 = 588 MeV and GΛ 2 0 = 4.88, which leads to Σ u = Σ d = −(239 MeV) 3 and quark effective masses M u = M d = 400 MeV [55]. A. Results for µ B = 0 and finite µ I We begin by stating the picture obtained from the nonlocal NJL model at vanishing baryon chemical potential.It is interesting to compare our results with LQCD calculations, which for µ B = 0 are free from the sign problem, even for nonzero values of the isospin chemical potential.

Figure 1 :
Figure 1: Normalized Σ and Π condensates as functions of the isospin chemical potential.

Figure 2 :
Figure 2: Squared speed of sound for µ B = 0. Dark and light gray bands correspond to the results for local and nonlocal NJL models, respectively.The dashed lines within the gray bands correspond to model parameterizations that lead to a quark-antiquark condensate of −(240 MeV)3 .LQCD data from Refs.[22,28] are also included for comparison.

Figure 3 :
Figure 3: Normalized values of the order parameters Σ and Π as functions of µ I /m π for nonlocal (left) and local (right) NJL models.

Finally
, in Fig. 6 we show how this picture evolves when we go forward to larger values of the chemical potential µ.To illustrate the situation we include some contour plots in which we take µ = 280 MeV and some representative values of µ I /m π .As in the previous cases, left (right) panels correspond to the nonlocal (local) NJL model.For values or µ I just above m π , in the case of the nonlocal model the metastable solution already exists (in fact, there is no critical value µ (sp) I for values of µ larger than about 270 MeV), while for the
Considering a µ − µ I phase diagram, for baryon chemical potentials lower than about 280 MeV and µ I < m π one finds a region of normal hadronic matter, in which chiral symmetry is spontaneously broken.Then, at µ I = m π there is a second order phase transition into a region in which one has a nonzero charged pion condensate.By increasing µ I the chiral condensate Σ gets progressively reduced, implying a smooth restoration of the U(1) I 3 A