Cold QCD at finite isospin density: confronting effective models with recent lattice data

We compute the QCD equation of state for zero temperature and finite isospin density within the Nambu-Jona-Lasinio model in the mean field approximation, motivated by the recently obtained Lattice QCD results for a new class of compact stars: pion stars. We have considered both the commonly used Traditional cutoff Regularization Scheme and the Medium Separation Scheme, where in the latter purely vacuum contributions are separated in such a way that one is left with ultraviolet divergent momentum integrals depending only on vacuum quantities. We have also compared our results with the recent results from Lattice QCD and Chiral Perturbation Theory.


I. INTRODUCTION
Quantum chromodynamics (QCD) is the fundamental theory of strong interactions. QCD has a remarkably rich phase structure with multiple facets which has been vividly explored over the years. Recently, with the imminent arrival of relativistic Heavy-Ion-Collision (HIC) experiments in FAIR and NICA, physical systems at finite baryon densities such as neutron stars have become the ideal subject for scrutiny in the heavy ion community [1,2]. However, systems with finite baryon densities are not easy to deal with theoretically, since in this region of QCD phase diagram, first-principle methods such as non-perturbative lattice calculations are not accessible due to the well known fermion "sign problem" [3,4]. For a recent review about the progress of lattice QCD in dealing with sign problem, see Ref [5].
In this work we focus on a new type of compact stars, where the pion condensates are considered to be the dom-inant constituents of the core under the circumstance of vanishing neutron density. Moreover this scenario is easily accessible through first principle methods unlike the study of compact star interiors with high baryon densities. This novel scenario was first identified as pion stars in Ref [11] and has recently been proposed through lattice QCD in Ref [46].
Though pion stars can be described as a subset of boson stars [47][48][49][50][51], they are free from hypothetical beyond standard model contributions usually associated with boson stars, such as QCD axion. Indeed, it can be proved in the framework of a dense neutrino gas that a Bose-Einstein condensate of positively charged pions can be formed [52]. Further exploration of the pion stars' Equation of State (EoS) revealed about its large mass and radius in comparison with neutron stars [46,53]. Recently studies in the similar line have also been done within the chiral perturbation theory [54].
Though there are also possibilities of pion condensation in the early universe driven by high lepton assymetry [52,55,56], in the current context we will consider the setting of compact stars with zero temperature. Further, the charged pion condensation requires accumulation of isospin charge at zero baryon density and zero strangeness. QCD with µ I = 0, µ B = µ s = T = 0 can be and is being realized well within lattice QCD and this new modified lattice result [46] in turn gives us the perfect platform for the consistency check of the effective models mimicking QCD, such as NJL model. As emphasized earlier, QCD with finite isospin chemical potential have already been explored through NJL model, albeit none in light of the new improved lattice results. Additionally the present study tries to rectify the regularization issues within NJL model to deal with the ultraviolet (UV) divergent momentum integrals. In the Traditional Regularization Scheme (TRS), commonly used in literature, the sharp UV cutoff Λ usually cuts important degrees of freedom near the Fermi surface leading to incorrect results, specially in scales of the order of Λ, e.g. µ I ∼ Λ [57,58]. On the other hand, the Medium Separation Scheme (MSS), coined in Refs [59,60], is based on a proper separation of medium effects from divergent integrals, originally having explicit medium dependence. This results in the disposal of all divergent integrals into the pure vacuum part i.e. µ I = 0 in the current context, as it should be. This scheme has already been successfully applied in the context of color superconductivity [57] and for quark matter with a chiral imbalance [59]. For a proper characterization of compact pion stars with high values of µ I (∼ Λ), as we will be dealing with in this work, the role of MSS becomes really important in this regard.
The paper is organized as follows. In section II we discuss the basic formalism of the two-flavor NJL model both within TRS and MSS. In section III we present our results obtained with the traditional regularization scheme and with the medium separation scheme, thermodynamic results are also presented and contrasted with other state of the art calculations. We conclude in section IV discussing the aftermath.
where ψ and m represent the quark fields and their current mass respectively and G is the scalar coupling constant of the model. τ 's are the generator matrices for the pseudoscalar interactions, which corresponds to the pionic excitations π 1 , π 2 , π 3 or equivalently π + , π − , π 3 , with τ ± = (τ 1 ± τ 2 )/ √ 2. For finite isospin chemical potential, the isospin symmetry group SU (2) explicitly breaks down to a subgroup U (1) I3 , third component of the isospin charge I 3 being the generator [24]. So within the context of the mean field approximation, for nonzero µ I one can consider the possibility of ψ iγ 5 τ 3 ψ = 0 as an ansatz, which further breaks the U (1) I3 symmetry. Now we can introduce the chiral condensate σ = −2G ψ ψ and pion condensates where the phase factor θ indicates the direction of the U (1) I3 symmetry breaking. Finally, for the present context of pion stars, we consider µ B = 0, such that µ u = −µ d = µ I . Collecting all these information, one can now obtain the thermodynamic potential within the mean field approximation as The physical values of the condensates vis-a-vis the ground state at finite isospin chemical potential is determined by minimizing Ω NJL (σ, ∆) with respect to the condensates σ and ∆, i.e. by solving the gap equations From these equations we obtain with the definitions In the following subsections we discuss in more details different ways of regularizing these integrals. The thermodynamic quantities, i.e. the pressure, the isospin density and the energy density of the system are then respectively given by Finally, the EoS within the two-flavor NJL model is given by the relation between P NJL and ε NJL .

A. TRS
TRS is the most common and used regularization scheme in the literature, as might be seen in some good reviews of the NJL model [61]. In this case we just perform the integrations in (2.8) and (2.9) up to a cutoff Λ, that becomes a model parameter. Therefore, the gap equations becomes This same procedure is used in Ω NJL , that becomes and also in the thermodynamic quantities. Specifically, the isospin density becomes Since NJL is nonrenormalizable, any physical quantity will depend on the scale of the model Λ. However, it is very important to keep in mind that cutoff dependent medium terms due to a naive regularization of the integrals may lead to results completely different from the ones obtained with a more careful treatment of divergences. MSS provides a tool to disentangle medium dependence from divergent contributions, so that only vacuum integrals need to be regularized. This scheme has been applied to the NJL model and successfully shows qualitative agreement with lattice simulations and more elaborated theories, as might be seen in Refs. [57,59,60].
The implementation of MSS starts by rewriting, for example, I ∆ given in Eq. (2.9) as (2.17) Using the identity (where M 0 is the vacuum mass, when µ I = ∆ = 0) we obtain, after two iterations, After some manipulations and performing the integration in x indicated in (2.17) we obtain with the definitions (2.23) where, in the last line of the equation above we have used the Feynman parametrization (2.25) Using similar steps one may write Using MSS the expression for the normalized thermodynamic potential becomes To obtain the expression for the isospin density we follow the same procedure used for the calculation of I ∆ and I σ , but due to its different divergency structure we need to iterate the identity (2.18) once more. The final expression is with the remaining definitions, (2.30)

.(2.32)
Note that integrals I 1 to I 6 are all finite, and must be performed up to infinite in k. This is the fundamental difference between TRS, where we cut the whole integral in the cutoff Λ and MSS, where all finite medium contributions are separated and performed for the whole momentum range.

III. RESULTS
The parameter set used for the purpose of the present study are m = 4.76 MeV, Λ = 659 MeV and G = 4.78 GeV −2 which we have obtained by fitting the same value of the pion mass as used by Lattice QCD [62], i.e. m π = 131.7 MeV, and other parameters as f π = 92.4 MeV and ψ ψ 1/3 = −250 MeV. This values corresponds to a vacuum mass M 0 303.5 MeV.  Figure 1 shows the variation of the pion condensate ∆ with µ I , scaled by the pion mass value. As might be seen from the plot, higher values of µ I (starting from µ I ∼ 1.5m π ) draw the differences between the two regularization processes. Notice that the values of ∆ are increasingly larger for TRS than MSS when µ I grows. At µ I ∼ Λ (i.e. µ I ∼ 5m π ) the difference between TRS and MSS goes up to 30 − 35 MeV. This difference in ∆ at higher values of µ I also justifies the use of the medium separation scheme, specially since we are working at the zero temperature limit.
In the following part of this section we shall discuss our results for different relevant thermodynamic quantities within the two-flavor NJL model, comparing each one with the corresponding recent Lattice QCD results [46] and Chiral perturbation theory [54] results for both Leading Order (LO) and Next to Leading Order (NLO). It is important to mention that in the present study we are using data sets collected through private communications [63]. In the χPT results used in this study the authors have used the Particle Data Group (PDG) value of the f π , i.e. √ 2f π = 130.2 (±1.7) MeV and for the pion mass m π = 135 MeV. Due to the uncertainty in the values of the low-energy constants [54,63] the uncertainty for the χPT-NLO results have also been presented.
In figures 2, 3 and 4, respectively, the variations of normalized pressure, isospin density and energy density are shown with respect to the isospin chemical potential Variations of the normalized isospin density (nI /m 3 π ) as a function of the normalized isospin chemical potential µI /mπ. The LQCD results [46] have been compared with the behavior of MSS and TRS within the NJL model (left panel) and with up to NLO results within χPT [63] (right panel). The plots are specifically zoomed into the region of interest, up to the value of µI for which LQCD data is available. The three lines for χPT-NLO depicts the uncertainty in the result due to the uncertainty in the low-energy constants [54,63]. Unlike the other thermodynamic quantities, here relatively fewer amount of lattice data points are shown with respective error bars. The dotted line represents the first order interpolation of the latter. scaled by m π . These plots have mainly focussed on the region where m π µ I 2m π as the region of interest, throughout which lattice QCD data was available 1 . In this range of µ I , the difference in results for TRS and MSS is relatively small, as evident from the plots. Comparing NJL results we can observe that TRS has an infinitesimally better agreement with current LQCD than 1 In general within Lattice QCD calculations, the maximum value of µ I is constrained by the value of the lattice spacing.
MSS. LO and NLO results within χPT have also been compared among others. Figure 2 distinctively shows the comparability between the NJL and LQCD results, specially in comparison with χPT results up to NLO. Note that for the χPT datasets used here, the value of pion mass used was taken as 135 MeV (particle data group).
Using instead a pion mass closer to the value adopted by LQCD, i.e, m π = 131±3 MeV and √ 2f π = 128±3 MeV, as it is made in the published version of Ref. [54], the agreement between LQCD and χPT is improved. Figures 3 and 4 show a typical behavior of LQCD data, which cross over the NJL TRS and MSS results around  [46] have been compared with the behavior of MSS and TRS within the NJL model (left panel) and with up to NLO results within χPT [63] (right panel). The plots are specifically zoomed into the region of interest, up to the value of µI for which LQCD data is available. The three lines for χPT-NLO depicts the uncertainty in the result due to the uncertainty in the low-energy constants [54,63].
µ I ∼ 1.5m π , though overall being largely in agreement. This cross over could be due to the current unavailability of larger number of lattice data for isospin density.
Normalized EoS is presented in figure 5 where we can notice the reflection of the behavior of figures 3 and 4 regarding the comparability of NJL and LQCD results. As it can be seen, within the limit of their uncertainties NLO χPT results are in better agreement with the LQCD results for the region P > 0.2m 4 π , whereas NJL (TRS and MSS) results are in better agreement in the lower region of P < 0.2m 4 π . Finally in figure 6 we consider the full spectrum of µ I , i.e. < 0 ≤ µ I ≤ Λ to emphasize the effect of the medium separation at higher values of µ I on the normalized thermodynamic quantities P NJL , n I NJL and ε NJL as well as the EoS. We interprete the parameter Λ as the scale of the model, trusting in results restricted by Λ. In general we use this Λ as an upper limit for the other relevant variables, e.g., temperature, external fields, chemical potentials etc, and the same idea was applied for µ I in this work. Though it is true that for µ I = Λ the regime of validity of our model ends, but we can see in Fig. 6 that the MSS results are different from TRS even for µ I < Λ. We have also plotted χPT results up to NLO in figure 6 but only up to µ I = 0.6Λ (∼ 3m π ). This is to emphasize the fact that those results cannot be trusted beyond µ I ∼ 3m π due to constraints on their validity [63].

IV. CONCLUSIONS
In conclusion, we would like to emphasize on the fact that both the TRS and the MSS regularization schemes within the NJL model show promising results in the front of thermodynamic quantities describing systems similar to pion stars, being largely in agreement with the LQCD results. Regions with higher values of µ I , where LQCD results are not available, we have predicted the pressure, isospin density, energy density and EoS both within TRS and MSS, highlighting the fact that MSS is more reliable in those regions due to its unique way of separating vacuum divergent effects from medium terms. In comparison with other effective theory results, i.e. χPT, our results within the mean field NJL model show a better agreement with LQCD results which prompts us to further investigate the phase diagram for the region with finite µ B and µ I which is inaccessible by LQCD due to the sign problem. Also as mentioned in section I, the possibility of pion condensation in light of early universe dictates further exploration in the T − µ I plane of the QCD phase diagram. Furthermore, χPT calculations for SU (3) at finite isospin have also appeared very recently in [65], which shows excellent agreement with lattice data for small values of µ I . Works in these directions within the NJL model are in progress.
Note added -While finishing the updated version of our paper we learned that a partially overlapping study was done by Zhen-Yan Lu, Cheng-Jun Xia and Marco Ruggieri [64]. χPT results up to NLO have also been presented up to µI = 0.6Λ. The three lines for χPT-NLO depicts the uncertainty in the result due to the uncertainty in the low-energy constants [63].