Hot QCD at finite isospin density: confronting SU(3) Nambu-Jona-Lasinio model with recent lattice data
Bruno. S. Lopes, Sidney S. Avancini, Aritra Bandyopadhyay, Dyana C. Duarte, Ricardo L. S. Farias
HHot QCD at finite isospin density: confronting SU(3) Nambu-Jona-Lasinio modelwith recent lattice data
Bruno. S. Lopes, Sidney S. Avancini, Aritra Bandyopadhyay,
3, 4
Dyana C. Duarte,
5, 6 and Ricardo L. S. Farias Departamento de F´ısica, Universidade Federal de Santa Maria, Santa Maria, RS 97105-900, Brazil Departamento de F´ısica, Universidade Federal de Santa Catarina, 88040-900 Florian´opolis, Santa Catarina, Brazil Guangdong Provincial Key Laboratory of Nuclear Science, Institute of Quantum Matter,South China Normal University, Guangzhou 510006, China Guangdong-Hong Kong Joint Laboratory of Quantum Matter,Southern Nuclear Science Computing Center, South China Normal University, Guangzhou 510006, China Institute for Nuclear Theory, University of Washington, Seattle, WA 98195, USA Departamento de F´ısica, Instituto Tecnol´ogico de Aeron´autica, 12228-900 S˜ao Jos´e dos Campos, SP, Brazil (Dated: February 8, 2021)Extending our recently published SU (2) results for zero temperature we now compute the QCDequation of state for finite isospin density within the three flavor Nambu-Jona-Lasinio model in themean field approximation, motivated by the recently obtained Lattice QCD results for both zeroand finite temperatures. Like our previous study, here also we have considered both the commonlyused Traditional cutoff Regularization Scheme and the Medium Separation Scheme. Our results arecompared with recent high-precision lattice simulations as well as previously obtained results in two-flavor Nambu-Jona-Lasinio model. The agreement between the lattice results and the predictionsfrom three-flavor NJL model is very good for low values of µ I (for both zero and finite temperatures).For larger values of µ I , the agreement between lattice data and the two-flavor NJL predictions issurprisingly good and better than with the three-flavor predictions. I. INTRODUCTION
As the fundamental theory of strong interactions, thephase structure of Quantum Chromodynamics (QCD)has been studied from different angles over the years.Aside from the well explored systems at finite tempera-tures and finite baryon densities, several new dimensionshave been added to the QCD phase diagrams: isospinchemical potential, magnetic field, electric field, rotationto name a few. Though near future relativistic Heavy-Ion-Collision (HIC) experiments in FAIR and NICA havebeen continuing to inspire studies of physical systems atlow temperatures and finite baryon densities such as neu-tron stars [1, 2], theoretical hurdles are still there, pre-dominantly due to the well known fermion “sign prob-lem” [3, 4] encountered by non-perturbative lattice cal-culations. Lattice QCD’s recent progress with the signproblem can be monitored in Ref [5].Among the relatively new features of the QCD phasediagram, finite isospin chemical potential ( µ I ) plays animportant role, specially because unlike finite baryonchemical potential it does not suffer from the sign prob-lem in lattice QCD based calculations. First bunch oflattice QCD results at finite temperature and isospindensity appeared in early 2000’s [6, 7] with dynamical u and d quarks, although with unphysical pion massesand/or an unphysical flavour content. This followed var-ious studies by other available theoretical tools yieldingqualitatively similar results. These studies include chiralperturbation theory ( χ PT) [8–20], Hard Thermal Loopperturbation theory (HTLPt) [21], Nambu-Jona-Lasinio(NJL) model [22–42] and its Polyakov loop extended ver-sion PNJL [43, 44], quark meson model (QMM) [45–48].Recently, early lattice QCD results have been modified by using an improved lattice action with staggered fermionsat physical quark masses and results for finite isospindensity are presented in Refs [49–52].In our recent work along the similar line [53], we fo-cused on a new type of compact stars known as pionstars [11, 54], where the pion condensates are consid-ered to be the dominant constituents of the core underthe circumstance of vanishing neutron density. In spiteof being a subset of boson stars [55–59], pion stars arefree from hypothetical beyond standard model contribu-tions like QCD axion. This gave us a scenario to workwith finite isospin density along with zero temperatureand zero baryon density which bypasses the sign prob-lem unlike systems with high baryon densities. Hence itis easily accessible through first principle methods [54]and through the pion stars’ Equation of State (EoS) wenow know about its large mass and radius in comparisonwith neutron stars [54, 60]. Our study in the said con-text of pion stars within two flavor NJL model showedbetter quantitative agreement with the lattice QCD re-sults than similar studies within the chiral perturbationtheory [18].Unlike our last study [53] where we have only con-sidered the setting of pion stars with zero temperature,in the present work we plan also to explore the systemswith finite temperature. Early universe with very hightemperature has been known to have possibilities of pioncondensation driven by high lepton asymmetry [61–63].Furthermore, in this work we have extended our two-flavor study within the three flavor NJL model. Whilethe two-flavor studies are sufficient to describe the pioncondensation at finite isospin density, a three-flavor studygives us the chance to explore the roles of the strangenessdegree of freedom and the U A (1) anomaly in the present a r X i v : . [ h e p - ph ] F e b context. Hence in our three flavor NJL Lagrangian, wewill also be considering the Kobayashi-Maskawa-’t Hooft(KMT) term, which mimics the U A (1) anomaly in theNJL model.Just like QCD systems with µ I (cid:54) = 0 , µ B = µ s = T = 0 [54], QCD systems with µ I (cid:54) = 0 , µ B = 0 , µ s (cid:54) =0 , T (cid:54) = 0 are also being explored well within latticeQCD [49, 52, 64–66]. Successful premises of this workhave already been provided by our previous study [53]which showed an exceptional quantitative agreement be-tween NJL and lattice QCD results. On the basis of thatand recent improved three flavor lattice QCD results atzero and finite temperature [49, 52, 54, 64–66] give us theperfect opportunity for the consistency check of the NJLmodel. As in our previous work [53] we have tried to rec-tify the regularization issues within NJL model to dealwith the cutting of important degrees of freedom nearthe Fermi surface because of a sharp ultraviolet (UV)cutoff [67–69]. Besides the commonly used TraditionalRegularization Scheme (TRS) we have used the MediumSeparation Scheme (MSS) [67, 70, 71], which properlyseparates the medium effects from divergent integrals.For systems with high values of µ I ( ∼ Λ) the role ofMSS becomes more and more important.The paper is organized as follows. In section II we dis-cuss the basic formalism of the three-flavor NJL modelboth within TRS and MSS. In section III we present anddiscuss our results obtained with the traditional regular-ization scheme and with the medium separation scheme,for both zero and finite temperature. Thermodynamicresults and the T − µ I phase diagram are also presentedand compared with other state of the art calculations. II. FORMALISM
We start with the partition function for the three-flavorNJL model at finite baryonic and isospin chemical poten-tial, given by Z NJL ( T, µ B , µ I , µ S ) = (cid:90) [ d ¯ ψ ][ dψ ] × exp β (cid:90) dτ (cid:90) d x (cid:0) L NJL + ¯ ψ ˆ µγ ψ (cid:1) , (2.1)where the quark chemical potential matrix in flavor spaceis ˆ µ = µ u µ d
00 0 µ s , (2.2) and µ u,d,s can be expressed in terms of the baryonic, theisospin and the strangeness chemical potential as µ u = µ B µ I ,µ d = µ B − µ I ,µ s = µ B − µ S , such that µ I = ( µ u − µ d ) / L NJL appearing in Eq. (2.1)is the NJL Lagrangian considering scalar and pseu-doscalar interactions, i.e. L NJL = ¯ ψ (cid:0) i /∂ − m (cid:1) ψ + G N f − (cid:88) α =0 (cid:104)(cid:0) ¯ ψλ α ψ (cid:1) + (cid:0) ¯ ψiγ λ α ψ (cid:1) (cid:105) − K (cid:2) det ¯ ψ (1 + γ ) ψ + det ¯ ψ (1 − γ ) ψ (cid:3) where ψ = ( u d s ) T and m = diag( m u , m d , m s ) representthe quark fields and their current mass respectively and G is the scalar coupling constant of the model from thefour-fermion interaction. The last term is the KMT termwhich represents the breaking of the flavor symmetry inthe chiral limit due to U A (1) anomaly. K is also knownas the U A (1) breaking strength.Next, in the mean field approximation we introducethe chiral condensates σ l = σ u/d = − G (cid:104) ¯ uu (cid:105) / (cid:104) ¯ dd (cid:105) , σ s = − G (cid:104) ¯ ss (cid:105) , (2.3)and the pseudoscalar pion condensate∆ = 2(2 G − Kσ s ) (cid:104) ¯ uiγ d (cid:105) , (2.4)where ∆ can be considered as real without loss of gener-ality as the related phase factor can be arbitrarily chosendue to the spontaneously broken U I (1) symmetry [29].In terms of these condensates, the thermodynamic po-tential for N f = 2 + 1 in the mean field approximation isgiven asΩ( σ l , σ s , ∆) = 2 σ l + σ s G + Kσ s σ l G + (cid:18) G + Kσ s G (cid:19) × ∆ (cid:0) G + Kσ s G (cid:1) − N c Λ (cid:90) k (cid:34) E + k + E − k + E sk + 2 T × ln (cid:110)(cid:16) e − βE − k (cid:17) (cid:16) e − βE + k (cid:17) (cid:16) e − βE sk (cid:17)(cid:111) (cid:35) (2.5)with E ± k = (cid:113)(cid:0) E lk ± µ I (cid:1) + ∆ with E lk = (cid:112) k + M l and E sk = (cid:112) k + M s and the symbol (cid:82) Λ k indicates three It is important to note here that in this work we are not consider-ing kaon condensation as we are working in the limit of vanishingbaryonic and strangeness chemical potentials, i.e. µ B = µ S = 0. momentum integrals that need to be regularized. Differ-ent effective masses M l and M s are defined as M l = m l + σ l + Kσ l σ s G ,M s = m s + σ s + Kσ l G + K (cid:0) G + Kσ s G (cid:1) , where m l = m u = m d and m s are the current quarkmasses.The ground state at finite temperature and isospinchemical potential is determined by minimizingΩ( σ l , σ s , ∆) with respect to σ l , σ s and ∆, i.e. bysolving ∂ Ω /∂σ l = ∂ Ω /∂σ s = ∂ Ω /∂ ∆ = 0.In the following subsections we discuss in more de-tails different ways of regularizing these integrals. Thethermodynamic quantities, i.e. the pressure, the isospindensity and the energy density of the system are thenrespectively given by P NJL = − Ω NJL ( σ l/s = σ l/s ; ∆ = ∆ ) , (2.6) (cid:104) n I (cid:105) NJL = ∂P NJL ∂µ I , (2.7) ε NJL = − P NJL + µ I (cid:104) n I (cid:105) NJL + T ∂P
NJL ∂T . (2.8)Finally, the interaction measure (or trace anomaly)within the NJL model is given by the relation between P NJL and ε NJL , i.e., I NJL = ε NJL − P NJL . (2.9) A. Regularization
Due to the nonrenormalizable nature of the NJLmodel, any physical quantity determined in its frame-work will be dependent on the scale of the model Λ.In the SU(2) version, the usual regularization schemesconsist in to determine Λ, the coupling constant G andcurrent quark mass m u = m d = m c that reproduce theempirical values of the pion mass m π , the pion decayconstant f π and the quark condensate (cid:10) ¯ ψψ (cid:11) . Since ouraim is to compare our results with lattice simulations wehave used two different sets of parameters, for T = 0 and T (cid:54) = 0, as can be seen in Tab. I.The SU(3) case is much more complicated; the pro-cedure is shown in details in appendix-A and the valuesobtained are shown in Tab. II. The two sets represent theparameters we use for two different cases, set-I for thecase of zero temperature and set-II for the case of finitetemperature, following the value of m π used by LatticeQCD respectively for both the cases.The fact that all physical quantities are dependent onΛ does not mean that we can just naively use this cutoffin all integrals, since it may lead to incorrect results. Inthis work we compare the results of two different schemes,namely the traditional regularization scheme (TRS) and TABLE I: Different parameter sets are listed, which have beenused in the present study for SU(2) case.Sets Input parameters Output parameters f π = 93 MeV Λ = 659 .
325 MeVI m π = 131 . G = 2 . / Λ (cid:104) ¯ ψψ (cid:105) / = 250 MeV m c = 4 .
757 MeV f π = 92 . .
325 MeVII m π = 135 . G = 2 . / Λ (cid:104) ¯ ψψ (cid:105) / = 250 MeV m c = 4 . f π = 93 MeV Λ = 574 .
68 MeV m π = 131 . G = 2 . / Λ I m K = 490 MeV K = 10 . / Λ m η = 950 MeV m s = 140 MeV m l = 5 . f π = 92 . .
431 MeV m π = 135 . G = 1 . / Λ II m K = 497 . K = 12 . / Λ m η = 957 . m s = 140 .
305 MeV m l = 5 . the medium separation scheme (MSS). The TRS is themost common in NJL studies, and consists only in toperform up to Λ the integrals that do not depend on thetemperature, e.g., the first three terms between brack-ets of the integral in Eq. (2.5) and its correspondents inthe gap equations and thermodynamic quantities, whilethermal integrals are performed up to infinity.By the other hand, MSS provides a tool to disentangleall the medium dependencies from divergent contribu-tions, so that only vacuum integrals remain to be regu-larized. This scheme has been applied to the NJL modeland successfully shows qualitative agreement with lat-tice simulations and more elaborated theories, as mightbe seen in Refs. [53, 67, 70, 71]. Let us start for examplefrom integral I ∆ of ∆ gap equation: I ∆ = (cid:88) j = ± (cid:90) Λ d k (2 π ) (cid:113) ( E lk + jµ ) + ∆ , (2.10)whose TRS version is obtained just by making the re-placement (cid:82) Λ d k (2 π ) → (cid:82) Λ0 dk k π . To use MSS we firstrewrite I ∆ = 1 π (cid:88) j = ± ∞ (cid:90) −∞ dx (cid:90) Λ d k (2 π ) x + ( E lk + jµ ) + ∆ , (2.11)where, to ease the notation, we made the replacement µ I / → µ . Using the identity1 x + ( E lk + jµ ) + ∆ = 1 x + k + M + M − ∆ − µ − M − jµE lk ( x + k + M ) (cid:2) x + ( E lk + jµ ) + ∆ (cid:3) . (2.12)Here M is the vacuum mass of light quarks, obtained inthe T = µ = ∆ = 0 limit. After two iterations of thisidentity we obtain (cid:88) j = ± x + ( E lk + jµ ) + ∆ = 2 x + k + M + 2 M ( x + k + M ) + 2 M + 8 µ ( E lk ) ( x + k + M ) + (cid:88) j = ± ( M − jµE lk ) ( x + k + M ) (cid:2) x + ( E lk + jµ ) + ∆ (cid:3) , (2.13)where we have defined M = M − ∆ − µ − M l . Aftersome manipulations and performing the integration in x indicated in (2.11) we obtain I MSS∆ = 2 I quad ( M ) − ( M l − M + ∆ − µ ) I log ( M )+ (cid:20) M + 4 µ M l )4 − µ M (cid:21) I + 2 I , (2.14)with the definitions I quad ( M ) = (cid:90) d k (2 π ) (cid:112) k + M , (2.15) I log ( M ) = (cid:90) d k (2 π ) k + M ) , (2.16) I = (cid:90) d k (2 π ) k + M ) , (2.17) I = 1532 (cid:88) j = ± (cid:90) d k (2 π ) (cid:90) dt (1 − t ) × ( M − jµE lk ) (cid:2) (2 jµE lk − M ) t + k + M (cid:3) . (2.18)A similar procedure can be used to obtain the integrals of other quantities: I σ l = (cid:88) j = ± (cid:90) Λ d k (2 π ) E lk E lk + jµ (cid:113) ( E lk + jµ ) + ∆ , (2.19) I n I = (cid:88) j = ± (cid:90) Λ d k (2 π ) j E lk + jµ (cid:113) ( E lk + jµ ) + ∆ , (2.20) I σ s = (cid:90) Λ d k (2 π ) E sk . (2.21)Since the steps to obtain these integrals for MSS are de-scribed in detail in previous references [53, 70, 71], herewe will just show the final results of each of these inte-grals: I MSS σ l = 2 I quad ( M ) − ( M l − M + ∆ ) I log ( M ) + I + 3 (cid:20) M µ ( M l − M − M ) (cid:21) I + 2 I , (2.22) I MSS n I = 2 µ ∆ I log ( M )3 µ (cid:20) M M ( M − M l ) + M l µ + 2 µ M (cid:21) I + 2 µI − µM l (cid:2) M + 4 µ M l (cid:3) I + 5 µ (cid:0) µ ( M − M l ) − M (cid:1) I + I , (2.23) I MSS σ s = I quad ( M s ) + M s − M s I log ( M s ) + I , (2.24)where M s is the vacuum strange quark mass, obtained inthe T = µ = ∆ = 0 limit, and the remaining definitions, I = 1516 (cid:88) j = ± (cid:90) d k (2 π ) ∞ (cid:90) t dt √ t × E lk jµ ( M − jµE lk ) (cid:2) ( k + M ) t + ( E lk + jµ ) + ∆ (cid:3) , (2.25) I = (cid:90) d k (2 π ) k + M ) , (2.26) I = (cid:90) d k (2 π ) k ( k + M ) , (2.27) I = 3532 (cid:88) j = ± (cid:90) d k (2 π ) ∞ (cid:90) t dt √ t × jE lk ( M − jµE lk ) (cid:2) ( k + M ) t + ( E lk + jµ ) + ∆ (cid:3) , (2.28) I = 34 (cid:90) d k (2 π ) ∞ (cid:90) tdt √ t × ( M s − M s ) [( k + M s ) t + k + M s ] . (2.29)Note that integrals I to I are all finite, and must beperformed up to infinity in k . This is the fundamentaldifference between TRS, where we cut the whole inte-gral in the cutoff Λ, and MSS, where all finite mediumcontributions are separated and performed for the wholemomentum range.Finally, the MSS expression for the normalized ther-modynamic potential readsΩ MSSNJL ( σ l , σ s , ∆) = 2 σ l + σ s G + Kσ s σ l G + (cid:18) G + Kσ s G (cid:19) ∆ (cid:0) G + Kσ s G (cid:1) − N c (cid:40) ˜ M I quad ( M ) + M s − M s I quad ( M s ) − (cid:16) ˜ M − µ ∆ (cid:17) I log ( M ) − M s − M s I log ( M s )+ (cid:90) d k (2 π ) (cid:34) ˜ M − µ ∆ E lk, ) − ˜ M E lk, − E lk, + E sk − E sk, − M s − M s E sk, + ( M s − M s ) E sk, ) + (cid:88) j = ± (cid:113) ( E lk + jµ ) + ∆ (cid:35)(cid:41) , (2.30)with the definitions ˜ M = ∆ + M l − M , E k, = (cid:112) k + M and E sk, = (cid:112) k + M s . III. RESULTS AND DISCUSSIONS
In this paper we have considered the SU(3) version ofthe NJL model at finite isospin imbalance incorporatingthe strange quark sector and the KMT determinant forboth the zero and finite temperature cases. Thus, wehave extended our previous study of the QCD equationof state at non-zero isospin density and zero tempera-ture within the SU(2) version of the Nambu-Jona-Lasiniomodel. Besides, in this work we have also considered forthe SU(2) model the effects of finite temperature in or-der to perform a complete comparison between the twoversions of the NJL model.The effects of the regularization scheme are discussedin details. We have used two alternative approaches forthe regularization of the nonrenormalizable NJL model,the first one, which we have named TRS (TraditionalRegularization Scheme), is the most frequently found inthe literature where the ultraviolet divergences are regu-larized through a sharp 3D cutoff, as shown in section-II.It is important to point out that the TRS approach doesnot disentangle finite medium contributions from the in-finity vacuum term and physically meaningful contribu-tions are usually discarded. The second scheme, whichwe named MSS (Medium Separation Scheme), is capableof disentangling exactly the vacuum divergent term fromthe finite medium ones and only the truly divergent vac-uum is regularized through a sharp 3D cutoff. It will be µ I / m π Σ ψψ & Σ π Σ ψψ , TRS Σ π , TRS Σ ψψ , MSS Σ π , MSS FIG. 1: (Color online) Chiral and pion condensates as func-tions of µ I at T = 0, evaluated according to Eqs. (3.1)and (3.2), for TRS and MSS. discussed in what follows how dependent are the observ-ables on the chosen regularization scheme and which oneis the more appropriated for each particular situation.The most important first principle approach to QCDin the non-perturbative regime is the lattice QCD simula-tion. Since for the finite isospin scenario no sign problemis found in the LQCD calculations, whenever possible,our results are compared with recent LQCD data. In or-der to make possible such comparisons, we have had to fitthe SU(2) and the SU(3) NJL model parameters accord-ing to the pion mass and pion decay constant adoptedin LQCD calculations. In tables-I and II our fitted pa-rameters are shown for the SU(2) and SU(3) NJL modelsrespectively. To compare our results with lattice simula-tions we used Set-I for zero temperature and Set-II forfinite temperature cases. The fitting procedure for theSU(3) version of the NJL model is more involved anddue to this fact we have included some details of thisparametrization procedure in the appendix-A.Next, we discuss our results for the SU(2) and SU(3)NJL models at finite isospin density and zero and fi-nite temperature using the TRS and MSS regularizationschemes. At this point, we would like to emphasize thefact that the results for finite isospin and zero tempera-ture within the SU(2) NJL model have been obtained inour previous paper[53]. However, they are shown here fora complete comparison between the NJL model versionsas we have extended the zero temperature results for theSU(3) case. The finite temperature extension, however,is a completely new addition in this present work, whichhas been done here for both SU(2) and SU(3) cases. A. Zero Temperature results
We start by showing our zero temperature SU(3) re-sults, Figure 1, for the chiral and pion condensates, which µ I / m π p N / m π TRS ( N f = 2)MSS ( N f = 2)TRS ( N f = 2+1)MSS ( N f = 2+1)LQCD FIG. 2: (Color online) Normalized pressure p N as a functionof µ I at T = 0, for SU(2) (solid and dot-dashed lines) andSU(3) (dashed and dotted lines) comparing TRS, MSS andlattice results from Ref. [54] (small dots). have been evaluated respectively according to the equa-tions, Σ ¯ ψψ = m l m π f π (cid:20) σ l − σ l G (cid:21) + 1 , (3.1)Σ π = m l m π f π ∆2 G + K σ s G , (3.2)as a function of the isospin chemical potential µ I forboth the TRS and MSS approaches. Here it is worth tomention that the chiral condensate in Eq. (3.1) does notinclude the contribution from the strange quarks. Wehave used the same definitions for these quantities asLQCD [49, 52], as we have compared our results againstthem for the case of finite temperature . Our resultsshow that the difference between both approaches in-creases at higher µ I , where the condensates calculated inthe MSS scheme are systematically lower than the corre-sponding ones calculated within the TRS scheme.The results for the normalized pressure are shown inFigure 2 as a function of the isospin chemical potential.The normalized pressure p N is defined as: p N = P NJL ( T, µ I ) − P NJL ( T, µ I = 0) , (3.3)where P NJL is defined in Eq. (2.6). It is apparent of thislatter figure that compared to the LQCD results [54],the SU(3)-NJL is better in the MSS scheme (dotted lines)than in the TRS scheme (dashed lines), at least, for µ I ≤ m π . It is also apparent from Figure 2 that the SU(2)-NJL results [53] agree slightly better with LQCD data Note that there is an apparent factor difference between ourdefinition of the condensates and the definition given in Refs [49,52], like it has been done in Ref. [20]. This has been done tocompensate for the same factor in our definitions of σ l and ∆. µ I / m π n I / m π TRS ( N f = 2)MSS ( N f = 2)TRS ( N f = 2+1)MSS ( N f = 2+1)LQCD FIG. 3: (Color online) Isospin density n I as a function of µ I at T = 0, for SU(2) (solid and dot-dashed lines) and SU(3)(dashed and dotted lines) comparing TRS, MSS and latticeresults from Ref. [54] (small dots). than the SU(3) ones and the differences between TRSand MSS are less important in this case.The results for the isospin density (see Eq. (2.7)) asa function of µ I at T = 0 are shown in Figure 3. Al-though the lattice data points are few, one can see thatfor small µ I the SU(3)-NJL model using the MSS schemeis in better agreement with LQCD results [54]. Never-theless, when µ I increases the SU(3)-NJL model in theTRS scheme seems to be closer to the LQCD results.As compared to the SU(2)-NJL model results, the MSSscheme is closer to lattice data for µ I /m π < .
6. Thus,for both SU(2) and SU(3) versions of NJL it seems thatfor larger isospin chemical potential the TRS scheme iscloser to lattice data. If one considers the overall trendof the lattice data, the SU(2)-NJL model seems to re-produce better the lattice data, as already noticed in ourdiscussion of the behavior of the pressure. We mentionhere that it would be highly desirable to get more latticedata in order to better distinguish the differences amongthe versions of NJL and regularization schemes.Next up in Figure 4, we show the scaled energy density(see Eq. (2.7)) as a function of the scaled µ I chemical po-tential for the SU(3)-NJL model and T = 0. The energydensity is scaled with its ideal or the Stefan-Boltzmannlimit for finite µ I and µ B = T = 0, given by ε SB ( µ I ) = N c N f π µ I . (3.4)A comparison with the perturbative QCD calculation [72]is performed for larger values of µ I . Some care is nec-essary when using NJL for larger µ I , since we have the We would like to mention that in this reference [72], µ I is definedas µ I = µ u − µ d , whereas in our case µ I = µ u − µ d . Hence, wehave rescaled our expression accordingly, by a factor 2. µ I / m π ε / ε S B TRS ( N f = 2+1)MSS ( N f = 2+1)pQCD FIG. 4: (Color online) Comparison of the energy densityscaled by the Stefan-Boltzmann limit for zero temperatureand finite isospin chemical potential. The pQCD results forhigher values of µ I /m π are obtained using the expressionsfrom Ref [72]. µ I / m π -0.10.00.10.20.30.4 ∆ I / m π TRS ( N f = 2)MSS ( N f = 2)TRS ( N f = 2+1)MSS ( N f = 2+1)LQCD T = 0 FIG. 5: (Color online) Normalized interaction measure (∆ I )as a function of µ I for SU(2) (solid and dot-dashed lines) andSU(3) (dashed and dotted lines) comparing TRS, MSS andlattice results from Ref. [54, 66] (small dots). Λ-cutoff as a natural momentum scale which makes themodel quantitatively trusty for µ I m π ≤ ∼
4, never-theless, only qualitative conclusions can be done extrap-olating such limit. However, it is clear from the latterfigure that the MSS scheme follows the trend of pQCDfor higher values of µ I . Moreover, one clearly sees in Fig-ure 4 that the Stefan-Boltzmann limit is expected to beachieved only in the MSS scheme.We finish our discussions on the zero temperature re-sults showing in Figure 5 the normalized interaction mea-sure ∆ I as a function of the scaled isospin chemical po-tential. The normalized interaction measure ∆ I is de-fined as: ∆ I = I NJL ( T, µ I ) − I NJL ( T, µ I = 0) , (3.5) T [GeV] Σ ψψ & Σ π Σ ψψ , TRS Σ π , TRS Σ ψψ , MSS Σ π , MSS µ I = 0.206 GeV FIG. 6: (Color online) SU(3) Chiral and pion condensates asfunctions of the temperature T with µ I = 0 .
206 GeV for TRSand MSS. where I NJL is defined in Eq. (2.9). As can be seenfrom Figure 5, the difference between the TRS (solid anddashed lines) and MSS (dot-dashed and dotted lines) re-sults for both SU(2) and SU(3) are small, but the MSSresults appear closer to the LQCD results [54, 66].At zero temperature, we observe the general trend thatthe agreement between lattice and NJL results is betterfor two-flavor at larger values of µ I and for three-flavor atlower values of µ I . Similar observations can also be foundin recent χ PT studies [19, 20]. In the next subsectionwe discuss the results at finite temperature and isospinchemical potential.
B. Finite temperature results
At finite temperature we begin with Figure 6, wherethe chiral and pion condensates as a function of the tem-perature calculated at fixed isospin chemical potential( µ I = 0 .
206 GeV) are shown. In the latter figure resultsfor the SU(3)-NJL model using both the MSS and TRSschemes are compared. One clearly sees that the meltingtemperature is larger in the MSS scheme compared tothe TRS one for both condensates. However, the pioncondensate melting is a second order phase transitionwith the critical temperature ∼
158 MeV for TRS and ∼
173 MeV for MSS. On the other hand, the correspond-ing chiral condensate behavior signals a crossover. Belowa certain temperature (around ∼
165 MeV for µ I = 206MeV) the TRS result for the chiral condensate supersedesthe MSS result, whereas after that temperature the TRSresult decreases more rapidly than the MSS result.Next we discuss about our findings for the thermody-namic quantities. In figure 7 the results are shown for thenormalized pressure, Eq.(3.3), as a function of the isospinchemical potential for three different temperatures, whichhave been chosen because they are available in LQCD µ I / m π p N / T TRS ( N f = 2)MSS ( N f = 2)TRS ( N f = 2+1)MSS ( N f = 2 + 1)LQCD T = 122 MeV µ I / m π p N / T TRS ( N f = 2)MSS ( N f = 2)TRS ( N f = 2+1)MSS ( N f = 2+1)LQCD T = 149 MeV µ I / m π p N / T TRS ( N f = 2)MSS ( N f = 2)TRS ( N f = 2+1)MSS ( N f = 2+1)LQCD T = 179 MeV FIG. 7: (Color online) Normalized pressure p N as a functionof µ I for different temperatures T , for TRS and MSS, com-paring SU(2) and SU(3) to lattice results from Refs. [49, 52,64, 65]. simulations [49, 52, 64, 65]. The results have been shownfor both SU(2) and SU(3)-NJL model within the TRSand MSS schemes. For larger values of µ I , the SU(3)-NJL model within the TRS scheme is closer to the LQCDresults, although the agreement is better for lower tem-peratures. For each of the plots we have also displayedan inset, where we show the results for smaller values of µ I / m π n I / T TRS ( N f = 2)MSS ( N f = 2)TRS ( N f = 2+1)MSS ( N f = 2+1)LQCD T = 122 MeV µ I / m π n I / T TRS ( N f = 2)MSS ( N f = 2)TRS ( N f = 2+1)MSS ( N f = 2+1)LQCD T = 149 MeV µ I / m π n I / T TRS ( N f = 2)MSS ( N f = 2)TRS ( N f = 2+1)MSS ( N f = 2+1)LQCD T = 179 MeV FIG. 8: (Color online) Isospin density n I as a function of µ I for different temperatures T , for TRS and MSS, comparingSU(2) and SU(3) to lattice results from Refs. [49, 52, 64, 65]. µ I (0 . m π < µ I < . m π ) and in this case we observethat the MSS and TRS schemes are closer to each other,and for the largest temperature considered in this workthe SU(3)-NJL using the MSS scheme is slightly better.Here, we notice a different behavior for finite tempera-ture when compared to the zero temperature case. For T = 0 the SU(2)-NJL results for the p N are in general inbetter agreement with LQCD results (for TRS and MSS) µ I / m π ∆ I / m π TRS ( N f = 2)MSS ( N f = 2)TRS ( N f = 2+1)MSS ( N f = 2+1) T = 122 MeVLQCD µ I / m π ∆ I / m π TRS ( N f = 2)MSS ( N f = 2)TRS ( N f = 2+1)MSS ( N f = 2+1) T = 149 MeVLQCD µ I / m π ∆ I / m π TRS ( N f = 2)MSS ( N f = 2)TRS ( N f = 2+1)MSS ( N f = 2+1) T = 168 MeVLQCD FIG. 9: (Color online) Normalized trace anomaly or interac-tion measure (∆ I ) as a function of µ I for different tempera-tures T , for TRS and MSS, comparing SU(2) and SU(3) tolattice results from Ref. [66]. than the SU(3) NJL model.In Figure 8, the isospin density as a function of isospinchemical potential is plotted for the same three temper-atures as discussed above. We can observe a change ofslope in the isospin density for T = 122 MeV and T = 149MeV which is related to the formation of pion conden-sate, when ∆ become nonzero. This change is not present in the curves correspondent to T = 179 MeV because thistemperature is high enough to prevent the pion conden-sate to be formed. In this case we have a similar be-havior as obtained for the normalized pressure, i. e., theSU(3)-NJL with the TRS scheme gives a better overallagreement with the LQCD results [49, 52, 64, 65]. Forthe lower temperature ( T = 122 MeV) considered here,the SU(2)-NJL in TRS scheme gives good results. Asbefore, the inset plots show that the MSS and TRS arevery similar for low isospin chemical potential and theSU(3)-NJL is the better model. At this point we onceagain stress the fact that the LQCD data for the isospindensity is not conclusive enough because of its scarcityand oscillating nature.Within the thermodynamic quantities we have alsoshown the variation of the normalized interaction mea-sure ∆ I (see Eq. (3.5)) as a function of isospin chemicalpotential for three different temperatures in Figure 9. Inthis case though, instead of T = 179 MeV, the highesttemperature we choose is T = 168 MeV, based on theavailability of LQCD results [66]. For lower temperature( T = 122 MeV) it is apparent from Figure 9 that bothTRS and MSS results for SU(2) and SU(3) fall withinthe domain prescribed by LQCD. At T = 149 MeV, forvery low µ I , all the NJL results fall within the LQCD do-main, however with increasing µ I , only SU(3) TRS resultappears closer to LQCD result. For the higher tempera-ture considered in Figure 9 ( T = 168 MeV), we can seethat the SU(2)-NJL results for TRS and MSS are morein agreement with the LQCD results. It is worth to men-tion here that for T = 168 MeV at higher values of µ I thepion condensate is zero in TRS, which is not the case ofMSS. The presence of a nonzero ∆ makes both pressureand energy density smaller, and the change of slope isrelated to the isospin density, as we discussed before (seeFigure 8).We conclude our discussion for finite temperaturewith the T − µ I phase diagrams shown in Figure 10for both SU(2) and SU(3). In each of the phase dia-grams we have considered TRS and MSS results withinNJL and compared them with the available LQCD re-sults [49, 52, 64, 65]. We notice that for both SU(2)and SU(3), TRS results appear closer to LQCD for pioncondensation and chiral crossover, compared to MSS. For µ I = 0, the critical temperature for the chiral phase tran-sition within NJL is higher than the same within LQCD,with maximum quantitative difference of ∼
16 MeV forSU(2) and ∼
22 MeV for SU(3). On the other hand, in-vestigating the pseudo triple point, beyond which boththe phase transitions coincide, we find that the pseudotriple points within MSS are closer to the LQCD resultscompared to TRS, specially for SU(3)-NJL.
IV. ACKNOWLEDGEMENTS
We thank Gergely Endrodi and Bastian B. Brandtfor useful discussions and also for providing the neces-0 µ I / m π [ G e V ] T χ TRS T χ MSS T χ LQCD T π TRS T π MSS T π LQCD
SU(2) µ I / m π [ G e V ] T χ TRS T χ MSS T χ LQCD T π TRS T π MSS T π LQCD
SU(3)
FIG. 10: (Color online) T − µ I phase diagram within NJL model, implementing both TRS and MSS and comparing SU(2) andSU(3) to lattice results from Refs. [49, 52, 64, 65]. sary lattice datasets, both at zero and finite tempera-tures. This work was partially supported by ConselhoNacional de Desenvolvimento Cient´ıfico e Tecnol´ogico(CNPq) under Grants No. 304758/2017-5 (R.L.S.F),No. 304518/2019-0 (S.S.A) and No. 136071/2018-0(B. S. L.); as a part of the project INCT-FNA (Insti-tuto Nacional de Ciˆencia e Tecnologia - F´ısica Nuclear eAplica¸c˜oes) No. 464898/2014-5 (S.S.A); U.S. DOE un-der Grant No. DE-FG02-00ER41132 and Funda¸c˜ao deAmparo `a Pesquisa do Estado de S˜ao Paulo (FAPESP)under Grant No. 2017/26111-4 (D.C.D); and Funda¸c˜aode Amparo `a Pesquisa do Estado do Rio Grande do Sul(FAPERGS), Grants No. 19/2551- 0000690-0 and No.19/2551-0001948-3 (R.L.S.F.). D.C.D. acknowledges thesupport of the Simons Foundation under the MultifariousMinds Program Grant No. 557037. A.B. acknowledgesthe support from Guangdong Major Project of Basic andApplied Basic Research No. 2020B0301030008 and Sci-ence and Technology Program of Guangzhou Project No.2019050001. Appendix A: NJL SU(3) Parameterization
In this appendix the procedure used for theparametrization of the NJL SU(3) model is discussed.Although we use standard techniques the complete ex-pressions are not usually given in a complete and friendlyway in the literature[73]. Moreover, having in mind toallow the reader to reproduce our calculations, the essen-tial expressions are given here. In order to restrict thenumber of free parameters, we set the u and d currentquark masses equal, i.e., m u = m d . Therefore, we ob-tain by solving the self-consistent gap equations that thecorresponding constituent u and d quark masses are alsoequal to each ohter: M u = M d . The free model param-eters are the coupling constants G and K , the current squark mass m s and the cutoff parameter Λ. The value of the m u = m d current quark mass is arbitrarily fixed. Theobservables that are used in the fitting procedure are thepion mass, the pion decay constant, the kaon mass andthe η (cid:48) meson mass.
1. Gap equations
The gap equations for the NJL SU(3) model are givenby M i = m i − Gσ i + 2 Kσ j σ k , (A1)with (ijk) being any cyclic permutation of (u,d,s), andthe condensate is defined as σ i = (cid:104) ¯ q i q i (cid:105) = − i Tr [ S i ( p )] = − i Tr 1 /p − M i . After the explicit calculation of the trace in Dirac andcolor spaces one obtains: σ i = (cid:104) ¯ q i q i (cid:105) = − M i I i , (A2)with I i = N c π (cid:90) p dpE i = N c π (cid:20) Λ (cid:15) Λ i − M i ln (cid:18) Λ + (cid:15) Λ i M i (cid:19)(cid:21) , (A3)where E i = (cid:112) p + M i and (cid:15) Λ i = (cid:112) Λ + M i , i=(u,d,s).
2. Pion and Kaon masses and decay constants
The dispersion relations for the pion and kaon massesare given by 1 − G π Π pπ ( Q ) (cid:12)(cid:12)(cid:12) Q = m π = 0 , (A4)1 − G K Π pK ( Q ) (cid:12)(cid:12)(cid:12) Q = m K = 0 , (A5)1where Π pπ = Π puu + Π pdd =2Π puu and Π pK = 2Π pus are thepseudo-scalar polarization loops for the pion and kaonmesons respectively, which can be evaluated from thegeneral expressionsΠ pij ( Q ) = 2 (cid:16) ( I i + I j ) − (cid:2) Q − ( M i − M j ) (cid:3) I ij (cid:17) , (A6)with I i already given in Eq. (A3) and I ij is given by I ij ( Q ) = N c π (cid:90) p dpE i E j E i + E j Q − ( E i + E j ) , (A7)whereas the modified couplings G π and G K for pions andkaons respectively are given by, G π = G − Kσ s , (A8) G K = G − Kσ u . (A9)For the pion decay constant we have f π = g π ¯ qq Q µ Q iN c (cid:90) d p Tr (cid:20) γ µ γ S u ( p + Q γ S u ( p + Q (cid:21) , after the explicit calculation of the trace in the last equa-tion, one obtains f π = − M u g π ¯ qq ( m π ) I uu ( m π ) (A10)where the coupling strength for the meson-quark-quarkinteraction is given by g − π ¯ qq ( m π ) = ∂ Π pπ ∂Q (cid:12)(cid:12)(cid:12) Q = m π = 2 ∂ Π puu ∂Q (cid:12)(cid:12)(cid:12) Q = m π . (A11)The last expression follows from the derivative ofEq. (A6): ∂ Π puu ∂Q = − I uu − Q ∂∂Q I uu . η and η (cid:48) mesons For the case of η and η (cid:48) meson, the inverse mesonicpropagator assumes a matrix form[73] D − = 12 K − − Π , (A12)where the effective coupling matrix K and the mesonicself energy Π for the η − η (cid:48) system are given by the ma-trices: Π = (cid:32) Π Π Π Π (cid:33) , K = (cid:32) K K K K , (cid:33) ,K − = 1det K (cid:32) K − K − K K (cid:33) (A13) where the determinant of K is given by det K = K K − K and the specific components are given by K = G + 13 K (2 σ u + σ s ) , (A14) K = G − K (4 σ u − σ s ) , (A15) K = K = − √ K ( σ u − σ s ) , (A16)and Π = 23 [2Π puu ( Q ) + Π pss ( Q )] , (A17)Π = 23 [Π puu ( Q ) + 2Π pss ( Q )] , (A18)Π = Π = 2 √
23 [Π puu ( Q ) − Π pss ( Q )] . (A19)So, finally the inverse propagator is given by D − = 12 det K (cid:32) A BB C (cid:33) , (A20)with A = K − K Π , (A21) B = − ( K + 2 det K Π ) , (A22) C = K − K Π , (A23)Diagonalizing D − we get D − = 12 det K O − (cid:32) D − η D − η (cid:48) (cid:33) O (A24)where the diagonalizing orthogonal matrix O is given by O = (cid:32) cos θ p sin θ p − sin θ p cos θ p (cid:33) , tan(2 θ p ) = 2 BA − C and the diagonal elements now represent the dispersionrelations for η and η (cid:48) mesons, i.e. D − η = ( A + C ) − (cid:112) ( C − A ) + 4 B , (A25) D − η (cid:48) = ( A + C ) + (cid:112) ( C − A ) + 4 B . (A26)The masses of the η and η (cid:48) meson can now be determinedfrom the equations D − η ( Q = m η ) = 0 , (A27) D − η (cid:48) ( Q = m η (cid:48) ) = 0 . (A28)At this point we want to note that m η can be evalu-ated from Eq. (A27) by using the integrals from Eqs. (A3)and (A7). But as η (cid:48) , in general, exists above the ¯ qq con-tinuum, Eq. (A28) has complex poles, which we can as-sume to be of the form, Q = Q = m η (cid:48) − i Γ, with Γbeing the width of the η (cid:48) -resonance. In the latter case2the calculation of A, B and C in Eq. (A26) can be readilydone making in Eq. (A6) the replacement: Q I ii → (cid:2) m η (cid:48) Re I ii + m η (cid:48) ΓIm I ii (cid:3) (A29)+ i (cid:2) m η (cid:48) Im I ii − m η (cid:48) ΓRe I ii (cid:3) , where terms of order Γ have been neglected. Then, aftersubstituting the latter expression in Eq. (A6) one obtainsΠ pii ( Q ) = Re Π pii ( Q ) + iIm Π pii ( Q ) , where Re Π pii ( Q ) = 4 I i − (cid:0) m η (cid:48) Re I ii + m η (cid:48) ΓIm I ii (cid:1) ,Im Π pii ( Q ) = 2 (cid:0) m η (cid:48) ΓRe I ii − m η (cid:48) Im I ii (cid:1) . (A30)The real and imaginary part of the integral I ii can beobtained from Eq. 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