The symmetry energy γ parameter of relativistic mean-field models
Mariana Dutra, Odilon Lourenço, Or Hen, Eliezer Piasetzky, Débora P. Menezes
aa r X i v : . [ nu c l - t h ] A p r The symmetry energy γ parameter of relativistic mean-field models Mariana Dutra
Departamento de Ciˆencias da Natureza, Universidade Federal Fluminense, 28895-532, Rio das Ostras, RJ, Brazil
Odilon Louren¸co
Universidade Federal do Rio de Janeiro, 27930-560, Maca´e, RJ, Brazil
Or Hen
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
Eliezer Piasetzky
School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel
D´ebora P. Menezes
Depto de F´ısica - CFM - Universidade Federal de Santa Catarina,Florian´opolis - SC - CP. 476 - CEP 88.040-900, Brazil (Dated: October 7, 2018)The relativistic mean-field models tested in previous works against nuclear matter experimentalvalues, critical parameters and macroscopic stellar properties are revisited and used in the evalua-tion of the symmetry energy γ parameter obtained in three different ways. We have checked thatindependent of the choice made to calculate the γ values, a trend of linear correlation is observedbetween γ and the symmetry energy ( S ) and a more clear linear relationship is established between γ and the slope of the symmetry energy ( L ). These results directly contribute to the arising of otherlinear correlations between γ and the neutron star radii of R . and R . , in agreement with recentfindings. Finally, we have found that short-range correlations induce two specific parametrizations,namely, IU-FSU and DD-ME δ , simultaneously compatible with the neutron star mass constraint of1 . M max /M ⊙ .
05 and with the overlap band for the L × S region, to present γ in the rangeof γ = 0 . ± . I. INTRODUCTION
Since the introduction of the first models in nuclearphysics, the main idea was to describe experimental data.Not all nuclear models can be applied to the descriptionof nuclear matter but they are relevant nevertheless. Rel-ativistic mean field (RMF) models were developed to de-scribe observables of nuclei, from which nuclear matterparameters can be extracted.A detailed analysis of 263 RMF models based on pureneutron and symmetric nuclear matter properties wasdone in Ref. [1] and only 35 of them were shown tosatisfy important nuclear constraints. In a subsequentwork, these models were used to analyse stellar proper-ties related to largely studied astrophysical quantities,namely, neutron star masses and radii of the canonicalneutron stars (obtained from observational data), thepossible onset of the Direct Urca process and sound veloc-ity constraints [2]. As a result, only 13 out of them pro-duced neutron stars with maximum mass in the range of1 . M max /M ⊙ .
05 [3, 4] as far as no hyperons wereconsidered, namely, one with density dependent cou-plings (DD-F) and one also incorporating scalar-isovector δ mesons (DD-ME δ ). The remaining parametrizations(BKA20, BKA22, BKA24, BSR8, BSR9, BSR10, BSR11,BSR12, FSUGZ03, IU-FSU, G2*) present constant cou-plings, nonlinear σ and ω terms, and cross terms involv-ing these fields. None of them could reproduce pulsars with 2 M ⊙ if hyperons were included. More recently, thesame models were revisited and their critical parameterswere obtained [5]. These critical parameters are the crit-ical temperature, critical pressure and critical density,at which nuclear matter is no longer unstable and theliquid-gas phase transition ceases to exist [6–12]. In thisinvestigation, the models were divided into 6 categories(BKA, BSR, FSU, G2*, Z271 and DD). More experimen-tal data is necessary, but so far, only two of them (Z271and DD) provided critical temperatures close to existingones. A clear correlation between the critical tempera-ture and the compressibility was obtained. More detailson these models are given along the paper.In the same context, the symmetry energy [13] andits slope are very important quantities and in the lastfifteen years, they were shown to be correlated with a se-ries of physical properties, which we comment next. Thesymmetry energy is related to the nuclei neutron skinthickness, which in turn is related to neutron star ra-dius: models that yield smaller neutron skin thickness inheavy nuclei, give rise to smaller neutron star radii [14].On the other hand, neutron skins are larger for modelswith higher slope [15]. Also, a strong correlation wasobserved between the neutron star radius and the vari-ation of the slope at sub-threshold densities [16]. Thesymmetry energy and the slope, however, can be easilycontrolled by the inclusion of a ω − ρ [15, 17–19] or a σ − ρ interaction [20] in non-linear models. The largerthe value of the ω − ρ interaction, for instance, the lowerthe values of the symmetry energy and its slope. On theother hand, for density dependent models a change in thedensity dependence of the ρ -meson coupling can modifythe symmetry energy and its slope.Besides the neutron skin thickness, the neutron starcrust-core properties are also correlated with the slopeof the symmetry energy, a fact that had already beenobserved in studies involving liquid-gas phase transitions,whose transition densities are approximately the same asthe ones obtained as the separating densities from thepasta phase to homogeneous matter [21].We next reanalyze the parametrizations studied inRef. [1], that we call consistent relativistic meanfield (CRMF) models from now on. In this context, theword “consistent” refers to those parametrizations thatwere shown to satisfy the nuclear matter constraints inRef. [1]. For these CRMF parametrizations, we evalu-ate the symmetry energy coefficient γ in three differentcases. Such a quantity is defined from S = S kin + S pot ,with S pot ( ρ ) ∝ ( ρ/ρ ) γ and was first analyzed in Refs.[22, 23]. Our aim is to look for possible correlationsbetween the γ parameter and some important nuclearmatter and neutron star properties, namely the symme-try energy, its slope, and the radii of 1.0 and 1.4 solarmass neutron stars. Within the assumptions made in thepresent work, the γ parameter fully defines the potentialpart of the symmetry energy and its density dependence.Theoretically, γ is sensitive to the nucleon-nucleon in-teraction at very short distances and can be extractedfrom the existing calculations. We also investigate whichparametrizations satisfy the ranges of γ recently obtainedin Refs. [24, 25]. It is worth pointing out that from the ex-perimental side, the γ parameter is not directly measuredbut, as most of the bulk nuclear matter properties, it canbe inferred from experiments. In order to be consistentwith previous studies, the ω − ρ and σ − ρ interactionsthat can be included in most models to cure their sym-metry energy and slope values are left aside. They arejust considered in the models that introduced them whenthey were proposed.This paper is organized as follows: the introduction isexhibited in Sec. 1. In Sec. 2, the formalism is introducedand three different forms of separate kinetic and potentialparts of the symmetry energy are presented. In Sec. 3 theresults are discussed. The summary is shown in Sec. 4. II. FORMALISM
The analysis performed in Ref. [1] pointed out to only35 parametrizations, out of 263 investigated, simulta-neously approved in seven distinct nuclear matter con-straints. These CRMF parametrizations had their bulkand thermodynamical quantities compared to respectivetheoretical/experimental data from symmetric nuclearmatter (SNM), pure neutron matter (PNM), and a mix-ture of both, namely, symmetry energy and its slope eval- uated at the saturation density, and the ratio of the sym-metry energy at ρ / ρ (MIX). Thesedetailed constraints are specified in Table I. TABLE I. Set of updated constraints (SET2a) used inRef. [1]. See that reference for more details.
Constraint Quantity Density Region RangeSM1 K at ρ −
270 MeVSM3a P ( ρ ) 2 < ρρ < P ( ρ ) 1 . < ρρ < . E PNM /ρ . < ρρ o < .
108 Band RegionMIX1a J at ρ −
35 MeVMIX2a L at ρ −
115 MeVMIX4 S ( ρ / J at ρ and ρ / − In Table II we present a brief compilation of the struc-ture and methods used in fitting the finite range RMFinteractions in accordance with the macroscopic con-straints, and the data used for the fittings. For full expla-nation and details, we address the readers to the originalpapers and for a complete description of the relativisticmean-field theory, to Ref. [26].Thirty out of the 35 parametrizations approved inmatching the constraints analyzed in Ref. [1] are oftype 4, i.e., the Lagrangian density includes nonlin-ear σ and ω terms and cross terms involving thesefields. They are: BKA20, BKA22, BKA24, BSR8, BSR9,BSR10, BSR11, BSR12, BSR15, BSR16, BSR17, BSR18,BSR19, BSR20, FSU-III, FSU-IV, FSUGold, FSUGold4,FSUGZ03, FSUGZ06, G2*, IU-FSU, Z271s2, Z271s3,Z271s4, Z271s5, Z271s6, Z271v4, Z271v5, and Z271v6.They are described by the following Lagrangian density, L NL = ψ ( iγ µ ∂ µ − M ) ψ + g σ σψψ − g ω ψγ µ ω µ ψ − g ρ ψγ µ ~ρ µ ~τ ψ + 12 ( ∂ µ σ∂ µ σ − m σ σ ) − A σ − B σ − F µν F µν + 12 m ω ω µ ω µ + C g ω ω µ ω µ ) − ~B µν ~B µν + 12 m ρ ~ρ µ ~ρ µ + 12 α ′ g ω g ρ ω µ ω µ ~ρ µ ~ρ µ + g σ g ω σω µ ω µ (cid:18) α + 12 α ′ g σ σ (cid:19) + g σ g ρ σ~ρ µ ~ρ µ (cid:18) α + 12 α ′ g σ σ (cid:19) , (1)with F µν = ∂ ν ω µ − ∂ µ ω ν and ~B µν = ∂ ν ~ρ µ − ∂ µ ~ρ ν . Thenucleon rest mass is M and the meson masses are m j ,for j = σ, ω, and ρ .Other four CRMF approved parametrizations aredensity dependent (DD): DD-F, TW99, DDH δ andDD-ME δ . Their Lagrangian density reads: L DD = ψ ( iγ µ ∂ µ − M ) ψ + Γ σ ( ρ ) σψψ − Γ ω ( ρ ) ψγ µ ω µ ψ − Γ ρ ( ρ )2 ψγ µ ~ρ µ ~τ ψ + Γ δ ( ρ ) ψ~δ~τ ψ − F µν F µν + 12 ( ∂ µ σ∂ µ σ − m σ σ ) + 12 m ω ω µ ω µ − ~B µν ~B µν + 12 m ρ ~ρ µ ~ρ µ + 12 ( ∂ µ ~δ∂ µ ~δ − m δ ~δ ) , (2)asdfas TABLE II. Structure of the RMF models and data used for fitting the finite range parametrizations considered in the presentwork. NL: nonlinear model. DD: density dependent model. NAP: number of adjusted parameters. AT: additional terms incomparison with the standard nonlinear σ − σ model with meson ρ included. Parametrization Type of model, n ◦ of parameters, NAP, AT Data used for fitting purposesBKA20, NL, 12, 10 Constraint properties of asymmetric nuclear matter for 26 differentBKA22, AT: ω , σ − ω , σ − ω , σ − ρ parametrizations:BKA24 [27] binding energies, charge radii for closed shell nuclei,neutron-skin thickness in the Pb nucleus: 0 .
20, 0 .
22, and 0 .
24 fm.BSR8 to NL, 14, 11 Binding energies: , O, , Ca, , Ni, Sr, Zr, , , Sn,BSR12 [28] AT: ω , σ − ω , σ − ω , σ − ρ , and Pb nuclei, σ − ρ , ω − ρ charge radii: O, , Ca, Ni, Sr, Zr,
Sn, and
Pb nucleineutron skin thickness:
Pbfree parameters: neutron-skin thickness ∆ R = 0 . , . , · · · , .
28 fmand the ω -meson self-coupling strength ξ = 0 . ξ = 0 . ω , σ − ω , σ − ω , σ − ρ , σ − ρ , ω − ρ FSU-III, NL, 10,7 Properties of asymmetric nuclear matter; the proton fraction inFSU-IV[29] AT: ω , ω − ρ β -stable npeµ matter; the core-crust transition density and pressurein neutron stars as predicted by FSUGold and IU-FSU;free parameters: the coupling constants between the isovector ρ meson (Λ v ) and the isoscalar σ and ω mesons (Λ s ):FSU-III: Λ v = 0 .
00 and Λ S = 0 . v = 0 .
00 and Λ s = 0 . Ca, Zr,AT: ω , ω − ρ , Sn,
PbFSUGold4 [31] NL, 10, 8, AT: ω , ω − ρ Adjusting the isovector parameters of the model g ρ and Λ v FSUGZ03, NL, 14, 12 Binding energies: , O, , Ca, , Ni, Sr, Zr,FSUGZ06 [32] AT: ω , σ − ω , σ − ω , σ − ρ , , , Sn, Pb σ − ρ , ω − ρ charge rms radii: O, , Ca, Ni, Sr, Zr,
Sn,
Pbneutron-skin thickness for
Pb nucleus: 0 . ± .
01 fmfree parameters: ζ and ξ corresponding to self-couplings for ω and ρ mesons: ζ = 0 . , .
06 and ξ = 0G2* [33] NL, 12, 10 Adjust the isovector-vector channel of the G2 parameter set.AT: ω , σ − ω , σ − ω , σ − ρ IU-FSU [34] NL, 10, 10 Change the isoscalar parameter to ξ = 0 . ω , ω − ρ refitting of the isoscalar parameters to maintain the saturationproperties of SNM of FSU;increase the isoscalar-isovector coupling constant to Λ = 0 . ω , σ − ρ free parameters: λ v = 0 and λ s = 0 . , . , . , . ω , ω − ρ free parameters are: λ s = 0 and λ v = 0 . , . , . Pb, spin-orbit splittingsTW99 [37] DD, 15, 12 Fix the density dependence of the couplings fromDirac-Brueckner calculations of nuclear matterbinding energies of symmetric nuclei ( O, Ca, Ni)and neutron-rich nuclei ( O, Ca, Zr,
Pb).DDH δ [38] DD, 20, 16 Reproduce bulk asymmetry parameter a = 33 . δ [39] DD, 24, 14 Finite nuclei and adjustment to ab initio calculations in infinitenuclear matter whereΓ i ( ρ ) = Γ i ( ρ ) f i ( x ); f i ( x ) = a i b i ( x + d i ) c i ( x + e i ) , (3)for i = σ, ω , and x = ρ/ρ . For the ρ coupling one hasΓ ρ ( ρ ) = Γ ρ ( ρ ) e − a ρ ( x − . (4)The Lagrangian density describing the DD-F and TW99parametrizations is the same as the one in Eq. (2) when the meson δ is not taken into account. For the DD-ME δ parametrization, the couplings in Eq. (3) are valid for i = σ, ω, ρ , and δ . Finally, the DDH δ model has thesame coupling parameters as in Eq. (3) for the mesons σ and ω , but functions f i ( x ) given by f i ( x ) = a i e − b i ( x − − c i ( x − d i ) , (5)for i = ρ, δ .Only one parametrization belongs to the nonlinearpoint coupling category, namely, the FA3 [40]. In thiskind of model, nucleons interact with each other with-out explicitly including mesons [41–44]. Here, we do notinvestigate such model since in Ref [2] we have shownit is not capable of generating a mass radius curve forneutron stars, due to a very particular behavior in thehigh-density regime, namely, a fall in the pressure versusenergy density ( E ) curve near E = 809 MeV/fm . Forthat reason, we have decided to discard this particularparametrization.All the details about the RMF approximation and re-lated equations of state (EoS) are given in Ref. [1] andwill not be repeated here. Only the formulae necessaryfor the understanding of the present analysis are definednext.The general definition of the symmetry energy readsas follows, S ( ρ ) = 18 ∂ ( E /ρ ) ∂y (cid:12)(cid:12)(cid:12)(cid:12) ρ,y =1 / = S kin ( ρ ) + S pot ( ρ ) , (6)where y = ρ p /ρ is the proton fraction of the system with ρ p being the proton density. By using such an expression,we compute the kinetic and potential contributions of thesymmetry energy slope L ( ρ ) = 3 ρ ∂ S ∂ρ = 3 ρ ∂ S kin ∂ρ + 3 ρ ∂ S pot ∂ρ = L kin ( ρ ) + L pot ( ρ ) . (7)If we consider the potential part of the symmetry en-ergy written as a power-law in density according to S pot ( ρ ) = S pot ( ρ/ρ ) γ ≡ S pot approx. ( ρ ) , (8)it is possible to express L as L = 3 ρ "(cid:18) ∂ S kin ∂ρ (cid:19) ρ = ρ + γρ S pot . (9)By using Eq. (7) at ρ = ρ and comparing it to Eq. (9),one can find γ as in Ref. [24], namely, γ = L − L kin S pot = L pot S pot , (10)where F kin,pot = F kin,pot ( ρ ), for F = S , L . In thatreference [24], the authors also introduced effects fromshort-range correlations (SRC) between proton-neutronpairs [48–50] in symmetric nuclear matter in order toprovide an analytical expression for the kinetic part of thesymmetry energy. From this expression, that we will alsouse in Sec. 2.3, they found the range of − ± . S kin , based on data from free proton-to-neutronratios measured in intermediate energy nucleus-nucleuscollisions. Such a range allowed the authors to predictthe values of γ = 0 . ± . γ valueis given in Ref. [25], where no short-range correlations in the kinetic part of the symmetry energy is taken intoaccount. In that case, the density dependence of thesymmetry energy was given by S ( ρ ) = S kin ( ρ ) + S pot ( ρ ) = a ( ρ/ρ ) / + b ( ρ/ρ ) γ , (11)with a = 12 MeV, b = 22 MeV, and γ possibly rangingfrom 0 . . γ = 0 . ± . A. Complete kinetic term (case 1)
Here we considered the complete kinetic term for thedifferent models. Within this assumption, the first termof the symmetry energy is the kinetic part and the re-maining is treated as the potential part. For the ki-netic part, the corresponding expressions for nonlinearand density dependent (with δ meson) RMF models are, S kini ( ρ ) = k F E ∗ F i (12)where i = NL, DD, with E ∗ F i = ( k F + M ∗ i ) / and M ∗ NL = M − g σ σ, M ∗ DD = M − Γ σ ( ρ ) σ, (13)for symmetric matter ( y = 1 / k F = (3 π ρ/ / .The potential part of the symmetry energy is writtenas S pot NL ( ρ ) = g ρ m ∗ ρ ρ, (14) S pot DD ( ρ ) = Γ ρ ρ m ρ − (Γ δ /m δ ) ( M ∗ DD ) ρ E ∗ F DD (cid:20) (cid:16) Γ δ m δ (cid:17) A DD (cid:21) , (15)where m ∗ ρ = m ρ + g σ g ρ σ (2 α + α ′ g σ σ ) + α ′ g ω g ρ ω , and (16) A DD = 2 π Z k F k dk [ k + ( M ∗ DD ) ] / = 3 (cid:18) ρ s M ∗ DD − ρE ∗ F DD (cid:19) . (17)The mean-field value of the vector field ω µ is ω , and ρ s is the scalar density.The respective expressions for the different contribu-tions of the symmetry energy slope, namely, L kin ( ρ ) and L pot ( ρ ), are obtained as indicated in Eq. (7), for this caseand the next ones. B. “Free” kinetic term (case 2)
In this case we have separated the really kinetic term ,the one without any dependence of the interaction withthe mesons, from the rest of the symmetry energy. Theexpressions in this case read S kin NL ( ρ ) = S kin DD ( ρ ) = k F E F , (18)with E F = ( k F + M ) / , for the kinetic part, and S pot NL ( ρ ) = k F E ∗ F NL − k F E F + g ρ m ∗ ρ ρ, (19) S pot DD ( ρ ) = k F E ∗ F DD − k F E F + Γ ρ ρ m ρ − (Γ δ /m δ ) ( M ∗ DD ) ρ E ∗ F DD (cid:20) (cid:16) Γ δ m δ (cid:17) A DD (cid:21) , (20)for the potential one. C. Short range correlations (case 3)
The idea here is to replace the kinetic part of the sym-metry energy by that one proposed in Ref. [24], where theauthors have considered S kin as composed by a free gasmodel term added to a correction term ∆ S kin that takesinto account short-range correlations between proton-neutron pairs in symmetric nuclear matter. Based on thisprocedure, we calculate the potential part of the symme-try energy as follows, S poti ( ρ ) = S i ( ρ ) − S kinSRC ( ρ ) , (21)where i = NL, DD. The expressions for the total sym-metry energy S i ( ρ ) are given by the sum of Eqs. (12)and (14) for the NL model, or Eqs. (12) and (15) for theDD one, by using the formulae of case 1. Exactly thesame expressions are found if the case 2 is taken into ac-count, i. e., if the sum of Eqs. (18) and (19) is performedfor the NL model, or the sum of Eqs. (18) and (20) is con-sidered for the DD model. Finally, the kinetic part of thesymmetry energy for the present case analysis, S kinSRC ( ρ ),is taken from Ref. [24] as S kinSRC ( ρ ) = (cid:16) / − (cid:17) k F M − ∆ S kin ( ρ ) , (22)with∆ S kin ( ρ ) = c k F M π " λ (cid:18) ρρ (cid:19) / − (cid:18) ρρ (cid:19) / + 3 ρ λρ , (23)where the parameters c = 4 .
48 and λ = 2 .
75 are alsotaken from Ref. [24].
III. RESULTS
Let’s start by revisiting the analysis of the correla-tion between the symmetry energy S = S ( ρ ) and itsslope L = L ( ρ ), both at saturation density, whose dataare shown respectively in columns 1 and 4 in Tables III,IV and V (that will be detailed latter) and are plottedin Fig. 1, where squares refer to those parametrizationswhich also satisfy the macroscopic stellar properties of1 . M max /M ⊙ .
05 from Refs. [3, 4]. This corre-lation has already been extensively investigated, for in-stance, in Refs. [19, 45–47] and only some of the pointsin Fig. 1 coincide with the overlap region of figure 2 inRef. [45] (gray band of our Fig. 1). This means that theaccepted range of values in Ref. [2] is broader than theoverlap of conditions shown in Ref. [45], namely, the over-lap among constraints from nuclear masses, neutron skinthickness of Sn isotopes, dipole polarizability of
Pb,giant dipole resonances, isotope diffusion in heavy ioncollisions, astrophysical observations, and neutron mat-ter constraints.
26 28 30 32 34 S (MeV) L ( M e V ) IU-FSUZ271s6 DD-ME δ DD-FZ271s5FSUGold4FSU-IV
FIG. 1. Slope as a function of the symmetry energy for theCRMF models (all points). The gray band was extracted fromRef. [45]. The squares represent parametrizations satisfyingalso the neutron star mass constraint of Refs. [3, 4].
We next obtain the γ values by using Eq. (10) for someCRMF parametrizations and then also compare our val-ues with the ranges proposed in Refs. [24, 25]. In ouranalysis, we assume that the potential part of the sym-metry energy can be written as in Eq. (8). Here, notall CRMF parametrizations are analyzed, but instead,only those in which the deviation defined by ∆( ρ ) = | S pot model ( ρ ) − S pot approx. ( ρ ) | /S pot model ( ρ ) is less than a certainvalue, with S pot approx. ( ρ ) = S pot ( ρ/ρ ) γ , see Eq. (8). Asfor each case one has different values for γ , the function S pot approx. ( ρ ) exhibits different density dependences for thesame parametrization. Therefore, each studied case pro-duces different values of ∆( ρ ) for the same parametriza-tion. Since experimental values of γ were extracted atsuprasaturation density regime, we decided to investigatethe values of ∆( ρ ) at a range of 1 ρ/ρ
4, and definethat only CRMF parametrizations, at this specific den-sity range, presenting ∆
15% are taken into account inour study. By considering this analysis, we ensure a goodagreement between the exact potential part of the sym-metry energy, S pot exact ( ρ ), of the CRMF parametrizationsand the approximated form given in Eq. (8).Our calculations were divided in the three differentcases presented in Sec. 2. For each one, we construct aspecific table where we use the symbol X to mark thosemodels in which the constraint 1 . M max /M ⊙ . to mark those parametrizations present-ing γ parameter in the range of γ = 0 . ± .
19, andthe symbol ⊠ to identify parametrizations for which γ = 0 . ± . γ parameter in cases 1 and 2 only with theexperimental range of γ = 0 . ± .
19, since such rangewas obtained without SRC effects, according to Ref. [25].Furthermore, we name the parameters calculated fromcases 1 and 2 respectively, as γ and γ .For case 3, the γ parameter obtained with SRC effectsis named γ . It is compared only with the experimentalrange of γ = 0 . ± .
05 because this range was proposedin Ref. [24] with SRC included in the analysis.
A. Case 1
The γ values calculated from case 1 ( γ ) are pre-sented in Table III. From this table we notice only 7parametrizations with γ in the range of γ = 0 . ± . γ and theisovector quantities at the saturation density. The resultsare depicted in Fig. 2.From Fig. 2(a), we observe a trend of linear correla-tion between γ and S . A quantitative measurement ofsuch finding can be given by calculation of the Pearson’scorrelation coefficient, defined as in Ref. [51]. Two dif-ferent quantities A and B are strongly correlated withina linear relationship the more the coefficient correlation C ( A, B ) is near to 1, or − S dependence of γ ,we found C ( S , γ ) = 0 . γ × L data,we noticed a better linear correlation than in the pre-vious case, since the correlation coefficient resulted in C ( L , γ ) = 0 . TABLE III. Symmetry energy and its slope, with the re-spective kinetic and potential parts, all of them at ρ = ρ ,obtained from the case 1 analysis for the CRMF parametriza-tions presenting ∆( ρ )
15% at a density range of 1 ρ/ρ X and is alsodefined in the text. Models S S kin S pot L L kin L pot γ (MeV) (MeV) (MeV) (MeV) (MeV) (MeV)BKA20 X .
24 16 .
58 15 .
66 75 .
38 48 .
47 26 .
91 0 . X .
17 17 .
44 15 .
73 78 .
79 52 .
12 26 .
67 0 . X .
19 17 .
54 16 .
65 84 .
80 52 .
09 32 .
70 0 . X .
08 17 .
47 13 .
61 60 .
25 52 .
78 7 .
47 0 . X .
61 17 .
57 14 .
05 63 .
89 52 .
41 11 .
49 0 . X .
72 17 .
63 15 .
09 70 .
83 53 .
09 17 .
74 0 . X .
69 17 .
47 16 .
22 78 .
78 51 .
89 26 .
89 0 . X .
00 17 .
47 16 .
53 77 .
90 52 .
30 25 .
60 0 . .
97 17 .
33 13 .
65 61 .
79 49 .
34 12 .
45 0 . .
24 17 .
35 13 .
90 62 .
33 49 .
41 12 .
92 0 . .
98 17 .
38 14 .
60 67 .
44 49 .
50 17 .
93 0 . .
74 17 .
39 15 .
35 72 .
65 49 .
48 23 .
17 0 . .
78 17 .
40 16 .
38 79 .
47 49 .
52 29 .
96 0 . .
54 17 .
40 17 .
14 88 .
03 49 .
10 38 .
93 0 . .
43 17 .
45 13 .
98 52 .
16 49 .
72 2 .
44 0 . .
56 17 .
45 15 .
11 60 .
44 49 .
72 10 .
72 0 . .
40 17 .
37 14 .
03 51 .
74 49 .
43 2 .
31 0 . X .
54 17 .
57 13 .
98 63 .
98 52 .
40 11 .
58 0 . .
18 17 .
35 13 .
83 62 .
42 49 .
42 13 .
00 0 . X .
30 17 .
94 13 .
36 47 .
21 54 . − . − . X .
39 16 .
61 13 .
77 69 .
68 46 .
31 23 .
37 0 . .
84 13 .
82 18 .
02 53 .
57 32 .
55 21 .
02 0 . .
20 13 .
82 17 .
38 47 .
81 32 .
55 15 .
25 0 .
40 50 60 70 80 90 L (MeV) -0.200.20.40.60.8 γ C(L , γ ) = 0.892C(L , γ ) = 0.95630 31 32 33 34 35 S (MeV) -0.200.20.40.60.8 γ C(S , γ )=0.667 (b)(a) FIG. 2. γ as a function of (a) symmetry energy, and(b) its slope, both at ρ = ρ , for the models displayedin Table III (all points). Squares represent parametriza-tions also satisfying the neutron star mass constraint of1 . M max /M ⊙ .
05 [3, 4]. The solid and dashed linesare fitting curves. still stronger in comparison to the one exhibited with allpoints. The correlation coefficient for the square pointsin Fig. 2(b) is C ( L , γ ) = 0 . B. Case 2
The γ values calculated here are presented in Ta-ble IV. Since in this case we have prevented the kineticpart of the symmetry energy from any influence of theeffective mass, and as a consequence of the scalar me-son effects, one can see here that S kin = k F / (6 E F ) dif-fers from each parametrization only due to the Fermienergy E F = ( k F + M ) / at the saturation point. As k F = (3 π ρ / / , and as for nuclear mean-field modelsthe saturation density is well established closely aroundthe value of ρ = 0 .
15 fm − , it becomes clear that theparametrizations analyzed according to Eqs. (18)-(20)present values of S kin in a very narrow band as onecan see from Table IV. For the same reason, the kineticpart of the symmetry energy slope, L kin , is also con-strained to a small range. Also from Table IV, we see thatfor the case 2 analysis, a large number of parametriza-tions, namely, 20 of them, have γ in the range of γ = 0 . ± . TABLE IV. Symmetry energy and its slope, with the re-spective kinetic and potential parts, all of them at ρ = ρ ,obtained from the case 2 analysis for the CRMF parametriza-tions presenting ∆( ρ )
15% at a density range of 1 ρ/ρ X and is alsodefined in the text. Models S S kin S pot L L kin L pot γ (MeV) (MeV) (MeV) (MeV) (MeV) (MeV)BKA20 X .
24 11 .
15 21 .
09 75 .
38 21 .
53 53 .
85 0 . X .
17 11 .
21 21 .
96 78 .
79 21 .
64 57 .
15 0 . X .
19 11 .
20 22 .
99 84 .
80 21 .
63 63 .
17 0 . X .
08 11 .
19 19 .
88 60 .
25 21 .
62 38 .
64 0 . X .
61 11 .
21 20 .
40 63 .
89 21 .
65 42 .
24 0 . X .
72 11 .
22 21 .
50 70 .
83 21 .
66 49 .
17 0 . X .
69 11 .
19 22 .
50 78 .
78 21 .
60 57 .
18 0 . X .
00 11 .
22 22 .
78 77 .
90 21 .
66 56 .
24 0 . .
97 11 .
13 19 .
85 61 .
79 21 .
49 40 .
30 0 . .
24 11 .
13 20 .
11 62 .
33 21 .
50 40 .
83 0 . .
98 11 .
17 20 .
81 67 .
44 21 .
57 45 .
87 0 . .
74 11 .
14 21 .
59 72 .
65 21 .
52 51 .
13 0 . .
78 11 .
19 22 .
60 79 .
47 21 .
60 57 .
87 0 . .
54 11 .
15 23 .
38 88 .
03 21 .
54 66 .
48 0 . .
89 11 .
26 22 .
64 71 .
72 21 .
73 49 .
99 0 . .
43 11 .
26 20 .
17 52 .
16 21 .
73 30 .
43 0 . .
40 11 .
22 20 .
18 51 .
74 21 .
66 30 .
07 0 . X .
54 11 .
21 20 .
33 63 .
98 21 .
65 42 .
33 0 . .
18 11 .
14 20 .
04 62 .
42 21 .
51 40 .
92 0 . X .
39 11 .
52 18 .
87 69 .
68 22 .
22 47 .
46 0 . .
08 11 .
27 22 .
81 76 .
62 21 .
75 54 .
87 0 . .
27 11 .
27 22 .
00 67 .
81 21 .
75 46 .
05 0 . .
53 11 .
27 21 .
26 60 .
18 21 .
75 38 .
43 0 . .
84 11 .
27 20 .
57 53 .
57 21 .
75 31 .
82 0 . .
20 11 .
27 19 .
93 47 .
81 21 .
75 26 .
05 0 . δ X .
18 11 .
44 20 .
74 51 .
43 22 .
08 29 .
35 0 . As a further study, we analyse here the effect of theabsence of the scalar interaction in the kinetic parts of thesymmetry energy and its slope in the possible correlationsof γ with S and L . The results are shown in Fig. 3.From this figure one can see that the trend of linearcorrelation between γ and S is worse when comparedwith case 1, since in case 2 one has C ( S , γ ) = 0 . γ × L is fa-vored when the kinetic parts of the symmetry energy andits slope are free from the scalar interaction effects. Thecorrelation coefficient in this case is C ( L , γ ) = 0 . C ( L , γ ) = 0 .
40 50 60 70 80 90 L (MeV) γ C(L , γ )=0.97730 31 32 33 34 35 S (MeV) γ C(S , γ )=0.639 (b)(a) FIG. 3. γ as a function of (a) symmetry energy, and (b) itsslope, both at ρ = ρ , for the models of Table IV (all points).Squares represent parametrizations also satisfying the neu-tron star mass constraint of 1 . M max /M ⊙ .
05 [3, 4].Solid lines: fitting curves.
C. Case 3
The use of Eqs. (21)-(23) along with Eq. (7), all ofthem evaluated at ρ = ρ , allows the calculation of γ from the definition given in Eq. (10). The results arepresented in Table V. In our procedure, only the sumof the kinetic and potential parts of the symmetry en-ergy matters. This sum does not change, and we extractthe potential part by subtracting from the total (exact)value, the kinetic part with SRC included, as indicatedin Eq. (21).
40 50 60 70 80 90 L (MeV) γ C(L , γ )=0.99430 31 32 33 34 35 S (MeV) γ C(S , γ )=0.689 (b)(a) FIG. 4. γ as a function of (a) symmetry energy, and (b) itsslope, both at ρ = ρ , for the models of Table V (all points).Squares represent parametrizations also satisfying the neu-tron star mass constraint of 1 . M max /M ⊙ .
05 [3, 4].Solid lines: fitting curves.
From Table V we see that S kinSRC , has a negative valuearound − . TABLE V. Symmetry energy and its slope, with the respec-tive kinetic and potential parts, all of them at ρ = ρ , ob-tained from the case 3 analysis for the CRMF parametriza-tions presenting ∆( ρ )
15% at a density range of 1 ρ/ρ X and ⊠ is alsodefined in the text. Models S S kinSRC , S pot L L kin L pot γ (MeV) (MeV) (MeV) (MeV) (MeV) (MeV)BKA20 X . − .
31 41 .
55 75 .
38 21 .
21 54 .
16 0 . X . − .
36 42 .
53 78 .
79 21 .
33 57 .
46 0 . X . − .
35 43 .
54 84 .
80 21 .
31 63 .
48 0 . X . − .
35 40 .
43 60 .
25 21 .
30 38 .
95 0 . X . − .
37 40 .
98 63 .
89 21 .
34 42 .
55 0 . X . − .
37 42 .
09 70 .
83 21 .
35 49 .
48 0 . X . − .
34 43 .
03 78 .
78 21 .
29 57 .
49 0 . X . − .
37 43 .
37 77 .
90 21 .
35 56 .
55 0 . . − .
29 40 .
26 61 .
79 21 .
17 40 .
62 0 . . − .
30 40 .
54 62 .
33 21 .
18 41 .
15 0 . . − .
33 41 .
31 67 .
44 21 .
25 46 .
18 0 . . − .
31 42 .
04 72 .
65 21 .
20 51 .
45 0 . . − .
34 43 .
13 79 .
47 21 .
29 58 .
19 0 . . − .
31 43 .
85 88 .
03 21 .
22 66 .
80 0 . . − .
40 43 .
30 71 .
72 21 .
43 50 .
30 0 . ⊠ . − .
40 40 .
83 52 .
16 21 .
43 30 .
73 0 . . − .
40 41 .
96 60 .
44 21 .
43 39 .
01 0 . ⊠ . − .
37 40 .
77 51 .
74 21 .
35 30 .
38 0 . X . − .
37 40 .
91 63 .
98 21 .
34 42 .
64 0 . . − .
30 40 .
48 62 .
42 21 .
19 41 .
24 0 . X ⊠ . − .
67 40 .
97 47 .
21 22 .
04 25 .
17 0 . X . − .
63 40 .
02 69 .
68 21 .
94 47 .
74 0 . . − .
41 43 .
49 76 .
62 21 .
45 55 .
18 0 . . − .
41 42 .
68 67 .
81 21 .
45 46 .
36 0 . . − .
41 41 .
94 60 .
18 21 .
45 38 .
74 0 . ⊠ . − .
41 41 .
25 53 .
57 21 .
45 32 .
12 0 . ⊠ . − .
41 40 .
61 47 .
81 21 .
45 26 .
36 0 . δ X ⊠ . − .
56 41 .
75 51 .
43 21 .
79 29 .
64 0 . tions between proton-neutron pairs in symmetric nuclearmatter introduced in Ref. [24], that produced the ex-pressions presented in Eqs. (22)-(23). Such a negativevalue for S kin of the CRMF parametrizations is indeedconsistent with the range of − ± . γ in the range of γ = 0 . ± . γ correlations. Itis clear from this figure that the linear dependence be-tween γ and the isovector bulk parameters is still morefavored when the short-range correlations are included inthe CRMF parametrizations. The correlation coefficientsobtained in this case are the closest to the unity, namely, C ( S , γ ) = 0 .
689 and C ( L , γ ) = 0 .
994 in comparisonwith the respective quantities regarding the cases 1 and 2.
D. Comments about the results
Regarding the correlations found mainly between γ and L , we remark that such result is not trivial, andthe reason can be given from an analysis of Eq. (10).From such equation we can write: γ = αL + β, (24) with α = 1 / (3 S pot ) and β = − L kin / (3 S pot ), i. e., alinear correlation between γ and L is obtained if α and β are (ideally) constant numbers. In our study,we investigate whether L kin and S pot are close enoughto constants for the sets of parametrization studied ineach case. If this is the case, variations ∆ α and ∆ β are close to zero. Since for each case studied we havethe highest and lowest values of S pot and L kin , it ispossible to calculate ∆ α = S pot ) high − S pot ) low , and∆ β = − ( L kin ) high S pot ) high + ( L kin ) low S pot ) low . The absolute values ofsuch calculations are shown in Table VI, and as one cansee, ∆ α and ∆ β are decreasing quantities if we analysesuch numbers from case 1 to 3. These behaviors explainthe increasing coefficient correlation shown in Figs. 2 to 4. TABLE VI. Absolute values of ∆ α and ∆ β . Calculations forthe three different cases analyzed. case | ∆ α | (MeV − ) | ∆ β | . . . . . . This result is a consequence of the data previously pre-sented in Tables III, IV and V. From these tables, onecan see that the values of L kin and S pot are closer to aconstant value in case 3 than in case 2. Also, these valuesare closer to a certain constant value in case 2 than incase 1.One can see from such a table that the closer resultsare obtained for the cases in which the scalar attractiveinteraction is not taken into account in the kinetic partof the symmetry energy, i. e., cases 2 and 3, being thelast one the case in which ∆ α and ∆ β are closer to zero.As a direct application of this specific correlation, wealso investigate whether γ , γ , and γ obtained from theCRMF parametrizations in the three different cases stud-ied and always calculated for symmetric nuclear matter,also correlates with neutron star radii. The motivation ofsuch study comes from the results presented in Ref. [52],in which authors found that for a class of 42 relativis-tic and Skyrme parametrizations, L linearly depends on R . and R . , namely, the radii of neutron stars present-ing M star = M ⊙ and 1 . M ⊙ , respectively. The R . and R . dependence of γ , γ , γ related to those CRMFparametrizations analyzed here is shown in Fig. 5. Inorder to obtain stellar macroscopic properties, the sameCRMF parametrizations are used, but now the modelsare subject to matter neutrality and β -equilibrium.In order to generate the neutron star radii, wehave joined the hadronic matter EoS from the CRMFparametrizations with those for electrons and muons. Af-ter that, the conditions of charge neutrality and chemi-cal equilibrium were taken into account and the Baym-Pethick-Sutherland (BPS) equation of state [53] for lowdensities was added to the EoS for hadrons and leptons.The resulting EoS was used as input to the Tolman-Oppenheimer-Volkoff equations [54]. We address the
12 13 14 R (km) C(R , γ ) = 0.750
12 13 14 R (km) -0.200.20.40.60.81 γ C(R , γ ) = 0.513 (b) CASE 1(a) CASE 1
12 13 14 R (km) C(R , γ ) = 0.914
12 13 14 R (km) -0.200.20.40.60.81 γ C(R , γ ) = 0.811 (d) CASE 2(c) CASE 2
12 13 14 R (km) C(R , γ ) = 0.922
12 13 14 R (km) -0.200.20.40.60.81 γ C(R , γ ) = 0.751 (f) CASE 3(e) CASE 3 FIG. 5. γ , γ , and γ as a function of the R . and R . neu-tron star radii for the CRMF parametrizations (all points).Squares represent parametrizations also satisfying the con-straint of 1 . M max /M ⊙ .
05 [3, 4]. reader to details regarding such calculations to Ref. [55],for instance.From Fig. 5, we can conclude that the CRMFparametrizations also present a linear behavior concern-ing γ , γ , γ and the neutron star radii. This resultis entirely compatible with the findings of Ref. [52]. Inthat paper, a linear correlation between L and the radiiwas found, and since in our study we have found a lineardependence for γ , γ , γ and L , according to Eq. (24),a direct consequence is the linear behavior described inFig. 5. Also as in Ref. [52], the correlations are strongerfor the R . neutron star radius as the correlation coef-ficients C ( R . , γ ) point out. Finally, as observed alongall investigations, the linear dependence is intensified incases 2 and 3 in which the effects of the scalar interactionare absent from the kinetic part of the symmetry energy.Another point we remark here concerns the gray bandof Fig. 1 for the CRMF models in the different casesstudied. In order to do that, we start by redrawingsuch a figure for the different models that reproduce γ = 0 . ± .
05, panel (a), and γ = 0 . ± .
19, panel (b)in Fig. 6.
29 30 31 32 33 34 35 S (MeV) L ( M e V ) Case 1Case 229 30 31 32 33 34 35 S (MeV) L ( M e V ) Case 1Case 3 (b)(a) γ = 0.25 ± 0.05 γ = 0.72 ± 0.19FSU-IVFSUGold4Z271s6IU-FSUZ271s5DD-ME δ FIG. 6. Slope as a function of the symmetry energy forthe models that produce the (a) lower and (b) higher γ ranges discussed in the last section. The gray band wasextracted from Ref. [45]. Orange points represent thoseparametrizations in which the neutron star mass constraintof 1 . M max /M ⊙ .
05 [3, 4] is verified.
It is worth noting that the CRMF parametrizations forwhich we have obtained the γ parameters from case 3are more compatible with the gray band proposed inRef. [45], i. e., the short-range correlations effects in-duce the CRMF parametrizations to present the γ pa-rameter inside the range of γ = 0 . ± .
05, simultane-ously being consistent with the overlap conditions ofRef. [45] obtained from many experimental and observa-tional data. For such a case, 6 parametrizations are insidethe overlap band, namely, IU-FSU, FSU-IV, FSUGold4,Z271s5, Z271s6, and DD-ME δ , with 2 of them, IU-FSUand DD-ME δ , also satisfying the neutron star mass con-straint of 1 . M max /M ⊙ .
05 [3, 4] and two of them,Z271s5 and Z271s6 yielding critical parameters close tothe existing proposition of experimental values, accord-ing to the findings of Ref. [5].
IV. SUMMARY
In summary, our calculations have shown that, in-dependently of the choice made to obtain the γ values(case 1, 2 or 3) for the CRMF models, a trend of linearcorrelation is observed between γ , γ , γ and S , anda more clear linear relationship is established regarding γ , γ , γ and the slope of the symmetry energy at thesaturation density, L . In cases 2 and 3, the last cor-relation is still more pronounced. Such effect arises dueto the absence of the attractive interaction in the kineticpart of the symmetry energy. Furthermore, the short-range correlations introduced in the case 3 analysis inten-sify the linear L dependence of γ as seen in Fig. 4(b).These results can be used to determine other linear cor-0relations of γ , γ , γ and the neutron star radii of R . and R . , as displayed in Fig. 5. Finally, specifically forcase 3, two specific parametrizations, namely, IU-FSUand DD-ME δ are shown to be compatible with the rangeof γ = 0 . ± .
05 [24], and simultaneously consistentwith the neutron star mass constraint of Refs. [3, 4], andother two, Z271s5 and Z271s6, simultaneously compati-ble with the range of γ = 0 . ± .
05 [24] and with prob-able critical parameters experimental values [5]. The fourparametrizations are consistent with the overlap band forthe L × S region described in Ref. [45], see Fig. 6.As a final remark, we remind the reader that we haveonly analyzed symmetric matter in this study for the cal-culation of the γ values, but the potential difference forneutrons and protons in neutron-rich matter and theirdensity dependence, for instance, can also be calculated.However, the results could not be compared with theexisting γ values. Furthermore, if we also want to inves-tigate the momentum dependence in asymmetric matter,single particle potentials, which are different for neu-trons and protons, have to be taken into account, see Ref. [56, 57]. To obtain this kind of dependence, onewould need either a theory that uses non-local interac-tions or a Thomas-Fermi calculation and both approachesare out of the scope of the present work. ACKNOWLEDGMENTS
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