Baryonic matter and the medium modification of the baryon masses
Nam-Yong Ghim, Ghil-Seok Yang, Hyun-Chul Kim, Ulugbek Yakhshiev
IINHA-NTG-02/2021
Baryonic matter and the medium modification of the baryon masses
Nam-Yong Ghim, ∗ Ghil-Seok Yang, † Hyun-Chul Kim,
1, 3, ‡ and Ulugbek Yakhshiev
1, 4, § Department of Physics, Inha University, Incheon 22212, Korea Department of Physics, Soongsil University, Seoul 06978, Korea School of Physics, Korea Institute for Advanced Study (KIAS), Seoul 02455, Korea Theoretical Physics Department, National University of Uzbekistan, Tashkent 700174, Uzbekistan
We investigate the properties of baryonic matter within the framework of the in-medium modifiedchiral soliton model by taking into account the effects of surrounding baryonic environment on theproperties of in-medium baryons. The internal parameters of the model are determined based onnuclear phenomenology at nonstrange sector and fitted by reproducing nuclear matter propertiesnear the saturation point. We discuss the equations of state in different nuclear environments such assymmetric nuclear matter, neutron and strange matters. We show that the results for the equationsof state are in good agreement with the phenomenology of nuclear matter. We also discuss how theSU(3) baryons masses undergo changes in these various types of nuclear matter.
Keywords: chiral soliton model, nuclear matter, neutron matter, strange matter, medium modification ofthe SU(3) baryons in different nuclear media.
I. INTRODUCTION
It is of paramount importance to understand how themasses of hadrons undergo changes in nuclear medium,since it is deeply rooted in the restoration of chiral sym-metry and even the quark confinement in quantum chro-modynamics (QCD) [1–4]. As discussed in Ref. [1], thechiral condensate is known to be modified in nuclear mat-ter, which reveals the mechanism as to how the sponta-neous broken chiral symmetry is restored as the nucleardensity increases. This also implies the changes of hadronmasses in it, since the dynamical quark mass arises asa consequence of the spontaneous breakdown of chiralsymmetry. Thus, understanding the medium modifica-tion of the nucleon mass has been one of the most sig-nificant issues well over decades [5]. Experimental dataalso indicate that the nucleon is modified in nuclei [6–11].This means that other baryons may also undergo changesin nuclear medium [12–18]. When one considers themedium modification of baryons, one should keep in mindthat nuclear matter itself is also affected self-consistentlyby the changes of baryons. However, it is very difficultto relate the medium modification of baryons to nuclearmedium consistently, even in the isolated case of normalnuclear matter.In the present work, we investigate the medium mod-ification of the low-lying SU(3) baryons in symmetricmatter, asymmetric matter, neutron matter, and strangebaryonic matter consistently, based on a pion mean-fieldapproach [19]. The general idea is based on the seminalpaper by E. Witten [20, 21]. In the large N c (the numberof colors) limit, the nucleon can be viewed as a state of N c valence quarks bound by the meson mean fields that ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] is produced self-consistently by the presence of the N c valence quarks, since the mesonic quantum fluctuationsare suppressed by the /N c factor. This approach hasbeen successfully applied for describing the various prop-erties of both light and singly heavy baryons in a unifyingmanner [22–31]. The main idea of the pion mean-field ap-proach is not to compute dynamical parameters withinthe chiral quark-soliton model [19, 32], which realizes thepion mean-field approach explicitly, but to fix all relevantdynamical parameters by using the experimental data.For example, the masses of the baryon decuplet can bepredicted by using the experimental data on those of thebaryon octet and the mass of the Ω baryon [23]. Actu-ally, this method was already used in the Skyrme modellong time ago [33].The pion mean-field approach can be also extended tothe description of light and singly heavy baryons in nu-clear medium. However, since the model is based on thequark degrees of freedom, one should consider the quarkchemical potential [34], which means that it is rather dif-ficult to connect the results from this approach directlyto the properties of the baryons in nuclear matter. Thus,we will follow a variational approach that was adopted inthe medium modified Skyrme models [35, 36]. In thesemodified Skyrme models various properties of the nucleonand ∆ isobar have been described in nuclear matter [36–39], and in finite nuclei [40–42]. The model enables onealso to investigate nuclear matter properties [43–45].Thus, we will show in this work how the pion mean-field approach can be extended to the investigation ofthe SU(3) baryon properties in both nuclear and strangebaryonic environments. This can be achieved by intro-ducing the density-dependent functionals as variationalparameters. The density functionals will be parametrizedand fitted completely in the SU(2) sector by taking intoaccount available experimental and empirical data, thelinear-response approximation being emphasized. Thisenables us to describe the strange baryonic matter andproperties of baryons in different media (isospin symmet- a r X i v : . [ nu c l - t h ] F e b ric, asymmetric and strange baryonic matter).The present paper is organized as follows. In Sec-tion II, we briefly review the pion mean-field approach,discussing the collective Hamiltonian and SU(3) baryonstates in free space. Then we will proceed to consider apossible modification of the model in order to take intoaccount the influence of the surrounding baryon environ-ment on the properties of a single baryon in medium. InSection III, we discuss the results for the binding energyin symmetric matter and determine the variational pa-rameters. The discussion of the properties of baryons innuclear and strange nuclear matter will be followed inSection IV. We will also show how to fit the remainingpart of the parameters. Then, we are able to discussthe properties of an arbitrary baryonic matter and thenumerical results for the medium modifications of SU(3)baryons. The final Section V is devoted to the summaryand conclusion of the present work and will give an out-look for future investigations. Some details of the modelare compiled in Appendix A. II. GENERAL FORMALISM
In the pion mean-field approach, the dynamics of thevalence and sea quarks generates the chiral-quark solitonwith hedgehog symmetry [21, 46–48]. Hedgehog sym-metry can be regarded as the minimal generalization ofspherical symmetry, which can keep the pion mean fieldseffectively [48]. We are able to derive the effective col-lective Hamiltonian by considering the zero-mode quan-tization with hedgehog symmetry, taking into accountthe rotational /N c corrections and the strange current-quark corrections from the explicit breaking of flavorSU(3) symmetry. Note that in the present approachthe presence of N c valence quarks constrains the righthypercharge Y (cid:48) = N c / , which picks up safely the low-est allowed representations such as the baryon octet ( )and decuplet ( ). On the other hand, Y (cid:48) is constrainedby the Wess-Zumino-Witten term in the SU(3) Skyrmemodel [49–51]. In this section, we will directly start fromthe collective Hamiltonian. For a detailed derivation, werefer to Ref. [52] (see also a review [32]). A. Collective Hamiltonian and SU(3) baryon state
If we consider both the explicit breakdowns of flavorSU(3) symmetry and isospin symmetry, we have four dif-ferent contributions to the collective Hamiltonian, givenas follows: H = M cl + H rot + H sb + H em , (1)where M cl , H rot , and H sb denote respectively the clas-sical soliton mass, the /N c rotational and symmetry-breaking corrections including the effects of isospin andflavor SU(3) f symmetry breakings [23, 53]. The last term H em stands for the term arising from the isospinsymmetry breaking caused by the electromagnetic self-energies [22]. We can neglect the modification of theelectromagnetic self-energies in nuclear matter [42]. Theclassical energy arises from the N c valence quarks in thepion mean fields and the sea quarks coming from thevacuum polarization in the presence of the N c valencequarks: E cl = N c E val + E sea . By minimizing E cl withrespect to the pion fields, we get the pion mean-field so-lution self-consistently, which yields the classical solitonmass M cl .The rotational /N c corrections, i.e., H rot , can be de-rived by the zero-mode collective quantization, since thezero modes are not at all small, one should take into ac-count them completely. Regarding the angular velocitiesof the chiral soliton as small parameters, we can expandthe quark propagator perturbatively in terms of the an-gular velocities, we find the rotational /N c term H rot as H rot = 12 I (cid:88) i =1 ˆ J i + 12 I (cid:88) p =4 ˆ J p . (2)This Hamiltonian depends on two moments of inertia I , and expressed in terms of the operators ˆ J i correspondingto the generators of the SU(3) group. I and I give thesplitting between different representations of the SU(3)group. The symmetry breaking part of the Hamiltonianhas the following form H sb = ( m d − m u ) (cid:32) √ α D (8)38 ( A )+ β ˆ T + 12 γ (cid:88) i =1 D (8)3 i ( A ) ˆ J i (cid:33) + ( m s − ¯ m ) (cid:16) α D (8)88 ( A ) + β ˆ Y + 1 √ γ (cid:88) i =1 D (8)8 i ( A ) ˆ J i (cid:33) , (3)where α , β , and γ depend on the moments of inertia thatare expressed as α = − (cid:18)
23 Σ πN m u + m d − K I (cid:19) ,β = − K I , γ = 2 (cid:18) K I − K I (cid:19) . (4)Here K , represent the anomalous moments of inertia ofthe soliton. m u , m d , and m s denote the current-quarkmasses of the up, down, and strange quarks, respectively.The ¯ m designates the average current-quark mass of theup and down quarks. The D ( R ) ab ( A ) indicate the SU(3)Wigner D functions in the representation R . The ˆ Y and ˆ T are the operators of the hypercharge and the thirdcomponent of the isospin, respectively.In the representation ( p, q ) of the SU(3) group, thesum of the generators can be expressed in terms of p and q (cid:88) i =1 J i = 13 (cid:2) p + q + p q + 3( p + q ) (cid:3) , (5)which yields the eigenvalues of the rotational collectiveHamiltonian H rot in Eq. (2) as follows: E ( p, q ) , J = 12 (cid:18) I − I (cid:19) J ( J + 1) − I + 16 I (cid:2) p + q + 3( p + q ) + p q (cid:3) (6)A corresponding eigenfunction is called the collectivewave function for a SU(3) baryon with the quantum num-bers of flavor F = ( Y, T, T ) and spin S = ( Y (cid:48) , J, J ) ψ ( R ) B ( A ) = (cid:112) dim ( R ) ( − J + Y (cid:48) / D ( R ) ∗FS ( A ) , (7)where D ( R ) ∗FS are again the Wigner D functions in a rep-resentation R and dim ( R ) designates the correspondingdimension of the representation R .Knowing the eigenvalues and eigenfunctions of theSU(3) baryon states, we can get their masses of whichthe explicit forms are presented in Appendix A. For adetailed formalism relating to the collective Hamiltonianand baryon states, we refer the reader to Ref. [23], whereall dynamical parameters such as I , I , K , K , α , β ,and γ are determined by using the experimental data in a“ model-independent way ”, so that we can avoid a specificdynamics of the chiral soliton models. We now turn tohow the model can be extended to nuclear medium. B. Solitons in nuclear matter
Since we have determined all the dynamical parametersby incorporating experimental information, we will followthe same strategy also in nuclear matter. We will fixthe density-dependent variational parameters by usingthe experimental and empirical data on the properties ofnuclear matter. So, we start from the average energy E ∗ per baryon in a baryonic system E ∗ A = ZM ∗ p + N M ∗ n + (cid:80) s =1 N s M ∗ s A , (8)where Z and N are the numbers of protons and neutrons,respectively, and N s is the corresponding number of thestrange baryons with the corresponding strangeness S , s = | S | . A stands for the total number of the baryons Asterisks “ ∗ ” in the superscripts denote in-medium modifiedquantities. A = Z + N + N + N + N . Having carried out a simplemanipulation, we can rewrite E ∗ /A as E ∗ A = M ∗ N (cid:32) − (cid:88) s =1 δ s (cid:33) + 12 δM ∗ np + (cid:88) s =1 δ s M ∗ s , (9)where M ∗ N = ( M ∗ p + M ∗ n ) / denotes the average mass ofnucleons, M ∗ np = M ∗ n − M ∗ p designates the mass differeneof the neutron and the proton in medium. In addition,we introduce the parameter for isospin asymmetry δ =( N − Z ) /A . δ s = N s /A represents the parameter forthe strangeness fraction with the corresponding value ofsubscript s . We can take s = 1 , or , depending on thehyperons with strangeness S we put.The binding energy per baryon in a baryonic mattercan be defined as the difference of the medium averageenergy per baryon E ∗ /A and the energy per baryon E/A for the noninteracting baryonic system. If one takes thenumber of the baryons to be infinity, which we can call itthe infinite baryon-matter approximation, we express thebinding energy per baryon in terms of the following exter-nal parameters: a normalized baryonic density λ = ρ/ρ ,the isospin asymmetry parameter δ , and the strangenessfraction parameter δ s with given s . Consequently, thebinding energy is then written as ε ( λ, δ, δ , δ , δ ) = E ∗ ( λ, δ, δ , δ , δ ) − EA = ∆ M N ( λ, δ, δ , δ , δ ) (cid:32) − (cid:88) s =1 δ s (cid:33) + 12 δ ∆ M np ( λ, δ, δ , δ , δ )+ (cid:88) s =1 δ s ∆ M s ( λ, δ, δ , δ , δ ) , (10)where ∆ M N = M ∗ N − M N denotes the isoscalar masschange whereas ∆ M np = M ∗ np − M np stands for theneutron-proton mass change in nuclear medium. Theyare explicitly expressed in terms of the in-medium mod-ified functionals of the chiral soliton ∆ M N = M ∗ cl − M cl + E ∗ (1 , / − E (1 , / − D ∗ − D ∗ + D + D , (11) ∆ M np = d ∗ − d ∗ − d + d , (12)where the explicit expressions for D , and d , in freespace are given in Appendix A (see Eqs. (A9)-(A12)). D , represent the linear m s corrections of flavor SU(3)symmetry breaking whereas d , denote the effects ofisospin symmetry breaking. They are related to themodel functionals to be discussed below through α , β and γ defined in Eq. (4). Note that for the mass dif-ferences of the hyperons ∆ M s = M ∗ s − M s ( s = 1 , , )we have the different expressions for the baryon octetand decuplet. For the moment, let us concentrate on thestrange baryonic medium made of the hyperons in thebaryon octet as in the case of the nonstrange baryons.Thus, we adopt the following expressions for ∆ M s ∆ M = M ∗ Λ + M ∗ Σ − M Λ + M Σ M ∗ cl − M cl + E ∗ (1 , / − E (1 , / , (13) ∆ M = M ∗ Ξ − M Ξ = M ∗ cl − M cl + E ∗ (1 , / − E (1 , / + D ∗ − D , (14) ∆ M = 0 . (15)We now discuss how we can modify the dynami-cal parameters of the pion mean-field approach in nu-clear medium. We follow the strategy presented inRefs. [44, 45] and assume that the dynamical parametersdiscussed in Subsection II A, i.e. M cl , I , and K , /I , ,will be modified as follows: M cl → M ∗ cl = M cl f cl ( λ, δ, δ , δ , δ ) , (16) I → I ∗ = I f ( λ, δ, δ , δ , δ ) , (17) I → I ∗ = I f ( λ, δ, δ , δ , δ ) , (18) ( m d − m u ) K , I , → E ∗ iso = ( m d − m u ) × K , I , f ( λ, δ, δ , δ , δ ) , (19) ( m s − ¯ m ) K , I , → E ∗ str = ( m s − ¯ m ) × K , I , f s ( λ, δ, δ , δ , δ ) , (20)where f cl , f , , , and f s represent the functions of nucleardensities for nuclear medium. We will parametrize thembased on information about nuclear matter in the nextsection. One should keep in mind that in general onecan consider density dependencies in a different manner,depending on the isospin splitting and the mass split-tings in different representations (see Eqs. (19) and (20)).As mentioned previously, we assume that the electromag-netic corrections to the neutron-proton mass difference isonly weakly affected by nuclear medium. So, we will notconsider them in the present work. We ignore also the ef-fects of isospin symmetry breaking coming from baryonson the binding energy except for the nucleons, assum-ing the small strangeness fraction up to normal nuclearmatter densities. III. NUCLEAR PHENOMENOLOGY
In the present section, we will discuss the results re-lated to symmetric nuclear matter, isospin asymmetricnuclear matter, and more general baryonic matter, oneby one. We first start with ordinary symmetric nuclearmatter.
A. Symmetric nuclear matter
We first consider isospin symmetric and non-strangeordinary nuclear matter with the external parameters δ = 0 and δ s = 0 . Then we can parametrize the den-sity functions such as f cl , f and f . In consequence,we are able to determine the values of the correspond-ing parameters phenomenologically. For example, theyare related to the properties of isospin-symmetric nuclearmatter near the saturation point, i.e., at the normal nu-clear matter density ρ ∼ (0 . − .
17) fm − . We remindthat in the case of isospin-symmetric nuclear matter thebinding energy per unit volume is given by E ≡ ε V ( ρ ) AV = ρ λ ε V ( λ ) . (21)Following the ideas presented in Refs [44, 45], we choosethe parametrization of the three medium functions f cl , f , and f , which are independent of the asymmetry pa-rameter δ and the strangeness fraction parameters δ s .Furthermore, we will parametrize them for simplicity ina linear density-dependent form f cl ( λ ) = (1 + C cl λ ) , f , ( λ ) = (1 + C , λ ) . (22)It is enough to employ this linear-density approximation,since the equations of state (EoS) for nuclear matter arewell explained. However, if the density of nuclear matterbecomes larger than the normal nuclear matter density,one may need to consider higher-order corrections to theparametrization we use. Nevertheless, we will computethe baryon properties as functions of the nuclear mat-ter density up to ρ to see how far the linear-densityapproximation works well.The properties of symmetric nuclear matter near thesaturation point can be related to the isoscalar nucleonmass change in the nuclear medium ε V ( λ ) = (cid:15) ( λ, , , ,
0) = ∆ M N ( λ ) . (23)This implies that α , β and γ will not be changed insymmetric nuclear medium. We will see that they willcome into play when we consider asymmetric nuclear andstrange baryonic matter. Then we can easily obtain thefollowing formula for the density dependence of the vol-ume energy ε V ( λ ) = M cl C cl λ − C λ I (1 + C λ ) − C λ I (1 + C λ ) . (24)We now proceed to calculate the properties of nuclearmatter near the saturation point λ = 1 by expandingthe volume energy with respect to the nuclear density.The expansion coefficients have clear physical meaningsrelated to the properties of nuclear matter at the satura-tion point. They are given as follows: a V = ε V (1) , P = ρ λ ∂ε V ( λ ) ∂λ (cid:12)(cid:12)(cid:12)(cid:12) λ =1 ,K = 9 λ ∂ ε V ( λ ) ∂λ (cid:12)(cid:12)(cid:12)(cid:12) λ =1 , (25)where a V denotes the value of the volume energy, P stands for that of the pressure, and K represents thecompressibility of nuclear matter at the saturation point.The value of the coefficient of the volume term a V is wellknown from the analysis of atomic nuclei according to thesemi-empirical Bethe-Weizsäker formula [54, 55]. So, wechoose the well-known value a V = −
16 MeV . The stabil-ity of nuclear matter requires the zero value of the pres-sure P = 0 at the saturation point. The compressibilityof nuclear matter within various approaches is found tobe K ∼ (290 ± MeV [56–61]. Based on a comprehen-sive reanalysis of recent data on the energies of the giantmonopole resonance (GMR) in even-even − Sn and , − Cd and earlier data on ≤ A ≤ nucleiin Ref. [62], the value of the compressibility can be takento be K ∼ (240 ± MeV [63]. Following this analysis,we choose K = 240 MeV in the present work. Thus,the three parameters in the density functions given inEq. (22) can be fitted to be C cl = − . , C = 0 . , C = − . . (26)Now we can predict the skewness of symmetric nuclearmatter, which is defined from the fourth coefficient in theseries of the volume energy: Q = 27 λ ∂ ε V ( λ ) ∂λ (cid:12)(cid:12)(cid:12)(cid:12) λ =1 = − (cid:18) C (1 + C ) I − C (1 + C ) I (cid:19) = −
182 MeV . (27)The result is consistent with those from other model cal-culations. For example, one can find similar results fromthe Hartree-Fock approach based on the Skyrme inter-actions [64] and the isospin- and momentum-dependentinteraction (MDI) model [65]. We want to emphasizethat these coefficients in the expansion of the volumeenergy can be used for understanding the properties ofsymmetric nuclear matter. B. Asymmetric nuclear matter
Since the asymmetric nuclear matter arises from theisospin symmetry breaking, Eq. (19) plays the key rolein describing the asymmetric nuclear matter. Followingthe strategy taken from Refs. [44, 45], we find that thedensity function f can be defined as a function of the normalized density λ and isospin asymmetry parameter δ in the following form f ( λ, δ ) = 1 + C num λ δ C den λ , (28)where C num and C den can be determined phenomeno-logically. This parametrization is also chosen under thesimple assumption: that is, if δ is zero or ρ is zero,then the value of f is equal to 1. Moreover, using theparametrized form given in Eq. (28), one can see thatvarious properties of asymmetric nuclear matter are welldescribed, e.g., the binding energy of asymmetric nuclearmatter will be given as a quadratic form with respect tothe asymmetry parameter δ . The nuclear symmetry energy is defined as the secondderivative of the binding energy with respect to δ : ε sym ( λ ) = 12! ∂ ε ( λ, δ, , , ∂δ (cid:12)(cid:12)(cid:12)(cid:12) δ =0 . (29)As in the case of the volume energy, we can expand ε sym around the saturation point λ = 1 as follows ε sym ( λ ) = a sym + L sym λ − K sym ( λ −
18 + · · · , (30)from which we obtain the value of the nuclear symmetryenergy at saturation point a sym , that of its slope parame-ter L sym , and the asymmetric part of the compressibility K sym . They are explicitly written as a sym = − C num ( b − r/ C den ) , (31) L sym = − C num ( b − r/ C den ) , (32) K sym = 81 C den C num ( b − r/ C den ) , (33)where b = ( m d − m u ) β and r = ( m d − m u ) γ .The value of the nuclear symmetry energy at the satu-ration point is known to be in the range ε sym (1) ∼ − MeV. So, we can take the average value a sym = 32 MeV.The correlation between the value of the symmetry en-ergy at the saturation density and that of its slop param-eter taken from the neutron skip thickness experimentsof Ni,
Sn, and
Pb indicates the tendency thatheavier the nucleus yields larger the value of L sym , whichcorresponds to that of a sym [66]. As a result, one canchoose L sym = 60 MeV for the asymmetric nuclear mat-ter. We mainly use these values of a sym and L sym in thecourse of the present calculation, if it is not specified oth-erwise. The empirical values of these two quantities will Note that there is another δ factor in Eq. (10). We will not use any other input data in the strangeness sector. adjust those of C num and C den , respectively. In order tocheck the stability of the present results for to neutronmatter ( δ = 1 ), however, we have analyzed the differentchoices for the a sym and L sym with small variations. Notethat the results are very insensitive to a sym in the rangeof its values discussed above. Thus, we will only show thevariations of L sym and the possible two choices of L sym in this work are listed in Table I. All the parameters are Table I. Possible sets of the parameters for the symmetryenergy. a sym [ MeV] L sym [MeV] C num C den Set I 32 60 65.60 0.60Set II 32 50 78.72 0.92 actually fitted in this way in relation to nuclear matterproperties at the saturation density, so the present modelcan be regarded as a simple model of nuclear matter withfive parameters. Using the values of the parameters forthe symmetry energy listed in Table I, we are able todiscuss the EoS for asymmetric nuclear matter, extrapo-lating to the low and high density regions, and to predictvarious properties of nuclear matter. In particular, em-ploying Set I, we can calculate the third coefficient inthe expansion of the symmetry energy, which leads to K sym = −
135 MeV . The following quantities, which arerelated to K sym , can be also determined as K τ = K sym − L sym = −
495 MeV ,K (0 , = K τ − QK L s = −
450 MeV . (34)The calculated values of K τ and K , are in good agree-ment with the results from other approaches. As anexample, we can compare the range of K , value withthat from the phenomenological momentum-independentmodel −
477 MeV ≤ K , ≤ −
241 MeV [67].Figure 1 draws illustratively the density dependence ofbinding energy per nucleon, given the different values ofthe asymmetry parameter δ . We find that the results arerather stable to the change of values of the parameters a sym and L sym . In particular, the present results changeonly slightly as the values of L sym are varied from 50 to60 MeV.It is natural that the neutron matter gets less boundrelatively to the symmetric matter, as already shown inFig. 1. The density dependence of the binding energy pernucleon in symmetric matter and neutron matter are inagreement with those from other models and phenomeno-logical ones. In particular, it is consistent with Akmal-Pandharipande-Ravenhall (APR) predictions [68] in therange of λ , where the simple linear-density approxima-tion is justified for the medium modification of the corre-sponding soliton functionals in nuclear matter. As λ in-creases, the present EoS becomes stiffer such that one canget the larger masses of neutron stars than the solar mass.However, as the density becomes higher than λ = 2 , the λ ( = ρ/ρ ) ε ( λ , δ ) ( M e V ) δ = 0 a sym = 32 MeV , L sym = 60 MeV ( δ = 1 ) a sym = 32 MeV , L sym = 50 MeV ( δ = 1 )APR predictions( δ = 1 )APR predictions( δ = 0 ) Figure 1. Binding energy per nucleon ε ( λ, δ ) ≡ ε ( λ, δ, , , as a function of the normalized nuclear matter density λ = ρ/ρ in unit of MeV. The blue solid curve depicts the sym-metric matter δ = 0 whereas the red solid and green dashedcurves illustrate those of neutron matter δ = 1 for the pos-sible two sets of symmetry energy parameters, respectively.The present results are compared with those given in APRpredictions [68] that are given by the yellow circles and boxes,respectively. linear density approximation may not be enough, whichrequires one to introduce higher-order nonlinear terms. λ ( = ρ/ρ ) ε s y m ( λ ) ( M e V ) a sym = 32 MeV , L sym = 60 MeV a sym = 32 MeV , L sym = 50 MeV APR predictionsIAS constraints
Figure 2. Nuclear symmetry energy ε sym ( λ ) as a functionof the normalized nuclear density λ = ρ/ρ in unit of MeV .The results with the possible two sets of parameters for thesymmetry energy are represented by the red solid and greendashed curves, respectively. The results are compared withthose from Ref. [68], which are marked by the blue circlesand those from the IAS constraints [69] shown by the shadedregion.
The nuclear symmetry energy plays a very importantrole in understanding the EoS of nuclear matter and,in particular, of the neutron matter. Figure 2 exhibitshow the nuclear symmetry energy depends on λ . Wepresent the results with the two sets of the parameters a sym and L sym listed in Table I. When the value of theslop parameter L sym gets smaller, the symmetry energybecomes slightly larger than that obtained by using thelarger value of L sym till the normal nuclear matter den-sity ( λ = 1 ), then it becomes smaller than that with L sym = 60 MeV. Note that, however, the present resultsare quite stable as the parameters vary, and are con-sistent with those obtained from other approaches andextracted data. In particular, the results are in goodagreement with APR predictions till the density reaches λ = 2 . At large nuclear matter densities the results of thesymmetry energy become smaller than the values of theAPR symmetry energy. The present results are also ingood agreement with the bounded values of the symme-try energy, obtained from the analysis of isobaric states(IAS) [69], which is represented by the shaded region inFig. 2. λ ( = ρ/ρ ) P ( λ , , , , ) ( M e V f m − ) (a) This workGMR Exp.Flow ExpFlow+20%Kaon Exp λ ( = ρ/ρ ) -1 P ( λ , , , , ) ( M e V f m − ) (b) a sym = 32 MeV, L sym = 60 MeV a sym = 32 MeV, L sym = 50 MeVPbNS 95%NS 68%QMC
Figure 3. Numerical results for the pressure P ( λ, δ, , , .In the upper (a) panel, the results are drawn for the symmet-ric nuclear matter ( δ = 0 ), compared with the data taken fromGMR [70, 71], flow [72], flow +20% [73, 74], and kaon [71, 75]experiments, whereas in the lower (b) panel those for the neu-tron matter ( δ = 1 ) are depicted, compared with the datafrom Pb experiment [76], NS [73], NS [73], and Quan-tum Monte-Carlo calculations [76–78]. In the neutron matter,the present results are obtained for the two sets of the param-eters: red solid and green dashed curves draw the results withSet I and Set II, respectively. For completeness, we present in the upper (a) andlower (b) panels of Fig. 3 the density dependence of the pressure in the symmetric nuclear matter and and in theneutron matter, respectively. The present results for thepressure are in good agreement with those obtained fromother approaches and the extracted data, in particular,in the range of ρ ∈ [0 , ρ ] .For example, in the upper (a) panel of Fig. 3 the resultof this work for the pressure P ( λ, , , , in the sym-metric matter ( δ = 0) , which is drawn in the blue solidcurve, is compared with the data extracted from variousexperiments. In particular, the present result is in goodagreement with the data in the range of . ≤ λ ≤ . extracted from the GMR experiments [70, 71] for heavynuclei, which are shown by the dashed curve. On theother hand, in Ref. [72] the flow experimental data on Au nuclei collision are analyzed, which are illustratedby the red-shaded region and correspond to the zero-temperature equation of state for the symmetric nuclearmatter. Additional studies are presented in Refs. [73, 74],which were extended to the range of the validity takinginto account the mass-radius relation of neutron starsfrom observational data. In the upper panel of Fig. 3data in the extended region are denoted as " Flow+20% ".One can see that our equations of state are consistentwith the newly predicted range. The EoS for the sym-metric nuclear matter in the range of . ≤ λ ≤ . canalso be constrained by the kaon production data fromhigh-energy nucleus-nucleus collision [71, 75]. They areshown in the green-colored region. The present resultslie also within that region and, in general, are consistentwith all the data extracted from different methods.In the lower (b) panel of Fig. 3, the results for thepressure are presented in the neutron matter ( δ = 1 ).We again depict the results for P ( λ, , , , with thetwo sets of the parameters a sym and L sym . They arerepresented by the red solid and green dashed curves, re-spectively in the lower panel of Fig. 3. One can see thatEoS obtained in the present work are quite stable withthe varying parameters that define the symmetric energy.We compare the results with those extracted from theseveral experiments. For example, the weighted averageof the experimental data on the neutron skin thickness in Pb is indicated by the red-colored star at subnucleardensity [76]. The studies in Ref. [73] provide a constraintfor the pressure values of neutron star matter from astro-physical observation data. In the lower panel of Fig. 3,this constraint is labeled as " NS 95% " and "
NS 68% " forthe two different confidence limits. There are also re-sults at the low-density region from the quantum MonteCarlo calculation (QMC) [76–78]. They are denoted bythe gray-shaded region. One can see that our results arein an excellent agreement with all extracted data in thedifferent ways in all density regions presented in the fig-ure.In short summary, this simple five-parametric modelfor nuclear matter within the framework of the model-independent chiral soliton approach describes the isospin-symmetric and neutron matter properties very well. Thisimplies that the meson mean-field approach quite suc-cessful not only for explaining various properties of lightand singly-heavy baryons in free space [23] but also fordescribing phenomenologically nuclear matter properties,based on minimal phenomenological information in thenonstrange sector.
C. Baryonic matter
We now proceed to baryonic matter properties in amore general case taking into account also the strangebaryons. So far, we have concentrated on the nonstrangesector and fitted our parameters according to the nu-clear phenomenology in non-strange sector. We havealso parametrized the influence of surrounding nuclearmatter to the in-medium nucleon properties in such away that the binding energy per nucleon appears as aquadratic term in the isospin asymmetry parameter δ .Following the strategy used in the nonstrange sector wecan parametrize the influence of baryonic matter withthe strangeness content. In doing that, we will considerthe simplicity as a guiding principle. Therefore, as a firststep we will not introduce any new parameter and tryto describe the strangeness-mixed baryonic matter. Forthat purpose, we expand the binding energy per baryoninto the series in the region with the small values of theisospin-asymmetry parameter δ and strangeness-mixingparameters δ s . Consequently, the series of the bindingenergy per nucleon at the small values of isospin asymme-try and hyperon mixture parameters δ and δ s ( i = 1 , , )given in Eq.(10) can be written as ε ( λ, δ,δ , . . . ) = ε V ( λ ) + ε sym ( λ ) δ + (cid:88) s =1 ∂ ε ( λ, δ, δ , . . . ) ∂ δ s (cid:12)(cid:12)(cid:12)(cid:12) δ = δ = ··· =0 δ s + 12 (cid:88) s,p =1 ∂ ε ( λ, δ, δ , . . . ) ∂ δ s ∂δ p (cid:12)(cid:12)(cid:12)(cid:12) δ = δ = ··· =0 δ s δ p + · · · , (35)where, for convenience of discussion, the terms of thestandard volume and symmetry energies for ordinary nu-clear matter are explicitly separated as the first and thesecond ones. It is obvious that the linear terms in δ areabsent due to the quadratic dependence of the bindingenergy per baryon on it.Next, assuming that the contributions of higher-orderterms in δ s are negligible, we can choose f s ’s similar to f (see Eq. (28)) as a linear form in δ s . Furthermore,we parametrize f s in such a way that there is no δ de-pendence. These parametrization will keep all our dis-cussions in the nonstrange sector intact. Then we havethe following forms of the remaining density functions f s ( λ, δ, δ , . . . ) = 1 + g s ( λ ) δ s . (36) Note that this is also the simplest choice.
We also assume that the third term in Eq. (35) is equalsto zero. This leads to the following form of g s g s ( λ ) = sg ( λ ) ,g ( λ ) = − M ∗ cl − M cl + E ∗ (1 , / − E (1 , / )3( m s − ˆ m ) × (cid:18) K I + K I (cid:19) − . (37)This final expression is a reasonable one, because thestrangeness content of nuclei is negligible and g s at smalldensities in Eq. (37) maximizes the energy for δ s = 0 .This choice is advantageous, since it allows one to fit allparameters in the SU(2) sector. As a result, we have nointernal density parameters in the SU(3) sector and wedo not need to relate this approach to the strange matterphenomenology. All results in the strangeness sector canbe considered as predictions in this simplified work.In the medium-modified SU(3) sector, we have only oneexternal free parameter, which is the fraction of strangematter. In order not to distinguish the species of strangematter, we introduce the strangeness-mixing parameter χ defining it as the following simple and reasonable way: δ s = sχ . So, we can discuss the strangeness effects byconsidering nonzero values of the free parameter χ . λ ( = ρ/ρ ) ε ( λ , δ , χ ) ( M e V ) δ = 0 , χ = 0 δ = 0 . , χ = 0 δ = 0 , χ = 0 . δ = 0 . , χ = 0 . Figure 4. Binding energy per nucleon ε ( λ, δ, χ ) as a func-tion of the normalized nuclear matter density λ = ρ/ρ . Theresults are drawn for the ordinary isospin symmetric mat-ter in the blue solid curve, the strangeness mixed isospin-symmetric matter in the red dashed one, the pure neutronmatter in the green dotted one and the strangeness-mixedisospin-asymmetric matter in the gray space-dashed one, re-spectively. The parameters of the symmetry energy are takenfrom Set I in Table I. The strangeness effects due to the surrounding environ-ment may come from the different combinations, e.g. theisospin-symmetric matter with the strangeness-mixingor the isopin-asymmetric matter with the strangenessmixing. The binding energies per nucleon for differentnuclear matters are presented in Fig. 4, which showsclearly how the binding energy undergoes modification as λ ( = ρ/ρ ) -1 P ( λ , δ , χ ) ( M e V f m − ) δ = 0 , χ = 0 δ = 0 . , χ = 0 δ = 0 , χ = 0 . δ = 0 . , χ = 0 . Figure 5. Pressure P ( λ, δ, χ ) as a function of the normalizednuclear matter density λ = ρ/ρ . Notations and parametersare the same as in Fig. 4. the strangeness content varies together with δ changed.For comparison, we again depict the binding energyfor symmetric matter in the blue solid curve, for theisospin-asymmetric matter in the red-dashed one, forthe strangeness-mixed isospin-symmetric matter in thegreen-dotted curve and the strangeness-mixed isospin-asymmetric matter in the gray space-dashed curve, re-spectively. One can see that the strangeness mixing leadsto the less bound system at subnuclear matter densitieswhile the binding energy per nucleon varies rather slowlyas λ increases, so that its magnitude becomes even largerthan those in both iso-symmetric and iso-asymmetric nu-clear matter at supranuclear matter densities, i.e. com-pare the blue solid curve and the green dotted one or thered dashed and grey dashed ones, respectively. At largedensities the strange matter may be a more favorablesystem so that strange quark stars are allowed to existwith a smaller mass due to the softening EoS in compar-ison with neutron stars. In general, the effect from theisospin asymmetric environment is much stronger thanthat from strangeness mixing. The results are consistentwith those from other approaches and model calculations.For example, see a recent review [13] about theoreticalapproaches to the production of hyperons, baryon reso-nance and hyperon matter in heavy-ion collision.These results also can be seen from the density depen-dence of the pressure shown in Fig. 5, where we draw theresults for the dependence of the pressure on λ for possi-ble four different cases, as discussed in the case of bindingenergy dependence on normalized nuclear matter density.One can see that the strangeness mixing will bring aboutthe softening of EoS, comparing the blue solid curve withthe green dotted ones or the red dashed and grey dashedones. IV. BARYONS MASSES IN DIFFERENTBARYON ENVIRONMENTS
We are now in a position to discuss how the masses ofthe SU(3) baryons undergo the changes in ordinary andstrangeness-mixed nuclear matter. Since all the mediumfunctions have been already fixed, we can study the mod-ification of the baryon masses in different nuclear media.While we have considered only the baryon octet in formu-lating the nuclear matter, we will investigate the mediummodifications of both the baryon octet and decuplet innuclear matter.The contributions from the surrounding baryon envi-ronment can be divided into two parts: the change ofthe classical soliton mass ∆ M cl = M ∗ cl − M cl (see Eq. (1))and that of quantum fluctuations ∆ M qf = ∆ M B − ∆ M cl ,where ∆ M B = M ∗ B − M B denotes the shift in the baryonmass. From the values of the parameters in Eq. (26),one can see that the classical soliton mass is homoge-neously dropped in nuclear matter. The soliton mass infree space in the present work is . MeV. Its change inthe medium at the normal nuclear matter density ( λ = 1 )is given as − . MeV. In the case of quantum fluctua-tions the situations is not at all trivial, because differentparts of the quantum fluctuations may behave in a dif-ferent way depending on the content of the surroundingbaryon environment.The masses of octet and decuplet member baryons inthe different baryon environments at normal nuclear mat-ter λ = 1 density are predicted and are listed in Table II.One can see that the change of quantum fluctuations and,consequently, the changes of baryons masses in the dif-ferent environments are different. In the symmetric or-dinary nuclear matter ( λ = 1 and δ = 0 ) the massesof the baryon octet and decuplet decrease as λ increases(compare third and fourth columns in Table II). Figure 6illustrates the density dependence of the mass shifts ofthe nucleon and ∆ isobar in the isospin symmetric nu-clear matter. As shown in Fig. 6, the mass shift of thenucleon decreases very slowly as λ increases. However, itis almost saturated in the vicinity of the normal nuclearmatter density and then starts to increase very slightly.On the other hand, the mass shift of the ∆ isobar falls offmonotonically as λ increases. Since the mass differenceof the nucleon and ∆ comes from the zero-mode quanti-zation of the chiral soliton, the medium modification ofthe zero-mode quantum fluctuation for the ∆ comes intoessential play. This fact makes the mass shift of ∆ turnout to be very different from that of the nucleon. Wefind the very similar results for the other members of thebaryon octet and decuplet.However, the situation is changed in isospin-asymmetric matter. The mass shift of the SU(3) baryonsin the isospin asymmetric environment depends on thethird component of baryons isospin. Thus, the massshifts of the baryons are more pronounced, in particu-lar, for the baryon with negative T . For example, themass shift of the proton in pure neutron matter ( δ = 1 )0 Table II. Masses of the baryon octet and decuplet both in free space and in the different baryon environments at normal nuclearmatter density λ = 1 . The parameters for the symmetry energy are taken from Set I in Table I. All the masses are given inunits of MeV. Baryon Exp Free space δ = 0 , χ = 0 δ = 1 , χ = 0 δ = 0 , χ = 0 . δ = 0 . , χ = 0 . p n Λ Σ + Σ Σ − Ξ Ξ − ∆ ++ ∆ + ∆ ∆ − Σ ∗ + Σ ∗ Σ ∗− Ξ ∗ Ξ ∗− Ω − λ ( = ρ/ρ ) ∆ M B ( λ , , ) ( M e V ) N ∆ Figure 6. The result for the mass shift ∆ M B = M ∗ B − M B ofthe nucleon B = N in the isospin symmetric nuclear matteris drawn in a solid curve, whereas that of the B = ∆ isobaris depicted in a dashed one. at normal nuclear matter density ( λ = 1 ) is obtained tobe − . MeV, while that of the neutron becomes pos-itive, i.e. +15 . MeV (see also Table II). This impliesthat the up and down quarks may undergo changes in adifferent manner.The results for the mass shifts of the baryon octet anddecuplet in the pure neutron matter are shown in the up-per and lower panels of Fig. 7, respectively. First of all,one can explicitly see that the effects of the isospin masssplitting are clearly shown in the isospin-asymmetric nu-clear environment. Depending on the charges of thebaryon octet, we can see that their mass shifts behave λ ( = ρ/ρ ) ∆ M B ( λ , , ) ( M e V ) (a) n, Ξ − p, Ξ Σ + Σ , ΛΣ − λ ( = ρ/ρ ) ∆ M B ( λ , , ) ( M e V ) (b) ∆ ++ ∆ − Σ ∗ + Σ +0 , Ω − Σ ∗− Ξ ∗ , ∆ + Ξ ∗− , ∆ Figure 7. Results for the mass shifts ∆ M B of the baryonoctet and the decuplet in the pure neutron matter ( δ = 1 )are drawn in the upper (a) and lower panels (b), respectively.The parameters for the symmetry energy are taken from Set Iin Table I. λ increases, whereasthe proton mass drops off as λ increases. In general,the masses of the members in the baryon octet with thenegative values of T rises in the neutron matter as λ increases. On the other hand, the masses of the octetbaryons with the positive values of T drop off as λ de-creases. However, while the mass of Ξ is identical tothat of the proton, those of Σ and Λ , which are alsoidentical each other, fall off slowly and then are saturatedas λ increases. This is originated from the fact that thein-medium functionals for the quantum fluctuations nearthe third component of isospin are quite sensitive to themedium effects and they are identical for the baryonsthat have the same isospin components.The general tendency and the effects of the isopin fac-tor can be seen also in the case of decuplet baryons, whichare given in the lower (b) panel of Fig. 7, but the massshifts are larger than those of the baryon octet. For ex-ample, the mass shift in the isospin averaged ∆ in normalnuclear matter is around two times smaller than the ∆ ++ mass shift in neutron matter at normal nuclear matterdensity λ = 1 . For example, one can see this by compar-ing the red dashed curve in Fig. 6 with the black solid onein the lower panel of Fig. 7. The isospin component factoris similar to the octet case and some baryons masses suchas Ξ ∗ and ∆ + have the identical dependence on λ . Dueto the isospin factor, the ∆ − mass in neutron matter re-mains almost constant which is seen from the red dashedcurve in the lower panel of Fig. 7. The present resultsare in qualitative agreement with those from Ref. [79].For completeness, we show also the mass changes ofthe baryons in the strangeness-mixed asymmetric envi-ronment. In Fig. 8, the results for the mass shifts of thenucleon and ∆ in strangeness-mixed asymmetric matterare presented as functions of λ , which we choose them asthe representatives of the baryon octet and decuplet, re-spectively. For the sake of the illustration, we include thechanges of isospin-averaged masses of the nucleon and ∆ by the black solid curves in the upper and lower panels,respectively. Figure 8 explicitly shows the isospin factorin nuclear matter, which are explained above. It also de-picts how the strength of the mass are changed due to theenvironment content. Comparing the Fig. 7 with Fig. 8,one can conclude that the changes in neutron matter arestronger than those in strange matter. In general, theresults in strangeness-mixed matter are rather similar tothose in pure neutron matter. V. SUMMARY AND OUTLOOK
In the present work, we have investigated the variousbaryonic matters such as the symmetric nuclear matter,pure neutron matter, and strangeness-mixed baryonicmatter, based on the meson mean-field approach or thegeneralized SU(3) chiral soliton model. All the dynami-cal parameters for the baryon masses in the model were λ ( = ρ/ρ ) ∆ M N ( λ , . , . ) ( M e V ) (a) Nnp λ ( = ρ/ρ ) ∆ M ∆ ( λ , . , . ) ( M e V ) (b) ∆∆ ++ ∆ + ∆ ∆ − Figure 8. Mass shifts M ∗ B − M B of the nucleon (upper (a)panel) and ∆ isobar (lower (b) panel) in the strangeness mixedisospin-asymmetric matter. determined by using the experimental data in free spaceand then we have introduced the parametrizations forthe density-dependent parameters. As a starting point,we took a " model-independent approach " for the SU(3)baryon properties in free space, which described success-fully the baryon masses of the baryon decuplet and otherproperties of baryons in free space [23, 24]. In the presentwork, the medium modifications of the model function-als were carried out by employing the linear density-dependent forms. Having determined the parameters forthe medium modification by using the empirical data re-lated to nuclear matter such as the binding energy per nu-cleon, the compressibility, and the symmetry energy, wewere able to describe the equation of states for various nu-clear environments including the nonstrange sector. Wefound that the present results were in good agreementwith the data extracted from the phenomenology andexperiments and with the results from other approaches.We also discussed the properties of the strangeness-mixedmatter and they also were in agreement with the phe-nomenology. Finally, we predicted the mass shifts ofthe baryon octet and decuplet in various baryonic en-vironments with different content of the isospin asymme-try and strangeness. We scrutinized the changes of the2masses of the baryons with different values of the thirdcomponents of isospin and found that the masses of thebaryons with negative charges show very different de-pendence on the nuclear matter density from those withpositive and null charges.Since we have formulated the equations of statesfor isospin-asymmetric and strangeness-mixing baryonicmatter, one can directly apply the present model to in-vestigate properties of neutron stars. The correspondinginvestigation is under way. ACKNOWLEDGMENTS
The present work was supported by Basic Science Re-search Program through the National Research Foun-dation (NRF) of Korea funded by the Korean gov-ernment (Ministry of Education, Science and Technol-ogy, MEST), Grant Numbers 2019R1A2C1010443 (Gh.-S. Y.), 2018R1A2B2001752 and 2018R1A5A1025563 (H.-Ch.K.), 2020R1F1A1067876 (U. Y.).
Appendix A: Masses of baryons in free space
The masses of baryon octet are expressed as M N = M cl + E (1 , , / + 15 (cid:18) c + 49 c (cid:19) T + 35 (cid:18) c + 227 c (cid:19) (cid:18) T + 14 (cid:19) − ( d − d ) T − ( D + D ) , (A1) M Λ = M cl + E (1 , , / + 110 (cid:18) c − c (cid:19) − D , (A2) M Σ = M cl + E (1 , , / + 12 (cid:18) T − (cid:19) c + 29 (cid:18) T − (cid:19) c − (cid:18) d + 12 d (cid:19) T + D , (A3) M Ξ = M cl + E (1 , , / + 45 (cid:18) c − c (cid:19) T − (cid:18) c − c (cid:19) (cid:18) T + 14 (cid:19) − ( d + 2 d ) T + D , (A4) where E (1 , , / can be obtained from Eq. (6). Themasses of the baryon decuplet are given by the follow-ing expressions M ∆ = M cl + E (3 ,
0) 3 / + 14 (cid:18) c + 863 c (cid:19) T + 563 T + 18 (cid:18) c − c (cid:19) − (cid:18) d − d (cid:19) T − (cid:18) D − D (cid:19) , (A5) M Σ ∗ = M cl + E (3 , , / + 14 (cid:18) c − c (cid:19) T + 563 c (cid:0) T − (cid:1) − (cid:18) d − d (cid:19) T , (A6) M Ξ ∗ = M cl + E (3 , , / + 14 (cid:18) c − c (cid:19) T − (cid:18) c + 863 c (cid:19) (cid:18) T + 14 (cid:19) − (cid:18) d − d (cid:19) T + (cid:18) D − D (cid:19) , (A7) M Ω = M cl + E (3 , , / − (cid:18) c − c (cid:19) + 2 (cid:18) D − D (cid:19) , (A8)where E (3 , , / can be obtained from Eq. (6). Here d , and D , are defined as d = ( m d − m u ) (cid:20) − α − β + 15 γ (cid:21) , (A9) d = ( m d − m u ) (cid:20) − α − γ (cid:21) , (A10) D = ( m s − ¯ m ) (cid:20) − α − β + 15 γ (cid:21) , (A11) D = ( m s − ¯ m ) (cid:20) − α − γ (cid:21) . (A12)The explicit forms of c and c , which denote the wave-function corrections, can be found in Ref. [23]. [1] E. G. Drukarev and E. M. Levin, Prog. Part. Nucl. Phys. , 77 (1991).[2] M. C. Birse, J. Phys. G , 1537 (1994).[3] G. E. Brown and M. Rho, Phys. Rept. , 333 (1996). [4] K. Saito, K. Tsushima and A. W. Thomas, Prog. Part.Nucl. Phys. , 1 (2007).[5] B. D. Serot and J. D. Walecka, Adv. Nucl. Phys. , 1(1986). [6] J. J. Aubert et al. [European Muon Collaboration], Phys.Lett. , 275 (1983).[7] S. Strauch et al. [Jefferson Lab E93-049 Collaboration],Phys. Rev. Lett. , 052301 (2003).[8] G. Agakishiev et al. [HADES Collaboration], Phys. Rev.C , 054906 (2014).[9] S. Malace, D. Gaskell, D. W. Higinbotham and I. Cloet,Int. J. Mod. Phys. E , 1430013 (2014).[10] K. J. Eskola, P. Paakkinen, H. Paukkunen and C. A. Sal-gado, Eur. Phys. J. C , 163 (2017).[11] T. Kolar et al. [A1 Collaboration], Phys. Lett. B ,135903 (2020).[12] F. Osterfeld, Rev. Mod. Phys. , 491 (1992).[13] H. Lenske, M. Dhar, T. Gaitanos and X. Cao, Prog. Part.Nucl. Phys. , 119 (2018).[14] R. Knorren, M. Prakash and P. J. Ellis, Phys. Rev. C
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