Relativistic Mean-Field Model with Scaled Hadron Masses and Couplings
aa r X i v : . [ nu c l - t h ] O c t Relativistic Mean-Field Modelwith Scaled Hadron Masses and Couplings
A.S. Khvorostukhin a , b ,V.D. Toneev a , c and D.N. Voskresensky d , c a Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region, Russia b Institute of Applied Physics, Moldova Academy of Science, MD-2028 Kishineu,Moldova c GSI, Plankstraße 1, D-64291 Darmstadt, Germany d Moscow Engineering Physical Institute,Kashirskoe Avenue 31, RU-115409 Moscow, Russia
Abstract
Here we continue to elaborate properties of the relativistic mean-field based model(SHMC) proposed in ref. [6] where hadron masses and coupling constants dependon the σ -meson field. The validity of approximations used in [6] is discussed. Weadditionally incorporate contribution of meson excitations to the equations of mo-tion. We also estimate the effects of the particle width. It is demonstrated that theinclusion of the baryon-baryon hole and baryon-antibaryon loop terms, if performedperturbatively, destroys the consistency of the model. In recent years there has been a great interest in the description of hadronicproperties of strongly interacting matter. It is based on the fact that var-ious experiments indicate modifications of hadron masses and widths in themedium (see for example [1]). As expected previously, these changes are possi-bly related to a partial chiral symmetry restoration in hot and/or dense nuclearmatter, cf. [2]. Later on it was realized that the connection between the chiralcondensate of QCD and hadronic spectral functions was not as direct as it wasoriginally envisaged. Nevertheless, the study of an in-medium modification ofhadrons is an essential point of scientific programs at new heavy ion facili-ties at FAIR (Darmstadt) [3], NICA (Dubna) [4] and low-energy campaign atRHIC (Brookhaven) [5].
Preprint submitted to Elsevier 13 November 2018 heoretical predictions for critical baryon density and temperature of thehadron-quark phase transition depend sensitively on the Equation of State(EoS) at high densities and temperatures. In [6] and here we focus on thestudy of the EoS of the hadronic matter. Any EoS of hadronic matter shouldsatisfy experimental information extracted from the description of global char-acteristics of atomic nuclei such as the saturation density, the binding energyper particle, the compressibility, the asymmetry energy and some other. Def-inite constraints on hadronic models of EoS are coming from the analysis ofdirect and elliptic flows in Heavy-Ion Collisions (HIC). In addition to theseconstraints astrophysical bounds on the high-density behavior of β -equilibriumneutron star matter should be applied, see [7].In [6], we constructed a phenomenological Relativistic Mean-Field (RMF)based model that allows one to calculate particle in-medium properties andthe EoS of hadronic matter in a broad density-temperature region. The valid-ity of this model was demonstrated for the description of heavy ion collisionsin a broad collision energy range. Microscopically based approaches, as theDirac–Brueckner–Hartree–Fock method, see [8], are very promising but needrather involved calculations. The model of ref. [6] is a generalization to finitetemperatures of the RMF model developed in [9] and applied in [7] (KVORmodel) for describing neutron star properties.Following ref. [9] we assume relevance of the (partial) chiral symmetry restora-tion at high baryon densities and/or temperatures [10] manifesting in theform of the Brown-Rho scaling hypothesis [11]: Masses and coupling constantsof all hadrons decrease with the density increase approximately in the sameway. In [6] we followed the simplest form of the scaling hypothesis and scaledthe quadratic (mass) terms of σ , ω , and ρ fields, as well as the nucleon mass,by a universal scaling function Φ which was assumed to be dependent on the σ mean field. In order to obtain a reasonable EoS, the meson-nucleon cou-pling constants were also scaled with the σ mean field treated as an orderparameter. Differences in scaling functions for the effective masses of the ω -and ρ -fields and their couplings to a nucleon allowed us to get an appropriatedensity-dependent behavior of both the total energy and the nuclear asym-metry energy, in agreement with the constrains obtained from neutron starmeasurements, cf. [7,12].Note that the idea of the dropping of the meson effective masses continues tobe ” a hot point” being extensively discussing in the literature. There existworks which simulate different modifications of the simplest form of the scalingtrying to find an optimal ansatz. E.g., the model [13] introduces a commondropping of the ω , ρ effective masses, whereas σ is treated differently, as purelyclassical field, i.e. the static space-independent order parameter. Scalings ofthe ω , ρ effective masses on the one hand and the nucleon effective mass onthe other hand are assumed to be different. Couplings are evaluated following2uark counting. As in [6] and in the given paper, ω , ρ mesons are assumedto be coupled only to the classical σ field, since in the quark model they aremade of a quark and an antiquark, which couple oppositely to the vectorfield. A support for the common dropping of the N, σ, ω, ρ masses comes fromlattice QCD in the strong coupling limit [14] where it was found that mesonmasses are approximately proportional to the equilibrium value of the chiralcondensate.There exist models, which do not accept the idea of the dropping of the ef-fective meson masses at all. E.g., most of the RMF models continue to usethe constant σ , ω , ρ effective masses. Some models introduce field interactionterms leading to an increase of the σ , ω , ρ effective masses with the increase ofthe nucleon density, e.g., see [15,16]. Ref. [17] suggests an increase rather thana decrease of the ρ meson mass with increase of the temperature, motivatingit by mixing of vector and axial mesons at finite temperature, that authorsconsider as an indication towards chiral symmetry restoration. Another mod-els simulate only the ρ width rather than a modification of the mass, althoughfrom general point of view a modification of the imaginary part of the self-energy of the resonance should stimulate a modification of the real part of theself-energy (effective mass), as a consequence of the Kramers-Kronig relation.In present paper, as in our previous paper [6], we avoid discussion of these in-teresting theoretical questions. Instead we will follow the Brown-Rho scalinghypothesis in its simplest form confronting further the results of the modelwith the HIC data. Besides the nucleon and meson σ , ω and ρ mean fields,we included low-lying non-strange and strange baryon resonances, meson ex-citations σ (600), ρ (770), ω (782) constructed on the ground of mean fields,and the (quasi)Goldstone excitations π (138), K (495), η (547) as well as theirhigh mass partners in the SU(3) multiplet K ∗ (892), η ′ (958), and ϕ (1020). Allcorresponding antiparticles are also comprised. Interactions with mean fieldsare incorporated as well. In ref. [18] it was shown that it is possible to re-produce particle scattering data when the lowest baryon octet and decoupletare assumed to be the only relevant degrees of freedom. Therefore we do notconsider higher resonances within our model.In order to construct a practical model in [6], we used several simplifica-tions. First, we assumed the validity of the quasiparticle approximation forall baryons and mesons. Second, we supposed that baryons and meson exci-tations interact only via σ , ω and ρ mean fields . Thus, the fermion-fermionhole and the fermion-antifermion loop diagrams for boson propagators andthe boson-fermion loop diagrams for fermion propagators were disregarded.Also, meson-meson excitation interactions were neglected. Thus, effectivelyexcitations were considered as an ideal gas of quasiparticles. Treating mesonexcitations perturbatively we have omitted their contribution in the equationsof motion. With this Scaled Hadron Mass-Coupling (SHMC) model we con-3tructed the EoS as a function of the temperature and the baryon density andused this EoS in a broad density-temperature region to describe properties ofhot and dense matter in heavy ion collisions.Note that the standard RMF models generalized to finite temperatures havebeen studied in the literature, e.g., see [19,20,21]. In [19] temperature depen-dence was included only into nucleon distributions. A general treatment ofmeson excitations has been considered within the imaginary time formalism[20] and a more convenient real time formulation [21]. We incorporate fluc-tuative terms expanding fields near their mean-field values. Simplifying weretain only quadratic fluctuations. Thus in our model the gas of excitationsinteracts only through mean fields. Within this approximation our results canbe reproduced using above mentioned finite temperature quantum field theorytechniques.In the present paper, we check the validity of different approximations as-sumed in [6] and consider several possibilities how the model can further beimproved. In sect. 2 we introduce the SHMC model of [6]. In Sect. 3, the pres-sure functional of the model is constructed and the equations of motion arederived. Boson excitation terms are incorporated in the equations of motionand a comparison is made with the perturbative treatment carried out in [6].Section 4 estimates the effects of finite particle widths. In Appendix A, wediscuss differences in two possible treatments of the σ meson field, first, asan order parameter (as in [6]) and second, as an independent variable, i.e.,considering σ on equal footing with other field variables ( ω and ρ ). AppendixB demonstrates problems which arise if the baryon loop terms are included.Fermion loop effects on the boson excitation masses are evaluated within aperturbation theory approach and arguments are given why these effects arenot included into the SHMC model.In reality the nucleon self-energies have a momentum dependence which is notso small. It manifests itself in high energy heavy-ion collisions [22] and affectsdifferent properties of atomic nuclei [23]. P-wave pion- and kaon-baryon inter-actions may significantly affect properties of the pion and kaon sub-systems,see [24,25,26] and refs. therein. As in [6] and in most of RMF models, here wecontinue to disregard the p-wave effects.A number of other important effects is not incorporated into our model. How-ever the full theoretical quantum field description of many strongly interactinghadron species can’t be constructed in any case. Using RMF based models andtheir generalizations one should always balance between a realistic and practi-cally tractable descriptions. Thus we postpone with further generalizations ofthe SHMC model. Further improvements of the model will be done after it willpass the check in actual hydrodynamical calculations of heavy ion collisionsin a broad energy regime, that is our future program.4 About the SHMC model
Following [9] we use the σ -field dependent effective masses of baryons m ∗ b /m b = Φ b ( χ σ σ ) = 1 − g σb χ σ σ/m b , b ∈ { b } (1)with the baryon set { b } = N (938), ∆(1232), Λ(1116), Σ(1193), Ξ(1318), Σ ∗ (1385), Ξ ∗ (1530), andΩ(1672)+ all antibaryons.The mass terms of the mean fields are m ∗ m /m m = | Φ m ( χ σ σ ) | , m ∈ { m } = σ, ω, ρ , (2)where g σb are the σb -coupling constants.For the sake of simplicity we scale all couplings g σb by a single scaling func-tion χ σ ( σ ), and all g ωb , g ρb by χ ω ( σ ) and χ ρ ( σ ) scaling functions, respectively.Therefore, all scaling functions depend only on σ [6]. The idea behind thatis as follows. The σ -field can be interpreted as an effective field simulating aresponse of the ud -quark condensate. The change of effective hadron massesand couplings is associated, namely, with a modification of the quark conden-sate in matter. Thus, we consider the σ -field as a composite field, like an orderparameter, whereas other meson fields are treated as fundamental fields. The σ excitations are then interpreted as fluctuations around the mean value of theorder parameter. Similarly, long-scale fluctuations are treated in the Landauphenomenological theory of phase transitions.To single out quasiparticles (excitations) from the mean fields, one should dothe following replacements in the Lagrangian: ω = ω cl0 + ω ′ , R = R cl0 + R ′ , ~ω = ~ω ′ and ~ρ = ~ρ ′ . Here ω cl0 , R cl0 are the mean (classical) field variables and ω ′ µ , ( ρ ′ ) µ , ( ρ ′± ) µ are responsible for new excitations, R = ρ . In [6] we con-structed a thermodynamic potential that besides mean-field terms includesthe contribution of ω and ρ excitations. By varying with respect to the fields ω cl0 , R cl0 we obtain equations of motion from where the ω cl0 ( σ ), R cl0 ( σ ) fields areextracted and put back into the thermodynamic potential. A similar proce-dure has been used in a number of works, e.g. in [13]. Then, in contrast with[13], supposing σ = σ cl + σ ′ , we expand the thus obtained effective potentialin σ ′ up to squared terms (contribution of σ ′ fluctuations) and, varying thethermodynamic potential in σ cl , derive the equation of motion for the resultingorder parameter.On the other hand, if the σ field was treated on equal footing with ω and5 , as it was done in the standard Walecka model, we would consider allthree fields as independent variables. The comparison between two choices isperformed in Appendix A.The dimensionless scaling functions Φ b and Φ m , as well as the coupling scalingfunctions χ m , depend on the scalar field in the combination χ σ ( σ ) σ . Therefore,we introduce the variable f = g σN χ σ σ/m N . (3)Following [9] we assume approximate validity of the Brown-Rho scaling ansatzin the simplest formΦ = Φ N = Φ σ = Φ ω = Φ ρ = 1 − f, (4)using χ σ = Φ σ . Thereby, in terms of σ one obtains Φ( σ ) = [1 + g σN σ/m N ] − .One could partially break the scaling, if it were required from comparison withthe data.We keep the standard expression for the nonlinear self-interaction (potential U ) of the RMF models, but now it is expressed in terms of the new variable f . Using (3) the potential U can be rewritten as follows: U = m N ( b f + c f ) = bm N ( g σN χ σ σ ) c ( g σN χ σ σ ) . (5)The presence of two additional parameters, ” b ” and ” c ”, allows one to accom-modate realistic values of the nuclear compressibility and the effective nucleonmass at the saturation density. Extra attention should be paid to the fact thatthe coefficient ” c ” must be positive to deal with the stable ground state. Valuesof the parameters used in our SHMC model can be found in [6]. In Fig. 1 wepresent the dependence of nucleon (cf. Fig. 1 of [6]) and antinucleon optical po-tentials on the single-particle energy. Comparison is presented with predictionsof the standard Walecka model (with only σ and ω mean fields). As it is seen,our model describes the nucleon optical potential in an optional way, betterthan the standard Walecka model. Differences in predictions of those modelsfor antinucleon optical potentials are drastic. A phenomenological value of anantiproton optical potential is limited within the range − ÷ −
350 MeV[28], in favor of the given model compared to the standard Walecka model.Predictions for antiprotons are very important in a light of future experimentsat FAIR.There are mean-field solutions of the baryon and σ, ω, ρ meson Lagrangian P b ∈{ b } L b + P m ∈{ m } L MF m [6]. To these terms we add the Lagrangian densityfor all meson excitations 6
200 0 200 400 600 800 1000-100-50050100 U op t , M e V -m N , MeV -200 0 200 400 600 800 1000-600-500-400-300-200-100 -m N , MeV Fig. 1. Energy dependence of the nucleon (left) and antinucleon (right) opticalpotentials. Solid lines – predictions of our model and dash lines, of the originalWalecka model. Shaded area shows uncertainties in extrapolation from finite nucleito cold nuclear matter [27]. L ex = X ex ∈{ ex } L ex , { ex } = π ± , (138); K ± , , ¯ K (495); η (547); (6) K ∗± , (892) , η ′ (958) , φ (1020); σ ′ , ω ′ , ρ ′ . The set { g } = ( π, K, η ) is often treated as (quasi)Goldstone (index ” g ”) bosonswithin the chiral SU(3) symmetrical models. Therefore, one may not scale theirmasses and couplings, as we have carried out for the mean fields σ, ω, ρ , cf.the set A for couplings in Fig. 8 (left) of ref. [6] . On the other hand, one mayobserve, cf. [6,9], that for the case of spatially homogeneous system the equa-tions for mean fields and thus their mean-field solutions do not change if onereplaces the σ , ω , R fields by the scaled fields χ σ σ , χ ω ω and χ ρ R , provided Φ b = Φ m = χ m , and χ ′ ρ = χ ρ ( χ ′ ρ is the scaling function of the ρ − ρ interaction g ρ , see [6]). If one wishes to extend this symmetry to the case when Goldstonesare included, in addition to the scaling of masses one should scale couplings, g ∗ mg = g mg χ m , cf. set B in Fig. 8 (right) of [6]. In [6], we tested both possibil-ities g ∗ mg = g mg and g ∗ mg = g mg χ m , and referred to them as versions withoutand with scaling, respectively. We include interaction of (quasi)Goldstoneswith mean fields (for K and η , for π it is small). As the result of this inter-action, at sufficiently large (overcritical) baryon densities there may appearmean field solutions for (quasi)Goldstone fields signaling of condensations ofthese fields. Values of critical densities are higher for the set B . Since there areno experimental indications of condensation of (quasi)Goldstone bosons in theheavy ion collision regimes, comparing our results with experimental data, asin [6], we will focus on the set B , where condensates do not occur. K ∗± , , η ′ , φ are assumed not to couple with mean fields, since there is no experimentalinformation for such a coupling.When the total Lagrangian is constructed, one can derive the equations ofmotion for every field. Even for low baryon density, the equations of motion7or σ , ω and ρ allow mean-field solutions σ , ω , ρ . Therefore, we use σ ≡ σ ; ω µ = ω δ ω ; ρ aµ = R δ a δ µ . (7)We assume that the system volume is sufficiently large and surface effectsmay be disregarded. Thus, only spatially homogeneous RMF solutions of theequations of motion are considered. The thermodynamic potential density Ω, pressure P , free energy density F ,energy density E and entropy density S are related as E = F + T S, F [ f, ω , R ] = X i µ i n i + Ω , Ω = − P, (8) µ i = ∂F∂n i . (9)Summation index i runs over all particle species; n i are particle densities.Chemical potentials µ i enter into the Green functions in the standard gaugecombinations ε i + µ i .Thermodynamic quantities (8) can be found from the energy-momentum ten-sor T µν which is defined by our Lagrangian. The energy density E and pressure P are given by the diagonal terms of this tensor E = h T i , P = 13 h T ii i . (10)In [6], the energy was chosen as a generating functional. Here we will use thepressure functional since it is more suitable to treat meson excitation effectsin the presence of the mean fields and baryon-loop contributions. The pressure can be presented as the sum of the mean σ -, ω -, ρ -field terms aswell as of contributions of baryons and of all meson excitations. So we have P [ f, ω , R ] = X m ∈{ m } P MF m [ f, ω , R ] + X b ∈{ b } P b [ f, ω , R ]+ P bos . ex . [ f, ω , R ] . (11)8he first two sums are included in every RMF model but with a smaller set { b } , whereas the boson excitation term P bos . ex . is constructed in [6] beyondthe scope of the RMF approximation and will be further elaborated here.Although in our treatment of the σ variable all terms in (11) are functions onlyof f and T , we also present them as functions of ω and R in such a way thatvalues of the ω ( f ) and R ( f ) mean fields can be found by minimization of thepressure. Then ω ( f ) and R ( f ) are plugged back in the pressure functionalthat becomes a function of f only. The equilibrium value of f can be foundby subsequent minimization of the resulting pressure in this field.In a self-consistent treatment, equations of motion for the mean fields render ∂∂ω P [ f, ω ] = 0 , ∂∂R P [ f, R ] = 0 , (12)and ddf P [ f, ω ( f ) , R ( f )] = ∂∂f P [ f, ω ( f ) , R ( f )] = 0 (13)with pressure P given by eq. (11). Since P bos . ex . [ f, ω , R ] depends on the meanfields, its minimization produces extra terms in the equations of motion forthe mean fields. In differentiating in (13) we used (12). This self-consistency ofthe scheme allows us to be sure of thermodynamic consistency of the model.In [6], excitations were treated perturbatively. Accordingly, we assumed that P bos . ex . = P bos . ex . [ f MF , ω MF0 , R
MF0 ], where f MF , ω MF0 , R
MF0 are found by mini-mization of the pressure without inclusion of the boson excitation term. Thus,equations of motion for mean fields that we used in [6] are: ∂∂ω X m ∈{ m } P MF m [ f, ω , R ] + X b ∈{ b } P b [ f, ω , R ] = 0 ,∂∂R X m ∈{ m } P MF m [ f, ω , R ] + X b ∈{ b } P b [ f, ω , R ] = 0 (14)and ddf X m ∈{ m } P MF m [ f, ω , R ] + X b ∈{ b } P b [ f, ω , R ] = ∂∂f X m ∈{ m } P MF m [ f, ω , R ] + X b ∈{ b } P b [ f, ω , R ] = 0 . (15)9quation (14) was used in differentiating in (15). Below in Figs. 2–5 we demon-strate how effects of a nonperturbative treatment of boson excitations, incor-porated in (12) and (13) (a self-consistent analysis) and neglected in (14),(15), affect results of the SHMC model.Actually, in [6] instead of varying the pressure at fixed chemical potentials µ i and the temperature T , we varied the energy density under the condition thatone should not vary it with respect to the particle occupation numbers. When E is varied, one should fix the particle densities n i and the entropy densities S i ,which is equivalent to fixed particle occupations in our quasiparticle approach.Two procedures mentioned are equivalent, provided baryon-baryon hole andbaryon-antibaryon excitation effects (loop contributions) are disregarded (aswe did in (12) – (15)). An attempt to incorporate the baryon loop correctionsinto our scheme has been done in Appendix B.Now let us consider partial contributions to the pressure in eq. (11). The contribution of the given baryon (antibaryon) species b ∈ { b } to thepressure is as follows: P b [ f, ω , R ] = 13 (2 s b + 1) ∞ Z d p p π f b ω b − t Qb n b µ ch ,p = | ~p | , ω b = q m ∗ b ( f ) + p . (16)The spin factor s b = 1 / N ) and hyperons, while s b = 3 / { b } to be used (taken from Table 1 of [6]) was fixed above, seeafter eq. (1). Little differences in masses of charged and neutral particles of thegiven species are ignored. Also we ignore small inhomogeneous Coulomb fieldeffects and put the electric potential V = 0. The charge chemical potential µ ch is then related to the isospin composition of the system. For the isospin-symmetric system, N = Z , one has µ ch = 0.The Fermi-particle (baryon/antibaryon) occupation f b = 1exp[( ω b − µ ∗ b ) /T ] + 1 (17)depends on the gauge-shifted values of the chemical potentials10 ∗ b = t b µ bar + t sb µ str + t Qb µ ch − g ωb χ ω ω − t b g ρb χ ρ R . (18)The baryon/antibaryon chemical potential of the b -species is µ b = t b µ bar , andthe corresponding strangeness term is µ sb = t sb µ str . Baryon quantum numbers t b , t bs , t b and t Qb are baryon charge, strangeness, isospin projection and electriccharge, respectively, and proper charge conjugated values for antiparticles aregiven in Table 1 in [6]. It is convenient to introduce the coupling ratios x mb = g mb /g mN , m ∈ { m } = σ, ω, ρ, (19)and, instead of χ m , other variables η m ( f ) = Φ m ( f ) /χ m ( f ) , (20)since the pressure depends namely on this sort of combinations rather thanon Φ m and χ m separately.In terms of these new variables the contribution of mean fields to the pressureis as follows: P MF σ [ f ] = − m N f C σ η σ ( f ) − U ( f ) , (21) P MF ω [ f, ω ] = m N η ω ( f )2 C ω [ g ωN χ ω ω ] , (22) P MF ρ [ f, R ] = m N η ρ ( f )2 C ρ [ g ρN χ ρ R ] . (23)Here the renormalized constants are C m = m N g mN m m . (24)The net baryon density is given by [6]:11 B ≡ X b ∈{ b } t b n b , n b = (2 s b + 1) ∞ Z d p p π f b , (25)where n b is the baryon (antibaryon) number density and occupation baryon(antibaryon) density is defined by eq. (17). On the other hand, for fixed baryonspecies the contribution to the baryon density should obey the thermodynamicconsistency condition n b = ∂P∂µ ∗ b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T . (26)Both quantities presented by (26) and (25) coincide provided contributions ofboson excitations do not depend on the baryon loop terms (see Appendix B).In this case thermodynamic consistency of the model is preserved, see belowin more detail.The isotopic charge density in the baryon sector is given by n tB = 2 X b ∈{ b } t b n b x ρb . (27)The isovector baryon density n tB plays the role of the source for the ρ -mesonfield ρ (3)0 = R . Therefore, for the iso-symmetrical matter ( N = Z ) one has n tB = 0 and P MF ρ = 0 .The net strangeness density of baryons and mesons reads n str = X i ∈{ b } , { ex } t si n i . (28)Bearing in mind applications of the model to high-energy heavy ion collisionsfrom AGS to RHIC energies we assume that all strange particles are trappedinside the fireball till the freeze-out. Therefore, the total strangeness is zero.Thus, we put n str = 0. This condition determines the value of the strangenesschemical potential µ str .Similarly, we may introduce the electric charge density n ch = X i ∈{ b } , { ex } t Qi n i . (29)The quantity n ch = ( Z/A ) n B determines the value of the charged chemicalpotential µ ch . For the symmetric matter, N = Z , ignoring Coulomb effectsone may put µ n = µ p and µ ch = 0. 12ur SHMC model pressure functional depends on four particular combina-tions of functions, η σ,ρ,ω ( f ) and U ( f ). Note that the dependence on the scalingfunction η σ can always be presented as part of the new potential U obtainedby means of the replacement U → U + m N f C σ (1 − η σ ( f )) , and vice versa, sothe potential U can be absorbed in the new quantity η σ . Thus actually only three independent functions enter into the pressure functional. Equation (11)together with eqs. (16), (21), (22), (23) demonstrates explicitly the equiva-lence of mean-field Lagrangians for constant fields with various parameters ifthey correspond to the same functions η ρ,ω ( f ) and η σ (either U ( f )) with thefield f related to the scalar field σ through eq. (3). In [6], we assumed η σ = 1.Here we accept the same choice. To find the total pressure (11), one should define the contribution of bosonicexcitations. Within our model and in agreement with [6] it is the sum of partialcontributions P bos . ex [ f, ω , R , T ] = P part σ + P part ω + P part ρ + P part π + P part K + P part η + P part K ∗ + P part η ′ + P part φ . (30)The pressure of the pion gas is P part π = P π + + P π + P π − = 13 ∞ Z d p p π (31) × " f π + ( ω π + ( p )) ω π + ( p ) + f π ( ω π ( p )) ω π ( p ) + f π − ( ω π − ( p )) ω π − ( p ) . Due to the absence of the ω → π decay the coupling g ∗ ωπ = 0. For N = Z ,the field R = 0 and dependence of pion spectra on g ∗ ρπ disappears. Also, asin [6], we suppose g ∗ σπ = 0 ignoring a small pion mass shift. Then for bothcharged and neutral pions we may use ω π ± ( p ) = ω π ( p ) = q m π + p . (32)The pressure of the kaon gas is given as follows P part K = P K + + P K + P K − + P ¯ K (33)= 13 ∞ Z d p p π " f K + ( ω K + ( p )) ω K + ( p ) + f K ( ω K ( p )) ω K ( p )
13 13 ∞ Z d pp π " f K − ( ω K − ( p )) ω K − ( p ) + f ¯ K ( ω ¯ K ( p )) ω ¯ K ( p ) , where ω K ± ( p ) = ± g ∗ ωK ω ± g ∗ ρK R + q m ∗ K + p , m ∗ K = m K − g ∗ σK σ (34)and g ∗ mK = g mK χ m for the parameters of the set B [6], we use here. Note thatin neglecting Coulomb effects the energy of K + coincides with that for K and the energy of K − coincides with the ¯ K energy.The η -contribution to the pressure is given by P part η = 13 ∞ Z d p p π f η ( ω η ( p )) ω η ( p ) , ω η = q m ∗ η + p (35)with m ∗ η = m η − X b ∈{ b } Σ ηb f π D ¯Ψ b Ψ b E / X b ∈{ b } κ ηb f π D ¯Ψ b Ψ b E (36)expressed in ref. [29] in terms of the total baryon scalar density P b ∈{ b } n sb = P b ∈{ b } D ¯Ψ b Ψ b E .In the above equations the Bose distributions of excitations are f i = 1exp[( q m ∗ + p − µ ∗ i ) /T ] − , (37) µ ∗ i = µ i + Q i µ ch − Q veci g ∗ ω i ω − Q veci g ∗ ρ i R ,i ∈ { bos . ex } = σ ′ , ω ′ , ρ ′ + , ρ ′ , ρ ′ − ; π + , π , π − ; K + , K , K − , ¯ K ; η ; K ∗ + , K ∗ , K ∗− , ¯ K ∗ ; η ′ ; ϕ . Here µ i = µ str for strange particles K and K ∗ and µ i = − µ str for theiranti-particles; Q i is the boson electric charge in proton charge units (ignoringCoulomb effects we put Q i = 0), Q veci = +1 for particle, Q veci = − Q veci = 0 for ”neutral” particles (with all zero charges includingstrangeness). We take couplings as in [6], g ∗ mi = 0 for all i except i = K, η .For η we select the parameter choice Σ ηb = 140 MeV, κ ηb = 0 .
2. As in [6] weassume m ∗ K ∗ = m K ∗ , m ∗ η ′ = m η ′ , m ∗ ϕ = m ϕ due to the absence of the corre-sponding experimental data. In this paper, we will focus on the considerationof the isospin-symmetrical matter; therefore, we may put R = 0.14he density of the gas of Bose excitations of the given species i is determinedby the integral n i = g i ∞ Z d p p π f i ( p ) , i ∈ { bos . ex } , (38)where g i is the corresponding degeneracy factor.Note that the p -wave pion and kaon terms can easily be included. For that oneneeds to replace ω π ( p ) and ω K ( p ) with more complicated expressions, see refs.[24,26]. In order to obtain the temperature-density dependent pion and kaonspectral functions one needs to calculate the pion and the kaon self-energies in-cluding baryon-baryon hole loops and baryon-baryon correlation effects. Theseeffects may result in appearance of the pion and antikaon condensates in denseand not too hot nuclear matter. Within the SHMC model we incorporate onlyinteractions of particles through mean fields. Even in this case many couplingconstants are not well fixed due to the lack of experimental data. Thus, wepostpone the inclusion of the p -wave interactions to the future work. A more nontrivial task is to fix the terms P part σ , P part ω , P part ρ . For the σ ′ , ω ′ , ρ ′ contributions to the pressure we use the standard ideal gas expressions witheffective masses determined as follows. Let us first focus on the contributionof the σ ′ -excitations. In order to get P part σ , we should expand total pressure P [ σ, ω ( σ ) , R ( σ )] in σ ′ = σ − σ cl . The term linear in σ ′ does not give acontribution due to subsequent requirement of the pressure minimum in σ cl .The quadratic term produces effective σ particle mass squared,( m part ∗ σ ) = − d P [ σ, ω ( σ ) , R ( σ )] dσ = − d P [ f, ω ( f ) , R ( f )] df dfdσ ! . (39)The first-order derivative dP/df = 0, as it follows from the equations of mo-tion. Since we deal with the strong interaction problem and the general solu-tion is impossible, one should use some approximations. First, keeping onlyquadratic terms in all thermodynamical quantities in boson fluctuating fieldswe disregard boson excitation contributions to the σ ′ , ω ′ , ρ ′ effective masses(higher order effects in boson fluctuating fields). Within this approximationwe neglect the P bos . ex . term in (39). Moreover, our aim is the constructionof a thermodynamically consistent model where the baryon density and theentropy density, calculated by using the thermodynamic relation (26) and re-lation S = ∂P∂T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) µ , (40)15oincide with quasiparticle expressions (25) and S = S bar + S bos . ex . , S bar = X b ∈{ b } g b Z d p b p b π [ − (1 − f b )ln(1 − f b ) − f b ln f b ] ,S bos . ex . = X i ∈{ bos . ex . } g i Z d p i p i π [(1 + f i )ln(1 + f i ) − f i ln f i ] , (41)namely, the latter expressions in ref. [6]. In order to keep thermodynamicconsistency of the model, we suppress the baryon contribution in (39). Thisterm is associated with taking into account of the baryon-antibaryon andbaryon-baryon hole loops. Problems arisen, if one includes baryon loops, arediscussed in Appendix B.Thus, the squared effective mass of the σ ′ excitation will be given by theexpression( m part ∗ σ ) ≃ − X m ∈{ m } d P MF m [ f, ω ( f ) , R ( f )] df dfdσ ! . (42)Using the relation χ σ = Φ = 1 − f we find that dfdσ = g σN m N (1 − f ) . (43)For the isospin-symmetrical nuclear matter N = Z , we obtain( m part ∗ σ ) = " U ′′ f + m N C σ g σN m N (cid:19) (1 − f ) . (44)One could use a different approximation introducing the effective mass termwith the help of the expression for the Hamiltonian (Lagrangian). In the lat-ter case, baryon terms are added to the meson mean field contribution andderivatives of the Hamiltonian are taken at fixed Ψ B . In this case( m part ∗ σ ) = * d Hdσ + = * d Hdf + dfdσ ! , (45)where H is the Hamiltonian and the averaging procedure is carried out over athermal equilibrium state after taking the derivatives. We used that D dHdσ E = 0,as it follows from the equations of motion. Neglecting the meson excitationterms in H we find 16 d Hdf + = − X m ∈{ m } d P MF m [ f, ω ( f ) , T ] df + * X b ∈{ b } d H b [ f, ω ( f ) , T ] df + = U ′′ f + m N C σ + C ω m N d df η ω ! X b x ωb n sb , (46)where the baryon (antibaryon) scalar density n sb is given by n sb = (2 s b + 1) ∞ Z dpp π m ∗ b ω b f b . Since in ref. [6] the value η − ω was chosen as a linear function of f , η − ω = (1 + zf ) / (1 + zf ( n )) , z = const, (47)( z = 0 .
65 in the KVOR-based SHMC model), we get P b ∈{ b } d H b [ f,ω ( f ) ,T ] df = 0,and both results (42) and (45) coincide. From here it is seen that in ourapproximation ( m part ∗ σ ) does not explicitly depend on the baryon variablesand the temperature. Obviously, within this approximation our model remainsthermodynamically consistent since meson excitation terms in the pressure donot contribute to the derivatives of the pressure over the baryon chemicalpotential (to obtain n B ) and over the temperature (to get entropy S ).Note that in [6] we introduced the squared effective mass of the σ ′ -excitationas the second derivative of the energy density. Since at the same time the con-tributions of all baryon particle-particle hole and particle-antiparticle (loop)terms were suppressed, this result coincides with expressions (42) and (45).Within our approximation the masses of vector particles are given by( m part ∗ ω i ) = * ∂ H∂ω i + = − X m ∈{ m } ∂ P MF m [ f, ω ] ∂ω i , (48)( m part ∗ ρ i ) = * ∂ H∂R i + = − X m ∈{ m } ∂ P MF m [ f, ω ] ∂R i , i = 0 , , , . In this case zero-components of one ω and three ρ -excitations prove to be thesame as those following from the mean-field mass terms m part ∗ ω = m ω | Φ ω ( f ) | , m part ∗ ρ = m ρ | Φ ω ( f ) | . (49)Moreover, there are excitations of two magnetic-like components of ω and sixmagnetic-like components of ρ . Their masses are given by eqs. (49) since thereare no ~ω and ~R mean fields, namely, expressions (49) were used in [6].17 hus, neglecting the baryon and boson excitation contributions to the effec-tive meson excitation masses we obtain a rather simple thermodynamicallyconsistent description.3.5 Some numerical results In Figs. 2, 3 we demonstrate the size of corrections which arise provided bosonexcitations are incorporated in the equations of motion. Particle excitationmasses are calculated following eq. (42) for σ meson excitation and (49) forthe nucleon, ω , and ρ excitations. Figure 2 demonstrates the ratio of theeffective masses to the bare masses for the nucleon- ω ′ - ρ ′ (left panel) and for σ ′ (right panel) as a function of the temperature at n B = 0 (thin curves) andat n B = 5 n (bold lines). Solid curves are calculations of the given paper,when boson excitations contribute to the equations of motion, whereas dashedcurves show perturbative calculations of ref. [6]. Although for T > ∼
120 MeVthere appear pronounced differences between both treatments of excitations,the qualitative behavior remains unchanged. Effective masses of all excitationsexhibit a similar behavior as functions of the temperature and the density, ina line with the mass-scaling hypothesis that we have exploited for the meanfields.Let us also compare our result for N , ω , ρ effective masses (Fig. 2 left) withthat previously obtained in the model of ref. [13], their Fig. 1 left (for N ) andright (for ω, ρ ). Shapes of the curves look similar. However in our case N , ω , ρ are scaled by one scaling function, whereas ref. [13] used different scalings for N and for ω, ρ . In their case the masses drop much stronger with the densityincrease but remain finite at high temperatures ( T > ∼
200 MeV), whereas inour case they drop to zero at T c ∼
210 MeV.The effect of the mentioned nonperturbative treatment of boson excitations onthermodynamic quantities is presented in Fig. 3. The temperature dependenceis shown for the reduced pressure calculated in the given work (solid lines) andin the perturbative treatment of boson excitations [6] at n B = 0 and n B = 5 n .One can see that the differences are rather noticeable only in the temperatureinterval 170 < T <
200 MeV, provided the baryon-meson couplings are notartificially suppressed. In the case ( n B = 0), when all g σb couplings except fornucleons are artificially suppressed by factors 2 / / the temperatureinterval, where solid and dash curves deviate from each other, is broader butthe value of the deviation is smaller. The curve for g σb suppressed by 1 / T c < T < ∼
230 MeV. As has beenmentioned in [6] one could fit the lattice data even in a broader region of As in Fig. 10 of [6], we suppress g σb -couplings, not g mb , as mistakenly indicatedthere. m * / m T, MeV n B =0n B =5 n
50 100 150 200 m * / m T, MeV
Fig. 2. Temperature dependence of the effective-to-bare mass ratio for nucleon- ω ′ - ρ ′ (left panel) and for σ ′ (right panel) for n B = 0 and n B = 5 n . The solid lines areour results, the dashed ones are perturbative calculations from [6]. To guide the eye,the horizontal dots show the m ∗ σ = 2 m π and m ∗ σ = m π thresholds.
50 100 150 200 25001234567891011 n B =5 n P / T T, MeV n B =0 Fig. 3. Temperature dependence of the reduced pressure. Solid lines show cal-culations of the given work, whereas dashed lines demonstrate the results of theperturbative treatment of boson excitations [6]. The curves labeled by 2 / / g σb couplings except for nucleons are suppressed byfactors 2 / /
10, respectively. Filled squares show the lattice QCD result forthe 2+1 flavor case [30]. temperatures (up to 500 MeV) if one introduced χ σ < Φ. A violation of theuniversality of the σ scaling would be also in a line with that we have used for ω and ρ , η ω = 1 and η ρ = 1, as required to describe properties of neutron stars,cf. [26]. However we will not elaborate such a possibility in the present work.Instead we simply suppress g σb , b = N . Hydrodynamic simulations of heavyion collisions are fully determined by the EoS to be described with quark and19luon degrees of freedom or with only hadron ones. With such an EoS (withsuppressed g σb ) a quark liquid would masquerade as a hadron one.
500 600 700 800406080100120140160180 T , M e V B , MeV Au+Au, b=2 fm
E=10.7 AGeV (S/N B =12) E=8 AGeV (S/N B =10.9) E=6 AGeV (S/N B =9.8) E=4 AGeV (S/N B =8.3) E=2 AGeV (S/N B =6.3) Fig. 4. Isentropic trajectories for central Au+Au collisions at different bombardingenergies calculated in the present paper (solid lines) and with the perturbativetreatment of boson excitations [6] (dashed lines). The results are presented for theSIS-to-AGS energies. Experimental points with error bars are taken from [31]. Thefreeze-out points marked by stars are obtained in [6]. Thin line corresponds to thefreeze-out curve in [31] while dash-dotted line is that from [32].
In Fig. 4 we show the isentropic trajectories for central Au+Au collisions atdifferent bombarding energies calculated in the present paper (solid lines) andwith the perturbative treatment of boson excitations from ref. [6] (dashedlines). As we can see from Fig. 4, for AGS energies the trajectories calcu-lated here almost coincide with those computed in ref. [6] where boson exci-tations were treated perturbatively. Then, as in [6] we extend our analysis tohigher bombarding energies. The entropy per baryon participants was calcu-lated in [35] within the 3-fluid hydrodynamic model assuming occurrence ofthe first order phase transition to a quark-gluon plasma. The energy rangefrom AGS to SPS was covered there. We use the
S/N B values for E lab =158,80, 40 and 20 AGeV, at which the particle ratios were measured by the NA49collaboration (cf. [31]). At the RHIC we put S/N B = 300 in accordance withan estimate in [34]. 20
100 200 300 400 500 600 700 800100150200 B =300 T , M e V B , MeV Fig. 5. Isentropic trajectories for central Au+Au collisions at different bombard-ing energies calculated in the present paper. Calculations are performed from AGSto RHIC energies for not suppressed couplings (solid lines) and with suppressed g σb couplings (except for nucleons) by a factor of 1 /
10 (dashed lines). The filleddiamonds and triangles are obtained from the 4 π particle ratios in [31] and [33],respectively. Filled circles are RHIC data based on the middle rapidity particle ra-tios [31]. Open circles, squares and triangles are the lattice 2-flavor QCD results [34]for S/N B = 30, 45 and 300, respectively. The thin line corresponds to the freeze-outcurve in [31]. Since there is a reasonable fit of the lattice data on the pressure for T > ∼
200 MeV with reduced meson-baryon coupling constants, see [6] and Fig. 3,for high collision energies we show in Fig. 5 both the results, when couplingsare not suppressed (solid curves) and also when all g σb couplings except fornucleons are suppressed by a factor of 1 /
10. As is seen, the decrease in g σb improves the high-temperature description (as compared to the case of notsuppressed couplings) for the lattice data at high temperatures not only forthe case µ b = 0 but also for µ b = 0. For low temperatures the solid anddashed curves (the curves for not suppressed and suppressed g σb couplings,respectively) are very close to each other. Also note that one should be rathercritical performing comparison with the existing lattice data, especially forthe case µ b = 0 (presented in Fig. 5 for 2 flavors). Doing this comparison wetentatively hope that results of future more realistic lattice calculations willnot deviate much from the existing ones.21n all cases our new calculations presented in Fig. 5 (if one additionally sup-presses couplings), as well as corresponding perturbative calculations [6], de-scribe reasonably the lattice data on the T − µ B trajectories and freeze outpoints.Concluding, our improvement of the model with keeping boson excitationterms in the equations of motion neither spoils nor improves the agreementwith the lattice results and with the thermodynamic parameters extracted atfreeze out. Thus, both perturbative and nonperturbative treatments of bo-son excitation effects can be used with equal success. However, one shouldnote that thermodynamic consistency conditions are fulfilled exactly in thenon-perturbative treatment of the given work, whereas in the perturbativetreatment of [6] they were satisfied only approximately. In the framework of our model we treated all resonances as quasiparticlesneglecting their widths. Excitations interact only with the mean fields. Thusimaginary parts of the self-energies as well as some contributions to the realparts are not taken into account. Definitely it is only a rough approximationcarried out just for simplification.At the resonance peak the vacuum ∆-isobar mass width is Γ max∆ ≃
115 MeV.In reality Γ ∆ is the temperature-, density- and energy-momentum-dependentquantity. For low ∆-energies the width is much less than Γ max∆ . A typical ∆energy is ω ∆ − m ∗ ∆ ∼ T . Thus for low temperature, T < ∼ ǫ F ( ǫ F is the nucleonFermi energy) the effective value of the ∆-width is significantly less thanΓ max∆ . At these temperatures the quasiparticle approximation does not workfor ∆’s but their contribution to thermodynamic quantities is small. Whenthe temperature is > ∼ m π there appears essential temperature contribution tothe width and the resonance becomes broader [36]. At such temperatures ∆’sessentially contribute to thermodynamic quantities. Only for temperatures T > ∼ Γ max∆ ( T ) the quasiparticle approximation becomes a reasonable approximation.The ρ - and σ -mesons also have rather broad widths. E.g., at a maximum the ρ -meson width is about 150 MeV. The observed enhancement of the dileptonproduction at CERN, in particular in the recent NA60 experiment [37] on µ + µ − production, can be explained by significant broadening of the ρ in matter[2], though decreasing of the ρ mass could also help in explanation of the data[38] . Besides, as was shown in [40] to be consistent with the QCD sum rules, As demonstrated in [38], the calculated large mass shift is mainly caused by theassumed temperature dependence of the in-medium mass. Inclusion of this tem- ρ mass should be takeninto account. Even more, one should be careful with interpretation of theNA60 experiment: Dileptons carry direct information on the ρ meson spectralfunction only if the vector dominance is valid but generally it is not the case[41].Also particles which have no widths in vacuum like nucleons acquire the widthsin matter due to collisional broadening. Their widths grow with the temper-ature increase, cf. [42]. As we have argued above in case with ∆ isobars, thequasiparticle approximation may become a reasonable approximation at suf-ficiently high temperature, if T > ∼ Γ( T ).Let us roughly estimate the effect of finite baryon and meson resonance widthson particle distributions and the EoS in order to understand how much and inwhich temperature-density regions these effects may affect results calculatedwithin our quasiparticle SHMC model.It is convenient to introduce single-particle spectral b A r and width b Γ r functions(operators), cf. [42], b A r = − b G R r ( q ) = − c M r + i b Γ r / , b Γ r = − b Σ R r , (50)where b G R r ( q ) is the full retarded Green function of the fermion and b Σ R r is theretarded self-energy. The quantity c M r = ( b G ,R ) − − Re b Σ R r (51)demonstrates the deviation from the mass shell: c M r = 0 on the quasiparticlemass shell in the matter, b G ,R is the free Green function. Following defini-tion (50) the spectral function of the fermion has dimensionality of m − andthe width, dimensionality m , whereas for bosons the spectral function hasdimensionality of m − and the width, dimensionality m .Let us start with the consideration of a fermion spin 1 / ∞ Z γ h b A fr(+) ( q , ~q ) + b A fr( − ) ( q , − ~q ) i dq π = 2 , (52) perature dependence modifies the scaling hypothesis originally claimed by Brownand Rho. Some arguments on what the proper mass-scaling predicts for dileptonproduction in HIC, e.g. NA60, were given in [39]. is the corresponding Dirac matrix, and subscripts ( ± ) specify particle andantiparticle terms. The trace is taken over spin degrees of freedom.Simplifying the spin structure, as it is seen from (50), we can present14 Tr γ b A fr(+) = A fr = e A r ω. (53)Dealing further with A fr = e A r ω we may not care anymore about the spinstructure. The ∆ spin 3 / n fr = N r ∞ Z πp dp (2 π ) ∞ Z dω π A fr f fr , f fr = 1 e ( ω − µ ∗ r ) /T + 1 . (54)Here µ ∗ r = µ ∗ N for a nucleon resonance such as ∆. The antiparticle density isobtained from (54) with the help of the replacement µ ∗ r → − µ ∗ r . The baryondensity of the given species is then n barr = n r ( µ ∗ r ) − n r ( − µ ∗ r ). N r is the degen-eracy factor.To do the problem tractable, instead of solving a complete set of the Dysonequations, we may select a simplified phenomenological expression for A fr (com-pare with [43]), e.g., A fr = 2 ξω [2 e Γ r ( s ) + 2 δ ]( s − m ∗ ) + [ e Γ r ( s ) + δ ] , ξ = const, s = ω − p > , (55)with δ → +0. The value δ → A fr = 2 ξω [2 e Γ r ( s )]( s − m ∗ ) + [ e Γ r ( s )] + 2 ξω · πδ ( s − ( m ∗ r ) ) θ ( s th − ( m ∗ r ) ) . (56)Here s th is the resonance threshold value of s . Note that within our ansatz,the spectral function depends only on the s -variable. It might be the caseonly for a dilute matter, when the density and temperature dependence ofthe width is rather weak. Furthermore, instead of a calculation of the density-temperature dependent part of the width, which actually can’t be performed inthe framework of our model, we vary the energy dependence and the amplitudeof the width thus simulating in such a way collision broadening effects.24or the decay of the resonance into two particles ( r → s -variable dependence of the width: e Γ r ( s ) = Γ m r F ( s ) p . m . ( s, m ∗ , m ∗ ) p . m . ( m , m , m ) ! α θ (cid:16) s − ( m ∗ + m ∗ ) (cid:17) , (57) p . m . ( s, m ∗ , m ∗ ) = ( s − ( m ∗ + m ∗ ) )( s − ( m ∗ − m ∗ ) )4 s . Here Γ = const , the width tends to zero at the threshold s → s th = ( m ∗ + m ∗ ) , α = l + 1 /
2, with α = 1 / s and α =3/2 for p resonance. An extraform-factor, F ( s ), is introduced to correct the high-energy behavior of thewidth.To simplify expression (57), we may expand e Γ r ( s ) near the threshold trans-porting remaining s -dependence to the form factor: e Γ r ( s ) = Γ F ( s ) m r s / − s / m r − s / α θ ( s − s th ) , (58)where one can take F = 11 + [( s − s th ) /s ] β (59)with s and β being constants. The parameters can be fitted to satisfy exper-imental data.The energy dependence of the width causes a problem. With a simple ansatzfor the behavior e Γ r ( s ) we get a complicated m ∗ ( s ) dependence, as it followsfrom the Kramers-Kronig relation. However, since m ∗ ( s ) is a smooth functionof s , one may ignore this complexity taking for simplicity m ∗ as a constant.The factor ξ is introduced to fulfill the sum-rule: ∞ Z d s π e A r = 1 , (60)that yields ξ ≃ O (Γ /m ∗ r ) (for m ∗ r ≫ Γ ), m ∗ r > m ∗ + m ∗ . In the caseof m ∗ r < m ∗ + m ∗ there appears an extra quasiparticle term in the spectralfunction, that contributes to the sum-rule.In Figs. 6 and 7, we present the ratio R ∆ = n res∆ /n qp∆ of the ∆-isobar density,calculated following eq. (54), to the quasiparticle density, as a function of thetemperature. The results are presented for the baryon density n B = 0 (in Fig.25 n r e s / n qp T, MeVs th1/2 = m *N
50 100 150 200 s =(500 MeV) s =(1000 MeV) n B =0 T, MeVs th1/2 = m *N +m Fig. 6. Ratio of the ∆ isobar density, calculated with the inclusion of the width,to that of the quasiparticle one, R ∆ = n res∆ /n qp∆ , as a function of temperature at n B = 0 for two values of the resonance threshold energy s / th . The parameters ofcalculation are presented in the figure.
6) and for n B = 5 n (in Fig. 7). Our aim here is to demonstrate the effect ofa finite particle width. Therefore, instead of searching for the best fit of thespectral function to available experimental data we vary the parameters toshow how strongly the density ratio may depend on them. We take into accountthe p -wave nature of the resonance and use Γ = 115 MeV. The thresholdquantities are chosen to be s / th = m ∗ N (left panels) and s / th = m ∗ N + m π (rightpanels). In vacuum s / = m N + m π . Thus, taking s / = m ∗ N + m π we simulatethe vacuum resonance placed in the mean field (in our model m ∗ π = m π ). With s / = m ∗ N we simulate the effect of in-medium off-shell pions (virtual pionscan be produced in matter at any energy). The form factor F is computedwith β = 3 and s = (500 MeV) (solid lines) and s = (1000 MeV) (dashlines) to present the dependence of R ∆ on the high-energy behavior of thewidth, which is not well defined even in vacuum.As we can see, in all examples the curves are rather flat in the temperaturerange T > ∼
50 MeV ÷
100 MeV. For n B = 0, at T ≃
170 MeV there appearsa slight bend associated with a sharp decrease in the nucleon effective massfor T > ∼
170 MeV. For n B = 5 n the bend is smeared. A substantial devia-tion of the R ∆ ratio from unity for T > ∼
50 MeV ÷
100 MeV in the case with26 n r e s / n qp T, MeVs th1/2 = m *N
50 100 150 200 s =(500 MeV) s =(1000 MeV) n B =5 n T, MeVs th1/2 = m *N +m Fig. 7. The same as in Fig. 6 but for n B = 5 n . the cut-off s = (1000 MeV) is due to a larger width at high resonance en-ergies (for large s ) compared to the case with the cut-off s = (500 MeV) .In the latter example, the quasiparticle approximation becomes appropriatefor T > ∼
100 MeV. On the contrary, for s = (1000 MeV) and n B = 0 thequasiparticle approximation does not work at all. For low temperatures, the R ∆ ratio is significantly higher than unity and for higher temperatures it be-comes essentially smaller than unity. The broader the width distribution is,the smaller R ∆ at large temperatures. The ratio dependence on the thresh-old value s / th = m ∗ N is rather pronounced for s = (1000 MeV) but it isonly minor for s = (500 MeV) . Thus, we conclude that taking into accountthe energy dependence of the width might be quite important for very broadresonances, when the width only slowly decreases with energy. If the widthdrops rather rapidly with the energy increase, the quasiparticle approximationbecomes appropriate for calculation of thermodynamic quantities already atnot too high temperature. The baryon density dependence of the ratio R ∆ isnot so pronounced (especially for s = (500 MeV) ), since in our parameter-ization the spectral function depends on the density only through the value m ∗ r ( n B ) and the choice of s th . For low temperatures ( T < ∼ ÷
100 MeV), the R ∆ ratio becomes significantly larger than unity. We also pay attention tothe shift of the reaction thresholds due to the dependence of the width on s and the threshold s th on the density and temperature. This point can be veryimportant for fitting of particle momentum distributions.27o demonstrate the effect of the finite resonance width on thermodynamiccharacteristics of the system, we calculate the energy density of the non-interacting resonances (however with width). Then E fr = N r ∞ Z πp dp (2 π ) ∞ Z ds π ω e A r f fr + ( µ r → − µ r ) , (61)where the degeneracy factor for ∆ is N r = 16.In Figs. 8 and 9, we show the ratios of the energy density for ∆’s (with andwithout width) to the total baryon energy density at n B = 0 and n B = 5 n ,respectively. The solid and dashed curves correspond to calculational resultsfor the width with s = (500 MeV) and s = (1 GeV) . The dash-dottedcurve is computed within the quasiparticle approximation. Calculations areperformed for two values of the threshold energies s / = m ∗ N (left panels) and s / = m ∗ N + m π (right panels).As was expected, in all cases the quasiparticle result is much closer to thatfor s = (500 MeV) than for s = (1 GeV) . In the former case, the differ-ences are almost negligible. For s = (1 GeV) the ratio remains smaller thanthe quasiparticle result, except for low temperatures. Differences in the ratioswith and without taking into account the width in all cases are not too no-ticeable. The density dependence of the ratio proves to be pronounced even inour model, although the density dependence of the width is not incorporatedexplicitly. Summarizing, for calculation of thermodynamic characteristics, onemay use the quasiparticle approximation for baryons in the whole temperatureinterval under consideration provided the high energetic tail of the resonancewidth-function is not too long. If a resonance has a long energetic tail, thenthe longer the width tail is, the more suppressed the ratio of the resonanceenergy to the total energy, as compared to the corresponding quasiparticleratio.For charged bosons the spectral function follows the sum-rule, cf. [45], ∞ Z ds π A br = 1 . (62)We again consider a dilute matter assuming that the spectral function dependsonly on the s -variable. Only in this case one may consider a single spectralfunction for vector mesons, like ω and ρ , whereas in the general case one shouldintroduce transversal and longitudinal components. As before, by the charge we mean any conserved quantity like electric charge,strangeness, etc. E / E B T, MeV s th1/2 = m *N
50 100 150 200 E res , s =(500 MeV) E res , s =(1000 MeV) E qp n B =0 T, MeV s th1/2 = m *N +m Fig. 8. Ratio of the ∆ isobar energy density, calculated with the inclusion of thewidth ( E ∆ ), to the total baryon energy density ( E B ), as a function of temperatureat n B = 0 for two values of the resonance threshold energy s / th . The parametersof calculation are presented in the figure. E qp ∆ is calculated within the quasiparticleapproximation. The Noether charged boson density (of given charge) is given by n br = (2 s r + 1) ∞ Z πp dp (2 π ) ∞ Z dω π ω A br f br , f br = 1 e ( ω − µ ∗ r ) /T − . (63)For practical calculations it is convenient to present the spectral function inthe form: A br = ξ [Γ br ( s )]( s − m ∗ ) + [Γ br ( s )] / ξ · πδ ( s − ( m ∗ r ) ) θ ( s th − ( m ∗ r ) ) (64)with ξ = const introduced to fulfill the sum-rule. Replacing e Γ r ( s ) = Γ br ( s ),we may use eq. (57) for e Γ r ( s ) with α = 1 / s and α = 3 / p resonance.In Figs. 10 and 11, the ratio R ρ = n ρ /n qp ρ of the ρ + density, calculated followingeq. (63), to the quasiparticle density is shown as a function of the temperature.Here we take into account the p -wave nature of the resonance and use two29 E / E B T, MeV s th1/2 = m *N
50 100 150 200 E res , s =(500 MeV) E res , s =(1000 MeV) E qp n B =5 n T, MeV s th1/2 = m *N +m Fig. 9. The same as in Fig. 8 but for n B = 5 n . values for the width: a small width Γ = 75 MeV (left panels) and a verylarge one Γ = 250 MeV (right panels). We use s / = 2 m π , thus taking m ∗ = m ∗ = m π . The results are presented for n B = 0 and n B = 5 n in Figs.10 and 11, respectively. These figures show only a moderate dependence of theratios R ρ on the value of the width at the resonance peak (on Γ ). At highenergies the energy dependence of the width is more pronounced (comparesolid and dash curves). In our model the density dependence of R ρ reflectsthe behavior of m ∗ ρ ( n B ). For n B = 0 the ratio R ρ becomes larger than unityfor T < ∼
100 MeV, whereas for n B = 5 n R ρ is larger than unity only for T < ∼ ÷
50 MeV. The ratio R ρ ( n B = 5 n ) < R ρ ( n B = 0) since the resonancemass m ∗ ρ ( n B ) for n B = 5 n is closer to the threshold value 2 m π than in thecase of n B = 0.Slight bends of the curves at T ∼
170 MeV are associated with dropping ofthe effective mass of ρ below the threshold. Then there arises a quasiparticlecontribution to the spectral function being added to the high-energy widthterm (second term in (64)). In general, the behavior of R ρ and R ∆ is simi-lar. The quasiparticle approximation is rather appropriate for T > ∼
100 MeV,provided the widths have not too long high-energy tails (see curves with s / = 500 MeV). Under these conditions our quasiparticle SHMC model workswell. If the resonance width has a very broad high-energy tail (see curves with s / = 1000 MeV), the deviation of the R r ratio (for the r-resonance) from30
50 100 150 2000,51,01,52,0 s =(500 MeV) s =(1000 MeV) n r e s / n qp T, MeV n B =0s th1/2 = 2m =75 MeV =250 MeV T, MeV
Fig. 10. Ratio of the ρ meson density calculated with the inclusion of the width tothe quasiparticle one, R ρ = n res ρ /n qp ρ , as a function of the temperature at n B = 0.The parameters of calculation are presented in the figure. unity is rather pronounced, even for T > ∼
100 MeV. Since in this case R r < T > ∼
170 MeV even without suppressing the coupling con-stants. The latter procedure was used in [6] to demonstrate a possibility to fitthe lattice results within our quasiparticle SHMC model.Broad hadronic resonances having a short lifetime are of a particular inter-est for dynamics at the late stage of relativistic heavy-ion collisions. As thesystem expands and cools, it will hadronize and chemically freeze out (van-ishing inelastic collisions, no creation of new particles). After some period ofhadronic elastic interactions, the system reaches the kinetic freeze out stage,when all hadrons stop interacting at all (vanishing even elastic collisions). Af-ter the stage of the kinetic freeze out, particles overcoming remaining meanfields free-stream towards the detectors, where measurements are performed.The lifetimes of the ρ meson and ∆-baryon are τ ρ ≃ / Γ ρ ≃ . f m/c and τ ∆ ≃ . f m/c , respectively, being small with respect to the lifetime ofthe expanding system. So short-lived resonances can decay and regenerate inscattering process all the way through the kinetic freeze out. The regenerationprocess depends on the hadronic cross sections of resonance daughters. Thusthe study of different resonances can provide an important probe of the timeevolution of the source from the beginning of the chemical freeze out to the31 s =(500 MeV) s =(1000 MeV) n r e s / n qp T, MeV n B =5 n s th1/2 = 2m =75 MeV
50 100 150 200 =250 MeV T, MeV
Fig. 11. The same as in Fig. 10 but for n B = 5 n . kinetic one. In-medium effects can modify properties of hadron quasiparticlesand resonances during this stage. In particular, it concerns finite widths andenergy-momentum relation evolving in time. Values of chemical potentials andthe temperature characterizing particle momentum distributions become com-pletely frozen at the kinetic freeze out. From this moment only the mean fieldscan survive and the hadron effective masses further evolve towards the baremasses with which particles reach detectors.In Fig. 12 the ratios of yields of resonances to the yields of stable particles withsimilar quark content, the ρ /π − and ∆ ++ /p ratios, are presented as a func-tion of temperature. For all four species considered, our calculations take intoaccount the feed-down from higher resonances n feedi = n i + P r n r Γ r → i / Γ r . Be-ing in an agreement with experiment, the statistical analysis of stable hadronratios at the chemical freeze-out [31] shows that these ratios are getting almostenergy-independent at √ s NN > ∼
100 GeV. This statement seems to be validalso for resonances provided they are treated within the quasiparticle approx-imation, as follows from the weak T dependence of the ideal gas (IG) model Those hadrons from the particle data table which originally were not included intothe SHMC model set are treated here as the ideal gas of resonances with vacuummasses and vanishing widths. It is a quasiparticle model with the vacuum masses for all hadrons. Mesons withmasses m i ≤ . m i ≤ .
20 130 140 150 160 170 1800,00,10,20,30,40,50,6 r a t i o s T, MeV / -++ /p s =(500 MeV) Fig. 12. The resonance ratios as a function of temperature. The solid lines are cal-culated within the SHMC model with accounting for the resonance width accordingto eqs. (54), (58). Parameters are the same as in Figs. 6 and 10 (right panels, solidlines). The dashed lines are calculated for the IG model. These results are obtainedat µ B = 20 MeV for the RHIC energy. Experimental data [48] for central d + Au collisions are plotted by straight dotted lines with error bars. results (see dashed lines in Fig. 12). As is seen in Fig. 12, the resonance ratioscalculated in the SHMC and IG models almost coincide with each other tilltemperature about 140 and 155 MeV for the ∆ ++ /p and ρ /π − ratios, respec-tively, and then the difference between them increases with T . The growth ofthe resonance yield in the SHMC model is mainly due to the dropping of theeffective mass at high temperatures. Dependence of the resonance abundanceon the value of the width and other parameters (at the same temperature) israther moderate (compare the solid and dashed curves in Figs. 6 and 10).Earlier, the resonance statistical treatment has been considered in refs. [46]and [47]. Both models are ideal gas ones but in the hadronization statisticalmodel [47] the vacuum (energy independent) resonance widths are included.At the same values T, µ B these models provide quite close results but if onerefers to the particular collision energy, their results differ due to makinguse of different approximations for the freeze out T and µ B as functions ofthe collision energy (see Fig. 5). In the hadronization statistical model [47],predictions were made only for the SPS energies.Very recently the short-lived resonances ∆ ++ and ρ have been measured bythe STAR Collaboration at RHIC in d + Au collisions [48]. The observed ρ /π − ++ /p ratios practically do not depend on the charged particle number atthe mid rapidity, dN ch /dη ( i.e. on centrality), and for central (20% centrality)collisions are 0 . ± .
050 and 0 . ± . T ∼
145 and ∼
160 MeV for ∆ ++ /p and ρ /π − , respectively. However, largeexperimental error bars do not allow to make preference to any specific model,though the IG model predictions are slightly but regularly below that for theSHMC model. One should remind that at the RHIC energy the chemical freezeout temperature is estimated as 177 MeV [47,48] and kinetic one is about 120MeV. The ρ /π − ratio was also measured in peripheral Au + Au (200 GeV)collisions [49] to be as large as ρ /π − = 0 . ± . stat ) ± . syst ). Notethat the existence of the difference between the experimental value of ρ /π − ratio and the statistical model result has been considered as a problem inref. [47]. Following their estimate, ρ /π − ratio is about 4 · − at the kineticfreeze out temperature 120 MeV, and it is 0.11 at the chemical freeze out T = 177 MeV. Our result ( ∼ .
06 at T = 120 MeV) strongly deviates fromthe statistical estimate [47] at the kinetic freeze out. At the chemical freezeout temperature, T = 177 MeV, predictions of the ρ /π − ratio practicallycoincide in all ideal gas based models remaining lower than the experimentalratio, whereas our SHMC model is able to reproduce experiment at T ∼ ÷
170 MeV. It is also of interest that the measured resonances have amass shift about 50-70 MeV [48] which is consistent with the SHMC modelresults at T ∼
160 MeV as presented in Fig. 2. Thus, one may infer that dueto a possible decay, regeneration and rescattering, the broad resonances, ifthey are treated within the statistical picture with taken into account theirin-medium mass shift, freeze at a somewhat lower temperature than that atthe chemical freeze out but this temperature is significantly higher than thevalue of T ≈
120 MeV, characterizing the kinetic freeze out [50]. A dynamicalconsideration of the freeze out is needed to draw more definite conclusions.
In this paper we made attempts to find several improvements of the SHMCmodel of [6].In [6] the boson excitation effects were assumed to be small. Therefore theircontribution to thermodynamic characteristics was calculated using pertur-bation theory in the fields of boson excitations. Thereby, boson excitationcontributions in the equations of motion were dropped. The approximation The temperature concept should be used with care for d + Au collisions. In exper-iments, temperature is usually associated with the inverse slope of transverse massdistributions. σ ,and ω - ρ -nucleon excitations turn to be minor for T < ∼ ÷
120 MeV and growwith the temperature increase. Corrections to thermodynamic characteristics,like total pressure, energy, entropy etc. , remain moderate even at higher tem-peratures. Qualitatively, one may conclude that all results of [6] remainedunchanged. With boson excitation terms incorporated into the equations ofmotion, our quasiparticle SHMC model fulfills exactly the thermodynamicconsistency conditions.Then we discussed possible effects of the resonance widths. We assumed vac-uum but energy-dependent widths of resonances. Under this assumption thewidth effects included do not change qualitative behavior of the system. Nev-ertheless, one should note that in dense and/or hot matter particle widthsmay acquire essential density- and temperature-dependent contributions thatmay significantly affect properties of the system, e.g., see [42], where for thecase of hot baryon-less system it is shown that width effects may completelysmear fermion distributions. We illustrated that the estimated yields of short-lived resonances to be important for the late stage of relativistic heavy-ioncollisions can be described sufficiently well if one treats those resonances inthe framework of our SHMC model with the vacuum widths at the freeze out.In the paper, the σ variable was considered as an order parameter. Then theeffective masses of σ ′ , ω ′ , ρ ′ excitations and the effective masses of nucleonshave a similar behavior as a function of the baryon density and tempera-ture. The effective masses remain rather flat functions of the temperature andsharply drop to zero only in the vicinity of T c . In the large temperature inter-val mentioned, the effective masses as a function of the density first decreasewith the density increase and then, at a very high density, begin to grow, cf.[6].In Appendix A we calculated what modifications of our model could be, ifthe σ - ω and ρ fields were treated on equal footing. In the latter case, thedensity-temperature behavior of the σ ′ excitation mass essentially differs fromthat of the ω ′ - ρ ′ excitation masses, namely, the σ ′ excitation mass drops tozero at n c ≃ . n and only slightly depends on T . Thus, if this model wasapplied to analyze heavy ion collisions, we would face with a problem of Bosecondensation of σ ′ excitations for n B > n c ∼ ÷ n in a broad temperatureinterval. Another unpleasant feature of a model version like this is a drasticdifference in the density behavior of the σ ′ effective mass and that of ω ′ - ρ ′ excitations. These are additional arguments in favor of the treatment of the σ variable as an order parameter (as we did earlier in ref. [6] and here in thepaper body). 35hen in Appendix B an attempt was made to incorporate the nucleon-nucleonhole and nucleon-antinucleon loop effects in our model. The loop terms men-tioned were calculated within the perturbative approach. If these terms aretaken into account, the σ ′ excitation effective mass exhibits rather unrealisticbehavior. Thus, the model completely loses its attractiveness if baryon loopsare included (at least within the perturbative approach). We have checked thatthis unpleasant feature is a common feature of many RMF models, includingthe original Walecka model (previously authors of ref. [51] arrived at a similarconclusion considering vacuum loop corrections in the Walecka model). In theFermi liquid approach to diminish contribution of the fermion loop terms oneincorporates the vertices corrected by a short-range baryon-baryon interactionintroduced with the help of the Landau-Migdal parameters, cf. [24,25,26]. Inthe framework of our SHMC model short-range correlation effects are simu-lated by the ω , σ , ρ exchanges and are non-local. Thereby, we do not includethese effects in the present work. Summarizing, either higher order fluctuationeffects should be included in all orders, that is a complicated problem, or theyshould be skipped within RMF based models. So we skipped these terms inour truncated scheme.Concluding, the SHMC model introduced in ref. [6] can be considered as areasonable model for application to the description of hadronic matter in abroad baryon density-temperature range, provided higher order fluctuationsof fermion fields are not included. Acknowledgements
We are very grateful to A. Andronic, Yu.B. Ivanov, E.E. Kolomeitsev, K.Redlich, and V.V. Skokov for numerous illuminating discussions, valuable re-marks, and constructive criticism. This work was supported in part by theDeutsche Forschungsgemeinschaft (DFG project 436 RUS 113/558/0-3), andthe Russian Foundation for Basic Research (RFBR grants 06-02-04001 and08-02-01003).
Appendix A. Treatment of the σ field as an independent variable. Asnoted in the paper, we continue to suppress contributions of baryon loops. Ifthe σ and ω fields are treated on equal footing, i.e. as independent variables(”ind.var.”), one can determine the effective σ ′ -mass using partial derivativesof the pressure,( m part ∗ σ ) . var . = − ∂ P m ∈{ m } P MF m [ f, ω ] ∂f dfdσ ! , (65)rather than full derivatives. 36t is convenient to present d P m ∈{ m } P MF m [ f, ω ( f )] df = ∂ P m ∈{ m } P MF m [ f, ω ] ∂f + δ σ ,δ σ ≃ ∂ P m ∈{ m } P MF m ∂ω ∂ω ∂f ! + 2 ∂ P m ∈{ m } P MF m ∂ω ∂f ∂ω ∂f . (66)Taking derivatives in (65) and (66) we use the exact equation of motion for f and ω , and suppress the boson excitation and baryon-loop contributions tothe pressure in the final expression.Taking partial derivatives we get − ∂ P m ∈{ m } P MF m [ f, ω ] ∂f ≃ m N C σ + U ′′ f − ω m ω , (67)and δ σ = − ω m ω , where η σ = 1 was used.If the effective σ ′ -excitation mass squared is determined using the Hamiltonianwe obtain * ∂ H [ f, ω ] ∂f + = m N C σ + U ′′ f − ω m ω + ω ∂ χ ω ∂f X b ∈{ b } g ωb t b n b , (68)that differs from (67) by the last term.In Fig.13, we show the ratio of the effective-to-bare masses of the σ ′ excitationas a function of the baryon density at different values of the temperature fortwo possible treatments of the σ -field, as an order parameter, see (42), andas an independent variable. In the latter case two expressions, (65) and (68),were used to calculate the value m part ∗ σ . The solid lines are evaluated followingeq.(65) for T = 50 ,
150 and 175 MeV (from the top to the bottom); and thedashed curves, using (68). In both the cases the effective σ ′ excitation mass,calculated following (65), drops to zero at n B ≈ . n (for T = 50 MeV)and for n B ≈ . n (for T = 150 MeV), respectively, which demonstrates amoderate temperature dependence of the effective mass up to T ∼
150 MeV.For higher temperatures the critical density significantly decreases. In all thecases, the differences between calculations following (65) and (68) are mi-nor; however, they significantly deviate from calculational results followingeq. (42) (dash-dotted curves). The behavior of the effective σ ′ excitation masscalculated using (65) and (68) is in contrast with the behavior of the ω ′ - ρ ′ - N effective masses which do not reach zero for all densities (see dash-dottedcurves). Since one of our main goals was to construct a model based on the37 m * / m n B /n T=50 MeVT=150 MeVT=175 MeV
Fig. 13. The effective σ ′ excitation mass as a function of density calculated usingeqs. (65) (solid curves), (68) (dash lines), and (42) (dash-dotted curves) for differenttemperatures. Straight dotted lines correspond to m σ = 2 m π and m σ = m π . idea of a rather similar behavior for all σ ′ - ω ′ - ρ ′ - N excitation masses, we refuseconsidering the σ field, as an independent variable, i.e., we refuse to considerit on equal footing with the ω and ρ fields. Therefore, instead of either eq. (65)or eq. (68), here we use eq. (42) treating the σ field as an order parameter. Appendix B. Inclusion of baryon loops.
In the general case, the σ ′ excitation mass is given by eq. (39), provided the σ field is treated as an order parameter. We will continue to keep only quadraticterms in fluctuating boson fields. Thus, we drop the boson excitation termin the pressure in expression (39) but we will keep the baryon term. So incontrast with (42), we use the expression( m part ∗ σ ) ≃ − " d P m ∈{ m } P MF m [ f, ω ( f )] df d P b ∈{ b } P b [ f, ω ( f )] df dfdσ ! . (69)An important difference between (69) and (45) (the latter expression yieldsthe same result as (42)) is that derivatives of the Hamiltonian are taken atfixed Ψ B , whereas the total pressure P depends on the baryon occupations,which should be varied. Thereby, (69) includes extra contributions from thebaryon-baryon hole and baryon-antibaryon excitations.Let us now calculate an additional, purely baryon contribution to the σ ′ -excitation mass incorporated in (69). Using that ∂f b ∂f = ∂f b ∂ω b " ∂ω b ∂f + g ωb ω ∂χ ω ∂f , ω b p ∂f b ∂p = ∂f b ∂ω b , (70)and (26), with the help of partial differentiations we find δ B ( m part ∗ σ ) = − dfdσ ! X b ∈{ b } ∂ P b [ f, ω ] ∂f , − X b ∈{ b } ∂ P b [ f, ω ] ∂f = m N X b ∈{ b } x σb " n sb m ∗ b − L b + X b ∈{ b } g ωb ω t b n b ∂ χ ω ∂f − X b ∈{ b } g ωb ω ∂χ ω ∂f ! B b + 2 m N X b x σb g ωb ω ∂χ ω ∂f e L b . (71)We also use the equations of motion and the relation (cid:16) ∂m ∗ b ∂f (cid:17) = − x σb m N .Here the quantities B b = L b + n sb m ∗ b , (72)and L b = N b ∞ Z dp π ω b f b (73)are the baryon-baryon hole and baryon-antibaryon loop-terms taken at zeroincoming energy and momentum.We also introduce a similar quantity 39 m * / m n B /n T=50 MeV
T=150 MeV n B /n n B /n T=175 MeV
50 100 150 200 2500,00,20,40,60,81,0 m * / m T, MeV n B =0
50 100 150 200 250
T, MeVn B =0.2 n
50 100 150 200 250
T, MeV n B =1 n Fig. 14. The ratio of the effective σ ′ excitation mass to the bare mass as a functionof density (top) and temperature (bottom) calculated by means of eqs. (42), (solidcurves), eq. (78) (dashed curves) and eq. (77) (dash-dotted lines), see (69). Thestraight dotted lines correspond to the level of m σ = 2 m π and m σ = m π . e L b = N b ∞ Z dp π m ∗ b t b f b . (74)Then we calculate an additional contribution δ = − d P [ f, ω ( f )] df + ∂ P [ f, ω ] ∂f
40 ( m part ∗ . ω ) ∂ω ∂f ! + 2 m ω ω ∂ Φ ω ∂f − X b ∈{ b } g ωb t b n b ∂χ ω ∂f + χ ω ω ∂χ ω ∂f X b ∈{ b } g ωb B b − m N χ ω X b ∈{ b } g ωb x σb e L b ∂ω ∂f , (75)that distinguishes full and partial derivative terms. Here ∂ω ∂f X b ∈{ b } g ωb m ω η ω B b = X b ∈{ b } g ωb m ω η ω χ ω " x σb m N e L b − B b g ωb ∂χ ω ∂f ω + X b ∈{ b } g ωb t b n b m ω ∂ [ η ω χ ω ] − ∂f (76)with ( m part ∗ . ω ) = ∂ P MF /∂ω .In [6], the baryon excitation loop L b contributions were suppressed. On theother hand, one may check that L b > n sb /m ∗ b , L b > e L b . Therefore, in [6] wedisregarded the term δ B ( m part ∗ σ ) as the whole, as well as the terms ∝ B b and e L b in (75) and (76). Thus in [6] eq. (42) is actually used, being compatiblewith (45).Incorporating the loop terms mentioned, one should use the following expres-sion ( m part ∗ σ ) = − X m ∈{ m } ∂ P MF m [ f, ω ] ∂f + δ B ( m part ∗ σ ) + δ dfdσ ! , (77)where partial contributions are given by (67), (71) and (75).If σ and ω are treated on equal footing, i.e. as independent variables (”ind.var.”),one should use( m part ∗ σ ) . var . = − ∂ P [ σ, ω ] ∂σ ≃ − ∂ P [ f, ω ] ∂f dfdσ ! . (78)The ratio of effective-to-bare masses of the σ ′ -excitation is shown in Fig. 14 asa function of the baryon density for three values of temperatures (top panel)and as a function of the temperature for three values of the baryon density n B (bottom panel) calculated by means of eq. (77), i.e., following (69). Resultsare compared with (78) and (42). As we can see, the inclusion of the baryonloop terms (performed within the perturbation theory) completely destroys41ll achievements of our SHMC model (besides the case n B = 0 when theseloop terms are suppressed).Thus, at the moment we can see no simple way to generalize the mean fieldbased the SHMC model which would include baryon-antibaryon and baryon– baryon hole fluctuations. References [1] V. Metag, Prog. Part. Nucl. Phys. (2008) 245 [arXiv:0711.4709].[2] R. Rapp and J. Wambach, Adv. Nucl. Phys. (2000) 1; R. Rapp,nucl-th/0608022.[3] P. Senger, J. Phys. G30 (2004) S1087; Acta Phys. Hung.
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