Properties of nuclear matter in relativistic Brueckner-Hartree-Fock model with high-precision charge-dependent potentials
aa r X i v : . [ nu c l - t h ] J u l Properties of nuclear matter in relativisticBrueckner-Hartree-Fock model with high-precisioncharge-dependent potentials
Chencan Wang and Hong Shen
School of Physics, Nankai University, Tianjin 300071, China
Jinniu Hu ∗ School of Physics, Nankai University, Tianjin 300071, ChinaStrangeness Nuclear Physics Laboratory,RIKEN Nishina Center, Wako, 351-0198, Japan
Ying Zhang
Department of Physics, Faculty of Science,Tianjin University, Tianjin 300072, ChinaStrangeness Nuclear Physics Laboratory,RIKEN Nishina Center, Wako, 351-0198, Japan (Dated: July 6, 2020) bstract Properties of nuclear matter are investigated in the framework of relativistic Brueckner-Hartree-Fock model with the latest high-precision charge-dependent Bonn (pvCD-Bonn) potentials, wherethe coupling between pion and nucleon is adopted as pseudovector form. These realistic pvCD-Bonn potentials are renormalized to effective nucleon-nucleon (
N N ) interactions, G matrices. Theyare obtained by solving the Blankenbecler-Sugar (BbS) equation in nuclear medium. Then, thesaturation properties of symmetric nuclear matter are calculated with pvCD-Bonn A, B, C poten-tials. The energies per nucleon are around − .
72 MeV to − .
83 MeV at saturation densities,0 .
139 fm − to 0 .
192 fm − with these three potentials, respectively. It clearly demonstrates thatthe pseudovector coupling between pion and nucleon can generate reasonable saturation propertiescomparing with pseudoscalar coupling. Furthermore, these saturation properties have strong cor-relations with the tensor components of N N potentials, i.e., the D -state probabilities of deuteron, P D to form a relativistic Coester band. The equations of state of pure neutron matter from pvCD-Bonn potentials are almost identical, since the prominent difference of pvCD Bonn potentials arethe components of tensor force, which provides very weak contributions in the case of total isospin T = 1. In addition, the charge symmetry breaking (CSB) and charge independence breaking (CIB)effects are also discussed in nuclear matter from the partial wave contributions with these high-precision charge-dependent potentials. In general, the magnitudes of CSB from the differencesbetween nn and pp potentials are about 0 .
05 MeV, while those of CIB are around 0 .
35 MeV fromthe differences between np and pp potentials. Finally, the equations of state of asymmetric nuclearmatter are also calculated with different asymmetry parameters. It is found that the effectiveneutron mass is larger than the proton one in neutron-rich matter. PACS numbers: 21.10.Dr, 21.60.Jz, 21.80.+a ∗ Electronic address: [email protected] . INTRODUCTION The infinite nuclear matter is a fundamental study subject in nuclear physics, whereprotons and neutrons compose a uniform many-body system in the nuclear matter. Dueto the translation invariance and rotation invariance, their wave functions are regarded asplane waves [1]. Although the nuclear matter is a hypothetical substance, the saturationproperties of symmetric nuclear matter can be extracted from experimental observations inthe central region of heavy nuclei [2, 3]. Furthermore, the equation of state of neutron-richmatter plays very important roles in the investigations of many astrophysical processes,such as, supernova explosion, neutron star cooling, binary neutron star merger, and soon [4–9]. Recently, with the worldwide development of radioactive facilities, many neutron-rich nuclei were discovered, where the isospin properties of nuclear matter are hopefullyextracted, i.e., the symmetry energy and its slope [10, 11]. Moreover, the central density ofcompact star is closed to five times of nuclear saturation density, which is far beyond thepresent experimental abilities. Therefore, the properties of nuclear matter from theoreticalresearches are highly demanded from both the investigations of neutron-rich nuclei andnuclear astrophysics [12–15].Due to the complexity of nucleon-nucleon (
N N ) potential, the study of nuclear matteris not as straightforward as the electron gas in condensed matter physics, although both ofthem are considered as uniform systems. The first calculation on the properties of nuclearmatter was achieved by Euler eighty years ago with second-order perturbation theory basedon Hartree-Fock approximation, where the
N N potential was chosen as a Gaussian func-tion [16]. With abundant experimental data of
N N scattering since 1940s, Jastrow proposedthat there was a very strong repulsive core at short-range distance between two nucleons infree space [17]. It means that the nuclear many-body system cannot be treated in the view-point of perturbation theory with the
N N interaction derived from the
N N scattering data,i.e., realistic
N N potential. Therefore, various nuclear many-body methods were developedto study the nuclear matter in the past seventy years.The strong repulsion of
N N potential at the short-range distance must be renormalizedin nuclear medium to generate the bound states of finite nuclei and saturation properties ofsymmetric nuclear matter. The earliest renormalization method was proposed by Brueckner et al. , where the repulsion can be removed by summations of all ladder diagrams included3n the nuclear medium
N N scattering process [18, 19]. The realistic
N N interaction will bereplaced by a density-dependent potential, G matrix. It can be used to describe the nuclearmany-body system in Hartree-Fock approximation reasonably. Meanwhile, a variationalmethod was shown by Jastrow through considering correlation functions to transfer the trialwave functions to the exact ones [20].With the rapid developments of high-precision N N potentials and the computationaltechniques, many advanced nuclear many-body methods with realistic
N N potentials weredeveloped in nonrelativistic framework, such as Brueckner-Hartree-Fock method [21–23],quantum Monte Carlo methods [24, 25], self-consistent Green’s function method [26],coupled-cluster method [27, 28], many-body perturbation theory [29–31], functional renor-malization group (FRG) method [32, 33], lowest order constrained variational method [34],and so on. These methods can more or less obtain the saturation behaviors of symmetricnuclear matter with modern high-precision
N N potentials, like Reid93 potential, Nijmegenpotential [35], AV18 potential [36], CD-Bonn potential [37], chiral N LO potentials [38, 39]and chiral N LO potentials [40–44]. However, all saturation properties from these calcula-tions cannot reproduce the empirical data,
E/A = − ± n = 0 . ± .
01 fm − only with two-body nuclear force. In order to reproduce the reasonable saturation proper-ties, the three-body nucleon force must be introduced in these nonrelativistic frameworks toprovide additional repulsion contributions [21, 45–47].In 1980s, the relativistic version of Brueckner-Hartree-Fock method was firstly proposedby Anastasio et al. [48], then developed by Horowitz et al. [49] and Brockmann et al. [50] Inthe relativistic Brueckner-Hartree-Fock (RBHF) model, a repulsive contribution is obtainedfrom the relativistic effect, which can properly describe the nuclear saturation properties withtwo-body realistic N N potential. Li et al. also verified that the contributions from three-body force and Z diagram, i.e. the nucleon-antinucleon excitation from the relativistic effectare partially in accord with each other [51], since the nucleon-antinucleon excitation affectsthe energy of nuclear matter in RBHF model via the second-order term of scalar meson,which can be regarded as one component of the microscopic three-body force generated bythe two-meson exchange between nucleon excitation states. Furthermore, the RBHF modelwas also applied to investigate the superfluity of nuclear matter, properties of neutron starand help to fit the free parameters of nuclear density functional theories [52–55]. Recently,Shen et al. realized a fully self-consistent calculation of RBHF model in finite nuclei system4nd extended this framework on the neutron drops [56–58]. The exact treatment for theangular integration of the center-of-mass momentum in asymmetric nuclear matter was alsoworked out by Tong et al. [59] within RBHF model.In RBHF model, the nuclear medium effect must be taken into account in the N N potential. Therefore, only few
N N interactions can be adopted, such as Bonn potentials [60,61]. With a large number of two-nucleon scattering data, many high-precision
N N potentialswere proposed based on the charge-dependent partial wave analysis from 1990s, as mentionedbefore, like Reid93 [35], AV18 [36], CD-Bonn potentials [37], and so on. In addition, thechiral
N N potentials derived from the chiral perturbation theory were also developed rapidly.The high-precision chiral
N N potentials, N LO potentials [40–44], were already presentedup to the fifth chiral expansion order. These state-of-the-art chiral potentials have beenwidely applied to describe the structures of finite nuclei and the saturation properties ofinfinite nuclear matter. When the three-body and four-body forces obtained from chiralperturbation theory systematically are included, the properties of light nuclei and nuclearmatter were reproduced perfectly below the breakdown scales [41, 45–47]. In such controlledhierarchy, the uncertainties from the few-body forces can be nicely estimated. We discussedsuch truncation errors and breakdown scale with Bayesian method for symmetric nuclearmatter and pure neutron matter with latest chiral potentials [62]. It was found that thebreakdown scale of these chiral potentials is around 600 MeV and the uncertainties fromhigh-order potentials increase with density. With such investigation, the properties of nuclearmatter below 0 . − should be believable for present chiral potentials. However, the studyof compact star requires the equation of state of nuclear matter above 0 . − . Therefore,it is very import to adopt an available many-body method and high-precision N N potentialsfor a better description of the nuclear matter especially in the high density region.In principle, the high-precision CD-Bonn potential with the same framework of Bonnpotential can be used in RBHF model. However, its pseudoscalar coupling scheme betweenpion and nucleon in relativistic framework will generate a very strong attractive contributionand thus can not reproduce the empirical saturation properties. Therefore, we attempted touse pseudovector coupling instead of the pseudoscalar one between pion and nucleon. NewpvCD-Bonn potentials were obtained by fitting the
N N phase shifts from the Nijmegenpartial wave analysis, which can be used in the RBHF model [63].In this work, properties of nuclear matter will be calculated in RBHF model with the latest5vCD-Bonn potentials. The exact angular integration of center-of-mass momentum will alsobe achieved. There are three pvCD-Bonn potentials with different tensor components, whoseeffects to saturation properties will be investigated. Furthermore, the charge symmetrybreaking (CSB) and charge independent breaking (CIB) effects also will be discussed, whichwere calculated by Sammarruca et al. in Brueckner-Hartree-Fock method with the originalCD-Bonn potential [64]. This paper is arranged as follows: in section II, the necessaryformulas of RBHF model for asymmetric nuclear matter will be introduced. In section III,properties of nuclear matter calculated by the new pvCD-Bonn potentials will be presented,including the equations of state of nuclear matter, the single-particle potentials, partial-wavecontributions to the potential energy, CSB and CIB effects in nuclear matter, and so on.In section IV, summaries and conclusions will be shown. The supplement derivations of in-medium Blankenbecler-Sugar (BbS) equation and the numerical details involved in practicalcalculations are given in the appendices.
II. THE RELATIVISTIC BRUECKNER-HARTREE-FOCK MODEL IN NU-CLEAR MATTER
In RBHF model, the single-nucleon energy in nuclear matter, E τ is given by a Diracequation with a single-particle potential U τ [50, 59],( α · p + βM τ + βU τ ) u τ ( p , s ) = E τ ( p ) u τ ( p , s ) , (1)where the subscript τ = p , n indicates proton or neutron. M τ is the nucleon mass and u τ ( p , s ) is the spinor solution of this Dirac equation with momentum p and spin s . Thesingle-particle potential in nuclear matter can be expressed as, U τ = U τ, s + γ U τ, v − γ · p U iτ, v , (2)which is decomposed into a scalar component U τ, s and a vector one U τ, v due to the trans-lational invariance and rotation invariance of infinite nuclear matter. The available inves-tigations showed that the momentum dependence of scalar and vector potentials are veryweak. Furthermore, the magnitude of the spacelike component of vector potential, U iτ, v , isnegligible comparing to the timelike one, U τ, v , and the scalar potential, U τ, s [54].Here, it must be emphasized that actually there are two schemes in RBHF model todetermine the Dirac structure of the nucleon self-energy. The first one is what we have6one following the framework of Brockmann and Machleidt [50], where the momentumdependence of the Dirac components of self-energy are neglected and the components arederived from the momentum dependence of the single-particle energies. The second one isevaluating the Dirac structure of the nucleon self-energy using a projection technique andkeep the momentum dependence [55]. These two schemes can yield rather similar resultsfor scalar and vector potentials in symmetric nuclear matter. However, as an example wemention that the isospin-dependent behavior of the effective nucleon masses in asymmetricnuclear matter, related to their scalar potentials, are completely opposite [65] . Therefore, inthis work, we will use the Brockmann-Machleidt scheme to discuss the properties of nuclearmatter with pvCD-Bonn potentials and investigate them in the future using the projectmethod.Therefore, the single-particle potential is approximately written as U τ ≈ U τ, s + γ U τ, v . (3)Within such approximation, the Dirac equation (1) in nuclear medium will be reduced as,( α · p + βM ∗ τ ) u τ ( p , s ) = E ∗ τ ( p ) u τ ( p , s ) (4)with effective nucleon mass and energy dressed in nuclear medium, M ∗ τ = M τ + U τ, s , E ∗ τ = E − U τ, v . (5)The wave function of Dirac equation in nuclear matter (4) can be solved analytically as aplane wave, u τ ( p , s ) = s E ∗ τ + M ∗ τ M ∗ τ (cid:18) σ · p M ∗ τ + E ∗ τ (cid:19) χ s , (6)where χ s stands the spin wavefunction for s state and E ∗ τ ( p ) = p p + M ∗ τ is the in-mediumon-shell single-particle energy. The normalization condition of spinor is ¯ u ( p , s ) u ( p , s ) = 1,here.The nucleon state vector can be expressed as | p , s i = u ( p , s ) and with its conjugatedvector h p , s | = u † ( p , s ). Hence, there will be an extra factor M ∗ /E ∗ to normalize thenucleon state due to the choice of ¯ u ( p , s ) u ( p , s ) = 1, M ∗ E ∗ h p , s | p , s i = 1 . (7)7he expectation value of single-particle potential can be evaluated within nucleon statevectors, U τ ( p ) = M ∗ τ E ∗ τ h p , s | βU τ | p , s i τ = M ∗ τ E ∗ τ U τ, s + U τ, v , (8)where the U τ, s and U τ, v are regarded as momentum independent. Their detailed valuesshould be determined by the N N interaction.In RBHF model, the realistic
N N interactions are replaced by effective G matricesdue to the nuclear medium effect, where the strong repulsion of realistic N N potentialat short-range distance is renormalized through summations of two-nucleon scattering lad-der diagrams. These diagrams can be contracted as an integral equation in free space,i.e., Bethe-Salpeter equation [66] in four-dimension space. It is usually reduced to a three-dimension equation to simplify the calculations. There are many reduction schemes, suchas, Blankenbecler-Sugar (BbS) equation [67], Thompson equation [68], Kadyshevsky equa-tion [69], and so on. Since the pvCD-Bonn potentials were obtained in the framework of BbSequation at free space [63], the G matrix in present RBHF model should be the solutions ofBbS equation in nuclear medium derived at appendix A, which is written as, G τ τ ( q ′ , q | P ) = V τ τ ( q ′ , q ) + Z d k (2 π ) V τ τ ( q ′ , k ) 2 W k W + W k Q τ τ ( k , P ) W − W k G τ τ ( k , q | P ) , (9)where q , k , and q ′ are initial, intermediate, and final relative momenta, respectively. P isthe two-nucleon center-of-mass momentum in nuclear matter rest frame. τ denotes the thirdcomponent of nucleon isospin. The transformations between nuclear matter rest frame andcenter-of-mass frame are k = p ′ − p , P = p + p ′ . (10)In BbS equation (9), V τ τ and G τ τ are related to the covariant amplitudes ¯ V τ τ and ¯ G τ τ with additional factors derived from the normalization convention in Eq. (7), which areexpressed as V τ τ = M ∗ τ E ∗ τ ¯ V τ τ M ∗ τ E ∗ τ , G τ τ = M ∗ τ E ∗ τ ¯ G τ τ M ∗ τ E ∗ τ . (11)To prevent the scattering states into the Fermi sea, a Pauli blocking operator Q τ τ ( k , P ) = (cid:26) | P + k | > k τ F and | P − k | > k τ F ) , , (12)is taken into account comparing to the free BbS equation, where k τF represents the Fermimomentum for nucleon τ . Furthermore, W = E ∗ τ ( P + q ) + E ∗ τ ( P − q ) and W k = E ∗ τ ( P + k ) + E ∗ τ ( P − k ) are the starting and intermediate energies respectively.8he equation of state (EOS) of nuclear matter is a function of baryon number density, n b and asymmetry parameter, α , n b = n p + n n , α = n n − n p n b , (13)where n p , n n are the baryon densities of proton and neutron. The averaged Fermi momentumis defined as k F = (3 π n b / , therefore Fermi momenta of proton and neutron are shownas k pF = (1 − α ) k F , and k nF = (1 + α ) k F . (14)When the Hartree-Fock approximation is applied, the single-particle potential of nucleonwith isospin τ is evaluated through G matrix U τ ( p ) = X τ ′ X ss ′ Z p ′ k τF d p ′ (2 π ) h p s, p ′ s ′ | G ττ ′ | p s, p ′ s ′ − p ′ s ′ , p s i . (15)In asymmetric nuclear matter, when charge symmetry breaking (CSB) and charge indepen-dence breaking (CIB) effects are considered, the G matrices are divided by G pp , G np , G pn ,and G nn . The corresponding single-particle potential for proton or neutron can be writtenas U τ = U τp + U τn . (16)At a given density, a self-consistent numerical calculation for singe-particle potential viaEq. (15) is started with initial scalar and vector potentials for proton and neutron. The G matrices are solved with two-body N N potential including the nuclear medium effect fromEq. (9). Then the new scalar and vector potentials can be extracted through Eq. (8). Withthe new scalar potentials, next-round calculation is worked out, until proton and neutronscalar potentials are converged at an acceptable accuracy. Finally, the energy per nucleonof nuclear matter at a fixed n b and α is evaluated by EA = 1 n b X τ,s Z p k τF d p (2 π ) M ∗ τ E ∗ τ h p , s | α · p + βM τ | p , s i − − α M p − α M n + 12 n b X ττ ′ X ss ′ Z p k τF d p (2 π ) Z p ′ k τ ′ F d p ′ (2 π ) h p s, p ′ s ′ | G ττ ′ | p s, p ′ s ′ − p ′ s ′ , p s i . (17)In practical calculations, the variables, p and p ′ , in integrals (15) and (17) are replacedby q and P . With further partial wave decomposition, G matrices are projected into LSJ representation. The solid angle dependence is removed in these integrals. The detailedexpressions for numerical calculations are explicitly given in Appendix B.9
II. RESULTS AND DISCUSSIONS1. The properties of symmetric nuclear matter and pure neutron matter
In our previous work [63], three charge-dependent Bonn potentials, named as pvCD-Bonn A, B, C, with pseudovector (PV) coupling between nucleon and pion were obtained byfitting the
N N scattering phase shifts driven from Nijmegen partial wave analysis. Thesethree potentials are almost identical except their tensor components due to the different πN coupling strengths. The D -state probabilities of deuteron, P D generated by pvCD-Bonn A,B, C potentials are 4 . , . . k F [fm −1 ] −25−20−15−10−50 E / A [ M e V ] (a) ABCABC α = 0.0 k F [fm −1 ] (b)α = 1.0 RBHFBHF FIG. 1: Equations of state of symmetric nuclear matter and pure neutron matter calculated byBHF and RBHF models with pvCD-Bonn A, B, C potentials. The panel (a) for symmetric nuclearmatter with the rectangular patch labeling the empirical saturation region. Panel (b) for pureneutron matter.
In panels (a) and (b) of Fig. 1, the energies per nucleon,
E/A as functions of Fermimomentum, k F , i.e., equations of state, for symmetric nuclear matter and pure neutronmatter, calculated by RBHF model are plotted within pvCD-Bonn potentials as solid curves,respectively. It can be found that saturation properties of symmetric nuclear matter frompvCD-Bonn A are closest to the empirical values shown as the rectangular area amongthree potentials. Its energy per nucleon, − .
83 MeV satisfies the value extracted from themass formula of finite nuclei, while the corresponding saturation density, n is 0 .
192 fm − ,that is higher than the normal one, 0 . ± .
01 fm − . On the other hand, the saturationdensity from pvCD-Bonn B potential, n = 0 .
158 fm − , however its energy per nucleonat saturation density is just − .
91 MeV. These differences between calculations of RBHF10odel and empirical values may be caused by that the non-nucleon degree of freedom, likethe ∆-isobar should be included in the
N N interaction as shown in recent works [70–72].The pvCD-Bonn A potential owns the weakest tensor component in three potentials andgenerates the largest saturation binding energy. On the whole, these results and conclusionsare similar with those from Bonn potentials by Brockmann and Machleidt [50]. For the pureneutron matter, the differences of energy per nucleon among three potentials are quite small.It is because that the tensor effect is very weak in pure neutron matter [73–75], where thetotal isospin of two neutron is T = 1 and contribution of tensor force is largely suppressed.Furthermore, the equations of state of symmetric nuclear matter and pure neutron mat-ter are also obtained in the nonrelativistic framework of BHF model with the same N N potentials, which are given as dashed curves in Fig. 1. At low density region, their ener-gies per nucleon are very similar with those in RBHF model. With the density increasing,the relativistic effect from the nucleon-antinucleon excitation, becomes obvious, providesstrong repulsive contributions, and leads to reasonable saturation properties of symmetricnuclear matter, which plays a similar role with the three-body force in the nonrelativis-tic ab initio approaches. Actually, Li et al. also confirmed that the three-body force andnucleon-antinucleon Z -diagram create the equivalent contributions in nuclear matter [51].Furthermore, the free nucleon mass in N N potential will be replaced by an effective nucleonmass, derived from the scalar potential to achieve the self-consistent RBHF calculation. n [fm −3 ] −20−15−10−5 E / A [ M e V ] A (4.22%)A (4.47%)B (5.10%)B (5.45%)C (5.53%)C (6.05%) (a) 0.1 0.15 0.2 0.25 n [fm −3 ] P D ( % ) (b) AAB BC C
Bonnp CD-Bonn
FIG. 2: A relativistic Coester band. The rectangle patch indicates empirical saturation region.The open patterns refer to the saturation properties from Bonn potentials, while the solid onescorrespond to those from pvCD-Bonn potentials.
In available investigations, the saturation properties of symmetric nuclear matter from11HF model included strong correlations with the strengths of tensor force in realistic two-body
N N interactions, which can be represented by the D -state probability of deuteron, P D . In general, the weaker tensor strength (smaller P D ) generates a larger saturation den-sity and deeper binding energy. This correlation relation was so-called ”Coester band” [76].In the panel (a) of Fig. 2, the saturation densities and the corresponding energies per nu-cleon of symmetric nuclear matter from pvCD-Bonn A, B, C potentials and Bonn A, B,C potentials in RBHF model are shown. There is a fine linear relationship between themwith different P D and this relativistic Coester band can cross over the empirical saturationregion. Generally speaking, the N N potential with lower P D , around 4% −
5% is preferredto generate relatively reasonable saturation properties. The correlation between D -stateprobability and saturation density is shown in the panel (b) with different potentials. Alarger tensor component results in a smaller saturation density. This is because that thetensor force provides the largely attractive contributions in low density region for symmetricnuclear matter and makes the saturation density go back, while the short-range correlationbecomes more important with density increasing [77].The pseudoscalar (PS) coupling and pseudovector (PV) coupling schemes between pionand nucleon from quantum field theory are equivalent for on-shell nucleon, since their cou-pling constants satisfy the relation, g π / M = f π /m π . However, their off-shell matrices havesignificant differences as shown in our previous work about pvCD-Bonn potentials [63]. Inpresent relativistic nuclear many-body methods, the PV coupling is adopted, which can sup-press the contributions from the antinucleon, i.e., pair suppression mechanism, and generatereasonable physical results. On the contrary, the PS coupling will drive a largely spuriousattraction [78, 79]. Therefore, in relativistic Hartree-Fock model [80–82] and RBHF model,the PV coupling interaction between pion and nucleon is required. In Fig. 3, the equationsof state of symmetric nuclear matter and pure neutron matter from RBHF model within theoriginal CD-Bonn potential by Machleidt [37] are plotted and are compared with the resultswithin pvCD-Bonn potentials. For symmetric nuclear matter, the saturation binding energyof CD-Bonn potential is about −
140 MeV. This extra attraction is obviously derived fromthe PS coupling. On the other hand, the equation of state of pure neutron matter from CD-Bonn potential is quite similar with those from pvCD-Bonn potentials. It is because thatthe contribution from pion interaction in one-boson-exchange potential is largely suppressedin pure neutron matter due the total isospin of two neutrons, T = 1, which almost does not12lay any role in total energy. n b [fm −3 ] −60−50−40−30−20−100 E / A [ M e V ] (a) (a) n b [fm −3 ] (b) pvCD-Bonn ApvCD-Bonn BpvCD-Bonn CCD-Bonn FIG. 3: The equations of state of symmetric matter and pure neutron matter from CD-Bonn andpvCD-Bonn potentials.
The calculations of RBHF model are usually complicated and time-consuming. To applythese results in other aspects easily, the equations of state of symmetric nuclear matter frompvCD-Bonn A, B, C potentials are better to be parameterized around the saturation density, n . It is suggested that the energy per nucleon can be expanded as the following functionin Ref. [83], E A ( n b ) = a (cid:18) n b n (cid:19) η + b (cid:18) n b n (cid:19) β ( α J = 0) . (18)Furthermore, the symmetry energy closed to the saturation density n also can be expressedas E sym ( n b ) = c (cid:18) n b n (cid:19) γ . (19)It can be approximately obtained from differences between the energies per nucleon of pureneutron matter and symmetric nuclear matter, E sym ( n b ) ≈ EA ( n b , α = 1) − EA ( n b , α = 0) . (20)The corresponding values of a , b, c and η, β, γ are obtained by fitting the numerical resultsof RBHF model with pvCD-Bonn A, B, C potentials, which are listed in Table I and shownin Fig. 4. These parameters are also consistent with the results from Bonn potentials workedout by Tong et al. [59] It can be found that the energy per nucleon and symmetry energyfrom RBHF model shown as open and solid circles are well parameterized by the fittingfunctions, Eqs. (18) and (20), denoted by the solid curves in Fig. 4.13 ABLE I: Parameters in Eqs. (18) and (20) for the equations of state of symmetric nuclear matterand symmetry energies from RBHF model with pvCD-Bonn A, B, C potentials. n [fm − ] a [MeV] η b [MeV] β c [MeV] γ pvCD-Bonn A 0.192 -20.86 0.64 4.03 3.28 36.75 0.73pvCD-Bonn B 0.158 -15.87 0.58 2.96 3.08 29.05 0.65pvCD-Bonn C 0.139 -13.06 0.52 2.34 2.95 25.12 0.59 b [fm ] 20 15 10 50 E / A [ M e V ] (a) pvCD-Bonn ApvCD-Bonn BpvCD-Bonn C b [fm −3 ]015304560 E s y m [ M e V ] (b) FIG. 4: The energies per nucleon and symmetry energies parameterized by Eqs. (18) and (20) forpvCD-Bonn potentials. The opened circles are the fitting data and solid ones are used to checkthe reliability of parameterizations.
The saturation properties, saturation density, n and corresponding energy per nucleon, E/A , incompressibility, K ∞ , symmetry energy, E sym , the slope of symmetry energy, L aresummarized in Table II for pvCD-Bonn A, B, C potentials from RBHF model. The resultsfrom Bonn potentials are also listed for comparison worked by Tong et al . [59] These empiricalobservables are strongly correlated to tensor components in N N potentials, i.e., the D -state probability of deuteron, P D . The incompressibilities, symmetry energies, and theirslopes at saturation densities satisfy the conventional constraints extracted from propertiesof finite nuclei within limits. Especially, the smaller values, L , are preferred by recentmeasurements about the neutron skin of finite nuclei and gravitational waves from binaryneutron star merger [5, 9]. Although G matrices were obtained by Thompson equation forBonn potentials, their P D dependence of saturation properties are accordance with thosederived by BbS equation for pvCD-Bonn potentials.14 ABLE II: The saturation properties of nuclear matter. The data of Bonn A, B, C are collectedfrom [59]. n refers to the saturation densities. P D n E/A K ∞ E sym L pvCD-Bonn A 4.22% 0.192 -16.83 315 36.8 80.5pvCD-Bonn B 5.45% 0.158 -12.91 206 29.1 56.7pvCD-Bonn C 6.05% 0.139 -10.72 151 25.1 44.5Bonn A 4.47% 0.180 -15.38 286 33.7 75.8Bonn B 5.10% 0.164 -13.44 222 29.9 63.0Bonn C 5.53% 0.149 -12.12 176 26.8 51.7Empirical 0.16 ± ± ±
20 31.7 ± ± In nuclear density functional theories, it was found that the slope of symmetry energyat saturation density has strong linear correlations with the neutron skins of
Pb andthe symmetry energy [10, 11]. In Fig. 5, the relation between E sym and L at saturationdensity are shown for pvCD-Bonn potentials and Bonn potentials. They also have thestrong linear correlation with different P D . In general, the lower P D provides a largersymmetry energy and a larger slope. Since the tensor force will suppress the depth ofbound state in symmetric nuclear matter. Recently, the behaviors of symmetry energy athigh density also attracted the wide attentions. The ASY-EOS experiment at GSI in 2016showed that the symmetry energy at 2 n E and 3 n E should be around 50 . − .
39 MeVand 64 . − .
74 MeV, respectively, where n E = 0 .
16 fm − is the empirical saturationdensity [86]. Furthermore, many theoretical works also presented their constraints on thedensity dependence of symmetry energy, like the chiral effective theory [87, 88]. Therefore,in Table III, symmetry energies at n E , 2 n E , and 3 n E are listed for pvCD-Bonn andBonn potentials. It is obvious that the density-dependent behaviors of symmetry energyfrom pvCD-Bonn A and Bonn A potentials satisfy the observations from the ASY-EOSexperiment. 15 E sym [MeV] L [ M e V ] C B ABC A pvCD-BonnBonn
FIG. 5: The relations between symmetry energies and their slopes at saturation densities forpvCD-Bonn and Bonn potentials.TABLE III: The values of symmetry energy at different densities obtained from pvCD-Bonn andBonn potentials. n b in unit fm − and symmetry energy in unit MeV. n b pvCD-Bonn A pvCD-Bonn B pvCD-Bonn C Bonn A Bonn B Bonn C0 .
16 32.17 29.29 27.29 30.87 29.41 28.100 .
32 53.36 45.96 41.08 51.92 47.77 43.790 .
48 71.74 59.82 52.19 70.37 63.45 56.77
2. The potentials of symmetric nuclear matter
The scalar and vector potentials are two important quantities in RBHF model to connectthe Dirac equation and G matrices through the nucleon single-particle potential, which alsodenote the attraction and repulsion of N N interaction at different ranges, respectively. InFig. 6, the scalar and vector potentials from pvCD-Bonn potentials in symmetric nuclearmatter and pure neutron matter are given in panel (a) and panel (b), respectively. In presentwork, they are assumed to be only density dependent and momentum independent, whichare extracted from Eqs. (8) and (15). The self-consistent calculation of RBHF model isdetermined by the convergence of proton and neutron scalar potentials. At low densityregions, U S and U V from three pvCD-Bonn potentials are almost identical. Their differencesamong the three potentials become obvious with density increasing. For vector potential,the pvCD-Bonn C provides more repulsive contribution both in symmetric nuclear matter16nd pure neutron matter, while for scalar potential, pvCD-Bonn C potential generates moreattractive component in pure neutron matter case. n b [fm −3 ] −600−400−2000200400600 U s a n d U v [ M e V ] U s U v α = 0.0 (a) pvCD-Bonn ApvCD-Bonn BpvCD-Bonn C n b [fm −3 ]U s U v α = 1.0 (b) FIG. 6: The scalar and vector potentials as functions of nucleon density in symmetric nuclearmatter and pure neutron matter.
The BbS equation actually was solved in partial wave
LSJ representation. In thesecalculations, the largest total angular momentum is taken up to J = 8. In Fig. 7, the maincontributions of partial waves to the potential energy of nucleon at isospin-triplet channels[ pp, nn, np ( T = 1)] are displayed for the symmetric nuclear matter from pvCD-Bonn Apotential. There are quite small differences among pp, nn, np ( T = 1), related to thecharge symmetry breaking (CSB) and charge independent breaking (CIB) effects of realistic N N potential [37]. These two effects will be discussed in detail later. The partial wavecontributions with J S channel generates most of the attraction, which represents the centralforce in N N interaction. Furthermore, D and P - F channels also provide the partiallybound energies. On the other hand, P and P channels give the repulsive contributions,where P channel has the stronger magnitude.In Fig. 8, the corresponding partial wave contributions from isospin-singlet channel tothe potential energy is also shown. There is only np potential due to the Pauli exclusionprinciple. The spin-triplet channel, S - D , provides the strongest attractive contributionand P channels generates the largest repulsive one. Especially, the S - D coupled channelmainly comes from the tensor force in N N potentials. There is a saturation point in thecontribution of S - D channel, whose saturation density is very closed to that of symmetricmatter. In fact, it can be found the contributions of each partial wave from three pvCD-17 .0 0.1 0.2 0.3 0.4 0.5 n b [fm −3 ] −10−505101520 E v [ M e V ] S P P D P − F J > 2pvCD-Bonn App 0.1 0.2 0.3 0.4 0.5 n b [fm −3 ] np (T = 1) S P P D P − F J > 2 0.1 0.2 0.3 0.4 0.5 n b [fm −3 ] nn S P P D P − F J > 2
FIG. 7: Partial wave contributions from isospin-triplet channels to the potential energy in sym-metric matter obtained by pvCD-Bonn A. n b [fm −3 ] −20−15−10−5051015 E v [ M e V ] P S − D D J > 2pvCD-Bonn Anp (T = 0)
FIG. 8: Partial wave contributions from isospin-singlet channel to the potential energy in symmetricmatter obtained by pvCD-Bonn A potential.
Bonn potentials are almost the same except the one from S - D channel. Therefore, theenergy from S - D channel determines the saturation properties of symmetric nuclearmatter. This is why the saturation properties of nuclear matter is so closely related withthe strength of tensor force, or alternatively, the D -state probability of deuteron, P D [77].To discuss the CSB and CIB effects of N N potentials in nuclear matter, the detailedvalues of partial wave contributions in symmetric nuclear matter from pvCD-Bonn potentialsare listed in Tables IV and V at empirical saturation density, n E = 0 .
16 fm − and n b = 0 . − , respectively. These partial wave contributions from pvCD-Bonn A, B, C potentials18re very similar in each channel, expect the coupled channels, S - D and P - F . It iseasy to understand these results, because pvCD-Bonn potentials can precisely describe thephase shifts obtained from the Nijmegen partial wave analysis. Their differences mainlycomes from the mixing parameters, ε and ε of S - D and P - F channels, which alsoare determined by the tensor force component of N N potentialIn symmetric nuclear matter, the Fermi momenta of proton and neutron are completelyidentical. The differences of partial wave contribution in the same channel can be derivedonly from the CSB or CIB effect. The charge symmetry of
N N potential is invariant undera transformation from proton-proton ( pp ) interaction to neutron-neutron ( nn ) interactionafter removing the Coulomb force between protons. The CSB effect in nuclear matter isregarded to the differences between energy contributions from pp and nn interactions. FromTables IV and V, it demonstrates that the CSB effect is mainly embodied in S channel.The energy differences from nn and pp potentials are about 0 . − .
06 MeV. In addition,the CIB effect is obtained by comparing the partial wave contributions from np with thosefrom pp and nn . In each isospin-triplet channel, the potential energy from np interactionhas the significant distinction with those from pp and nn interactions. The largest differenceis around 0 .
35 MeV. Therefore, in nuclear matter with pvCD-Bonn potentials, the CIBeffect has a more obvious signature comparing to the CSB effect. Actually, there was alsoa similar conclusion for the
N N singlet scattering length and effective range at S channelin Refs. [36, 37].
3. The properties of asymmetric nuclear matter
The asymmetric nuclear matter with different fractions of protons and neutrons are veryimportant for the investigations of compact star and supernova simulations [12, 13, 89].The Pauli operators in the medium BbS equation will become more complicated due to thedistinguished Fermi integration spheres of proton and neutron. The detailed formulas aboutthe evaluation of asymmetric nuclear matter are given in the appendix B. The equations ofstate of asymmetric nuclear matter with different asymmetry parameters, α , are plotted inFig. 9 from pvCD-Bonn potentials. With the neutron numbers increasing, the equation ofstates of asymmetric nuclear matter are not saturated and not bound above α = 0 .
8. Thedifferences of three equations of state among pvCD-Bonn A, B, C potentials also quickly19
ABLE IV: The partial wave contributions derived by nn, pp, np interactions from pvCD-Bonnpotentials at n b = 0 .
16 fm − in the unit MeV.pvCD-Bonn A pvCD-Bonn B pvCD-Bonn C pp np nn total pp np nn total pp np nn total S -5.28 -5.44 -5.32 -16.04 -5.28 -5.46 -5.32 -16.06 -5.26 -5.45 -5.30 -16.01 P -0.26 -0.18 -0.27 -0.71 -0.31 -0.23 -0.31 -0.85 -0.32 -0.24 -0.32 -0.88 P P S − D -17.50 -17.50 -14.72 -14.72 -12.72 -12.72 D -0.68 -0.66 -0.69 -2.03 -0.67 -0.65 -0.68 -2.00 -0.67 -0.65 -0.68 -2.00 D -3.15 -3.15 -3.16 -3.16 -3.17 -3.17 P − F -2.14 -2.07 -2.15 -6.46 -2.06 -1.98 -2.08 -6.22 -2.05 -1.98 -2.06 -6.19 F F D − G nn, pp, np interactions from pvCD-Bonnpotentials at n b = 0 .
32 fm − in the unit MeV.pvCD-Bonn A pvCD-Bonn B pvCD-Bonn C pp np nn total pp np nn total pp np nn total S -6.45 -6.74 -6.50 -19.69 -6.54 -6.88 -6.60 -20.02 -6.54 -6.89 -6.60 -20.03 P P P S − D -19.44 -19.44 -11.67 -11.67 -7.18 -7.18 D -1.24 -1.22 -1.24 -3.70 -1.21 -1.19 -1.22 -3.62 -1.23 -1.21 -1.24 -3.68 D -6.01 -6.01 -6.22 -6.22 -6.25 -6.25 P − F -4.03 -3.89 -4.05 -11.97 -3.92 -3.77 -3.94 -11.63 -3.93 -3.79 -3.94 -11.66 F F D − G reduced for larger α .The energy per nucleon in asymmetric nuclear matter is regarded to be expanded as apolynomial with a variable α around α = 0, EA ( n b , α ) = E A ( n b ) + E sym ( n b ) α + O ( α ) , (21)where the coefficient in second term, E sym ( n b ) is defined as the symmetry energy. In Fig. 10,20 n b [fm −3 ] −20020406080100 E / A [ M e V ] pvCD-Bonn A (a) α = 0.0α = 0.4α = 0.6α = 0.8α = 1.0 0.1 0.2 0.3 0.4 0.5 n b [fm −3 ] pvCD-Bonn B 0.1 0.2 0.3 0.4 0.5 n b [fm −3 ] pvCD-Bonn C FIG. 9: The equations of state of asymmetric nuclear matter calculated by RBHF model withdifferent α from pvCD-Bonn A, B, C potentials. the energy differences ∆ E = EA ( n b , α ) − EA ( n b , α are plotted in presentcalculations. It is a suitable way to check the expansion convergence of α in Eq. 21. Inthe panel (a) of this figure, energy differences from three pvCD-Bonn potentials at empiricalsaturation density n E = 0 .
16 fm − have almost the linear relations with α . It demonstratesthat the neglect of higher terms about α in the expansion of asymmetric nuclear matter isreasonable. In panel (b), the validity of such linear relation is checked at different baryondensities n b = 0 . , . , . , .
30 fm − with pvCD-Bonn A potential, which still workwell. α Δ E Δ [ M e V ] n b = 0.16Δfm (a) pvCD-Bonn ApvCD-Bonn BpvCD-Bonn C α (b) −3 FIG. 10: The energy differences ∆ E at fixed baryon density as functions of α . In panel (a), the∆ E from pvCD-Bonn A, B, C potentials at fixed density n E = 0 .
16 fm − . In panel (b), ∆ E frompvCD-Bonn A potential at n b = 0 . , . , . , .
30 fm − . .0 0.2 0.4 0.6 0.8 1.0 p/k F −100−80−60−40−20 U τ ( p ) [ M e V ] P otonNeut on(a) n b = 0.16 fm −3 pvCD-Bonn A 0.0 0.2 0.4 0.6 0.8 1.0 p/k F −120−80−40040 ProtonNeutron (b) n b = 0.32 fm −3 α = 0.0α = 0.2α = 0.4α = 0.6 FIG. 11: The neutron (upper) and proton (lower) single-particle potentials in asymmetric nuclearmatter at different asymmetry parameter obtained by pvCD-Bonn A potential.
In Fig. (11), the neutron and proton single-particle potentials as functions of momentumare displayed with different asymmetric parameters at n E = 0 .
16 fm − (panel (a)) and n b = 0 .
32 fm − (panel (b)). In symmetric nuclear matter, these potentials for neutron andproton are identical. With the fractions of neutron increasing, the neutron single-particlepotential become more repulsive, while the case of proton is opposite. It means that theproton obtains more attractive contribution from the N N potential. Therefore the effectiveneutron mass is larger than the proton one in neutron-rich matter. The differences betweenneutron and proton single-particle potentials become smaller with momentum for a fixed α and increase with the nucleon density. This behavior of nucleon single-particle potential iscompletely consistent with the work using the Bonn potentials by Sammarruca [54]. IV. SUMMARY AND OUTLOOK
Properties of nuclear matter were investigated in relativistic Brueckner-Hartree-Fock(RBHF) model with the latest charge-dependent nucleon-nucleon potentials, pvCD-BonnA, B, C, where the coupling scheme between pion and nucleon is taken as pseudovectorform. These three potentials have different tensor components. Furthermore, the center-of-mass momentum related to G matrix was exactly integrated in present work without theconventional angle-averaged approximation.Firstly, the equations of state of symmetric nuclear matter and pure neutron matterwith three pvCD-Bonn potentials were obtained. Their saturation densities and saturation22nergies were closed to the empirical data for symmetric nuclear matter. These saturationproperties are strongly related to the tensor components of N N potentials, which can bepresented by the D -state probability of deuteron, P D . Generally speaking, the smallertensor component provides larger saturation density and more attractive binding energy.They could be summarized as a relativistic Coester band with the results from pvCD-Bonn and Bonn potentials. Furthermore, they were also compared to the results fromnonrelativistic framework. The relativistic effect provides more repulsive contribution andgenerates reasonable saturation properties. For the pure neutron matter, the equations ofstate from pvCD-Bonn potentials almost were identical, since the tensor contribution is veryweak in the isopin T = 1 case.The original CD-Bonn potential with pseudoscalar (PS) coupling was also applied tocalculate the properties of nuclear matter. It was confirmed that the PS coupling betweenpion and nucleon provides a too much attractive contribution and generates over-boundstate for symmetric nuclear matter, while it did not influence the pure neutron matter.With these equations of state, the additional properties, such as incompressibility, symmetryenergy and its slope at saturation density were evaluated. These properties also satisfied theconstraints extracted from the finite nuclei experiments. The symmetry energies at highernuclear densities, such as twice or three times empirical saturation density from pvCD-Bonnpotentials were accordance with recent results from ASY-EOS experiment at GSI laboratory.Through discussing the partial wave contributions to potential energy, it was found thatthe differences among three pvCD-Bonn potentials for the symmetric nuclear matter mainlycame from the coupled channel S - D , since their phase shifts were only distinguished bythe mixing parameters ε . In addition, the charge symmetry breaking (CSB) and chargeindependent breaking (CIB) effects in nuclear matter were also investigated. The CSBeffect derived by the difference between pp and nn potentials embodied in S channelabout 0 . − .
06 MeV. The CIB effect from np to pp or nn potentials appeared in eachisospin-triplet channel and was more obvious than the CSB effect. The equations of state ofasymmetric nuclear matter were also calculated with pvCD-Bonn potentials. They were notbound together when the asymmetry parameters were larger than 0 .
8. The magnitude ofneutron single-particle potential was higher than that from proton in neutron-rich matter,which leads to the fact that the neutron effective mass in nuclear medium is larger than theproton one. 23he RBHF model is a very powerful ab initio method in relativistic framework, whichcan explain the saturation properties of nuclear matter reasonably by using only two-body
N N potential. With the newly developed high-precision charge-dependent Bonn potentials,pvCD-Bonn A, B, C, more investigations will be done in nuclear physics, such as propertiesof neutron star, the superfluity of nucleon in medium, and the saturation mechanism ofnuclear matter in future.
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China(Grant No. 11775119, No. 11405116, and No. 11675083), the Natural Science Foundation ofTianjin, and China Scholarship Council (Grant No. 201906205013 and No. 201906255002).
Appendix A: In-medium Blanckenbecler-Sugar equation
In the conventional RBHF model, the G matrix was solved via the in-medium Thompsonequation, since the Bonn potentials were obtained by fitting the N N scattering data withThompson equation in free space [50, 61]. The pvCD-Bonn potentials were generated byBlanckenbecler-Sugar (BbS) equation to keep the consistency with the original CD-Bonnpotential. Therefore, the in-medium BbS equation must be derived in this work, which hasbeen mentioned in the appendix A of Ref. [50]. The procedure of nucleon-nucleon scatteringis dominated by Bethe-Salpter (BS) equation, which is written as T = V + VGT , (A1)where T is an invariant amplitude for N N scattering and G is a two-nucleon propagator.The BS equation in Eq. (A1) is defined in four-dimension space, explicitly, at center-of-massframe, T ( q ′ , q | P ) = V ( q ′ , q | P ) + Z d k (2 π ) V ( q ′ , k | P ) G ( k | P ) T ( k, q | P ) , (A2)where q, k, q ′ are initial, intermediate, and final relative four-momenta, respectively. P =( p + p ) / G in Eq.(A1) is given as G ( k | P ) = ip / − M + iǫ ip / − M + iǫ , (A3)24here M and M denote the nucleon masses, p = ( E , p ) and p = ( E , p ) are four-momenta of two one-shell nucleons, respectively.Actually, the BS equation is very difficult to be solved in four-dimension space for nu-merical calculation. To simplify BS equation, the propagator G must be reduced into three-dimension space as g . It should reserve unitary and covariant properties of original G in thisprocess. Therefore, g and G should have the same discontinuity across the branch cut,Im g ( k | P ) = Im G ( k | P ) = − π ( p / + M )( p / + M ) δ (+) ( p − M ) δ (+) ( p − M ) , (A4)where δ (+) ( p i − M i ) means that only the on-shell nucleons are involved (exclusion of anti-nucleon).It is more convenient to express Im g in center-of-mass frame,Im g ( k | P ) = − π ( p / + M )( p / + M ) δ ( p − E ) δ ( p − E )4 E E = − π M M E E Λ (1)+ ( p )Λ (2)+ ( p ) δ (2 P − W k ) δ ( k − E / E / − π W k M M E E Λ (1)+ ( p )Λ (2)+ ( p ) δ ( s ′ − W k + 4 P ) δ ( k − E / E /
2) (A5)with W k = E + E and the immediate total energy, s ′ = 4 P − P . The projection operatoris defined by Λ ( i )+ ( p ) = p/ + M i M i = X λ u i ( p , λ )¯ u i ( p , λ ) , (A6)where λ represents the eigenvalue of spin operator. Furthermore, the imaginary part (A5), g ( k, s ) can be constructed by the dispersion integral, g ( k, s ) = 1 π Z + ∞ M ds ′ s ′ − s − iǫ Im g ( k, s ′ ) . (A7)Here, we consider that the starting energy is written as W = E P + q + E P − q and s = W − P .Therefore, the four-dimension propagator, (A3) can be reduced to three-dimension one as G ( k | P ) → g ( k, s ) = 2 πδ ( k − E / E / g ( k , s ) (A8)with different choices for ¯ g ( k , s ). After integrating (A7), the BbS propagator is obtainedexplicitly [67], ¯ g BbS ( k , s ) = 2 W k M M E E Λ (1)+ ( P + k )Λ (2)+ ( P − k ) W − W k + iǫ . (A9)25he in-medium effects must be taken into account in the BbS propagator for nuclear many-body system with Pauli operator Q ,¯ g BbS ( k , s ) = 2 W k M ∗ M ∗ E ∗ E ∗ Λ (1)+ ( P + k ) Q ( k , P )Λ (2)+ ( P − k ) W − W k . (A10)We take this propagator into BS equation (A2) in three-dimension space, T ( q ′ , q | P ) = V ( q ′ , q | P )+ Z d k (2 π ) W k M ∗ M ∗ E ∗ E ∗ V ( q ′ , k | P ) Q ( k , P ) Λ (1)+ ( P + k )Λ (2)+ ( P − k ) W − W k T ( k , q ′ | P ) . (A11)With expressions of Dirac spinor, the invariance amplitude, T and N N potential, V can bewritten as Lorentz scalars,¯ V ( q ′ , q ) = ¯ u ( P + q )¯ u ( P − q ) V ( q ′ , q | P ) u ( P + q ) u ( P − q ) , ¯ T ( q ′ , q | P ) = ¯ u ( P + q )¯ u ( P − q ) T ( q ′ , q | P ) u ( P + q ) u ( P − q ) . (A12)Thus the BbS equation in nuclear medium, Eq. (A11), is rewritten as¯ T ( q ′ , q | P ) = ¯ V ( q ′ , q ) + Z d k (2 π ) M ∗ M ∗ E ∗ E ∗ ¯ V ( q ′ , k ) 2 W k W + W k Q ( k , P ) W − W k ¯ T ( k , q | P ) . (A13)The in-medium scattering amplitude and N N potential can be redefined with the normal-ization condition of spinor (7), G ( q ′ , q | P ) = M ∗ E ∗ ¯ T ( q ′ , q | P ) M ∗ E ∗ and V ( q ′ , q ) = M ∗ E ∗ ¯ V ( q ′ , q ) M ∗ E ∗ . Finally the Eq. (A11) can be simplified as a more compact form, G ( q ′ , q | P ) = V ( q ′ , q ) + Z d k (2 π ) V ( q ′ , k ) 2 W k W + W k Q ( k , P ) W − W k G ( k , q | P ) . (A14) Appendix B: The detailed formulas for asymmetric nuclear matter
In BbS equation, three momenta, q ′ , q , and P must be treated. When the asymmetricnuclear matter is considered, the integrals about these momenta become very complicated,especially for the Pauli operator. In conventional calculations of RBHF model, the Paulioperator Q τ τ in the propagator is replaced by its average over solid angle with differentcases [59].For the case τ τ = pp or nn : 26 (a) 0 < P k τF : Q av ττ = k < Γ , k − Γ P k Γ k < k τF + P, k > P + k τF , (B1) • (b) P > k τF : Q av ττ = 0 . with Γ = p k τ F − P .For the case τ τ = np or pn : • (a) 0 P ( k nF − k pF ) / Q av np = k < k nF − P, ( k + P ) − k n F P k k nF − P k < k nF + P, k > P + k nF . (B2) • (b) ( k nF − k pF ) / P ( k nF + k pF ) / Q av np = k < Γ , k − Γ P k Γ k < k pF + P, ( k + P ) − k n F P k k pF + P k < k nF + P, k > k nF + P. (B3) • (c) P > ( k pF + k nF ) / Q av np = 0 , with Γ = q ( k p F + k n F ) / − P .With this approximation, the solid angle dependence is removed in the integral of Eq. (9)at partial wave representation, G α J τ τ ,ℓ ℓ ( q ′ , q | P ) = V α J τ τ ,ℓ ℓ ( q ′ , q )+ X ℓ ′ Z k dk V α j τ τ ,ℓ ℓ ′ ( q ′ , k ) 2 W k W + W k Q av τ τ ( k, P ) W − W k G α J τ τ ,ℓ ′ ℓ ( k, q | P ) , (B4)where α j indicate six possible | LSJ i states with a fixed total angular momentum J . At thesame time, the single-particle potential (8) is transformed into center-of-mass frame, andthen decomposed into partial wave | LSJ i states. Its explicit expression is shown as [52], U τ ( p ) = X τ = p,n Z kτ F + p dq · q C τ ( p, q ) t Tτ τ X j,α j (2 J + 1) G α j τ τ ( q i | P av τ ( p, q )) . (B5)27he coefficients t Tτ τ are concerned with different isospins, t nn = t pp = 1 , t pn = t pn = 12 , otherwise : t Tτ τ = 0 . The factor C τ ( p, q ) are related with the Fermi momentum of nucleon, C τ = ( < q k τ F − p , k τ F − ( p − q ) pq k τF − p < q k τF + p . (B6)The averaged total momentum in Pauli operator is given by p k F : P av τ = p p + q < q k τF − p , p p + k τ F − pq k τF − p < q k τF + p . (B7) p > k F : P av τ = 12 q p + k τ F − pq p − k τF < q k τF + p . (B8)otherwise P av τ = 0.When the total energy of nuclear matter is evaluated, the relevant integrals can also beperformed in center-of-mass frame and taken as the partial-wave decomposition [59],12 n b X τ τ X ss ′ Z p k τ F d p (2 π ) Z p ′ k τ F d p ′ (2 π ) h ps, p ′ s ′ | G τ τ ( W τ τ ) | ps, p ′ s ′ i = 1(2 π ) n b X τ τ Z ( k τ F + k τ ) / d q Z | P + q | k τ F | P − q | k τ F d P t Tτ τ X J,α j (2 J + 1) G α j τ τ ( q, q | P ) . (B9)The solid angle integration for center-of-mass momenta in last line of this equation is dividedinto following cases.For the case of τ τ = pp, nn , Z d Ω P = P k τ F − q, Γ − P P q k τF − q < P Γ , P > Γ , (B10)with Γ = p k τ F − q .For the case of τ τ = np, pn , Z d Ω P = P k pF − q, k p F − ( q − P ) P q k pF − q < P k nF − q, Γ − P P q k nF − q < P Γ , P > Γ , (B11)28ith Γ = q ( k p + k n ) / − q . [1] M. Baldo (Ed.), Nuclear Methods and Nuclear Equation of State in International Review ofNuclear Physics, Vol. (World Scientific Publishing Company, 1999).[2] P. Kl¨upfel, P.-G. Reinhard, T. J. Burvenich, and J. A. Maruhn, Phys. Rev. C , 034310(2009).[3] X. Roca-Maza and N. Paar, Prog. Part. Nucl. Phys. , 96 (2018).[4] M. Oertel, M. Hempel, T. Kl¨ahn, and S. Typel. Rev. Mod. Phys. , 015007 (2017).[5] B. P. Abbott et al. , (LIGO Scientific and Virgo Collaboration), Phys. Rev. Lett. , 161101(2017).[6] B. P. Abbott et al. , (Virgo, Fermi-GBM, INTEGRAL, and LIGO Scientific Collaboration),Astrophys. J. , L13 (2017).[7] B. P. Abbott et al. , Astrophys. J. , L12 (2017).[8] A. Goldstein et al. , Astrophys. J. , L14 (2017).[9] B. P. Abbott et al. , (LIGO Scientific and Virgo Collaboration), Phys. Rev. Lett. , 161101(2018).[10] B.A. Li, L.W. Chen, and C.M. Ko, Phys. Rep. , 113 (2008).[11] B. A. Li, P. G. Krastev, D. H. Wen, and N. B. Zhang, Eur. Phys. J. A , 117 (2019).[12] H. Shen, H. Toki, K. Oyamatsu, and K. Sumiyoshi, Nucl. Phys. A , 435 (1998).[13] H. Shen, H. Toki, K. Oyamatsu, and K. Sumiyoshi, Prog. Theor. Phys. , 1013 (1998).[14] H. Shen, Phys. Rev. C , 035802 (2002).[15] H. Shen, H. Toki, K. Oyamatsu and K Sumiyoshi, Astrophys. J. Suppl. , 20 (2011).[16] H. Euler, Z. Phys. , 553 (1937).[17] R. Jastrow, Phys. Rev. , 165 (1951).[18] K. A. Brueckner, C. A. Levinson, and H. M. Mahmound, Phys. Rev. , 217 (1954).[19] H. A. Bethe, Phys. Rev. , 1353 (1956).[20] R. Jastrow, Phys. Rev. , 1479 (1955).[21] Z. H. Li, U. Lombardo, H.-J. Schulze, W. Zuo, L. W. Chen, and H. R. Ma, Phys. Rev. C ,047304 (2006).
22] M. Baldo, C. Maieron, J. Phys. G , R243 (2007).[23] M. Baldo and G. F. Burgio, Prog. Part. Nucl. Phys. , 203 (2016).[24] A. Akmal, V. R. Pandharipande, and D. G. Ravenhall, Phys. Rev. C , 1804 (1998).[25] J. Carlson, S. Gandolfi, F. Pederiva, Steven C. Pieper, R. Schiavilla, K.E. Schmidt, and R.B.Wiringa, Rev. Mod. Phys. , 1067 (2015).[26] W. H. Dickhoff and C. Barbieri, Prog. Part. Nucl. Phys. , 377 (2004).[27] G. Hagen, T. Papenbrock, M. Hjorth-Jensen, and D. J. Dean, Rep. Prog. Phys. , 096302(2014).[28] G. Hagen, T. Papenbrock, A. Ekstr¨om, K. A. Wendt, G. Baardsen, S. Gandolfi, M. Hjorth-Jensen, and C. J. Horowitz, Phys. Rev. C , 014319 (2014).[29] A. Carbone, A. Polls, and A. Rios, Phys. Rev. C , 044302 (2013).[30] A. Carbone, A. Rios, and A. Polls, Phys. Rev. C , 054322 (2014).[31] C. Drischler, V. Som` a , and A. Schwenk, Phys. Rev. C , 025806 (2014).[32] M. Drews and W. Weise, Phys. Rev. C , 035802 (2015).[33] M. Drews and W. Weise, Prog. Part. Nucl. Phys. , 69 (2017).[34] M. Modarres, J. Phys. G: Nucl. Part. Phys. , 1349 (1993).[35] V. G. J. Stoks, R. A. M. Klomp, C. P. F. Terheggen, and J. J. de Swart, Phys. Rev. C ,2950 (1994).[36] R. B. Wiringa, V. G. J. Stoks, and R. Schiavilla, Phys. Rev. C , 38 (1995).[37] R. Machleidt, Phys. Rev. C , 024001 (2001).[38] D. R. Entem and R. Machleidt, Phys. Rev. C , 041001(R) (2003).[39] E. Epelbaum, W. Gl¨ockle, and U.-G. Meißner, Nucl. Phys. A , 362 (2005).[40] E. Epelbaum, H. Krebs, and U.-G. Meißner, Eur. Phys. J. A , 53 (2015).[41] E. Epelbaum, H. Krebs, and U.-G. Meißner, Phys. Rev. Lett. , 122301 (2015).[42] D. R. Entem, N. Kaiser, R. Machleidt, and Y. Nosyk, Phys. Rev. C , 014002 (2015).[43] D. R. Entem, R. Machleidt, and Y. Nosyk, Phys. Rev. C , 024004 (2017).[44] P. Reinert, H. Krebs, and E. Epelbaum, Eur. Phys. J. A , 86 (2018).[45] J. Hu, Y. Zhang, E. Epelbaum, Ulf-G. Meissner, and J. Meng, Phys. Rev. C , 034307(2017).[46] F. Sammarruca, L. E. Marcucci, L. Coraggio, J. W. Holt, N. Itaco, R. Machleidt,arXiv:1807.06640[47] D. Logoteta, Phys. Rev. C , 045803 (2019).
48] M. R. Ansatasio, L. S. Celenza, W. S. Pong, and C. M. Shakin, Phys. Rep. , 327 (1983).[49] C. J. Horowitz and B. D. Serot, Nucl. Phys. A , 613 (1987).[50] R. Brockmann and R. Machleidt, Phys. Rev. C , 1965 (1990).[51] Z. H. Li, U. Lombardo, H.-J. Schulze, and W. Zuo, Phys. Rev. C , 034316 (2008).[52] D. Alonso and F. Sammarruca, Phys. Rev. C , 054305 (2003).[53] P. G. Krastev and F. Sammarruca, Phys. Rev. C , 025808 (2006).[54] F. Sammarruca, Int. J. Mod. Phys. E , 1259 (2010).[55] E. N. E. Dalen and H. Muether, Int. J. Mod. Phys. E , 2077 (2010).[56] S. Shen, J. Hu, H. Liang, J. Meng, P. Ring, and S. Zhang, Chin. Phys. Lett. , 102103(2016).[57] S. Shen, H. Liang, J. Meng, P. Ring, and S. Q. Zhang, Phys. Rev. C , 014316(2017).[58] S. Shen, H. Liang, W. Long, J. Meng, and P. Ring, Prog. Part. Nucl. Phys. , 103713(2019).[59] H. Tong, X. L. Ren, P. Ring, S. H. Shen, S. B. Wang, and J. Meng, Phys. Rev. C , 054302(2018).[60] R. Machleidt, K. Holinde, and Ch. Elster, Phys. Rep. , 1 (1987).[61] R. Machleidt, Adv. Nucl. Phys. , 189 (1989).[62] J. Hu, P. We, and, Y. Zhang, Phys. Lett. B , 134982 (2019).[63] C. Wang, J. Hu, Y. Zhang, and H. Shen, Chin, Phys. C , 114107 (2019).[64] F. Sammarruca, L. White, and B. Chen, Eur. Phys. J. A , 181 (2012).[65] E. N. E. Dalen and H. M¨uther, Phys. Rev. C , 014319 (2010).[66] E. E. Salpeter and H. A. Bethe, Phys. Rev. , 1232 (1951).[67] R. Blankenbecler and R. Sugar, Phys. Rev. , 1051 (1966).[68] R. H. Thompson, Phys. Rev. D , 110 (1970).[69] V. G. Kadyshevsky, Nucl. Phys. B , 125 (1968).[70] A. Deltuva, R. Machleidt, and P. U. Sauer, Phys. Rev. C , 024005 (2003).[71] D. Logoteta, I. Bombaci, and A. Kievsky, Phys. Rev. C , 064001 (2016).[72] A. Ekstr¨om, G. Hagen, T. D. Morris, T. Papenbrock, and P. D. Schwartz, Phys. Rev. C ,024332[73] J. Hu, H. Toki, W. Wen, and H. Shen, Phys. Lett. B , 271 (2010).[74] Y. Wang, J. Hu, H. Toki, and H. Shen, Prog. Theo. Phys. , 739 (2012).
75] Y. Zhang, P. Liu, and J. Hu, arXiv:1910.11765.[76] F. Coester, S. Cohen, B. Day, and C. M. Vincent, Phys. Rev. C , 769 (1970).[77] J. Hu, H. Toki, and Y. Ogawa, Prog. Theor. Exp. Phys. , (2013).[78] B. D. Serot and J. D. Walecka, Adv. Nuc. Phys. , 1 (1986).[79] C. Fuchs, T. Waindzoch, A. Faessler, and D. S. Kosov, Phys. Rev. C , 2022 (1998).[80] A. Bouyssy, J. -F Mathiot, N. Van Giai, and S. Marcos, Phys. Rev. C , 380 (1987).[81] W. H. Long, N. Van Giai, and J. Meng, Phys. Lett. B , 150 (2006).[82] W. H. Long, H. Sagawa, N. V. Giai, and J. Meng, Phys. Rev. C , 034314 (2007).[83] S. Gandolfi, J. Carlson, and S. Reddy, Phys. Rev. C , 032801(R) (2012).[84] P. Danielewicz and J. Lee, Nucl. Phys. A
36 (2009).[85] U. Garg and G. Col´o, Prog. Part. Nucl. Phys. , 55 (2018).[86] P. Russotto et al. , Phys. Rev. C , 034608 (2016).[87] J. W. Holt and N. Kaiser, Phys. Rev. C , 034326 (2017).[88] Y. Lim and J. W. Holt, Eur. Phys. J. A , 209 (2019).[89] H. Tong, P. W. Zhao, and J. Meng, arXiv:1903.05938., 209 (2019).[89] H. Tong, P. W. Zhao, and J. Meng, arXiv:1903.05938.