Nuclear fourth-order symmetry energy and its effects on neutron star properties in the relativistic Hartree-Fock theory
Zhi Wei Liu Zhuang Qian Ruo Yu Xing Jia Rui Niu Bao Yuan Sun
nucl-th








School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000, China


School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000, China


School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000, China


School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000, China


School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000, China




Adopting the density dependent relativistic mean-field (RMF) and relativistic Hartree-Fock (RHF) approaches, the properties of the nuclear fourth-order symmetry energy S_4 are studied within the covariant density functional (CDF) theory. It is found that the fourth-order symmetry energies are suppressed in RHF at both saturation and supranuclear densities, where the extra contribution from the Fock terms is demonstrated, specifically via the isoscalar meson-nucleon coupling channels. The reservation of S_4 and higher-order symmetry energies in the nuclear equation of state then affects essentially the prediction of neutron star properties, which is illustrated in the quantities such as the proton fraction, the core-crust transition density as well as the fraction of crustal moment of inertia. Since the Fock terms enhance the density dependence of the thermodynamical potential, the RHF calculations predict systematically smaller values of density, proton fraction and pressure at the core-crust transition boundary of neutron stars than density dependent RMF ones. In addition, a linear anti-correlation between the core-crust transition density ρ_t and the density slope of symmetry energy L is found which is then utilized to constrain the core-crust transition density as ρ_t[0.069, 0.098] fm^-3 with the recent empirical information on L. The study clarifies the important role of the fourth-order symmetry energy in determining the properties of nuclear matter at extreme isospin or density conditions.



  21.30.Fe, 
  21.60.Jz, 
  21.65.Ef, 
  26.60.-c, 
  26.60.Gj, 
  97.60.Gb  


                              
Nuclear fourth-order symmetry energy and its effects on neutron star properties in the relativistic Hartree-Fock theory
    Bao Yuan Sun [sunby@lzu.edu.cn]
    December 30, 2023
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§ INTRODUCTION

The study on nuclear equation of state (EoS), especially its properties at extreme conditions, is not only a longstanding goal in nuclear related science but in astrophysics <cit.>. In recent years, facilities for radioactive ion beam (RIB) are developing competitively around the terrestrial laboratories, which have made great progress in exploring the nuclear EoS at both supranuclear and subnuclear densities <cit.>. In particular, the isospin-asymmetric part of the EoS, namely the nuclear symmetry energy, is proved to be a crucial issue in understanding the physics of several terrestrial experiments and astrophysical observations, such as neutron skin thickness <cit.>, dipole excitation modes of stable or exotic nuclei <cit.>, isospin diffusion and π^+/π^- ratio in heavy-ion collisions at intermediate energies <cit.>, parity violating electron scattering <cit.>, as well as the radius, the moment of inertia, the stability of matter and cooling mechanism of neutron stars <cit.>.

Theoretically, the nuclear symmetry energy is introduced by expanding the binding energy per nucleon in a Taylor series with respect to the isospin asymmetry, and is usually approximated to its second-order term S_2 for convenience. Although a number of phenomenological and microscopic nuclear models have been devoted to constrain the symmetry energy around the saturation density ρ_0 and its density dependence <cit.>, large uncertainties still remains <cit.>. Recently, a data collective analysis constrained the symmetry energy S_2(ρ_0) at the saturation density as S_2(ρ_0)=31.7±3.2 MeV, and its density slope parameter as L=58.7±28.1 MeV retaining a relatively large error bar <cit.>, implying the necessity to improve the work further in both experimental and theoretical sides.

Recently, it has been indicated that the higher-order terms of symmetry energy than S_2 may become non-negligible and should be considered carefully under some extreme physical conditions <cit.>. Especially, the inclusion of the fourth-order term of nuclear symmetry energy S_4 beyond the parabolic approximation<cit.> in the EoS could exert great impact on the description of neutron star properties, such as the core-crust transition density <cit.>, the proton fraction and the critical density for the direct URCA (DUrca) process <cit.>. However, because of the absence of experimental information, there is still large uncertainty in constraining the magnitude of S_4, even at the saturation density ρ_0. It is found that the magnitude of S_4(ρ_0) is generally smaller than 2 MeV within the non-relativistic <cit.> and relativistic mean-field models <cit.>, as well as the chiral pion-nucleon dynamics <cit.>. Alternatively, the analysis from quantum molecular dynamics model<cit.> and the extraction from an extended nuclear mass formula <cit.> predict significantly large values of S_4(ρ_0), for example 20.0±4.6 MeV in latter one. In addition, the kinetic part ε_k of the energy density functional has been identified as a particularly good indicator of the short-range correlated (SRC) nucleon pairs <cit.>. Theoretically, this SRC pairs can happen due to the tensor part of the nucleon-nucleon interaction <cit.>, which consequently is revealed to reduce the kinetic part of S_2(ρ_0) <cit.> and increase that of S_4(ρ_0) <cit.> significantly.

The covariant density functional (CDF) theory, based on the meson exchange diagram of nuclear force, has achieved great successes in describing the bulk properties of nuclear matter, the ground state and excitation properties of finite nuclei <cit.>. During the past decades, the CDF theory without Fock terms, namely the relativistic mean field (RMF) theory, occupied the CDF market with various versions of Lagrangian <cit.>. Since the Fock terms are ignored in the RMF theory, the important degrees of freedom in the meson exchange diagrams, such as the pseudo-vector π-couplings, are dropped. Moreover, the nonlocal potential as well as the tensor part of nuclear force cannot be self-consistently taken into account. With increasing computer technology, the relativistic Hartree-Fock (RHF) theory <cit.>, also referred as the CDF theory with Fock terms, achieved success in terms of the density-dependent meson-nucleon coupling <cit.>. Significant improvements were obtained by the RHF theory in exploring nuclear structure <cit.>, nuclear excitation and decay modes <cit.> as well as nuclear matter and neutron star porperties <cit.>.

In fact, with the inclusion of the Fock terms in the CDF theory, it is realized that isoscalar meson-nucleon coupling except for the isovector one also plays a vital role in studying the isospin properties of nuclear matter, such as the nuclear symmetry energy and the neutron-proton effective mass splitting <cit.>. After including the Λ hyperons into the β-equilibrium nuclear matter, the symmetry energy at high densities is suppressed enormously due to the extra suppression effect originating from the Fock channel, leading to a relatively small predicted value of the neutron-star radius <cit.>. Additionally, it was recognized that the Fock diagrams of the meson-nucleon couplings can take the important ingredient of nuclear force — the tensor force into account naturally <cit.>, which softens the density-dependent behavior of the symmetry energy and consequently raises the threshold density for the direct URCA process that cools the neutron star rapidly <cit.>. Furthermore, a sizable reduction of the kinetic part of S_2 at the supranuclear density region is found in the RHF calculations compared to the RMF ones, regarded partly as the effect of the nuclear tensor force <cit.>.

The studies demonstrated the Fock terms are of great importance when talking about isospin related physics in the CDF theory, and it is interesting to investigate its effects further on higher-order symmetry energy rather than S_2 and the corresponding influence on neutron star properties, which founds the motivation of this work. In the following, we will briefly introduce the theoretical framework of the RHF theory for nuclear matter in Section II. Then in Section III we discuss in detail the density-dependent behavior of nuclear fourth-order symmetry energy S_4 and its properties at saturation density within the RHF theory and the effects of S_4 on the neutron star properties, including the proton fraction, the core-crust transition as well as the moment of inertia utilized to describe the glitch phenomenon, are given later. Finally, a summary is given in Section IV.



§ THEORETICAL FRAMEWORK

In this section, the main formalism of the CDF theory with the inclusion of the Fock terms will be briefly recalled for nuclear matter, which is then utilized to extract various order of the symmetry energy. For more details of the CDF theory especially the RHF theory for nuclear matter, we refer the reader to Refs. <cit.>.

Via the meson exchange diagrams of nuclear force, the CDF theory is started from an effective Lagrangian density which can be deduced from the collaboration with the degrees of freedom of nucleon and mesons, while the photon field is ignored for uniform nuclear matter systems. Following the standard procedure <cit.>, the energy density functional (EDF) is then obtained by taking the expectation value of the Hamiltonian with respect to the Hartree-Fock ground state, which consists of three parts,

  
    ε_kin   =∑_psτ u̅(p,s,τ) (γ·p+M) u(p,s,τ),
    ε_ϕ^D
           =1/2∑_p_1s_1τ_1∑_p_2s_2τ_2 u̅(p_1,s_1,τ_1)u̅(p_2,s_2,τ_2) Γ_ϕ(1,2)
           ×1/m_ϕ^2 u(p_2,s_2,τ_2)u(p_1,s_1,τ_1),
    ε_ϕ^E
           =-1/2∑_p_1s_1τ_1∑_p_2s_2τ_2 u̅(p_1,s_1,τ_1)u̅(p_2,s_2,τ_2) Γ_ϕ(1,2)
           ×1/m_ϕ^2+q^2 u(p_1,s_1,τ_1)u(p_2,s_2,τ_2),


where ε_kin denotes the kinetic EDF, and ε_ϕ^D and ε_ϕ^E correspond to the Hartree (direct) and Fock (exchange) terms of the potential EDF, where ϕ = σ, ω, ρ, π represents various meson-nucleon coupling and Γ_ϕ(1,2) are corresponding interaction vertices. The Dirac spinors u(p,s,τ) depend on the momentum p, spin s, and isospin τ of nucleon,

    u(p,s,τ)=(E^*+M^*/2E^*)^1/21σ·p^*/E^*+M^* χ_sχ_τ.

Here χ_s and χ_τ stand for the spin and isospin wave functions, respectively. The starred quantities, which obey the effective relativistic mass-energy relation E^*2=M^*2+p^*2, are defined as:

  
    M^*
            = M + Σ_S(p),
    p^*
            = p   + p̂Σ_V(p),
    
        E^*
            = E  - Σ_0(p),


where Σ_S is the scalar self-energy, Σ_0 and Σ_V are the time and space components of the vector self-energy, respectively, and p̂ is the unit vector along p.

Substituting Eq. (<ref>) into Eq. (<ref>), the kinetic EDF is then expressed as

    ε_kin
      =1/π^2∑_i=n,p∫^k_F,i_0p^2dp[MM̂+pP̂],

where the hatted quantities are introduced by

    M̂   =M^*/E^*,    P̂   =p^*/E^*.

ε_kin can be divided again according to the self-energy so as to study the influence of various meson-nucleon coupling channels quantitatively <cit.>,

    ε_kin   =1/π^2∑_i=n,p∫^k_F,i_0p^2dp1/E^*
       ×[M^2+p^2
      +∑_ϕ MΣ_S^D,ϕ+∑_ϕ(MΣ_S^E,ϕ+pΣ_V^E,ϕ)]
       ≡ ε_kin^M + ε_kin^p + ε_kin^D + ε_kin^E,

where ε_kin^M and ε_kin^p correspond to the contributions from the rest mass and the momentum, respectively, ε_kin^D denotes the contribution from the Hartree (direct) terms of the scalar self-energy Σ_S^D,ϕ, and ε_kin^E represents the total contribution from the Fock (exchange) terms of the scalar self-energy  Σ_S^E,ϕ and the space component of the vector self-energy Σ_V^E,ϕ.

The EoS of asymmetric nuclear matter at zero temperature is defined by its binding energy per nucleon E(ρ,δ), where ρ=ρ_n+ρ_p denotes the baryon density, and δ≡(ρ_n-ρ_p)/(ρ_n+ρ_p) is the isospin asymmetry with ρ_n/p the neutron/proton density. Conventionally, due to the difficulty of analytical extraction, the various order of nuclear symmetry energies can be approximately extracted by expanding the zero-temperature EoS in a Taylor series with respect to the δ. The convergence of such an isospin-asymmetry expansion has been acknowledged in the self-consistent mean-field calculations, such as the CDF theory (at the first Hartree-Fock level) adopted in this work, but broken overall when second-order perturbative contributions are involved in many-body theory <cit.>. Within this approximation, the EoS is then expressed as

  
    E(ρ,δ)
            = E_0(ρ) + ∑_n=1^∞ S_2n(ρ)δ^2n,
    
        S_2n(ρ)
            = 1/(2n)!∂^2nE(ρ,δ)/∂δ^2n|_δ=0,


where E_0(ρ)=E(ρ,δ=0) denotes the EoS of symmetric nuclear matter, and the coefficients S_2n(ρ) give the 2n-order symmetry energy, where n=1 presents the density-dependent symmetry energy S_2(ρ) and n=2 the fourth-order symmetry energy S_4(ρ), respectively,

  
    S_2(ρ)    = 1/2!∂^2E(ρ,δ)/∂δ^2|_δ=0,
    
        S_4(ρ)    = 1/4!∂^4E(ρ,δ)/∂δ^4|_δ=0.


The odd-order terms of the expansion are discarded in Eq. (<ref>) owing to assuming the charge-independence of nuclear force and neglecting the Coulomb interaction in the infinite nuclear matter. The density slope parameter L is used to reflect the density dependence of S_2(ρ) at saturation density ρ_0, which is defined as

    L=3ρ_0∂ S_2(ρ)/∂ρ|_ρ=ρ_0.

In addition, the various order of terms in isospin asymmetry S_2n(ρ) can be decomposed according to the separation of the EDF, namely,

  
    S_2n   = S_2n,pot + S_2n,kin,
    
        S_2n,pot    = 1/(2n)!∂^2n/∂δ^2n1/ρ.[∑_ϕ(ε_ϕ^D + ε_ϕ^E)]|_δ=0
        ≡ S_2n,pot^D + S_2n,pot^E,
    
        S_2n,kin    = 1/(2n)!∂^2n(ε_kin/ρ)/∂δ^2n|_δ=0
        ≡ S_2n,kin^M + S_2n,kin^p + S_2n,kin^D + S_2n,kin^E,


where S_2n,pot and S_2n,kin denote the potential and kinetic part of S_2n. Additionally, S_2n,pot^D (S_2n,pot^E) corresponds to the contributions from the Hartree (Fock) terms to S_2n,pot, while S_2n,kin^M, S_2n,kin^p, S_2n,kin^D and S_2n,kin^E define the corresponding contributions to S_2n,kin from Eq. (<ref>), respectively.



§ RESULTS AND DISCUSSION


In this work, the properties of nuclear fourth-order symmetry energy, and correspondingly its effects on several related quantities in neutron stars are studied in the CDF theory. Much attention is paid to the role of the Fock terms. The calculations are performed by using the RHF functionals PKA1<cit.>, PKO1<cit.>, PKO2 and PKO3<cit.>, in comparison with the RMF functionals PKDD<cit.>, TW99<cit.>, DD-ME1<cit.> and DD-ME2<cit.>. These functionals have been adopted in the description of nuclear matter and finite nuclei successfully, taking the advantage of the density-dependent meson-nucleon couplings, by which the medium effects of nuclear force in a nuclear many-body system are taken into account effectively. Notice that due to the limitation of the approach itself, the π and ρ-tensor couplings are missing in four selected RMF functionals, while the RHF ones PKO1, PKO2, PKO3 contains the π couplings, and PKA1 contains both. The applications of the RHF theory to the physics of nuclear matter and neutron stars have addressed essential role of the Fock terms, and one can find the details in Refs. <cit.>. For the symmetry energies in Eq. (<ref>), the seven-point finite difference method is adopted in the practical calculations. The convergence and stability of the numerical results have been checked carefully, and the step size of isospin asymmetry δ is suggested as 0.01.



 §.§ Properties of nuclear fourth-order symmetry energy




  §.§.§ Density dependence of nuclear fourth-order symmetry energy

It is generally agreed that the effects of nuclear fourth-order symmetry energy S_4(ρ) become non-negligible at high densities and at extreme isospin like in the interior of neutron stars, while its density dependent behavior is still poorly known. From the calculations of the selected CDF functionals, in Fig. <ref> are shown the density dependence of S_4(ρ). It is seen that all CDF models predict the similar curves at around and lower than the saturation density ρ_0, but the distinct deviation between the RHF and the RMF functionals occurs at the supranuclear density region. While the fourth-order symmetry energies in the RMF models increase monotonously with respect to the density ρ, those in the RHF turn to decrease beyond ρ≳ρ_0, even being negative when ρ[0.2, 0.5] fm^-3, and arise monotonously again after reaching the minimum value. The results are illustrated further by dividing the S_4(ρ) into its potential part S_4,pot and kinetic part S_4,kin, as shown in the middle and right panels of Fig. <ref>, respectively. It is found that the density dependence of S_4,pot and S_4,kin becomes stronger in the RHF models than RMF ones. Specifically, within the RHF a clear reduction of S_4,pot at ρ[0.1, 0.5] fm^-3 appears, which actually dominates the trend of the total fourth-order symmetry energy, leading to the values of S_4(ρ) in RHF lower systematically than those in RMF.





To clarify the origin of the difference of S_4,pot between two kinds of CDF models, it is convenient to take the results in Fig. <ref>(b) apart into their contributions from the Hartree and Fock channels according to Eq. (<ref>), namely S_4,pot^D and S_4,pot^E, as shown in Fig. <ref>(a). For convenience, only the results from the RHF functional PKO1 and the RMF one TW99 are displayed, while the other functionals actually do not change the analysis and the conclusion. It is seen that the divergence on the density dependence of S_4,pot results mainly from the Hartree part S_4,pot^D, which can be explained by the differences in the magnitude of meson-nucleon coupling constants and their density dependence between the RHF and RMF functionals. In fact, due to the extra interaction brought about by the Fock terms, the balance between the nuclear attractions and the repulsions is changed, which sequentially causes the difference in the coupling constants. Aside from the contribution of Hartree terms, the contribution from the Fock terms S_4,pot^E can not be ignored as well, by giving a strongly suppressed contribution (maximum as ⋍-4.7 MeV) to S_4,pot in a fairly broader density region. By separating S_4,pot^E further in terms of meson-nucleon coupling channels, as plotted in Fig. <ref>(b), one can find such an extra contribution from the Fock diagrams is mainly due to the isoscalar coupling channels, with a remarkably negative contribution from the ω-meson coupling channel compensated by a positive contribution from the σ-meson one. Because the isovector meson-nucleon couplings decrease rapidly with respect to the density, as the case in present RHF functionals, the contributions to S_4,pot^E from the ρ- and π-mesons are relatively weak. The results demonstrate again the importance of the isoscalar mesons via Fock diagram to the symmetry energy and correspondingly the essential role from the isospin-triplet components of the exchange potential EDF <cit.>.



Similar as S_4,pot, the kinetic part of nuclear fourth-order symmetry energy S_4,kin can be divided further according to Eq. (<ref>), as shown in Fig. <ref>(a). In the RHF calculation, S_4,kin^M+p+D exhibits stronger density dependence than that in RMF when ρ≳0.2 fm^-3, with a rapid growing up and then a very slow dropping down as the density increases. Such deviation between two functionals results from two aspects, for S_4,kin^M+S_4,kin^p the E^* plays a dominate role, while for S_4,kin^D the role of the Dirac mass M^* is partly involved, as seen in Eq. (<ref>). In fact, the starred quantity E^* corresponds to the Landau mass in the CDF theory if the momentum dependence of the self-energy is left out. Therefore, it is the isospin dependence of the effective mass, including both the Dirac mass and the Landau mass, that accounts for the kinetic part of S_4(ρ). Furthermore, extra contribution S_4,kin^E from Fock diagrams is found, which provides a relatively small and positive value around the saturation density but an explicit suppression when ρ≳0.2 fm^-3. From Fig. <ref>(b), it is revealed that the density dependence of S_4,kin^E is dominantly ascribed to the isoscalar ω-meson coupling channel. Hence, the significant role of the Fock terms in the isospin properties of nuclear matter is demonstrated in both potential and kinetic part of the fourth-order symmetry energy.



  §.§.§ Nuclear fourth-order symmetry energy at saturation density

To get compared with the constraints from other approaches, it is useful to discuss the properties of nuclear fourth-order symmetry energy S_4(ρ) at saturation density ρ_0. Table <ref> shows the values of symmetry energies and their potential and kinetic components at ρ_0 with the different CDF functionals. It is worth noting that the values of S_4(ρ_0) with the RHF functionals are systematically smaller than those in the RMF ones. While S_4(ρ_0) predicted in RMF is located around 0.65 MeV, it is estimated to be about 0.350.58 MeV in RHF, in which the functional PKA1, due to the inclusion of extra ρ-tensor coupling, gives the smallest value of S_4(ρ_0)=0.352 MeV. Thus, it is expected that the involvement of the Fock terms in the CDF theory reduces the fourth-order symmetry energy at ρ_0, and the results are still in agreement with those from the density functional theory <cit.> and the chiral pion-nucleon dynamics <cit.>, i.e., in general less than 2 MeV, but smaller in magnitude than the latest prediction by an extended nuclear mass formula <cit.>.

2column
