The fourth-order symmetry energy $S_4(\rho)$ of neutron-rich nuclear matter: exchange-channel structure across covariant density functionals, kinetic short-range-correlation content, and neutron-star crust consequences

The energy per nucleon of cold nucleonic matter expanded in the isospin asymmetry $\delta$ contains, beyond the familiar quadratic symmetry energy $S_2(\rho)$, a fourth-order term $S_4(\rho)$. Liu, Qian, Xing, Niu and Sun (Phys. Rev. C 97, 025801, 2018) found within covariant density functional (CDF) theory that $S_4(\rho)$ is systematically suppressed in relativistic Hartree-Fock (RHF) functionals, which include Fock (exchange) diagrams, relative to Hartree-only relativistic mean-field (RMF) functionals, and that this suppression shifts neutron-star predictions (proton fraction, core-crust transition density and pressure, crustal moment-of-inertia fraction). They explicitly flagged as open whether this Fock-induced suppression survives in nonlinear and point-coupling CDF variants, what the role of the tensor force in the kinetic part $S_{4,\mathrm{kin}}$ is (whose modification by short-range correlations the paper also discusses but does not settle), and how the conclusions change under a realistic dynamical (random-phase-approximation) treatment of the crust-core transition. These questions remain open: the above-saturation density dependence of $S_4$ has so far been mapped only within the density-dependent meson-exchange subset of CDF theory, where it turns over and becomes negative for $\rho\approx0.2$-$0.5$ fm$^{-3}$ in RHF while rising monotonically in RMF, so whether the nonlinear and point-coupling variants reproduce this behavior remains untested; the factor-of-few discrepancy in $S_{4,\mathrm{kin}}(\rho_0)$ between short-range-correlation estimates ($7.18\pm2.52$ MeV) and mean-field values ($0.5$-$2$ MeV) is unreconciled, and no RHF-based dynamical crust-core study exists. The earlier claim of a very large $S_4(\rho_0)\approx20$ MeV from nuclear masses was later shown to be unreliable because of unconstrained high-order surface tension.

Background

Nuclear matter with neutron-proton asymmetry $\delta=(\rho_n-\rho_p)/\rho$ has an energy per nucleon whose isospin-dependent part is conventionally expanded as $E_{\mathrm{sym}}(\rho,\delta)=S_2(\rho)\delta^2+S_4(\rho)\delta^4+S_6(\rho)\delta^6+\cdots$, where $\rho$ is the baryon density and $S_{2n}$ are the even-order symmetry energies; truncating after $S_2\delta^2$ is the parabolic approximation. Because neutron-star matter reaches $\delta\approx0.8$-$0.95$ in beta equilibrium, the quartic term $S_4(\rho)$ and higher orders measurably affect the proton fraction (and hence neutrino cooling channels such as the direct Urca process, the sequence of nucleon beta decay and its inverse that becomes allowed once the proton fraction crosses a threshold and is then the fastest cooling channel), the location of the boundary between the liquid core and the solid inner crust, and the fraction of the star's moment of inertia residing in the crust, $\Delta I/I$, which feeds the interpretation of pulsar glitches (sudden spin-ups attributed to angular-momentum transfer from the neutron superfluid in the inner crust). The magnitude of $S_4$ at saturation density $\rho_0\approx0.16$ fm$^{-3}$ is predicted to be small, below about 1 MeV in mean-field models: Skyrme and Gogny energy density functionals (nonrelativistic functionals based, respectively, on zero-range and finite-range effective interactions; Pu, Zhang and Chen, 2017) and nonlinear relativistic mean-field (RMF) models (Cai and Chen, 2012). Chiral pion-nucleon calculations give a somewhat larger value, about 1.5 MeV (Kaiser, 2015), while many-body perturbation theory (Wellenhofer, Holt and Kaiser, 2016) showed the asymmetry expansion can diverge at moderate $\delta$, so each higher-order coefficient must be handled with care. In covariant density functional (CDF) theory, nucleons interact through exchanges of mesons ($\sigma$, $\omega$, $\rho$, $\pi$, possibly $\delta$); RMF functionals keep only the direct (Hartree) terms, while relativistic Hartree-Fock (RHF) functionals also keep the exchange (Fock) diagrams, which embed, among other effects, the nuclear tensor force through the $\pi$ and $\rho$ couplings. Liu, Qian, Xing, Niu and Sun (2018) compared the density-dependent meson-exchange RMF functionals (PKDD, TW99, DD-ME1, DD-ME2) with the RHF functionals (PKA1, PKO1, PKO2, PKO3) and found $S_4(\rho_0)\approx0.35$-$0.58$ MeV in RHF, systematically suppressed relative to RMF, with the suppression dominated by the isoscalar $\sigma$ and $\omega$ exchange channels; the kinetic part $S_{4,\mathrm{kin}}(\rho_0)\approx1.09$-$2.12$ MeV in RHF exceeds the free-Fermi-gas value of $0.45$ MeV, which the authors attributed partly to the tensor force in the exchange diagrams. They determined the core-crust transition with the thermodynamical method, a long-wavelength approximation that neglects Coulomb, surface and finite-size effects; dynamical methods based on the random phase approximation (RPA), which test the stability of uniform matter against cluster formation, typically lower the transition density by $0.005$-$0.015$ fm$^{-3}$. Two observational anchors frame the astrophysical stakes: the crustal moment of inertia must satisfy $\Delta I/I\geq0.016$ to power Vela pulsar glitches without entrainment, or $\Delta I/I\geq0.07$ once the entrainment of neutron superfluid by the crustal lattice (Chamel, 2013) is included, and whether crustal superfluidity alone suffices remains debated (Andersson, Glampedakis, Ho and Espinoza, 2012; Piekarewicz, Fattoyev and Horowitz, 2014; Watanabe and Pethick, 2017). On the empirical side, Wang and Chen (2017) extracted $S_4(\rho_0)=20.0\pm4.6$ MeV from nuclear masses, far above all mean-field predictions, while Wan, Xu, Ren and Liu (2018) obtained a quartic coefficient of $3.63$ MeV from double beta-decay energies; the origin of such large finite-nucleus values versus the small model predictions became itself part of the problem.

Problem Statement

The open problem is to determine the fourth-order symmetry energy $S_4(\rho)$ of neutron-rich nuclear matter over the density range from the sub-saturation crust-core transition region to supra-saturation densities — its magnitude, density dependence, kinetic/potential decomposition, and coupling-channel content — robustly across the covariant density functional (CDF) family, and to establish the resulting neutron-star consequences. The single scientific objective is a variant-independent, quantitatively constrained characterization of how Fock (exchange) terms and short-range/tensor correlations shape $S_4(\rho)$ and, through it, the properties of the neutron-star crust. Concretely, in the formulation left open by Liu, Qian, Xing, Niu and Sun (2018): (1) compute $S_4(\rho)$ with its coupling-channel decomposition in nonlinear and point-coupling CDF functionals that include Fock terms, and test whether the Fock-induced suppression of $S_4$ and the associated RHF-versus-RMF shifts in the core-crust transition density $\rho_t$, proton fraction and transition pressure $P_t$ are generic features of CDF theory or artifacts of the density-dependent meson-exchange subset; (2) resolve the kinetic discrepancy $S_{4,\mathrm{kin}}(\rho_0)\approx7.18\pm2.52$ MeV from short-range-correlation momentum distributions versus $\approx0.45$-$2.12$ MeV from free-gas and mean-field/RHF treatments, clarifying the role of short-range correlated nucleon pairs and the tensor force and how they interplay with the mean-field exchange contributions; and (3) propagate the constrained $S_4(\rho)$ (and higher orders where relevant) into neutron-star observables using a more realistic dynamical treatment of the core-crust transition — random-phase-approximation type including finite-size, Coulomb and surface effects beyond the thermodynamical long-wavelength approximation — and quantify the impact on the crustal moment-of-inertia fraction $\Delta I/I$ relevant to pulsar-glitch energetics. Resolving any one of (1)-(3) alone is a valid partial result, but the task is complete only when all three are established.

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