Unquantified meson retardation and rho-tensor correlations in relativistic Hartree-Fock compact-star matter, and their joint treatment with hyperon/Delta exchange contributions to the equation of state, maximum masses, and direct-Urca thresholds

In covariant density functional theory, the relativistic Hartree-Fock (RHF) approximation adds Fock (exchange) diagrams to the Hartree relativistic mean field, producing momentum-dependent nonlocal nucleon self-energies that reshape the dense-matter equation of state (EoS), composition, and direct-Urca cooling thresholds of neutron stars. The founding RHF compact-star calculations acknowledged that several exchange-sector ingredients were omitted: rho-meson tensor correlations and meson-retardation effects (the static-Yukawa approximation to the exchanged meson propagator) in density-dependent RHF, and the hyperon and Delta(1232) exchange channels. Later work quantified the hyperon and Delta exchange contributions and tensor-coupling effects on Urca emissivity, but no modern RHF compact-star calculation is known to include meson retardation — its neglect is justified only by a 1989 near-saturation estimate — rho-tensor effects have not been mapped within refitted calibrated functionals for Urca onset thresholds, and no self-consistent joint treatment exists. The problem is to complete and quantify this exchange sector for compact-star matter.

Background

Covariant density functional theory describes nuclear and neutron-star matter as nucleons interacting through the exchange of mesons (scalar $\sigma$, vector $\omega$, isovector $\rho$, pion $\pi$) generated by an effective Lagrangian. At the mean-field ('Hartree') level — relativistic mean field (RMF) theory — only direct meson fields contribute, giving local, momentum-independent nucleon self-energies. The relativistic Hartree-Fock (RHF) approximation adds the Fock ('exchange') diagrams, which produce momentum-dependent, nonlocal self-energies and bring pion-exchange and tensor pieces of the interaction into covariant functionals. In the density-dependent RHF (DDRHF) version the meson-nucleon couplings are functions of baryon density, fitted to finite-nucleus data (e.g. the PKO parameterizations), with rearrangement terms added to keep the thermodynamics consistent.

Sun, Long, Meng and Lombardo (Phys. Rev. C 78, 065805, 2008) computed neutron-star properties in DDRHF for the first time: relative to Hartree RMF, the exchange terms stiffen the density dependence of the symmetry energy, raise proton fractions, and shift the onset of the direct Urca process — the fast neutrino-cooling reactions $n\rightarrow p+e^-+\bar\nu_e$ (and their inverse), kinematically allowed once the proton fraction exceeds $x^{\mathrm{DU}}\approx 14.8\%$ when muons are present — to critical stellar masses near $1.45\,M_\odot$ for PKO2/PKO3, with maximum masses $\sim 2.5\,M_\odot$. That calculation, however, omitted several ingredients of the exchange sector: the tensor coupling of the $\rho$ meson (a $\sigma^{\mu\nu}$-type vertex supplementing the vector coupling) was explicitly not enclosed; meson retardation — the energy dependence of the exchanged meson propagator, approximated by the static Yukawa form $D_\phi=(m_\phi^2+\boldsymbol{q}^2)^{-1}$, which affects only the Fock terms — was neglected; and hyperons (strange baryons expected in inner cores) were excluded altogether. Miyatsu, Katayama and Saito (Phys. Lett. B 709, 242, 2012) studied neutron matter in an RHF model that does include $\omega$ and $\rho$ tensor couplings as well as in-medium baryon structure from the quark-meson coupling (QMC) model, in which each baryon is a bound state of three quarks polarized by the surrounding scalar field; they found the tensor coupling vital for reaching a $\sim 2.0\,M_\odot$ maximum mass, and listed the $\Delta$(1232) isobar (spin-3/2 baryon resonance), kaon exchange, and baryon mixing in the Fock diagrams, plus meson condensates, as future work. The open problem raised jointly by these sources is the quantitative completion of this exchange sector for compact-star matter — determining which of its omitted Fock ingredients, among meson retardation, rho-tensor correlations, and the hyperon/Delta exchange channels, change the EoS, maximum masses, and cooling thresholds, and by how much. Miyatsu et al.'s further future-work items — kaon exchange and baryon mixing in the Fock diagrams, and meson condensates or quark matter — introduce additional degrees of freedom or phase structure rather than omitted meson-exchange ingredients of the founding DDRHF calculations, and lie outside the present problem.

Problem Statement

Determine, within self-consistent relativistic Hartree-Fock theory for beta-equilibrated (weak-interaction chemical equilibrium among neutrons, protons, electrons, and muons, extended in the joint treatment (iii) below to include the hyperon and Delta(1232) states), charge-neutral compact-star matter, the quantitative impact of the exchange-sector ingredients omitted from the founding calculations — rho-meson tensor correlations, meson-retardation effects in the Fock propagators, and hyperon/Delta exchange channels — on the dense-matter equation of state, composition, maximum stellar masses, and direct-Urca thresholds. Because the hyperon and Delta(1232) exchange contributions and tensor-coupling effects on Urca emissivity have since been quantified in separate studies, the surviving open core is: (i) the retardation contribution, i.e. the effect of energy-dependent (rather than static Yukawa) meson propagators in the Fock self-energies at supranuclear densities, currently justified only by a 1989 near-saturation estimate of 'a few percent'; (ii) the effect of rho-tensor correlations within refitted, calibrated density-dependent functionals on the EoS and on direct-Urca onset thresholds (as distinct from emissivities); and (iii) a self-consistent joint treatment of these ingredients together with the hyperon/Delta exchange channels. Kaon exchange in the Fock diagrams and meson condensates, also listed as future work by Miyatsu et al. (2012), introduce additional degrees of freedom and phase structure rather than omitted exchange-sector ingredients of the founding calculations, and are therefore outside the scope of this problem. Quantifying ingredients (i) and (ii) requires establishing, in self-consistent calculations, how the equation of state, maximum stellar masses, particle fractions, and direct-Urca onset densities and threshold masses change when each is included alone and jointly with the others, so as to determine whether the remaining omissions are subdominant corrections or systematic effects at neutron-star-core densities; the hyperon/Delta exchange channels, already quantified standalone in the cited studies, enter only through the joint treatment (iii).

References

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