Coupled nuclear and leptonic longitudinal collective modes in neutron star matter : a covariant Vlasov approach

Abstract: A covariant relativistic approach based on the Vlasov equation is used to study collective modes in neutron-star matter. The analysis is carried out within relativistic mean-field models describing charge-neutral and $β$-equilibrated matter composed of neutrons, protons, electrons, and muons. We investigate the conditions under which nuclear collective excitations couple to electron and muon plasmon modes, a phenomenon relevant for neutron stars and supernova matter. The study is undertaken considering relativistic mean field models with different isoscalar and isovector properties. It is shown that the nuclear-leptonic coupling can be sufficiently strong to modify the onset of nuclear collective modes and to affect their isoscalar or isovector character.

• The paper extends a covariant relativistic Vlasov framework to derive coupled longitudinal-mode dispersion relations for beta-equilibrated npeμ matter using the NL3, NL3ωρ, and FSU2H RMF models.
• Leptons influence nuclear dynamics exclusively through electromagnetic coupling to protons, altering mode onsets, damping, and isospin character while generating distinct electron- and muon-plasmon branches.
• The predicted spectra vary strongly with density, momentum, proton fraction, and symmetry-energy stiffness, with implications for neutrino opacity, cooling, and supernova transport, although hyperons and finite temperature remain excluded.

Overview

This paper extends a covariant relativistic Vlasov framework to the study of longitudinal collective modes in charge-neutral, $\beta$-equilibrated neutron-star matter containing neutrons, protons, electrons, and muons (npe$\mu$ matter). Building on earlier work on asymmetric nuclear matter (2606.00810), the authors derive dispersion relations for coupled nuclear and leptonic excitations within three relativistic mean-field (RMF) parameterizations — NL3, NL3$\omega\rho$, and FSU2H — chosen to span different stiffnesses of the isoscalar equation of state and of the symmetry energy. The central result is that nuclear–leptonic coupling can be strong enough to modify the onset and isospin character of nuclear collective modes, with direct consequences for neutrino transport in neutron-star interiors.

Formalism

The starting point is an RMF Lagrangian with $\sigma$-$\omega$-$\rho$ meson exchange, nonlinear scalar self-interactions ($\kappa$, $\lambda$), a quartic vector term ($\xi$), and an $\omega$-$\mu$0 mixing term ($\mu$1) that softens the symmetry energy at high density. The covariant Vlasov equation is written for each species $\mu$2, with the equilibrium distributions taken as zero-temperature Fermi steps. Linearizing around equilibrium and Fourier transforming yields density fluctuations expressed through Lindhard-type functions $\mu$3, where $\mu$4 and $\mu$5.

The field equations for the perturbed meson and photon fields close the system, producing a $\mu$6 matrix equation for the density fluctuations $\mu$7, $\mu$8, $\mu$9, $\omega\rho$0. A key structural finding is that the determinant factorizes: only the proton–proton component of the hadronic subspace is renormalized by the leptonic sector, via an effective interaction

$\omega\rho$1

where $\omega\rho$2 are Coulomb couplings. This compact form makes explicit that leptons enter the nuclear dynamics exclusively through their electromagnetic coupling to protons.

Model dependence of composition

The proton fraction in $\omega\rho$3 equilibrium tracks the density dependence of the symmetry energy. NL3, with its stiff symmetry energy ($\omega\rho$4 MeV), predicts the largest proton fractions, exceeding $\omega\rho$5 at $\omega\rho$6 for npe$\omega\rho$7 matter, whereas NL3$\omega\rho$8 and FSU2H saturate near $\omega\rho$9. At subsaturation densities the ordering reverses, with FSU2H ($\sigma$0 MeV) giving the largest proton fraction. Muon inclusion raises the proton fraction at high densities by altering chemical equilibrium.

Collective-mode spectrum across densities

At saturation density, all three models exhibit two well-separated branches: a high-energy plasmon-like mode dominated by electron oscillations and lower-energy proton-like nuclear modes. The plasmon ceases to propagate beyond a momentum cutoff, while the nuclear modes persist up to model-dependent limits of roughly 30 MeV (NL3), 80 MeV (NL3$\sigma$1), and above 90 MeV (FSU2H). Quantitatively, NL3 gives the softest response with maximum mode energies of 7–8 MeV, versus $\sigma$2–19 MeV for the stiffer models. Notably, when dynamical electrons are included, the plasmon-like proton mode of np matter converts into a sound-like mode, confirming the mechanism discussed by Baldo et al. In npe$\sigma$3 matter, a second muonic plasmon branch appears without qualitatively altering the nuclear sector.

At $\sigma$4, a neutron-dominated zero-sound mode emerges in NL3 but not in the softer models, confirming that its propagation requires a sufficiently stiff EOS and acquires isoscalar character above roughly twice saturation density. At $\sigma$5, NL3 produces the richest spectrum — neutron-like and proton-like modes each appearing as Landau-damped/undamped pairs, with the neutron-like mode switching between isoscalar and isovector character as a function of $\sigma$6 — whereas FSU2H shows only a single plasmon-like mode strongly coupled to the proton mode, reflecting its soft high-density symmetry energy.

The density scan at fixed $\sigma$7 MeV identifies several systematic behaviors: a lone low-density mode marking the crust region below 0.1 fm$\sigma$8; cessation of proton-mode propagation above 0.3–0.4 fm$\sigma$9 in npe matter due to electron coupling; and earlier termination (0.25–0.3 fm$\omega$0) in npe$\omega$1 matter where muons couple to both electron and proton modes. FSU2H supports no propagating neutron-like collective mode at any density considered.

Coupling conditions and isospin character

Analysis of the sound velocity $\omega$2 relative to the neutron particle-hole continuum shows that the soundlike lepton branch couples to nuclear modes when these are predominantly isovector and the momentum transfer is small. At $\omega$3 MeV the plasmon–nucleon coupling is strong; at $\omega$4 MeV it shifts to densities above 3–4$\omega$5 for NL3 and higher still for NL3$\omega$6. At the coupling point the nucleon mode has isovector character, switching to isoscalar just above it. The stiff symmetry energy of NL3 enhances lepton densities and promotes earlier, stronger coupling, while the smaller proton fractions of NL3$\omega$7 shift or suppress it. Consequently, the onset density of the nuclear isovector mode depends sensitively on both proton fraction and momentum transfer.

Limitations and open questions

The authors state plainly that hyperonic degrees of freedom, expected to become relevant at the highest densities treated here, are excluded and deferred to future work. All results are obtained at zero temperature within nonlinear RMF models; the quantitative details are model dependent, and the authors caution that density-dependent RMF parameterizations may exhibit modified momentum-transfer dependence. Whether the qualitative features survive the inclusion of hyperons or finite-temperature effects relevant to supernova cores remains open.

Conclusion

By extending the covariant Vlasov formalism to fully $\omega$8-equilibrated npe$\omega$9 matter, this work demonstrates that the longitudinal collective spectrum of neutron-star matter is jointly shaped by the stiffness of the isoscalar EOS, the density dependence of the symmetry energy, and the electromagnetic coupling to degenerate leptons. The factorized structure of the dispersion relation isolates the leptonic influence to the proton channel, yet this channel suffices to alter mode onsets and isospin character. Since plasmon properties govern neutrino-pair emission and coherent scattering off density fluctuations, the model-dependent mode spectra reported here bear directly on neutrino opacity calculations and hence on neutron-star cooling and supernova dynamics.