Neutrino emission from spin waves in neutron spin-triplet superfluid

The linear response of a neutron spin-triplet superfluid onto external weak axial-vector field is studied for the case of $^{3}P_{2}$ pairing with a projection of the total angular momentum $m_{j}=0$. The problem is considered in the BCS approximation discarding Fermi-liquid effects. The anomalous axial-vector vertices of neutron quasiparticles possess singularities at some frequencies which specify existence of undamped spin-density waves in the Cooper condensate. The spin waves are of a low excitation energy and are kinematically able to decay into neutrino pairs through neutral weak currents. We evaluate the neutrino emissivity from the spin wave decays in the bulk neutron superfluid in old neutron stars. This calculation predicts significant energy losses from within a neutron star at lowest temperatures when all other mechanisms of neutrino emission are killed by the neutron and proton superfluidity.

💡 Research Summary

The paper investigates a previously overlooked neutrino‑emission channel that can dominate the cooling of old neutron stars when both neutron and proton superfluidity have suppressed all conventional neutrino processes. The authors focus on the $^{3}P_{2}$ neutron pairing state with magnetic quantum number $m_{j}=0$, which is believed to be the most favorable configuration in the dense core of a neutron star.

Using the BCS approximation and deliberately neglecting Fermi‑liquid corrections, they calculate the linear response of the superfluid to an external weak axial‑vector field. The response is expressed through ordinary axial vertices (simply the Pauli matrices) and anomalous vertices that arise from the pairing interaction. The anomalous vertices satisfy Dyson‑type integral equations, which can be reduced to a compact form after a renormalization of the pairing interaction.

A key result is that the anomalous vertex function $f(\omega,T)$ possesses poles at frequencies $\omega_{s}$ where the denominator $\chi(\omega,T)$ vanishes. These poles correspond to collective spin‑density excitations—spin waves—in the condensate. By solving the dispersion relation $\chi(\omega_{s},T)=0$, the authors first obtain a rough estimate $\omega_{s}\simeq\Delta/\sqrt{5}\approx0.45\Delta$, showing that the spin‑wave energy lies below the pair‑breaking threshold $2\Delta$ and therefore can decay into a neutrino–antineutrino pair without being Landau‑damped.

A more accurate treatment retains the full angular dependence of the order‑parameter vector $\bar{\mathbf b}(\mathbf n)$. Introducing the dimensionless variables $y=\Delta(T)/T$ and $\Omega=\omega/(2\Delta)$, they numerically evaluate the integral equation for a wide range of temperatures. Near the critical temperature $T_{c}$ the dimensionless frequency is $\Omega_{s}\approx0.21$ ($\omega_{s}\approx0.42\Delta$); as the temperature drops, $\Omega_{s}$ smoothly approaches $\approx0.15$ ($\omega_{s}\approx0.30\Delta$). Thus the spin wave remains a low‑energy mode even at $T\ll T_{c}$.

The neutrino emissivity is then derived from the imaginary part of the weak axial‑vector polarization tensor $\Pi^{\rm weak}{\mu\nu}$. The spin‑wave contribution appears as a $\delta$‑function at $\omega=\omega{s}$, reflecting the decay of an undamped collective mode into a neutrino pair. The final emissivity can be written in the compact form

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