Coupled self-consistent treatment of induced-interaction vertex corrections and self-energy effects in the proton $^1S_0$ pairing gap of neutron-star matter

In beta-equilibrated neutron-star matter, the $^1S_0$ proton pairing gap controls whether the proton component becomes superconducting and thereby whether it can suppress direct Urca (dURCA) neutrino cooling. Guo, Dong, Shang, Zhang, Zuo, Colonna and Lombardo (Nucl. Phys. A 986, 18, 2019) computed this gap with a Babu-Brown induced interaction in the Landau limit and third-order G-matrix self-energy effects, finding tiny gaps of order $10^{-2}$ MeV, but they explicitly left open the fully consistent problem: vertex corrections and self-energy effects due to the coupling to the collective modes have never been treated on the same footing within one coupled diagrammatic scheme. This problem asks for such a coupled self-consistent calculation, a question that remains open in the literature up to 2026.

Background

When a neutron star cools below roughly $10^9$ K, its degenerate proton component can undergo Bardeen-Cooper-Schrieffer (BCS) pairing — the standard theory in which fermions near the Fermi surface bind into Cooper pairs and open an energy gap $\Delta$ in the excitation spectrum — in the spin-singlet, zero-angular-momentum $^1S_0$ channel. In beta-equilibrated matter (neutrons, protons, electrons, and muons in chemical equilibrium), the proton fraction is only a few percent, set by the equation of state; the proton Fermi momentum $k_F^p$ is therefore small and the pairing occurs in the outer core. The gap magnitude matters astrophysically because superconductivity suppresses neutrino-emitting processes: in particular the direct Urca (dURCA) process — the fast neutrino cooling reaction $n\to p+e^-+\bar\nu$ (and its inverse), which activates only when the proton fraction exceeds a threshold — is exponentially damped once the proton gap opens, so the size and density range of the gap feed directly into neutron-star cooling simulations and into interpretations of observed cooling (e.g. the Cassiopeia A remnant).

A bare two-body force alone is known to be insufficient, and medium corrections enter in two physically distinct ways. First, particle-hole (ph) polarization of the surrounding neutron-proton medium screens the pairing interaction: the Cooper pair can exchange collective density and spin-density fluctuations, producing an induced interaction $V_{ind}$ on top of the bare pairing force, in the sense of the Babu-Brown induced-interaction theory (the quasiparticle interaction generated by exchange of collective modes in a Fermi liquid). For the dilute proton impurity in neutron-rich matter this vertex correction is expected to be attractive at low density and repulsive at higher density, since the isospin character of the medium's collective modes changes along the beta-stability trajectory; this strong isospin dependence was mapped by Zhang, Cao, Lombardo and Schuck (Phys. Rev. C 93, 044329, 2016), whose Landau-parameters-based $V_{ind}$ is the one adopted by the source paper. Second, self-energy effects dress the single-particle motion: the quasiparticle enters the gap equation with an effective mass $m^*$ and a quasiparticle strength (wave-function renormalization factor) $Z<1$, which in the source paper were extracted from the self-energy $\Sigma(k,E)$ computed to third order of the G-matrix in a Brueckner-Hartree-Fock (BHF) calculation — the standard self-consistent summation of two- and three-body-force ladder diagrams — using the Argonne $v_{18}$ two-body force (2BF) plus a meson-exchange three-body force (3BF).

Guo et al. (Nucl. Phys. A 986, 18, 2019; arXiv:1810.02709) combined these ingredients: the induced interaction was inserted into a linearized gap equation in the Landau limit — evaluating $V_{ind}$ only at the Fermi surface, $V_{ind}\simeq V_{ind}(k_F,k_F)$, implemented through a narrow momentum window around $k_F$, because the Landau-limit induced interaction provides no off-diagonal momentum-space matrix elements — while $m^*$ and $Z$ came from a separate third-order G-matrix calculation. The vertex correction enhanced the gap by over 20% at low density, the self-energy effects reduced it substantially, and the final gaps (with both self-energy and induced interaction) were tiny, $0.027$–$0.036$ MeV over $\rho\simeq 0.08$–$0.12\ {\rm fm}^{-3}$ ($0.016$–$0.021$ MeV without the induced interaction), with the pairing density range remaining below the dURCA activation threshold, so that proton pairing cannot quench dURCA cooling. The precursor calculation of Baldo and Schulze (Phys. Rev. C 75, 025802, 2007), using RPA (random-phase approximation, the ring-diagram resummation of ph propagators) screening, had likewise found strong suppression; the review of Sedrakian and Clark (Eur. Phys. J. A 55, 167, 2019) still displays the proton gap inside a large model spread and notes that gap calculations generally adopt the decoupling approximation of normal-state single-particle energies. The authors of the source paper themselves state that their result should be taken with caution because a consistent calculation including, on the same footing, vertex and self-energy corrections due to the coupling to collective modes is still lacking: in their work the polarization and the self-energy were computed in two separate schemes and simply multiplied together ($Z^2$ times the induced interaction), a product that a genuinely coupled treatment could change qualitatively.

Problem Statement

Compute the proton $^1S_0$ pairing gap $\Delta_p$ as a function of density (equivalently of the proton Fermi momentum $k_F^p$) in beta-equilibrated neutron-star matter in a single coupled diagrammatic scheme in which the induced-interaction vertex corrections and the self-energy effects are treated on the same footing through their mutual coupling to the collective (density and spin-density) modes of the medium. Concretely, the polarization insertions entering the induced interaction $V_{ind}$ and the self-energy $\Sigma$ entering the quasiparticle spectrum (effective mass $m^*$ and strength $Z$) must be generated from the same propagators and interactions, with the pairing gap, the ph propagators, and the single-particle spectrum determined self-consistently and without double counting, replacing the source calculation's separate treatments in which the screening was simply multiplied by $Z^2$. For definiteness the coupled calculation is to be carried out with the microscopic input of the source: the Argonne $v_{18}$ two-body force together with a meson-exchange three-body force, the Landau parameters of Zhang et al. (their particle-hole interaction kernel) as the input from which the induced interaction is generated, and the source's beta-equilibrium equation of state fixing the proton fraction (and hence $k_F^p$) along the density trajectory; the gap curve is to be reported over the outer-core density range $\rho\simeq 0.05$–$0.16\ {\rm fm}^{-3}$. To count as resolving the coupled problem, $V_{ind}$ must be regenerated from the medium's own particle-hole ring (RPA-type) propagators with its full momentum dependence retained — including the off-diagonal momentum-space matrix elements that the source's narrow-window Landau-limit evaluation $V_{ind}\simeq V_{ind}(k_F,k_F)$ discards — so that simply adopting the fixed Fermi-surface value of Zhang et al. does not qualify; the pairing gap must feed back into the particle-hole propagators and the self-energy, and the gap equation, the ph propagators, and the quasiparticle spectrum ($m^*$, $Z$) must be iterated to mutual convergence. A complete answer reports $\Delta_p(\rho)$ obtained in the coupled scheme together with its difference from the decoupled treatment in which the screening is multiplied by $Z^2$, so that the effect of treating the vertex and self-energy corrections on the same footing is quantified.

References

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