Mean-field proton-neutron pairing correlations with the Gogny D1S energy density functional

TL;DR

This paper investigates proton-neutron pairing correlations using the Gogny D1S energy density functional within a generalized HFB framework, revealing stability issues and the tendency of the functional to favor no proton-neutron pairing in studied nuclei.

Contribution

It extends the HFB method to include proton-neutron mixing with Gogny functionals and compares stability and energy behaviors against other interactions.

Findings

Gogny D1S functional shows instabilities with proton-neutron mixing due to zero-range density dependence.

Stable solutions are obtained with the B1 interaction, unlike Gogny D1S.

Energy minima correspond to no proton-neutron pairing in studied nuclei.

Abstract

We study proton-neutron pairing correlations within the Hartree-Fock-Bogoliubov (HFB) framework using Gogny-type energy density functionals. By allowing for proton-neutron mixing in the quasi-particle transformation, both isovector ($T=1$) and isoscalar ($T=0$) pairing channels are explicitly included at the mean-field level. The \texttt{TAURUS} code has been extended to treat density-dependent Gogny interactions in this generalized HFB scheme. We examine the numerical behavior of the widely used Gogny D1S functional and compare it with calculations performed using the Hamiltonian-based Brink-Boecker B1 interaction supplemented by a zero-range spin-orbit term. When proton-neutron mixing is included and large single-particle spaces are employed, instabilities are observed for Gogny D1S due to the zero-range density-dependent term contribution to the proton-neutron pairing field, whereas stable solutions are obtained with the B1 interaction. Constrained HFB calculations performed in reduced configuration spaces allow us to explore total energy curves as functions of proton-neutron pairing collective coordinates in selected $sd$-shell nuclei. In all cases studied, the self-consistent minima correspond to vanishing proton-neutron pairing, with energy increasing rapidly as proton-neutron pairing correlations are introduced. These results provide insight into the behavior of Gogny functionals under generalized HFB conditions and offer useful guidance for future developments.