From bare interactions, low-energy constants, and unitary gas to nuclear density functionals without free parameters: Application to neutron matter

Abstract

We further progress along the line of Ref. [D. Lacroix, Phys. Rev. A 94, 043614 (2016)] where a functional for Fermi systems with anomalously large $s$-wave scattering length ${a}_{s}$ was proposed that has no free parameters. The functional is designed to correctly reproduce the unitary limit in Fermi gases together with the leading-order contributions in the $s$- and $p$-wave channels at low density. The functional is shown to be predictive up to densities $\ensuremath{\sim}0.01 {\mathrm{fm}}^{\ensuremath{-}3}$ that is much higher densities compared to the Lee-Yang functional, valid for $\ensuremath{\rho}<{10}^{\ensuremath{-}6} {\mathrm{fm}}^{\ensuremath{-}3}$. The form of the functional retained in this work is further motivated. It is shown that the new functional corresponds to an expansion of the energy in $({a}_{s}{k}_{F})$ and $({r}_{e}{k}_{F})$ to all orders, where ${r}_{e}$ is the effective range and ${k}_{F}$ is the Fermi momentum. One conclusion from the present work is that, except in the extremely low-density regime, nuclear systems can be treated perturbatively in $\ensuremath{-}{({a}_{s}{k}_{F})}^{\ensuremath{-}1}$ with respect to the unitary limit. Starting from the functional, we introduce density-dependent scales and show that scales associated with the bare interaction are strongly renormalized by medium effects. As a consequence, some of the scales at play around saturation are dominated by the unitary gas properties and not directly by low-energy constants. For instance, we show that the scale in the $s$-wave channel around saturation is proportional to the so-called Bertsch parameter ${\ensuremath{\xi}}_{0}$ and becomes independent of ${a}_{s}$. We also point out that these scales are of the same order of magnitude than those empirically obtained in the Skyrme energy density functional. We finally propose a slight modification of the functional such that it becomes accurate up to the saturation density $\ensuremath{\rho}\ensuremath{\simeq}0.16 {\mathrm{fm}}^{\ensuremath{-}3}$.