Incorporating Brueckner-Hartree-Fock correlations in the Density Matrix Expansion approach

TL;DR

This paper enhances the density matrix expansion approach by incorporating Brueckner-Hartree-Fock correlations through a contact-term series, improving the modeling of nuclear many-body interactions with low-momentum forces.

Contribution

It introduces a method to include BHF correlations into the density matrix expansion using a series of contact terms, bridging HF and BHF approaches with effective interactions.

Findings

The difference between the G-matrix and V_NN can be modeled by contact terms.

The contact term coefficients depend on the potential and renormalization group resolution.

The approach enables HF-level density matrix expansion to account for BHF correlations.

Abstract

Recently, a microscopically motivated nuclear energy density functional was derived by applying the density matrix expansion to the Hartree-Fock (HF) energy obtained from long-range chiral effective field theory two- and three-nucleon interactions. However, the HF approach cannot account for all many-body correlations. One class of correlations is included by Brueckner-Hartree-Fock (BHF) theory, which gives an improved definition of the one-body HF potential by replacing the interaction by a reaction matrix $G$. In this paper, we find that the difference between the $G$-matrix and the nucleon-nucleon potential $V_{\mathrm{NN}}$ can be well accounted for by a truncated series of contact terms. This is consistent with renormalization group decoupling generating a series of counterterms as short-distance physics is integrated out. The coefficients $C_{n}$ of the power series expansion $\sum C_{n}q^{n}$ for the counterterms are examined for two potentials at different renormalization group resolutions and at a range of densities. The success of this expansion for $G-V_{\mathrm{NN}}$ means we can apply the density matrix expansion at the HF level with low-momentum interactions and density-dependent zero-range interactions to model BHF correlations.