# Toward a systematic strategy for defining power counting in the construction of the energy density functional theory

## TL;DR

This paper introduces a systematic scheme for defining power counting in energy density functional theory using effective field theory principles, improving the construction of nuclear interactions beyond mean-field approximations.

## Contribution

It develops a new power counting scheme for EDF based on effective field theory, incorporating renormalization and counter interactions up to next-to-leading order.

## Key findings

- Constructed a Skyrme-like interaction consistent with renormalizability.
- Demonstrated the scheme's effectiveness with three different scenarios.
- Achieved satisfactory results in modeling the equation of state of nuclear matter.

## Abstract

We propose a new scheme for constructing an effective-field-theory-based interaction to be used in the energy-density-functional (EDF) theory with specific assumptions for defining a power counting. This procedure is developed through the evaluation of the equation of state (EOS) of symmetric and pure neutron matter going beyond the mean-field scheme and using a functional defined up to next-to-leading order (NLO), that we will call NLO EDF. A Skyrme-like interaction is constructed based on the condition of renormalizibility and on a power counting on $k_F/\Lambda_{hi}$, where $k_F$ is the Fermi momentum and $\Lambda_{hi}$ is the breakdown scale of our expansion. To absorb the divergences present in beyond mean-field diagrams, counter interactions are introduced for the NLO EDF and determined through renormalization conditions. In particular, three scenarios are explored and all of them lead to satisfactory results. These counter interactions contain also parameters which do not contribute to the EOS of matter and may eventually be determined through future adjustments to properties of some selected finite nuclei. Our work serves as a simple starting point for constructing a well-defined power counting within the EDF framework.