Testing Variational Perturbation Theory for Effective Actions Using the Gaudin-Yang Model

TL;DR

This paper evaluates Variational Perturbation Theory (VPT) against traditional methods using the exactly solvable Gaudin-Yang model, demonstrating VPT's superior performance in handling many-body problems across various densities.

Contribution

It introduces a comparative analysis of VPT with other perturbative methods using an exactly solvable model, highlighting VPT's advantages for effective action calculations.

Findings

VPT outperforms ordinary perturbation and inversion methods at second order.

Hubbard-Stratonovich auxiliary fields pose challenges in the approaches.

The study provides a pathway for extending VPT to nuclear physics applications.

Abstract

The background field formalism based on effective actions is a compelling framework for developing an effective field theory for nuclear density functional theory. Among the challenges in carrying out this development is handling both the particle-hole and pairing channels beyond the mean-field level, which includes how to incorporate collective degrees of freedom. Here we use the exactly solvable one-dimensional Gaudin-Yang model as a theoretical laboratory to explore candidate approaches. We compare Variational Perturbation Theory (VPT) to ordinary many-body perturbation theory and the inversion method, all to second order in their respective expansions, and verify issues with Hubbard-Stratonovich auxiliary fields. VPT outperforms the other approaches at this level over a wide range of densities. The next steps to extend this approach toward nuclei are outlined.