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

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.