Addressing energy density functionals in the language of path-integrals I: comparative study of diagrammatic techniques applied to the (0 + 0)-D O(N)-symmetric $$\varphi ^{4}$$-theory

The energy density functional (EDF) method is currently the only microscopic theoretical approach able to tackle entire nuclear chart. Nevertheless, it suffers from limitations resulting its empirical character and deteriorating reliability. This paper part of a larger program that aims at formulating EDF as an effective field theory (EFT) in order overcome these limitations. A relevant framework achieve this path-integral (PI) formulation quantum (QFT). latter indeed provides wide variety treatments many-body problem well suited deal with non-perturbative interactions exploit Lagrangian EFT starting point. While developing formalism general setting, we present comparative study such techniques applied toy model, i.e. (0+0)-D $O(N)$-symmetric $\varphi^{4}$-theory. More specifically, our focus will be on following diagrammatic techniques: loop expansion (LE), optimized perturbation (OPT) self-consistent (SCPT). With methods, notably address spontaneous breakdown $O(N)$ symmetry care especially since breakings (SSBs) play paramount role current implementations approach.