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The present paper is the second of a series of publications that aim atinvestigating relevant directions to turn the nuclear energy density functional(EDF) method as an effective field theory (EFT). The EDF approach has knownnumerous successes in nuclear theory over the past decades and is currently theonly microscopic technique that can be applied to all atomic nuclei. However,the phenomenological character of the EDF method also comes with importantlimitations, such as the lack of an explicit connection with quantumchromodynamics (QCD). As was argued in the first paper of this series,reformulating the EDF framework as an EFT would enable us to overcome theselimitations. In particular, path-integral (PI) techniques are suited to achievesuch a purpose as they allow to design numerous non-perturbative approximationsand can take Lagrangians possibly derived from EFTs of QCD as inputs. In ourprevious paper, we have illustrated such technical features for diagrammatic PItechniques in a study of the (0+0)-D $O(N)$-symmetric $\varphi^{4}$-theory. Inthe present work, we consider another class of PI techniques, i.e. functionalrenormalization group (FRG) approaches, that we apply to the same toy model.Despite our explicit interest for the nuclear many-body problem, the presentedstudy is also directed towards FRG practitioners from various fields: technicaldetails are provided for FRG techniques based on 1-particle-irreducible (1PI),2-particle-irreducible (2PI) and 2-particle-point-irreducible (2PPI) effectiveactions (EAs), coined respectively as 1PI-, 2PI- and 2PPI-FRGs, and thetreatment of the $O(N)$ symmetry is also addressed thoroughly. Connectionsbetween these various FRG methods are identified as well.

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