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