  Fermionic functional renormalization group (FRG) is applied to describe the
superfluid phase transition of the two-component fermionic system with
attractive contact interaction. Connection between the fermionic FRG approach
and the conventional Bardeen-Cooper-Schrieffer (BCS) theory with Gorkov and
Melik-Barkhudarov (GMB) correction are clarified in details in the weak
coupling region by using the renormalization group flow of the fermionic
four-point vertex with particle-particle and particle-hole scattering
contributions. To go beyond the BCS+GMB theory, coupled FRG flow equations of
the fermion self-energy and the four-point vertex are studied under an Ansatz
concerning their frequency/momentum dependence. We found that the fermion
self-energy turns out to be substantial even in the weak couping region, and
the frequency dependence of the four-point vertex is essential to obtain the
correct asymptotic-ultraviolet behavior of the flow for the self-energy. The
superfluid transition temperature and the associated chemical potential are
calculated in the region of negative scattering lengths.
