  We consider a system of fermions with local interactions on a lattice
(Hubbard model) and apply a novel extension of the Laplace's method
(saddle-point approximation) for evaluating the corresponding partition
function. There, we introduce dual free bosonic fields, with a propagator
corresponding to an effective (renormalized) interaction with Maki-Thompson and
Aslamazov-Larkin type corrections and beyond, and demonstrate that the
superconducting pairing originates as an instability of the effective
interaction. We derive the corresponding Bethe-Salpeter equation (instability
criterion) and show that the interaction enters the equation only in its
effective form to all orders, including the exchange part of the self-energy.
An important implication of this result is that the effective interaction
always remains finite, even at phase-transition points, directly contradicting
the often used assumption of linear relationship between the interaction and
susceptibility, established within the random-phase approximation. By analyzing
the Bethe-Salpeter equation in the context of unconventional superconductivity,
we find that the presence of a flat band close the Fermi level, found in
materials such as twisted bilayer graphene, has a twofold favorable impact
persisting beyond the weak-coupling approximation: a reduced kinetic energy
cost of the gap formation and an increased anisotropy of the effective
interaction, favoring a momentum dependent order parameter.
