  We present a consistent fusion of functional renormalization group and
mean-field theory which explicitly introduces a bosonic field via a
Hubbard-Stratonovich transformation at the critical scale, at which the order
sets in. We show that a minimal truncation of the flow equations, that neglects
order parameter fluctuations, is integrable and fulfills fundamental
constraints as the Goldstone theorem and the Ward identity connected with the
broken global symmetry. To introduce the bosonic field, we present a technique
to factorize the most singular part of the vertex, even when the full
dependence on all its arguments is retained. We test our method on the
two-dimensional attractive Hubbard model at half-filling and calculate the
superfluid gap as well as the Yukawa couplings and residual two fermion
interactions in the ordered phase as functions of fermionic Matsubara
frequencies. Furthermore, we analyze the gap and the condensate fraction for
weak and moderate couplings and compare our results with previous functional
renormalization group studies, and with quantum Monte Carlo data. Our formalism
constitutes a convenient starting point for the inclusion of order parameter
fluctuations by keeping a full, non simplified, dependence on fermionic momenta
and/or frequencies.
