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We present a consistent fusion of functional renormalization group andmean-field theory which explicitly introduces a bosonic field via aHubbard-Stratonovich transformation at the critical scale, at which the ordersets in. We show that a minimal truncation of the flow equations, that neglectsorder parameter fluctuations, is integrable and fulfills fundamentalconstraints as the Goldstone theorem and the Ward identity connected with thebroken global symmetry. To introduce the bosonic field, we present a techniqueto factorize the most singular part of the vertex, even when the fulldependence on all its arguments is retained. We test our method on thetwo-dimensional attractive Hubbard model at half-filling and calculate thesuperfluid gap as well as the Yukawa couplings and residual two fermioninteractions in the ordered phase as functions of fermionic Matsubarafrequencies. Furthermore, we analyze the gap and the condensate fraction forweak and moderate couplings and compare our results with previous functionalrenormalization group studies, and with quantum Monte Carlo data. Our formalismconstitutes a convenient starting point for the inclusion of order parameterfluctuations by keeping a full, non simplified, dependence on fermionic momentaand/or frequencies.

Accessing the ordered phase of correlated Fermi systems: vertex bosonization and mean-field theory within the functional renormalization group