  The aim of this work is to investigate the occurrence of two different
dynamical symmetry breakings in two levels of the description of fermion-scalar
field system by means of a set of gap equations and of a background field
effective action. For that, we consider the Yukawa model, as a toy model for
interactions between non-massive fermions intermediated by a self-interacting
real scalar field. This model has at stakes two symmetries at the classical
level that, as we know, might be spontaneously or dynamically broken with mass
generation for the particles. To explore the spontaneous symmetry breakings the
auxiliary field method is used to produce coupled renormalized quantum gap
equations and then we write the effective action with quantum contributions by
external background fields method. We brought to light how the renormalization
procedure affects the physical gaps and discuss the auxiliary fields with their
link with the vertices.
