  The aim of this work is to investigate the occurrence of two different
spontaneous symmetry breakings {at} two levels of the description of
fermion-scalar field model, by means of a set of gap equations and {with} 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. The auxiliary field method is
considered and it produces coupled renormalized gap equations. The effective
action is then written with quantum contributions by {the} external background
{field} method. We brought to light how the renormalization procedure affects
the physical gaps, investigate its properties, and discuss the connection
between the auxiliary fields not only to define composite states but also to
compute the effective action.
