  We study the details of the explicit and spontaneous symmetry breaking of the
constrained 3BF action representing the Standard Model coupled to
Einstein-Cartan gravity. First we discuss how each particular constraint breaks
the original symmetry of the topological 3BF action. Then we investigate the
spontaneous symmetry breaking and the Higgs mechanism for the electroweak
theory in the constrained 3BF form, in order to demonstrate that they can
indeed be performed in the framework of higher gauge theory. A formulation of
the Proca action as a constrained 3BF theory is also studied in detail.
