  Spontaneous symmetry breaking of interacting fermion systems constitutes a
major challenge for many-body theory due to the proliferation of new
independent scattering channels once absent or degenerate in the symmetric
phase. One example is given by the ferro/antiferromagnetic broken symmetry
phase (BSP) of the Hubbard model, where vertices in the spin-transverse and
spin-longitudinal channels become independent with a consequent increase in the
computational power for their calculation. Here we generalize the formalism of
the non-perturbative Two-Particle-Self-Consistent method (TPSC) to treat broken
SU(2) magnetic phases of the Hubbard model, providing with a efficient yet
reliable method. We show that in the BSP, the sum-rule enforcement of
susceptibilities must be accompanied by a modified gap equation resulting in a
renormalisation of the order parameter, vertex corrections and the preservation
of the gap-less feature of the Goldstone modes. We then apply the theory to the
antiferromagnetic phase of the Hubbard model in the cubic lattice at
half-filling. We compare our results of double occupancies and staggered
magnetisation to the ones obtained using Diagrammatic Monte Carlo showing
excellent quantitative agreement. We demonstrate how vertex corrections play a
central role in lowering the Higgs resonance with respect to the quasi-particle
excitation gap in the spin-longitudinal susceptibility, yielding a well visible
Higgs-mode.
