Geometric minimization of softly broken potentials

We study the minimization of the potentials of multi-Higgs models with symmetries that are softly broken. The powerful method of geometric minimization enables analytic minimization of the potentials of multi-Higgs models invariant under large symmetries. When these symmetries are softly broken, the method needs to be adapted. We propose a useful generalization that considers the effect of the soft-breaking terms to the quadratic part of the potential, by applying the procedure to restricted orbit spaces. We exemplify our novel methodology by finding and classifying the minima for an S_4 multi-Higgs model that is softly broken with specific terms.