Boundedness from below of SU(n) potentials

Abstract

Vacuum stability requires that the scalar potential is bounded from below. Whether or not this is true depends on the scalar quartic interactions alone, but even so the analysis is arduous and has only been carried out for a limited set of models. Complementing the existing literature, this work contains the necessary and sufficient conditions for two SU(n) invariant potentials to be bounded from below. In particular, expressions are given for models with the fundamental and the 2-index (anti)symmetric representations of this group. A sufficient condition for vacuum stability is also provided for models with the fundamental and the adjoint representations. Finally, some considerations are made concerning the model with the gauge group SU(2) and the scalar representations $\boldsymbol{1}$, $\boldsymbol{2}$ and $\boldsymbol{3}$; such a setup is particularly important for neutrino mass generation and lepton number violation.