Bounded-from-below conditions for $A_4$-symmetric 3HDM

Deriving necessary and sufficient conditions for a scalar potential to be bounded from below (BFB) is a difficult task beyond the simplest cases. Recently, a set of BFB conditions was proposed for the $A_4$-invariant three-Higgs-doublet model (3HDM). However, that set of conditions relied on numerical scan, and a complete analytic proof was lacking. Here, we fill this gap. We prove that the conjectured BFB conditions are indeed necessary and sufficient within the neutral Higgs subspace. We bypass technically challenging direct algebraic computations with a novel technique that relies on an auxiliary function, which is related to the Higgs potential but which is easier to analyze. This technique may finally be sufficient to tackle the more involved case of the original Weinberg's 3HDM model.