Recognizing symmetries in a 3HDM in a basis-independent way

Higgs doublets may come in three generations. The scalar sector of the resulting three-Higgs-doublet model (3HDM) may be constrained by global symmetry groups G leading to characteristic phenomenology. There exists the full list of symmetry groups G realizable in the 3HDM scalar sector and the expressions for G-symmetric scalar potentials written in special bases where the generators of G take simple form. However recognizing the presence of a symmetry in a generic basis remains a major technical challenge, which impedes efficient exploration of the 3HDM parameter space. In this paper, we solve this problem using the recently proposed approach, in which basis-independent conditions are formulated as relations among basis-covariant objects. We develop the formalism and derive basis-independent necessary and sufficient conditions for the 3HDM scalar sector to be invariant under each of the realizable symmetry group. We also comment on phenomenological consequences of these results.