Classifying accidental symmetries in multi-Higgs doublet models

The potential of n-Higgs doublet models (nHDMs) contains a large number of SU(2)_L-preserving accidental symmetries as subgroups of the symplectic group Sp(2n). To classify these, we introduce prime invariants and irreducible representations in bilinear field space that enable us to explicitly construct accidentally symmetric nHDM potentials. We showcase the classifications of symmetries and present the relationship among the theoretical parameters of the scalar potential for the following: (i) the two Higgs doublet model (2HDM), and (ii) the three Higgs doublet model (3HDM). We recover the maximum number of 13 accidental symmetries for the 2HDM potential, and for the first time, we present the complete list of 40 accidental symmetries for the 3HDM potential.