Symmetries of symmetries and geometrical CP violation

Abstract

We investigate transformations which are not symmetries of a theory but nevertheless leave invariant the set of all symmetry elements and representations. Generalizing from the example of a three Higgs doublet model with $\Delta (27)$ symmetry, we show that the possibility of such transformations signals physical degeneracies in the parameter space of a theory. We show that stationary points only appear in multiplets which are representations of the group of these so-called equivalence transformations. As a consequence, the stationary points are amongst the solutions of a set of homogeneous linear equations. This is relevant to the minimization of potentials in general and sheds new light on the origin of calculable phases and geometrical CP violation.

20+9 pages, 1 figure; v1: minor changes, added clarification, matches the published version

References in corpus (5)

• Symmetry breaking patterns in 3HDM
• Discrete symmetries in the three-Higgs-doublet model
• Stability of the normal vacuum in multi-Higgs-doublet models
• Properties of the general NHDM. I. The orbit space
• Geometrical CP violation in the N-Higgs-doublet model

Cited by in corpus (5)

• When the C in CP does not matter: anatomy of order-4 CP eigenstates and their Yukawa interactions
• Spontaneous CP violation in multi-Higgs potentials with triplets of $\Delta (3n^2)$ and $\Delta (6n^2)$
• Breaking classical Lie groups to finite subgroups - an automated approach
• CP and other Symmetries of Symmetries
• $\mathcal{C}\mathcal{P}$ violation with an unbroken $\mathcal{C}\mathcal{P}$ transformation