Group-theoretic condition for spontaneous CP violation

We formulate the necessary conditions for a scalar potential to exhibit spontaneous CP violation. Associated with each complex scalar field is a U(1) symmetry that may be explicitly broken by terms in the scalar potential (called spurions). In order for CP-odd phases in the vacuum to be physical, these phases must be related to spontaneously broken U(1) generators that are also explicitly broken by a sufficient number of inequivalent spurions. In the case where the vacuum is characterized by a single complex phase, our result implies that the phase must be associated with a U(1) generator that is broken explicitly by at least two inequivalent spurions. A suitable generalization of this result to the case of multiple complex phases has also been obtained. These conditions may be used both to distinguish models capable of spontaneous CP violation and as a model building technique for obtaining spontaneously CP-violating deformations of CP-conserving models. As an example, we analyze the generic two Higgs doublet model, where we also carry out a complete spurion analysis. We also comment on other models with spontaneous CP violation, including the chiral Lagrangian, a minimal version of the Nelson-Barr model, and little Higgs models with spontaneous CP violation.