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.

Figure 1

Regions of the parameter space of Dashen’s model parameter space of Dashen’s model, where spontaneous $CP$ violation occurs [cf., Eq. (38)]. A point in this parameter space corresponds to $(x,y)\equiv (m_u/m_s,m_d/m_s)$. The size of the phase ${\theta }_u$ is shown, with maximum values depicted in dark (blue) and minimum values in light (yellow). In these regions, ${\theta }_d$ also acquires a nonzero value, as explained in the text. The value of ${\theta }_d$ at the point ($x$, $y$) is equal to the value of ${\theta }_u$ at the point ($y$, $x$).