We propose the use of basis invariants, valid for any choice of charge parity ($\mathcal{CP}$) transformation, as a powerful approach to studying specific models of $\mathcal{CP}$ violation in the presence of discrete family symmetries. We illustrate the virtues of this approach for examples based on $A_4$ and $\mathrm{\Delta }(27)$ family symmetries. For $A_4$, we show how to elegantly obtain several known results in the literature. In $\mathrm{\Delta }(27)$ we use the invariant approach to identify how explicit (rather than spontaneous) $\mathcal{CP}$ violation arises, which is geometrical in nature, i.e. persisting for arbitrary couplings in the Lagrangian.

• Received 24 February 2015

DOI:https://doi.org/10.1103/PhysRevD.92.036007