Invariant approach to CP in family symmetry models

We propose the use of basis invariants, valid for any choice of CP transformation, as a powerful approach to studying specific models of 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) CP violation arises, which is geometrical in nature, i.e. persisting for arbitrary couplings in the Lagrangian.