$CP$-odd invariants for multi-Higgs models: Applications with discrete symmetry

$CP$-odd invariants provide a basis independent way of studying the $CP$ properties of Lagrangians. We propose powerful methods for constructing basis invariants and determining whether they are $CP$ odd or $CP$ even, then systematically construct all of the simplest $CP$-odd invariants up to a given order, finding many new ones. The $CP$-odd invariants are valid for general potentials when expressed in a standard form. We then apply our results to scalar potentials involving three (or six) Higgs fields which form irreducible triplets under a discrete symmetry, including invariants for both explicit as well as spontaneous $CP$ violation. The considered cases include one triplet of Standard Model (SM) gauge singlet scalars, one triplet of SM Higgs doublets, two triplets of SM singlets, and two triplets of SM Higgs doublets. For each case, we study the potential symmetric under one of the simplest discrete symmetries with irreducible triplet representations, namely $A_4$, $S_4$, $\mathrm{\Delta }(27)$ or $\mathrm{\Delta }(54)$, as well as the infinite classes of discrete symmetries $\mathrm{\Delta }(3n^2)$ or $\mathrm{\Delta }(6n^2)$.

Figure 1

Example diagrams corresponding to small invariants.

Figure 2

Examples of contraction matrices for small invariants. All contraction matrices are symmetric except for the CPI.

Figure 3

Examples for contraction matrices of CPIs for spontaneous $CP$ violation. We draw each of the VEVs (as opposed to a single vertex for the whole $W$ tensor).