Thomas Neder

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 $A_4$, $S_4$, $\Delta(27)$ or $\Delta(54)$, as well as the infinite classes of discrete symmetries $\Delta(3n^2)$ or $\Delta(6n^2)$.

$CP$-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 $\mathrm{\Delta }(6n^2)$.

Generalised CP transformations are the only known framework which allows to predict Majorana phases in a flavour model purely from symmetry. For the first time generalised CP transformations are investigated for an infinite series of finite groups, $\Delta(6n^2)=(Z_n\times Z_n)\rtimes S_3$. In direct models the mixing angles and Dirac CP phase are solely predicted from symmetry. $\Delta(6n^2)$ flavour symmetry provides many examples of viable predictions for mixing angles. For all groups the mixing matrix has a trimaximal middle column and the Dirac CP phase is 0 or $\pi$. The Majorana phases are predicted from residual flavour and CP symmetries where $\alpha_{21}$ can take several discrete values for each $n$ and the Majorana phase $\alpha_{31}$ is a multiple of $\pi$. We discuss constraints on the groups and CP transformations from measurements of the neutrino mixing angles and from neutrinoless double-beta decay and find that predictions for mixing angles and all phases are accessible to experiments in the near future.

Experimentally viable lepton mixing parameters can be predicted in so-called direct flavour models with Majorana neutrinos using $\Delta(6n^2)$ groups as a flavour group. In direct models, in which the flavour group is broken to a $Z_2\times Z_2$ subgroup in the neutrino sector, mixing angles and Dirac CP phase are purely predicted from symmetry. General predictions of direct models with $\Delta(6n^2)$ flavour groups are that all mixing angles are fixed up to a discrete choice and that the Dirac CP phase is $0$ or $\pi$; Furthermore, the middle column of the mixing matrix is trimaximal which yields the sum rule $\theta_{23}=45^\circ \mp \theta_{13}/\sqrt{2}$ depending on the Dirac phase. These predictions of lepton mixing parameters are compatible with recent global fit results or will be tested experimentally in the near future. It is the first time that such predictions have been obtained model-independently for an infinite series of groups.