The invariant approach is a powerful method for studying CP violation for specific Lagrangians. The method is particularly useful for dealing with discrete family symmetries. We focus on the CP properties of unbroken $Delta(27)$ invariant Lagrangians
with Yukawa-like terms, which proves to be a rich framework, with distinct aspects of CP, making it an ideal group to investigate with the invariant approach. We classify Lagrangians depending on the number of fields transforming as irreducible triplet representations of $Delta(27)$. For each case, we construct CP-odd weak basis invariants and use them to discuss the respective CP properties. We find that CP violation is sensitive to the number and type of $Delta(27)$ representations.
I propose the use of CP-odd invariants, which are independent of basis and valid for any choice of CP transformation, as a powerful approach to study CP in the presence of flavour symmetries. As examples of the approach I focus on Lagrangians invaria
nt under $Delta(27)$. I comment on the consequences of adding a specific CP symmetry to a Lagrangian and distinguish cases where several $Delta(27)$ singlets are present depending on how they couple to the triplets. One of the examples included is a very simple toy model with explicit CP violation with calculable phases, which is referred to as explicit geometrical CP violation by comparison with previously known cases of (spontaneous) geometrical CP violation.
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 exampl
es based on $A_4$ and $Delta(27)$ family symmetries. For $A_4$, we show how to elegantly obtain several known results in the literature. In $Delta(27)$ 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.
In terms of its eigenvector decomposition, the neutrino mass matrix (in the basis where the charged lepton mass matrix is diagonal) can be understood as originating from a tribimaximal dominant structure with small deviations, as demanded by data. If
neutrino masses originate from at least two different mechanisms, referred to as hybrid neutrino masses, the experimentally observed structure naturally emerges provided one mechanism accounts for the dominant tribimaximal structure while the other is responsible for the deviations. We demonstrate the feasibility of this picture in a fairly model-independent way by using lepton-number-violating effective operators, whose structure we assume becomes dictated by an underlying $A_4$ flavor symmetry. We show that if a second mechanism is at work, the requirement of generating a reactor angle within its experimental range always fixes the solar and atmospheric angles in agreement with data, in contrast to the case where the deviations are induced by next-to-leading order effective operators. We prove this idea is viable by constructing an $A_4$-based ultraviolet completion, where the dominant tribimaximal structure arises from the type-I seesaw while the subleading contribution is determined by either type-II or type-III seesaw driven by a non-trivial $A_4$ singlet (minimal hybrid model). After finding general criteria, we identify all the $mathbb{Z}_N$ symmetries capable of producing such $A_4$-based minimal hybrid models.
We furnish a supersymmetric extension of the Standard Model with a flavour discrete symmetry $A_5$ under which the lepton fields transform as an irreducible triplet. Additional (`flavon) superfields are used to break $A_5$ into a $mathbb{Z}_2 times m
athbb{Z}_2$ subgroup in the charged-lepton sector and another $mathbb{Z}_2$ subgroup in the neutrino sector. The first column of the resulting lepton mixing matrix is predicted and has entries which are related to the golden ratio. Using the observed $theta_{13}$ as input, our model predicts a solar mixing angle $theta_{12}$ in very good agreement with experiment; it also predicts a correlation between the atmospheric mixing angle $theta_{23}$ and the $CP$-violating Dirac phase $delta$.
These are the proceedings of the 2nd Workshop on Flavor Symmetries and Consequences in Accelerators and Cosmology, held 30 June 2012 - 4 July 2012, Dortmund, Germany.
We consider in detail the non-renormalisable scalar potential of three Higgs doublets transforming as an irreducible triplet of Delta(27) or Delta(54). We start from a renormalisable potential that spontaneously leads to a vacuum with CP-violating ph
ases independent of arbitrary parameters - geometrical CP violation. Then we analyse to arbitrarily high order non-renormalisable terms that are consistent with the symmetry and we demonstrate that inclusion of non-renormalisable terms in the potential can preserve the geometrical CP-violating vacuum.
We consider how, for quasi-degenerate neutrinos with tri-bi-maximal mixing at a high-energy scale, the mixing angles are affected by radiative running from high to low-energy scales in a supersymmetric theory. The limits on the high-energy scale that
follow from consistency with the observed mixing are determined. We construct a model in which a non-Abelian discrete family symmetry leads both to a quasi-degenerate neutrino mass spectrum and to near tri-bi-maximal mixing.
