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Register a new userInvariant approach to CP in family symmetry models
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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 $Delta(27)$ 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.
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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.
We use a new weak basis invariant approach to classify all the observable phases in any extension of the Standard Model (SM). We apply this formalism to determine the invariant CP phases in a simplified version of the Minimal Supersymmetric SM with only three non-trivial flavour structures. We propose four experimental measures to fix completely all the observable phases in the model. After these phases have been determined from experiment, we are able to make predictions on any other CP-violating observable in the theory, much in the same way as in the Standard Model all CP-violation observables are proportional to the Jarlskog invariant.
We study the spontaneous $CP$ violation through the stabilization of the modulus $tau$ in modular invariant flavor models. The $CP$-invaraiant potentential has the minimum only at ${rm Re}[tau] = 0$ or 1/2. From this prediction, we study $CP$ violation in modular invariant flavor models. The physical $CP$ phase is vanishing. The important point for the $CP$ conservation is the $T$ transformation in the modular symmetry. One needs the violation of $T$ symmetry to realize the spontaneous $CP$ violation.
CP-odd invariants, independent of basis and valid for any choice of CP transformation are a powerful tool in the study of CP. They are particularly convenient to study the CP properties of models with family symmetries. After interpreting the consequences of adding specific CP symmetries to a Lagrangian invariant under $Delta(27)$, I use the invariant approach to systematically study Yukawa-like Lagrangians with an increasing field content in terms of $Delta(27)$ representations. Included in the Lagrangians studied are models featuring explicit CP violation with calculable phases (referred to as explicit geometrical CP violation) and models that automatically conserve CP, despite having all the $Delta(27)$ representations.
We show how the SUSY flavour and CP problems can be solved using gauged SU(3) family symmetry previously introduced to describe quark and lepton masses and mixings, in particular neutrino tri-bimaximal mixing via constrained sequential dominance. The Yukawa and soft trilinear and scalar mass squared matrices and kinetic terms are expanded in powers of the flavons used to spontaneously break the SU(3) family symmetry, and the canonically normaliz
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