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 $\Delta(27)$ 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.Comment: 4 pages plus references. v2: to be published in PR

Invariant approach to CP in unbroken Δ(27)\Delta(27)

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.Comment: 24 pages, 1 figure. v2: to be published in NP

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 $A_4$, $S_4$, $\Delta(27)$ or $\Delta(54)$, as well as the infinite classes of discrete symmetries $\Delta(3n^2)$ or $\Delta(6n^2)$.Comment: 54 pages, 39 diagrams, minor changes, version accepted in PR

Measuring stellar differential rotation with high-precision space-borne photometry

We introduce a method of measuring a lower limit to the amplitude of surface differential rotation from high-precision, evenly sampled photometric time series. It is applied to main-sequence late-type stars whose optical flux modulation is dominated by starspots. An autocorrelation of the time series was used to select stars that allow an accurate determination of starspot rotation periods. A simple two-spot model was applied together with a Bayesian information criterion to preliminarily select intervals of the time series showing evidence of differential rotation with starspots of almost constant area. Finally, the significance of the differential rotation detection and a measurement of its amplitude and uncertainty were obtained by an a posteriori Bayesian analysis based on a Monte Carlo Markov Chain approach. We applied our method to the Sun and eight other stars for which previous spot modelling had been performed to compare our results with previous ones. We find that autocorrelation is a simple method for selecting stars with a coherent rotational signal that is a prerequisite for successfully measuring differential rotation through spot modelling. For a proper Monte Carlo Markov Chain analysis, it is necessary to take the strong correlations among different parameters that exist in spot modelling into account. For the planet-hosting star Kepler-30, we derive a lower limit to the relative amplitude of the differential rotation of \Delta P / P = 0.0523 \pm 0.0016. We confirm that the Sun as a star in the optical passband is not suitable for measuring differential rotation owing to the rapid evolution of its photospheric active regions. In general, our method performs well in comparison to more sophisticated and time-consuming approaches.Comment: Accepted to Astronomy and Astrophysics, 15 pages, 13 figures, 4 tables and an Appendi

Towards a complete A4×SU(5)A_4 \times SU(5) SUSY GUT

We propose a renormalisable model based on $A_4$ family symmetry with an $SU(5)$ grand unified theory (GUT) which leads to the minimal supersymmetric standard model (MSSM) with a two right-handed neutrino seesaw mechanism. Discrete $\mathbb{Z}_9\times \mathbb{Z}_6$ symmetry provides the fermion mass hierarchy in both the quark and lepton sectors, while $\mathbb{Z}_4^R$ symmetry is broken to $\mathbb{Z}_2^R$, identified as usual R-parity. Proton decay is highly suppressed by these symmetries. We discuss both the $A_4$ and $SU(5)$ symmetry breaking sectors, including doublet-triplet splitting, Higgs mixing and the origin of the $\mu$ term. The model provides an excellent fit (better than one sigma) to all quark and lepton (including neutrino) masses and mixing with spontaneous CP violation. With the $A_4$ vacuum alignments, $(0,1,1)$ and $(1,3,1)$, the model predicts the entire PMNS mixing matrix with no free parameters, up to a relative phase, selected to be $2\pi/3$ from a choice of the nine complex roots of unity, providing a direct link between neutrino oscillations and leptogenesis.Comment: 30 pages, 17 figures, 7 tables; Minor changes, references added, version accepted in JHE