A to Z of Flavour with Pati-Salam

We propose an elegant theory of flavour based on $A_4\times Z_5$ family symmetry with Pati-Salam unification which provides an excellent description of quark and lepton masses, mixing and CP violation. The $A_4$ symmetry unifies the left-handed families and its vacuum alignment determines the columns of Yukawa matrices. The $Z_5$ symmetry distinguishes the right-handed families and its breaking controls CP violation in both the quark and lepton sectors. The Pati-Salam symmetry relates the quark and lepton Yukawa matrices, with $Y^u=Y^{\nu}$ and $Y^d\sim Y^e$. Using the see-saw mechanism with very hierarchical right-handed neutrinos and CSD4 vacuum alignment, the model predicts the entire PMNS mixing matrix and gives a Cabibbo angle $\theta_C\approx 1/4$. In particular it predicts maximal atmospheric mixing, $\theta^l_{23}=45^\circ\pm 0.5^\circ$ and leptonic CP violating phase $\delta^l=260^\circ \pm 5^\circ$. The reactor angle prediction is $\theta^l_{13}=9^\circ\pm 0.5^\circ$, while the solar angle is $34^\circ \geq \theta^l_{12}\geq 31^\circ$, for a lightest neutrino mass in the range $0 \leq m_1 \leq 0.5$ meV, corresponding to a normal neutrino mass hierarchy and a very small rate for neutrinoless double beta decay.Comment: 38 pages, 28 figures, published versio

RK(∗)R_{K^{(*)}} and the origin of Yukawa couplings

We explore the possibility that the semi-leptonic $B$ decay ratios $R_{K^{(*)}}$ which violate $\mu - e$ universality are related to the origin of the fermion Yukawa couplings in the Standard Model. Some time ago, a vector-like fourth family (without a $Z'$) was used to generate fermion mass hierarchies and mixing patterns without introducing any family symmetry. Recently the idea of inducing flavourful $Z'$ couplings via mixing with a vector-like fourth family which carries gauged $U(1)'$ charges has been proposed as a simple way of producing controlled flavour universality violation while elegantly cancelling anomalies. We show that the fusion of these two ideas provides a nice connection between $R_{K^{(*)}}$ and the origin of Yukawa couplings in the quark sector. However the lepton sector requires some tuning of Yukawa couplings to obtain the desired coupling of $Z'$ to muons.Comment: Minor corrections to phenomenology section to match published version. 19 pages, 4 figure

Testing constrained sequential dominance models of neutrinos

Constrained sequential dominance (CSD) is a natural framework for implementing the see-saw mechanism of neutrino masses which allows the mixing angles and phases to be accurately predicted in terms of relatively few input parameters. We analyze a class of CSD($n$) models where, in the flavour basis, two right-handed neutrinos are dominantly responsible for the "atmospheric" and "solar" neutrino masses with Yukawa couplings to $(\nu_e, \nu_{\mu}, \nu_{\tau})$ proportional to $(0,1,1)$ and $(1,n,n-2)$, respectively, where $n$ is a positive integer. These coupling patterns may arise in indirect family symmetry models based on $A_4$. With two right-handed neutrinos, using a $\chi^2$ test, we find a good agreement with data for CSD(3) and CSD(4) where the entire PMNS mixing matrix is controlled by a single phase $\eta$, which takes simple values, leading to accurate predictions for mixing angles and the magnitude of the oscillation phase $|\delta_{CP}|$. We carefully study the perturbing effect of a third "decoupled" right-handed neutrino, leading to a bound on the lightest physical neutrino mass $m_1 \lesssim 1$ meV for the viable cases, corresponding to a normal neutrino mass hierarchy. We also discuss a direct link between the oscillation phase $\delta_{CP}$ and leptogenesis in CSD($n$) due to the same see-saw phase $\eta$ appearing in both the neutrino mass matrix and leptogenesis.Comment: 34 pages, 15 figures. Version to be published in J.Phys G. Note the change in title. Clarifying comments added. Previous versions: 32 pages, 15 figures. Improved discussion of chi squared analysis, new plots added. // 29 pages, 13 figures. Minor changes and discussion about the origin of the vacuum alignments added to an Appendi

Lepton Mixing Predictions including Majorana Phases from Δ(6n2)\Delta(6n^2) Flavour Symmetry and Generalised CP

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.Comment: 16 pages, 8 figures; references added; clarification in section 2.3 added; results are unchange

A model of quarks with Delta (6N^2) family symmetry

We propose a first model of quarks based on the discrete family symmetry Delta (6N^2) in which the Cabibbo angle is correctly determined by a residual Z_2 times Z_2 subgroup, and the smaller quark mixing angles may be qualitatively understood from the model. The present model of quarks may be regarded as a first step towards formulating a complete model of quarks and leptons based on Delta (6N^2), in which the lepton mixing matrix is fully determined by a Klein subgroup. For example, the choice N=28 provides an accurate determination of both the reactor angle and the Cabibbo angle.Comment: 12 pages, 5 figures; reference adde

Neutrino Mass and Mixing with Discrete Symmetry

This is a review article about neutrino mass and mixing and flavour model building strategies based on discrete family symmetry. After a pedagogical introduction and overview of the whole of neutrino physics, we focus on the PMNS mixing matrix and the latest global fits following the Daya Bay and RENO experiments which measure the reactor angle. We then describe the simple bimaximal, tri-bimaximal and golden ratio patterns of lepton mixing and the deviations required for a non-zero reactor angle, with solar or atmospheric mixing sum rules resulting from charged lepton corrections or residual trimaximal mixing. The different types of see-saw mechanism are then reviewed as well as the sequential dominance mechanism. We then give a mini-review of finite group theory, which may be used as a discrete family symmetry broken by flavons either completely, or with different subgroups preserved in the neutrino and charged lepton sectors. These two approaches are then reviewed in detail in separate chapters including mechanisms for flavon vacuum alignment and different model building strategies that have been proposed to generate the reactor angle. We then briefly review grand unified theories (GUTs) and how they may be combined with discrete family symmetry to describe all quark and lepton masses and mixing. Finally we discuss three model examples which combine an SU(5) GUT with the discrete family symmetries A4, S4 and Delta(96).Comment: 99 pages, 13 figures, review article, updated to include the results from the latest global fit