Quark and leptonic mixing patterns from the breakdown of a common discrete flavor symmetry

Abstract

Assuming the Majorana nature of neutrinos, we recently performed a scan of leptonic mixing patterns derived from finite discrete groups of order less than 1536. Here we show that the 3 groups identified there as giving predictions close to experiment also contain another class of Abelian subgroups that predict an interesting leading-order quark mixing pattern where only the Cabibbo angle is generated at leading order. We further broaden our study by assuming that neutrinos are Dirac particles and find 4 groups of order up to 200 that can predict acceptable quark and leptonic mixing angles. Since large flavor groups allow for a multitude of leading-order mixing patterns, we define a measure that is suitable to compare the predictivity of a given flavor group, taking this fact into account. We give the result of this measure for a wide range of discrete flavor groups and identify the group ðZ 18 Â Z 6 Þ 2S 3 as being most predictive in the sense of this measure. We further discuss alternative measures and their implications.

Mixing patterns from the groups Σ(nφ)

“…5 We find that the two smaller groups Σ(36×3) and Σ(72×3) only give rise to a few new mixing patterns, while the larger groups lead to several new patterns, since they have a richer structure of subgroups. Generically one sees (as already suggested by the scans of groups making use of the computer program GAP [12,13,15]) that only very few patterns are well-compatible with the experimental data of all three different mixing angles [3], if only small corrections are admitted, i.e. for each group one to two.…”

“…3 The group Σ(36) is among the finite groups which can appear as symmetry group of the scalar sector of a three Higgs doublet model without leading to a potential with a continuous symmetry [21]. 4 This is in contrast to the recently performed scans of groups with an order smaller than 200 [15], an order smaller than 512 [12] or an order smaller than 1536 [13] that had the scope to only figure Among these four groups only Σ(360 × 3) has Klein subgroups and is thus suitable as G l , if neutrinos are assumed to be Majorana particles. Σ(36 × 3), Σ(72 × 3) and Σ(216 × 3) instead are only appropriate as G l , if neutrinos are Dirac particles or only one of the Z 2 symmetries of the neutrino mass matrix is contained in G l .…”

A model of quark and lepton mixing

“…In several of these models quarks are included via SU(5) unification, but typically vacuum alignment does not determine the quark mixing angles. However, in a purely symmetry approach, the direct approach has been extended to the quark sector, where a subgroup of the discrete family symmetry is used to constrain also the quark mixing angles, in analogy with the procedure followed for the Klein symmetry in the neutrino sector [21][22][23], but no realistic model has been proposed. In some such approaches [22,23], the symmetry groups can be quite large, for example ∆(6n 2 ) for large values of n [24].…”

A to Z of Flavour with Pati-Salam

“…In such models, it has been shown that the vacuum alignment completely breaks the A 4 symmetry, and such models are therefore referred to as "indirect" models [26]. Such "indirect" models are highly predictive and do not require such large discrete groups as the "direct" models where the Klein symmetry of the neutrino mass matrix is identified as a subgroup of the family symmetry [27][28][29].…”