On discrete Minimal Flavour Violation

Abstract

We investigate the consequences of replacing the global flavour symmetry of Minimal Flavour Violation (MFV) SU(3)_QxSU(3)_UxSU(3)_Dx..., by a discrete D_QxD_UxD_Dx.. symmetry. Goldstone bosons resulting from the breaking of the flavour symmetry generically lead to bounds on new flavour structure many orders of magnitude above the TeV-scale. The absence of Goldstone bosons for discrete symmetries constitute the \emph{primary} motivation of our work. Less symmetry implies further invariants and renders the mass flavour basis transformation observable in principle and calls for a hierarchy in the Yukawa matrix expansion. We show, through the dimension of the representations, that the (discrete) symmetry in principle does allow for additional Delta F = 2 operators. If though the Delta F = 2 transitions are generated by two subsequent Delta F = 1 processes, as for example in the Standard Model, then the four crystal-like groups Sigma(168) ~ PSL(2,7), Sigma(72phi), Sigma(216phi) and especially Sigma(360phi) do provide enough protection for a TeV-scale discrete MFV scenario. Models where this is not the case have to be investigated case by case. Interestingly Sigma(216phi) has a (non-faithful) representation corresponding to an A4-symmetry. Moreover we argue that the, apparently often omitted, (D)-groups are subgroups of an appropriate Delta(6g^2). We would like to stress that we do not provide an actual model that realizes the MFV scenario nor any other theory of flavour.

25 pages (15 pages main text) added reference and edited appendix A.2.1. concerning (C) & (D) groups

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