On discrete Minimal Flavour Violation

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.