Discrete Minimal Flavour Violation.

We investigate the consequences of replacing the global flavor symmetry of minimal flavor violation (MFV) SU(3)Q×SU(3)U×SU(3)D×⋯ by a discrete DQ×DU×DD×⋯ symmetry. Goldstone bosons resulting from the breaking of the flavor symmetry generically lead to bounds on new flavor structure many orders of magnitude above the TeV scale. The absence of Goldstone bosons for discrete symmetries constitute the primary motivation of our work. Less symmetry implies further invariants and renders the mass-flavor 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 ΔF=2 operators. If though the ΔF=2 transitions are generated by two subsequent ΔF=1 processes, as, for example, in the standard model, then the four crystal-like groups Σ(168)≈PSL(2,F7), Σ(72φ), Σ(216φ) and especially Σ(360φ) 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 Σ(216φ) has a (nonfaithful) representation corresponding to an A4 symmetry. Moreover we argue that the, apparently often omitted, (D) groups are subgroups of an appropriate Δ(6g2). We would like to stress that we do not provide an actual model that realizes the MFV scenario nor any other theory of flavor.