Solving the flavour problem in supersymmetric Standard Models with three Higgs families

We show how a non-Abelian family symmetry Δ27 can be used to solve the flavour problem of supersymmetric Standard Models containing three Higgs families such as the Exceptional Supersymmetric Standard Model (E6SSM). The three 27-dimensional families of the E6SSM, including the three families of Higgs fields, transform in a triplet representation of the Δ27 family symmetry, allowing the family symmetry to commute with a possible high energy E6 symmetry. The Δ27 family symmetry here provides a high energy understanding of the Z2H symmetry of the E6SSM, which solves the flavour changing neutral current problem of the three families of Higgs fields. The main phenomenological predictions of the model are tri-bi-maximal mixing for leptons, two almost degenerate LSPs and two almost degenerate families of colour triplet D-fermions, providing a clear prediction for the LHC. In addition the model predicts PGBs with masses below the TeV scale, and possibly much lighter, which appears to be a quite general and robust prediction of all models based on the D-term vacuum alignment mechanism.