Supergravity Higgs inflation and shift symmetry in electroweak theory

We present a model of inflation in a supergravity framework in the Einstein frame where the Higgs field of the next to minimal supersymmetric standard model (NMSSM) plays the role of the inflaton. Previous attempts which assumed non-minimal coupling to gravity failed due to a tachyonic instability of the singlet field during inflation. A canonical Kahler potential with minimal coupling to gravity can resolve the tachyonic instability but runs into the η-problem. We suggest a model which is free of the η-problem due to an additional coupling in the Kähler potential which is allowed by the Standard Model gauge group. This induces directions in the potential which we call K-flat. For a certain value of the new coupling in the (N)MSSM, the Kåhler potential is special, because it can be associated with a certain shift symmetry for the Higgs doublets, a generalization of the shift symmetry for singlets in earlier models. We find that K-flat direction has H ⁰_u = -H_d^0*. This shift symmetry is broken by interactions coming from the superpotential and gauge fields. This flat direction fails to produce successful inflation in the MSSM but yields a more interesting model in the NMSSM, even though it does not pass existing cosmological constraints. We point out that, in building more sophisticated models of this type, one may also need to take into account their implications for axion searches or other elementary particle constraints.