Starobinksy Inflation, Gravitational Contact Terms and the Induced BEH Boson Mass

Abstract

The Starobinsky model of inflation remains consistent with observation, forty years after its introduction. It provides a well motivated origin for the scalar inflaton, the "scaleron" with a mass of $O(10^{13})$ GeV, emerging as a graviton degree of freedom from $R^2$ corrections to Einstein-Hilbert gravity. However the coupling of such a heavy state to the BEH ("Higgs") scalar is problematic as its quantum loop corrections can induce an unacceptably large contribution to the radiatively induced BEH scalar mass. The calculation of these corrections is normally done by Weyl transforming to the Einstein frame, yet at the quantum level Weyl transformations are fraught with ambiguities. However the recent realization that there exist "gravitational contact interactions" largely sidesteps these ambiguities. Such contact terms are necessarily present, coming from t-channel graviton exchange interactions, and they show that the theory defined in the "Jordan Frame" is identical to the theory in the Einstein frame, with additional Planck-scale suppressed interactions that take on the form of a Weyl transformation. This avoids ambiguous nonlinear field redefinitions, and reliable loop calculations are possible leading to a consistent low energy theory in an expansion in $1/M^2$. Taking account of the contact terms we study the radiative corrections to the BEH mass in the mixed Higgs/$R^2$ model with explicit scale breaking, and in an extension of the model in which exact scale symmetry is spontaneously broken.