Quantum corrections in Higgs inflation: the real scalar case

We present a critical discussion of quantum corrections, renormalisation, and the computation of the beta functions and the effective potential in Higgs inflation. In contrast with claims in the literature, we find no evidence for a disagreement between the Jordan and Einstein frames, even at the quantum level. For clarity of discussion we concentrate on the case of a real scalar Higgs. We first review the classical calculation and then discuss the back reaction of gravity. We compute the beta functions for the Higgs quartic coupling and non-minimal coupling constant. Here, the mid-field regime is non-renormalisable, but we are able to give an upper bound on the 1-loop corrections to the effective potential. We show that, in computing the effective potential, the Jordan and Einstein frames are compatible if all mass scales are transformed between the two frames. As such, it is consistent to take a constant cutoff in either the Jordan or Einstein frame, and both prescriptions yield the same result for the effective potential. Our results are extended to the case of a complex scalar Higgs.