Abstract
We present a critical discussion of quantum corrections,renormalisation, and the computation of the beta functions and theeffective potential in Higgs inflation. In contrast with claims inthe literature, we find no evidence for a disagreement between theJordan and Einstein frames, even at the quantum level. For clarity ofdiscussion we concentrate on the case of a real scalar Higgs. Wefirst review the classical calculation and then discuss the backreaction of gravity. We compute the beta functions for the Higgsquartic coupling and non-minimal coupling constant. Here, themid-field regime is non-renormalisable, but we are able to give anupper bound on the 1-loop corrections to the effective potential. Weshow that, in computing the effective potential, the Jordan andEinstein frames are compatible if all mass scales are transformedbetween the two frames. As such, it is consistent to take a constantcutoff in either the Jordan or Einstein frame, and both prescriptionsyield the same result for the effective potential. Our results areextended to the case of a complex scalar Higgs.
Highlights
In the paradigm of Higgs inflation the standard model (SM) Higgs doublet plays the role of the inflaton, and provides an almost-constant de Sitter vacuum energy to exponentially expand the universe during its initial stages [1, 2]
To compare the large-field beta function with the beta functions calculated in the small field regime, we find an expression for βλvalid for small field: βλsmall
One never encounters the problem that the divergences cannot be reabsorbed in a finite number of counter terms, something that we found to break down at higher order in the mid-field regime when calculating in the Einstein frame
Summary
In the paradigm of Higgs inflation the standard model (SM) Higgs doublet plays the role of the inflaton, and provides an almost-constant de Sitter vacuum energy to exponentially expand the universe during its initial stages [1, 2]. The classical analysis of Higgs inflation is well understood: one transforms to the Einstein frame to make the gravity sector canonical, redefines the Higgs degree of freedom (in unitary gauge) to obtain a canonical kinetic term, and uses the resulting potential to compute slow roll parameters in the usual way. This gives a connection between the quartic coupling in the Higgs potential and the new non-minimal coupling, and parameters of the CMB, ns and r. These are within current 1σ bounds from Planck [33]
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