Abstract
We show that the dynamics of the Higgs field during inflation is not affected by metric fluctuations if the Higgs is an energetically subdominant light spectator. For Standard Model parameters we find that couplings between Higgs and metric fluctuations are suppressed by \U0001d4aa(10−7). They are negligible compared to both pure Higgs terms in the effective potential and the unavoidable non-minimal Higgs coupling to background scalar curvature. The question of the electroweak vacuum instability during high energy scale inflation can therefore be studied consistently using the Jordan frame action in a Friedmann-Lemaître-Robertson-Walker metric, where the Higgs-curvature coupling enters as an effective mass contribution. Similar results apply for other light spectator scalar fields during inflation.
Highlights
Background solutionsThe equations of motion for a classical homogeneous and isotropic background obtained from the action (3.4) are given by φ + 3Hφ (1 ξ α2) 2ξα φh ̇ Mpl+ ∂φVtot h + 3Hh + ∂hVtot = 0= 3Mp2lH2, where we have defined Vtot (4.4)We concentrate on inflaton dominated solutions where the Higgs is a dynamically irrelevant spectator, + U (h)Note that the last term on the left hand side of eq.(4.3) is small due to (3.3)
The analysis of the section (4.3) shows that even if one were to quantize all fluctuations the resulting effective potential can be approximated by quantising only the matter fields as defined in the Jordan frame action (2.1), where the error is given by the small terms in (4.19)
What we found was that the corrections from quantum gravity were negligible with a suppression of 10−7 or more and that the quantization of the matter fields could well be made in the Einstein or Jordan frame
Summary
We revisit the stability analysis of the EW vacuum during inflation relaxing the assumption of a fixed gravitational background in the computation of the effective Higgs potential. These slow roll suppressed terms can be neglected in our analysis which verifies that the assumption of minimally coupled inflaton sector is self-consistent. It should be noted that even if there was a non-minimal inflaton coupling ξφRφ2, it can be removed at any given scale by a conformal transformation. This generates a ξφ dependent non-renormalisable Higgs-inflaton coupling in the Einstein frame. While ξ(μ) can be renormalized to zero at any given scale μ0, radiative corrections from Higgs loops in a curved spacetime drive ξ(μ) to non-zero values when moving away from μ0. We concentrate on the range 0 < ξ < 1/6 where the fate of the EW vacuum is a non-trivial interplay between the positive contribution to the effective Higgs mass from the non-minimal coupling and negative contribution (on large enough energy scales) from the quartic self-coupling λ
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