Abstract
Recently, a novel mechanism to address the hierarchy problem has been proposed \cite{Graham:2015cka}, where the hierarchy between weak scale physics and any putative `cutoff' $M$ is translated into a parametrically large field excursion for the so-called relaxion field, driving the Higgs mass to values much less than $M$ through cosmological dynamics. In its simplest incarnation, the relaxion mechanism requires nothing beyond the standard model other than an axion (the relaxion field) and an inflaton. In this note, we critically re-examine the requirements for successfully realizing the relaxion mechanism and point out that parametrically larger field excursions can be obtained for a given number of e-folds by simply requiring that the background break exact de Sitter invariance. We discuss several corollaries of this observation, including the interplay between the upper bound on the scale $M$ and the order parameter $\epsilon$ associated with the breaking of dS symmetry, and entertain the possibility that the relaxion could play the role of a curvaton. We find that a successful realization of the mechanism is possible with as few as $\mathcal O (10^3)$ e-foldings, albeit with a reduced cutoff $M \sim 10^6$ GeV for a dark QCD axion and outline a minimal scenario that can be made consistent with CMB observations.
Highlights
M2 g where the ellipses denote higher order terms3 in gφ, and h is the (φ dependent) vacuum expectation value of the Higgs
We discuss several corollaries of this observation, including the interplay between the upper bound on the scale M and the order parameter associated with the breaking of de Sitter (dS) symmetry, and entertain the possibility that the relaxion could play the role of a curvaton
We find that a successful realization of the mechanism is possible with as few as O(103) e-foldings, albeit with a reduced cutoff M ∼ 106 GeV for a dark QCD axion and outline a minimal scenario that can be made consistent with CMB observations
Summary
The relaxion mechanism depends on the background cosmology in several crucial respects, and so it pays to re-examine how one can satisfy all that is required of the relaxion field in as general terms as possible. The primary requirement is for enough field excursion to have been executed over the course of early universe evolution. We demonstrate that this is an artefact of deriving various bounds without accounting for finite changes in the Hubble factor during inflation. Incorporating these corrections greatly ameliorates the situation in that φ executes a parametrically larger excursion on a quasi de Sitter background
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