Relaxing the Electroweak Scale: the Role of Broken dS Symmetry

TL;DR

The paper addresses the electroweak hierarchy problem using the relaxion mechanism and shows that allowing the inflationary background to break de Sitter symmetry (nonzero ε) parametrically enhances the relaxion's field excursion per e-fold, dramatically reducing the required number of e-folds from the dS limit. The attractor analysis yields an explicit solution on FRW backgrounds and a bound ΔN_min ≳ \\log\left(1 + 2\\epsilon_0 \\frac{3 H_0^2}{g^2}\\right)^{\\frac{1}{2\\epsilon_0}} with ε0>0, which can yield ΔN_min ~ O(10^3) for ε0 ~ 10^{-2}. Observational considerations necessitate a curvaton mechanism to produce the observed CMB perturbations in significant parts of parameter space, potentially with the relaxion acting as the curvaton; the paper also analyzes concrete realizations, finding a dark QCD-like sector with Λ ~ 100 GeV and M ~ 10^5–10^6 GeV as a natural minimal setup, while CHAIN-like variants may extend M to ~10^9 GeV at the cost of additional tuning. Overall, breaking dS symmetry broadens the viable relaxion parameter space, reduces inflationary constraints, and connects the mechanism to testable dark-sector phenomenology and cosmological observables through curvaton dynamics and potential collider signatures.

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 $ε$ 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.

Figures (2)

• Figure 1: Schematic behaviour of the relaxion potential above and below $\phi = M^2/g$.
• Figure 2: Left: Upper bounds on $M$ as function of $\epsilon_0$. The solid red (blue) contours correspond to $H_0 = \Lambda = 1$ GeV (100 GeV) respectively, the region consistent with (\ref{['rhophi2']}) lies within the triangular shapes. Dashed (dotted) lines correspond to $H_0 = 10^{-5} \Lambda$ ($H_0 = 10^{5} \Lambda$). Right: Allowed region for $\Lambda = 100$ GeV, with grey lines showing $\log_{10} {\cal N}_{\rm min}$. The light (darker) orange shaded region has $T_{RH} <\hbox{TeV}$ ($T_{RH} < 100$ MeV), and we choose $H_0$ such that $H_0 \leq \sqrt[3]{\Lambda^4/M}$ is satisfied for $M \leq 10^9$ GeV. The parameter $g$ is set everywhere by $g M^3 = \Lambda^4$. In the region to the right of the vertical dashed line, a curvaton field is needed to satisfy (\ref{['cmb']}).