Musings on cosmological relaxation and the hierarchy problem

TL;DR

The paper evaluates cosmological relaxation as a solution to the electroweak hierarchy problem, focusing on the Graham–Kaplan–Rajendran two-field framework with an axion and the Higgs. By deriving the combined axion–Higgs potential and slow-roll conditions, it identifies three regimes in parameter space, showing that a hierarchically small Higgs vev can arise in two regimes but only under tight parameter relations; the full two-field dynamics can modify the naive single-field predictions. A key result is that the Higgs vev scales roughly as ⟨h⟩ ∼ (c1+c2) g f M^2/κ under certain conditions, implying that achieving v ≪ M requires small g (and thus ε = g/M), which induces tuning that can be more severe than in the SM when the cutoff is raised. The work highlights both the potential for a simple vev–parameter relationship and the substantial challenges in embedding the mechanism in a UV-complete theory or inflationary sector, noting possible phenomenological signatures in non-QCD axion models and the need for further exploration of embedding strategies.

Abstract

Recently Graham, Kaplan and Rajendran [1] proposed cosmological relaxation as a mechanism for generating a hierarchically small Higgs vacuum expectation value. Inspired by this we collect some thoughts on steps towards a solution to the electroweak hierarchy problem and apply them to the original model of cosmological relaxation [1]. To do so, we study the dynamics of the model and determine the relation between the fundamental input parameters and the electroweak vacuum expectation value. Depending on the input parameters the model exhibits three qualitatively different regimes, two of which allow for hierarchically small Higgs vacuum expectation values. One leads to standard electroweak symmetry breaking whereas in the other regime electroweak symmetry is mainly broken by a Higgs source term. While the latter is not acceptable in a model based on the QCD axion, in non-QCD models this may lead to new and interesting signatures in Higgs observables.

Figures (8)

• Figure 1: Sketch of what appears to be natural versus situations one may consider fine-tuned.
• Figure 2: When trying to increase the cutoff scale, fine-tuning may become worse.
• Figure 3: Sketch of different ways in which a parameter region can be small. The middle and right panels show situations which are still "tuned" but one can hope for an easier embedding.
• Figure 4: Plot of the effective axion potential for $c_1=4.0$, $c_2=1.0$, $M=0.001 M_{pl}$, $g=0.002 M$ and $f=80 M$.
• Figure 5: (a): Numerical result for the evolution of $\tilde{\phi}$ according to the full two-field model (red) and according to the single-field approximation (green, dashed). (b): evolution of $h$ for the full two-field model. Here we used $c_1=8.0$, $c_2=1.0$, $M=0.01 M_{pl}$, $g=0.0015 M$, $H=0.02 M$, $\lambda=0.5$, $\kappa=0.5 M^3$ and $f= 15 M$. The initial conditions are $\tilde{\phi}(0)= - 2 M^2/g$, $h(0)=10^{-6} M$ and $\dot{\tilde{\phi}}(0) = \dot{h}(0)=0$. For the two-field case the axion evolution stops before $\tilde{\phi}$ becomes positive, i.e. before the Higgs mass becomes tachyonic.