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Relaxion Monodromy and the Weak Gravity Conjecture

Luis E. Ibanez, Miguel Montero, Angel Uranga, Irene Valenzuela

TL;DR

The paper presents a monodromic relaxion framework in which a Minkowski 3-form and its quantized 4-form flux generate a multi-branched potential that enables trans-Planckian axion excursions while preserving a discrete shift symmetry. The relaxion-Higgs coupling arises via a Kaloper-Sorbo-type 4-form interaction, with the tiny coupling g induced by a seesaw relation g ≈ Λ_k^2/f, where Λ_k is a low fundamental scale tied to the 4-form flux. Constraints from membrane nucleation and the Weak Gravity Conjecture are used to bound the allowed parameter space, with gravity effects potentially suppressing tunneling and imposing upper limits on the cutoff M (e.g., M ≲ 10^9 GeV in generic cases, tighter for QCD-like realizations). The work further discusses a string-theory embedding and highlights the obstacles in obtaining the required hierarchical mass scales, outlining directions for consistent UV completions of monodromic relaxion models. Overall, it provides a theoretically motivated route to reconcile relaxion dynamics with quantum gravity constraints and points to concrete avenues for UV completion.

Abstract

The recently proposed relaxion models require extremely large trans-Planckian axion excursions as well as a potential explicitly violating the axion shift symmetry. The latter property is however inconsistent with the axion periodicity, which corresponds to a gauged discrete shift symmetry. A way to make things consistent is to use monodromy, i.e. both the axion and the potential parameters transform under the discrete shift symmetry. The structure is better described in terms of a 3-form field $C_{μνρ}$ coupling to the SM Higgs through its field strength $F_4$. The 4-form also couples linearly to the relaxion, in the Kaloper-Sorbo fashion. The extremely small relaxion-Higgs coupling arises in a see-saw fashion as $g\simeq F_4/f$, with $f$ being the axion decay constant. We discuss constraints on this type of constructions from membrane nucleation and the Weak Gravity Conjecture. The latter requires the existence of membranes, whose too fast nucleation could in principle drive the theory out of control, unless the cut-off scale is lowered. This allows to constrain relaxion models on purely theoretical grounds. We also discuss possible avenues to embed this structure into string theory.

Relaxion Monodromy and the Weak Gravity Conjecture

TL;DR

The paper presents a monodromic relaxion framework in which a Minkowski 3-form and its quantized 4-form flux generate a multi-branched potential that enables trans-Planckian axion excursions while preserving a discrete shift symmetry. The relaxion-Higgs coupling arises via a Kaloper-Sorbo-type 4-form interaction, with the tiny coupling g induced by a seesaw relation g ≈ Λ_k^2/f, where Λ_k is a low fundamental scale tied to the 4-form flux. Constraints from membrane nucleation and the Weak Gravity Conjecture are used to bound the allowed parameter space, with gravity effects potentially suppressing tunneling and imposing upper limits on the cutoff M (e.g., M ≲ 10^9 GeV in generic cases, tighter for QCD-like realizations). The work further discusses a string-theory embedding and highlights the obstacles in obtaining the required hierarchical mass scales, outlining directions for consistent UV completions of monodromic relaxion models. Overall, it provides a theoretically motivated route to reconcile relaxion dynamics with quantum gravity constraints and points to concrete avenues for UV completion.

Abstract

The recently proposed relaxion models require extremely large trans-Planckian axion excursions as well as a potential explicitly violating the axion shift symmetry. The latter property is however inconsistent with the axion periodicity, which corresponds to a gauged discrete shift symmetry. A way to make things consistent is to use monodromy, i.e. both the axion and the potential parameters transform under the discrete shift symmetry. The structure is better described in terms of a 3-form field coupling to the SM Higgs through its field strength . The 4-form also couples linearly to the relaxion, in the Kaloper-Sorbo fashion. The extremely small relaxion-Higgs coupling arises in a see-saw fashion as , with being the axion decay constant. We discuss constraints on this type of constructions from membrane nucleation and the Weak Gravity Conjecture. The latter requires the existence of membranes, whose too fast nucleation could in principle drive the theory out of control, unless the cut-off scale is lowered. This allows to constrain relaxion models on purely theoretical grounds. We also discuss possible avenues to embed this structure into string theory.

Paper Structure

This paper contains 20 sections, 100 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Axion monodromy
  3. A minimal relaxion monodromy model
  4. Seesaw-like scales and stability
  5. Coupling a multi-branched axion to the SM
  6. Membrane nucleation and the Weak Gravity Conjecture
  7. Membranes and monodromy
  8. WGC and membranes
  9. Constraints on the relaxion
  10. Constraints on the relaxion parameter space
  11. WGC constraints
  12. Monodromy relaxions and string theory
  13. Conclusions
  14. Axion monodromy in the dual 2-form view
  15. Analysis of the bubbles
  16. ...and 5 more sections

Figures (2)

  • Figure 1: Multi-branched structure of a typical axion monodromy model.
  • Figure 2: Plot of constraint $B>1$ including the $w(q)$ dependence, in blue, and of the constraints coming from coming from eqs.\ref{['barriers-form']} and \ref{['energydens']}, plotted in orange. The blue area corresponds to the region surviving to the tunneling constraints in the parameter space spanned by the cut-off scale $M$ and the parameter $q\simeq M^2/(HM_p)$ (left) or the Hubble scale (right).