[Paper Review] Shift-Invariant Orders of an Axionlike Particle

Quentin Bonnefoy, Christophe Grojean+1 more|arXiv (Cornell University)|Jan 1, 2022

Advanced Topics in Algebra1 citations

TL;DR

This paper introduces a complete set of flavor-invariant order parameters to quantify the breaking of axion shift symmetry in axion-like particle (ALP) couplings to Standard Model fermions. By identifying Jarlskog-like invariants derived from Yukawa coupling matrices, the authors provide measurable, gauge-invariant quantities that capture perturbative shift-breaking effects. The key result is that these invariants close under renormalization group evolution and that CP conservation eliminates all but one such invariant, significantly constraining ALP phenomenology.

ABSTRACT

It is generally believed that global symmetries, in particular axion shift symmetries, can only be approximate. This motivates us to quantify the breaking of the shift invariance that characterizes the couplings of an axion-like particle (ALP), and to identify proper order parameters associated to this breaking. Focusing on the flavorful effective Yukawa couplings to Standard Model fermions, we work out explicit conditions for them to maintain an exact axion shift symmetry. Those conditions are given in terms of Jarlskog-like flavor-invariants and can be directly evaluated from the values of the different Yukawa couplings. Therefore, they represent order parameters for the breaking of the axion shift symmetry. We illustrate this construction by matching the axion EFT to UV models, and by showing that the renormalization group running closes on those shift-breaking flavor-invariants, as it should on any complete set of order parameters. Furthermore, the study of the invariants' CP-parities indicate that all but one are CP-odd, hence the assumption of CP conservation suffices to cancel all but one sources of shift-breaking in the theory. We also investigate similar conditions in the low-energy EFT below the electroweak scale, and comment on relations inherited from a UV completion which realizes the electroweak symmetry linearly. Finally, we discuss the order parameter associated to the non-perturbative shift-breaking induced by the axion-gluons coupling, which is also flavorful.

Motivation & Objective

• To systematically quantify the breaking of axion shift symmetry in Yukawa couplings to SM fermions.
• To identify gauge-invariant, flavor-invariant order parameters that measure the degree of shift symmetry breaking.
• To demonstrate that these invariants form a complete set that closes under renormalization group evolution.
• To clarify the role of CP violation in generating shift-breaking, showing that CP conservation eliminates all but one source of breaking.
• To extend the analysis to non-perturbative shift-breaking via axion-gluon couplings and their RG running.

Proposed method

• Constructs flavor-invariant order parameters using complex Yukawa coupling matrices in the quark and lepton sectors, analogous to Jarlskog invariants in the CKM matrix.
• Derives explicit conditions—linear in the Yukawa matrices—for exact axion shift symmetry, expressed as trace conditions on combinations like $ \text{Tr}[(\tilde{Y}_f \tilde{Y}_f^\dagger + \text{h.c.})^2] $.
• Uses effective field theory (EFT) at the SMEFT level to describe ALP couplings, with a non-redundant operator basis capturing leading-order interactions.
• Performs renormalization group evolution (RGE) analysis on the shift-breaking invariants, showing they close under RG flow.
• Analyzes the non-perturbative shift-breaking from the axion-gluon coupling via the topological term, identifying a separate order parameter.
• Matches the EFT to UV models (e.g., axiflavon, two-Higgs doublet, weakly-broken PQ) to verify consistency and physical realization.

Experimental results

Research questions

• RQ1 What are the complete, gauge-invariant, and flavor-invariant order parameters that quantify perturbative shift symmetry breaking in ALP-fermion couplings?
• RQ2 How do these order parameters evolve under the renormalization group, and do they form a closed set under RG flow?
• RQ3 What is the role of CP violation in generating shift-breaking, and can CP conservation eliminate most sources of breaking?
• RQ4 How do the shift-breaking invariants relate to UV completions such as axiflavon or two-Higgs-doublet models?
• RQ5 What is the nature of non-perturbative shift-breaking from the axion-gluon coupling, and how does it behave under RG evolution?

Key findings

• The paper identifies a complete set of linear flavor-invariant order parameters for shift symmetry breaking in ALP-fermion couplings, derived from the structure of Yukawa matrices.
• These invariants are explicitly calculable from the values of the Yukawa couplings and are closed under renormalization group evolution, confirming their role as physical order parameters.
• All but one of the shift-breaking invariants are CP-odd, so assuming CP conservation eliminates all but one source of perturbative shift-breaking.
• The non-perturbative shift-breaking from the axion-gluon coupling is captured by a separate, gauge-invariant order parameter related to the topological term.
• The RGE for the axion-gluon coupling coefficient $ C_g $ receives anomalous contributions from field redefinitions that mix with the axial anomaly, and these are fully accounted for in the analysis.
• The analysis confirms that the EFT framework with these invariants is consistent with UV models, including those with linearly realized electroweak symmetry.