On Gauge Invariance and Minimal Coupling

TL;DR

This paper critically reevaluates the role of minimal coupling as an organizing principle in effective field theories (EFTs) for Higgs and beyond-the-Standard-Model physics. It argues that minimal coupling is ill-defined and often non-unique at the operator level, and demonstrates through diverse examples (QED/QCD, chiral perturbation theory, bound states, and pseudo-Goldstone Higgs models) that many non-minimal operators can arise at tree level and are not systematically loop-suppressed. The authors emphasize that EFT power counting, gauge invariance, and symmetry structure—not minimal coupling—should govern operator hierarchies and RG running, with significant implications for interpreting Higgs processes and the operator basis in EFT analyses. The work cautions against over-reliance on minimal coupling in pseudo-Goldstone Higgs theories and reinforces a framework where higher-dimension operators are organized by standard EFT methods and symmetries, rather than by a dubious minimal-coupling criterion.

Abstract

The principle of minimal coupling has been used in the study of Higgs boson interactions to argue that certain higher dimensional operators in the low-energy effective theory generalization of the Standard Model are suppressed by loop factors, and thus smaller than others. It also has been extensively used to analyze beyond-the-standard-model theories. We show that in field theory, and even in quantum mechanics, the concept of minimal coupling is ill-defined and inapplicable as a general principle, and give many pedagogical examples which illustrate this fact. We also clarify some related misconceptions about the dynamics of strongly coupled gauge theories. Many arguments in the literature on Higgs boson interactions that use minimal coupling, particularly in pseudo-Goldstone Higgs theories, are inherently flawed.

Figures (2)

• Figure 1: (a) Gauge boson contribution to the pseudo-Goldstone boson mass $m_H^2 H^\dagger H$ (b) Higgs-gauge interaction $H^\dagger H F_{\mu \nu}F^{\mu \nu}$.
• Figure 2: Meson exchange contribution to $\pi-\pi$ scattering.