The effective action for gauge bosons

TL;DR

This paper develops a gauge-invariant effective field theory for gauge bosons, providing explicit Wilson coefficients for various operators induced by massive particles, and reveals redundancies in the operator basis at one loop.

Contribution

It introduces a gauge-invariant matching procedure for EFT of gauge bosons applicable to various gauge groups, with explicit analytic expressions for Wilson coefficients and analysis of operator redundancies.

Findings

Explicit Wilson coefficients for dimension four, six, and eight operators.

Gauge invariance maintained off-shell during matching.

Identification of operator redundancies at one loop.

Abstract

By treating the vacuum as a medium, H. Euler and W. Heisenberg estimated the non-linear interactions between photons well before the advent of Quantum Electrodynamics. In a modern language, their result is often presented as the archetype of an Effective Field Theory (EFT). In this work, we develop a similar EFT for the gauge bosons of some generic gauge symmetry, valid for example for $SU(2)$, $SU(3)$, various grand unified groups, or mixed $U(1)\otimes SU(N)$ and $SU(M)\otimes SU(N)$ gauge groups. Using the diagrammatic approach, we perform a detailed matching procedure which remains manifestly gauge invariant at all steps, but does not rely on the equations of motion hence is valid off-shell. We provide explicit analytic expressions for the Wilson coefficients of the dimension four, six, and eight operators as induced by massive scalar, fermion, and vector fields in generic representations of the gauge group. These expressions rely on a careful analysis of the quartic Casimir invariants, for which we provide a review using conventions adapted to Feynman diagram calculations. Finally, our computations show that at one loop, some operators are redundant whatever the representation or spin of the particle being integrated out, reducing the apparent complexity of the operator basis that can be constructed solely based on symmetry arguments.