A local and renormalizable framework for the gauge-invariant operator $A^2_{\min}$ in Euclidean Yang-Mills theories in linear covariant gauges

TL;DR

This paper demonstrates that the gauge-invariant operator $A^2_{min}$ in Euclidean Yang-Mills theories can be localized with an auxiliary field, leading to a renormalizable framework that preserves gauge invariance and gauge parameter independence.

Contribution

The authors introduce a local, renormalizable formulation of the non-local operator $A^2_{min}$ using an auxiliary Stueckelberg field, ensuring gauge invariance and all-order renormalizability.

Findings

The localized operator $A^2_{min}$ is renormalizable to all orders.

The anomalous dimension of $A^2_{min}$ is gauge parameter independent.

The framework maintains gauge invariance in the renormalization process.

Abstract

We address the issue of the renormalizability of the gauge-invariant non-local dimension-two operator $A^2_{\rm min}$, whose minimization is defined along the gauge orbit. Despite its non-local character, we show that the operator $A^2_{\rm min}$ can be cast in local form through the introduction of an auxiliary Stueckelberg field. The localization procedure gives rise to an unconventional kind of Stueckelberg-type action which turns out to be renormalizable to all orders of perturbation theory. In particular, as a consequence of its gauge invariance, the anomalous dimension of the operator $A^2_{\rm min}$ turns out to be independent from the gauge parameter $\alpha$ entering the gauge-fixing condition, being thus given by the anomalous dimension of the operator $A^2$ in the Landau gauge.