Gauge parameter dependence in the background field gauge and the construction of an invariant charge

TL;DR

This paper investigates the gauge parameter dependence in the background field gauge, demonstrating the unavoidable introduction of a normalization gauge parameter and constructing a gauge-invariant charge at 1-loop order.

Contribution

It introduces a method to control gauge parameter dependence using enlarged BRS transformations and constructs a gauge-invariant charge, highlighting limitations in independence from the normalization gauge parameter.

Findings

A normalization gauge parameter $\xi_o$ is essential for Green functions.

A gauge parameter independent combination of 2-point functions can be constructed at 1-loop order.

The constructed RG-invariant charge does not satisfy the differential equation for $\xi_o$, indicating residual dependence.

Abstract

By using the enlarged BRS transformations we control the gauge parameter dependence of Green functions in the background field gauge. We show that it is unavoidable -- also if we consider the local Ward identity -- to introduce the normalization gauge parameter $\xi_o$, which enters the Green functions of higher orders similarly to the normalization point $\kappa$. The dependence of Green functions on $\xi_o$ is governed by a further partial differential equation. By modifying the Ward identity we are able to construct in 1-loop order a gauge parameter independent combination of 2-point vector and background vector functions. By explicit construction of the next orders we show that this combination can be used to construct a gauge parameter independent RG-invariant charge. However, it is seen that this RG-invariant charge does not satisfy the differential equation of the normalization gauge parameter $\xi_o$, and, hence, is not $\xi_o$-independent as required.