A note on the implications of gauge invariance in QCD

TL;DR

This paper explores how gauge invariance affects scattering amplitudes in QCD, comparing it with QED, and derives a generalization of a key gauge invariance rule for photons to the context of gluons in QCD.

Contribution

It provides the first derivation of the most general gauge invariance condition for QCD scattering amplitudes analogous to the QED rule.

Findings

Derived the general gauge invariance condition for QCD amplitudes.

Compared gauge invariance implications between QED and QCD.

Highlighted differences in polarization vector transformations.

Abstract

We compare and contrast the implications of gauge invariance for the structure of scattering amplitudes in QED and QCD. We derive the most general analogue for QCD of the famous QED rule that the scattering amplitude must be invariant if the polarization vectors $\epsilon^{{\mu}_j}(k_j)$ of any number of photons undergo the replacement $ \epsilon^{{\mu}_j}(k_j) \rightarrow \epsilon^{{\mu}_j}(k_j) + c \, k_j^{\mu_j} $, where $c$ is an arbitrary constant.