The Nielsen Identities for the Two-Point Functions of QED and QCD

TL;DR

This paper derives Nielsen identities for two-point functions in QED and QCD, demonstrating gauge independence of key quantities and providing formal solutions for propagators in terms of gauge-dependent Green functions.

Contribution

It presents a comprehensive derivation of Nielsen identities for QED and QCD two-point functions, including explicit one-loop calculations and formal solutions for propagators.

Findings

Photon self-energy gauge independence demonstrated

Quark mass renormalisation gauge independence in on-shell schemes

Explicit one-loop contributions to gluon and ghost identities

Abstract

We consider the Nielsen identities for the two-point functions of full QCD and QED in the class of Lorentz gauges. For pedagogical reasons the identities are first derived in QED to demonstrate the gauge independence of the photon self-energy, and of the electron mass shell. In QCD we derive the general identity and hence the identities for the quark, gluon and ghost propagators. The explicit contributions to the gluon and ghost identities are calculated to one-loop order, and then we show that the quark identity requires that in on-shell schemes the quark mass renormalisation must be gauge independent. Furthermore, we obtain formal solutions for the gluon self-energy and ghost propagator in terms of the gauge dependence of other, independent Green functions.