Diagrammatic structures of the Nielsen identity

TL;DR

This paper proves the Nielsen identity diagrammatically in an abelian gauge theory using the $ar{R}_\xi$ gauge, with implications for more complex models and resummation techniques.

Contribution

It introduces a diagrammatic proof of the Nielsen identity in an abelian model and discusses generalizations to non-abelian, finite temperature, and resummation contexts.

Findings

Diagrammatic proof of Nielsen identity in abelian gauge theory.

Method to separate bulk and ghost contributions in diagrams.

Potential for extending the approach to complex models and resummation techniques.

Abstract

The $\Gamma$-function, or the effective potential of a gauge field theory should comply with the Nielsen identity, which implies how the effective potential evolves as we shift the gauge-fixing term. In this paper, relying on an abelian toy model, we aim at proving this identity in a diagrammatic form with the $\overline{R}_{\xi}$ gauge. The basic idea is to find out the ghost chain after partially differentiating the diagram by the $\xi$ parameter, and shrink the waists of the diagram into points to separate the bulk-part and $C$-part of the diagrams. The calculations can be generalized to the models implemented with non-abelian groups, multiple Higgs and fermion multiplets, and to the finite temperature cases. Inspired by this, we also suggest that when resumming the super-daisy diagrams, one can deduct some irrelevant terms at the connections between the daisy ringlets to fit the Nielsen identity up to arbitrary $\hbar$ orders.