A covariant representation of the Ball-Chiu vertex

TL;DR

This paper presents a unified, compact, off-shell representation of the three-gluon vertex in nonabelian gauge theory using string-inspired methods, revealing structural insights and connections to effective actions and higher-loop behavior.

Contribution

It introduces a string-inspired formalism that unifies scalar, spinor, and gluon loop contributions to the one-loop vertex, extending Bern-Kosower rules off-shell and linking to the Ball-Chiu decomposition.

Findings

Derived a compact off-shell vertex representation applicable to all loop contributions.

Confirmed Bern-Kosower rules hold off-shell, enabling broader applications.

Predicted the persistence of the antisymmetric coefficient's vanishing at higher loops.

Abstract

In nonabelian gauge theory the three-gluon vertex function contains important structural information, in particular on infrared divergences, and is also an essential ingredient in the Schwinger-Dyson equations. Much effort has gone into analyzing its general structure, and at the one-loop level also a number of explicit computations have been done, using various approaches. Here we use the string-inspired formalism to unify the calculations of the scalar, spinor and gluon loop contributions to the one-loop vertex, leading to an extremely compact representation in all cases. The vertex is computed fully off-shell and in dimensionally continued form, so that it can be used as a building block for higher-loop calculations. We find that the Bern-Kosower loop replacement rules, originally derived for the on-shell case, hold off-shell as well. We explain the relation of the structure of this representation to the low-energy effective action, and establish the precise connection with the standard Ball-Chiu decomposition of the vertex. This allows us also to predict that the vanishing of the completely antisymmetric coefficient function S of this decomposition is not a one-loop accident, but persists at higher loop orders. The sum rule found by Binger and Brodsky, which leads to the vanishing of the one-loop vertex in N=4 SYM theory, in the present approach relates to worldline supersymmetry.