Piecewise Linear Wilson lines

TL;DR

This paper introduces an algorithm for simplifying piecewise Wilson lines in QCD, reducing computational complexity and enabling easier comparison of different path structures for gauge-invariant calculations.

Contribution

It develops a novel algorithm to express piecewise path-ordered exponentials as integrals over segments, facilitating analysis of Wilson line topologies and their gauge invariance properties.

Findings

Reduces the number of diagrams needed for calculations.

Enables easy switching between different Wilson line structures.

Facilitates testing universality properties of non-perturbative objects.

Abstract

Wilson lines, being comparators that render non-local operator products gauge invariant, are extensively used in QCD calculations, especially in small-$x$ calculations, calculations concerning validation of factorisation schemes and in calculations for constructing or modelling parton density functions. We develop an algorithm to express piecewise path ordered exponentials as path ordered integrals over the separate segments, and apply it on linear segments, reducing the number of diagrams needed to be calculated. We show how different linear path topologies can be related using their colour structure. This framework allows one to easily switch results between different Wilson line structures, which is especially useful when testing different structures against each other, e.g. when checking universality properties of non-perturbative objects.