The Non-Abelian Exponentiation theorem for multiple Wilson lines

TL;DR

This paper extends the non-Abelian exponentiation theorem to multiple Wilson lines, demonstrating that soft gluon corrections exponentiate with fully connected colour factors, aiding higher-order calculations in gauge theories.

Contribution

It generalizes the non-Abelian exponentiation theorem from Wilson loops to multiple Wilson lines, introducing a new formalism with effective vertices for connected colour factors.

Findings

Proves exponentiation with connected colour factors for multiple Wilson lines.

Develops a new formalism with effective vertices for emissions.

Catalogues three-loop examples for soft anomalous dimension calculations.

Abstract

We study the structure of soft gluon corrections to multi-leg scattering amplitudes in a non-Abelian gauge theory by analysing the corresponding product of semi-infinite Wilson lines. We prove that diagrams exponentiate such that the colour factors in the exponent are fully connected. This completes the generalisation of the non-Abelian exponentiation theorem, previously proven in the case of a Wilson loop, to the case of multiple Wilson lines in arbitrary representations of the colour group. Our proof is based on the replica trick in conjunction with a new formalism where multiple emissions from a Wilson line are described by effective vertices, each having a connected colour factor. The exponent consists of connected graphs made out of these vertices. We show that this readily provides a general colour basis for webs. We further discuss the kinematic combinations that accompany each connected colour factor, and explicitly catalogue all three-loop examples, as necessary for a direct computation of the soft anomalous dimension at this order.