Recursion for Wilson-line Form Factors

TL;DR

This paper introduces a recursion method for efficiently computing Wilson-line form factors in gauge theories, leading to new compact expressions for splitting functions in QCD, validated at tree-level.

Contribution

It develops a novel recursion relation approach for Wilson-line form factors using a complex momentum shift, enabling efficient calculations and new compact splitting function formulas.

Findings

Derived recursion relations for Wilson-line form factors.

Obtained new compact expressions for QCD splitting functions.

Validated methods with $1\to 2$ and $1\to 3$ splitting functions.

Abstract

Matrix elements of Wilson-line dressed operators play a central role in the factorization of soft and collinear modes in gauge theories. When expressed using spinor helicity variables, these so-called form factors admit a classification starting from a Maximally Helicity Violating configuration, in close analogy with gauge theory amplitudes. We show that a single-line complex momentum shift can be used to derive recursion relations that efficiently compute these helicity form factors at tree-level: a combination of lower point form factors and on-shell amplitudes serve as the input building blocks. We obtain novel compact expressions for the $1\to 2$ and $1\to 3$ splitting functions in QCD, which also serves to validate our methods.