Relating amplitude and PDF factorisation through Wilson-line geometries

TL;DR

This paper investigates the connection between amplitude and PDF factorisation in perturbative QCD using Wilson-line geometries, confirming the structure of soft singularities and their role in resummation for various high-energy processes.

Contribution

It provides a detailed diagrammatic analysis of the anomalous dimensions governing soft singularities in PDFs, linking geometric Wilson-line configurations to physical resummation applications.

Findings

Confirmed Casimir scaling of the anomalous dimension.

Explicit two-loop computation of the PDF anomalous dimension.

Unified picture of singularities at cusps and along lightlike lines.

Abstract

We study long-distance singularities governing different physical quantities involving massless partons in perturbative QCD by using factorisation in terms of Wilson-line correlators. By isolating the process-independent hard-collinear singularities from quark and gluon form factors, and identifying these with the ones governing the elastic limit of the perturbative Parton Distribution Functions (PDFs) -- $\delta(1-x)$ in the large-$x$ limit of DGLAP splitting functions -- we extract the anomalous dimension controlling soft singularities of the PDFs, verifying that it admits Casimir scaling. We then perform an independent diagrammatic computation of the latter using its definition in terms of Wilson lines, confirming explicitly the above result through two loops. By comparing our eikonal PDF calculation to that of the eikonal form factor by Erdogan and Sterman and the classical computation of the closed parallelogram by Korchemsky and Korchemskaya, a consistent picture emerges whereby all singularities emerge in diagrammatic configurations localised at the cusps or along lightlike lines, but where distinct contributions to the anomalous dimensions are associated with finite (closed) lightlike segments as compared to infinite (open) ones. Both are relevant for resumming large logarithms in physical quantities, notably the anomalous dimension controlling Drell-Yan or Higgs production near threshold on the one hand, and the gluon Regge trajectory controlling the high-energy limit of partonic scattering on the other.