Relating amplitude and PDF factorisation through Wilson-line geometries

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.

arXiv comment53 pages, 5 figures; references added, typos fixed, journal version

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• Cusp and collinear anomalous dimensions in four-loop QCD from form factors
• The Four Loop QCD Rapidity Anomalous Dimension
• Subleading Power Resummation of Rapidity Logarithms: The Energy-Energy Correlator in $\mathcal{N}=4$ SYM
• Soft corrections to inclusive deep-inelastic scattering at four loops and beyond
• Disentangling the Regge cut and Regge pole in perturbative QCD
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• Cwebs beyond three loops in multiparton amplitudes
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• The SAGEX Review on Scattering Amplitudes, Chapter 15: The Multi-Regge Limit
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• Renormalization of the next-to-leading-power $\gamma \gamma \to h$ and $gg\to h$ soft quark functions
• Next-to-leading power two-loop soft functions for the Drell-Yan process at threshold