Ultraviolet divergences and factorization for coordinate-space amplitudes

TL;DR

This paper analyzes ultraviolet divergences in coordinate-space scattering amplitudes with Wilson lines, classifies singularities, and develops a subtraction scheme to factorize amplitudes into gauge-invariant components, confirming renormalizability.

Contribution

It introduces a coordinate-space classification of singularities and a nested subtraction method for factorization, extending momentum-space techniques to coordinate space.

Findings

Classified singular regions in coordinate space for scattering amplitudes.

Developed a subtraction scheme enabling factorization into hard, jet, and soft factors.

Confirmed the multiplicative renormalizability of Wilson line products.

Abstract

We consider the coordinate-space matrix elements that correspond to fixed-angle scattering amplitudes involving partons and Wilson lines in coordinate space, working in Feynman gauge. In coordinate space, both collinear and short-distance limits produce ultraviolet divergences. We classify singularities in coordinate space, and identify neighborhoods associated unambiguously with individual subspaces (pinch surfaces) where the integrals are singular. The set of such regions is finite for any diagram. Within each of these regions, coordinate-space soft-collinear and hard-collinear approximations reproduce singular behavior. Based on this classification of regions and approximations, we develop a series of nested subtraction approximations by analogy to the formalism in momentum space. This enables us to rewrite each amplitude as a sum of terms to which gauge theory Ward identities can be applied, factorizing them into hard, jet and soft factors, and to confirm the multiplicative renormalizability of products of lightlike Wilson lines. We study in some detail the simplest case, the color-singlet cusp linking two Wilson lines, and show that the logarithm of this amplitude, which is a sum of diagrams known as webs, is closely related to the corresponding subtracted amplitude order by order in perturbation theory. This enables us to confirm that the logarithm of the cusp can be written as the integral of an ultraviolet-finite function over a surface. We study to what extent this result generalizes to amplitudes involving multiple Wilson lines.