Ultraviolet divergences and factorization for coordinate-space amplitudes

Ozan Erdoğan and George Sterman
Phys. Rev. D 91, 065033 – Published 27 March 2015

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.

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  • Received 4 December 2014

DOI:https://doi.org/10.1103/PhysRevD.91.065033

© 2015 American Physical Society

Authors & Affiliations

Ozan Erdoğan1 and George Sterman2

  • 1Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, United Kingdom
  • 2C.N. Yang Institute for Theoretical Physics and Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794–3840, USA

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Issue

Vol. 91, Iss. 6 — 15 March 2015

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