Soft gluons at large angles in hadron collisions

TL;DR

This work develops a unified framework for single-logarithmic soft radiation in two-scale QCD observables involving four partons, introducing the fifth form factor F_X that encodes cross-channel colour exchange. By formulating the problem in an s-channel colour basis and diagonalising the soft anomalous dimension, the authors provide explicit expressions for F_X in terms of cross-channel Casimirs T_t^2 and T_u^2, and they apply the formalism to gg→gg, deriving eigenvalues, eigenvectors, and several important limits (90°, large-N, Regge). A key result is the cancellation of the divergent non-Abelian Coulomb phase, yielding a finite, physically meaningful S_X that governs SL soft corrections, along with an unexpected symmetry linking external scattering angles to internal colour structure. The findings enhance the resummation of global observables and clarify the role of inter-jet soft radiation in hadron collisions, with concrete results for gluon-gluon scattering and clear limits for practical use.

Abstract

A general discussion is presented of the single logarithmic soft factor that appears in two scale QCD observables in processes involving four partons. We treat it as the ``fifth form factor'', accompanying the four collinear singular Sudakov form factors attached to colliding and outgoing hard partons. The fifth form factor is expressed in terms of the Casimir operators (squared colour charges) of irreducible representations in the crossing $t$- and $u$-channels. As an application we revisit the problem of large angle radiation in $gg\to gg$ and give a relatively simple solution and interpretation of the results. We found an unexpected symmetry of the soft anomalous dimension under exchange of internal and external variables of the problem whose existence calls for explanation.