A Forest Formula to Subtract Infrared Singularities in Amplitudes for Wide-angle Scattering

TL;DR

This paper develops a systematic forest formula in momentum space to subtract infrared singularities in wide-angle scattering amplitudes, generalizing previous analyses and ensuring all IR divergences cancel pairwise.

Contribution

It introduces a new forest formula for IR subtraction in momentum space applicable to any wide-angle QCD amplitude with multiple external legs.

Findings

IR singularities are systematically canceled in the subtraction terms.

The formalism generalizes Sudakov form factors to multiple external momenta.

Amplitude can be reorganized into a factorized form using the forest formula.

Abstract

For any hard QCD amplitude with massless partons, infrared (IR) singularities arise from pinches in the complex planes of loop momenta, called pinch surfaces. To organize and study their leading behaviors in the neighborhoods of these surfaces, we can construct approximation operators for collinear and soft singularities. A BPHZ-like forest formula can be developed to subtract them systematically. In this paper, we utilize the position-space analysis of Erdogan and Sterman for Green functions, and develop the formalism for momentum space. A related analysis has been carried out by Collins for the Sudakov form factors, and is generalized here to any wide-angle kinematics with an arbitrary number of external momenta. We will first see that the approximations yield much richer IR structures than those of an original amplitude, then construct the forest formula and prove that all the singularities appearing in its subtraction terms cancel pairwise. With the help of the forest formula, the full amplitude can also be reorganized into a factorized expression, which helps to generalize the Sudakov form factor result to arbitrary numbers of external momenta. All our analysis will be on the amplitude level.