Singular terms of helicity amplitudes at one-loop in QCD and the soft limit of the cross sections of multi-parton processes

TL;DR

This work presents a general Born-level algorithm to extract all singular terms of one-loop helicity amplitudes in QCD for multi-parton processes, leveraging tree-level amplitudes and color algebra within D-dimensional regularization. The authors explicitly compute the singular parts of all 2→3 subprocess color subamplitudes and derive the soft limit of 2→4 cross sections, establishing universal soft/collinear structures and demonstrating cancellations with Bremsstrahlung terms. They provide compact spin-dependent and spin-independent soft-coefficient functions psi_mn that can be used as inputs for angular ordering in Monte Carlo shower programs, enhancing the treatment of coherent soft gluon emission in hadronic collisions. The results advance perturbative QCD calculations by enabling direct, Born-based determination of singular terms and by supplying practical soft-limit formulas for multi-jet final states. This has immediate phenomenological relevance for improving initial conditions in parton-shower simulations.

Abstract

We describe a general method that enables us to obtain all the singular terms of helicity amplitudes of n-parton processes at one loop. The algorithm uses helicity amplitudes at tree level and simple color algebra. We illustrate the method by calculating the singular part of the one loop helicity amplitudes of all $ 2\to 3$ parton subprocesses. The results are used to derive the soft gluon limit of the cross sections of all $2\to 4$ parton scattering subprocesses which provide a useful initial condition for the angular ordering approximation to coherent multiple soft gluon emission, incorporated in existing Monte Carlo simulation programs.