# Triple (and quadruple) soft-gluon radiation in QCD hard scattering

## TL;DR

This paper analyzes the complex patterns of soft-gluon radiation in QCD, revealing new color correlations and corrections beyond existing models, with implications for understanding multi-gluon emissions in high-energy particle collisions.

## Contribution

It computes the tree-level triple soft-gluon current for both massless and massive partons, introducing the concept of color quadrupole correlations and corrections to Casimir scaling.

## Key findings

- Triple soft-gluon radiation produces color quadrupole correlations.
- Color quadrupole interactions break Casimir scaling symmetry.
- First correction of order 1/N_c^2 to the eikonal formula is derived.

## Abstract

We consider the radiation of three soft gluons in a generic process for multiparton hard scattering in QCD. In the soft limit the corresponding scattering amplitude has a singular behaviour that is factorized and controlled by a colorful soft current. We compute the tree-level current for triple soft-gluon emission from both massless and massive hard partons. The three-gluon current is expressed in terms of maximally non-abelian irreducible correlations. We compute the soft behaviour of squared amplitudes and the colour correlations produced by the squared current. The radiation of one and two soft gluons leads to colour dipole correlations. Triple soft-gluon radiation produces in addition colour quadrupole correlations between the hard partons. We examine the soft and collinear singularities of the squared current in various energy ordered and angular ordered regions. We discuss some features of soft radiation to all-loop orders for processes with two and three hard partons. Considering triple soft-gluon radiation from three hard partons, colour quadrupole interactions break the Casimir scaling symmetry between quarks and gluons. We also present some results on the radiation of four soft gluons from two hard partons, and we discuss the colour monster contribution and its relation with the violation (and generalization) of Casimir scaling. We also compute the first correction of ${\cal O}(1/N_c^2)$ to the eikonal formula for multiple soft-gluon radiation with strong energy ordering from two hard gluons.