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The soft-gluon current at one-loop order

S. Catani, M. Grazzini

TL;DR

The paper derives a process-independent factorization formula for the soft limit of one-loop QCD amplitudes by introducing a generalized soft-gluon current that incorporates quantum corrections. It provides an explicit, gauge-invariant expression for the one-loop soft current J^(1) that exhibits purely non-abelian colour correlations and reduces, via color conservation, to a tree-level-like structure for two- and three-hard-parton squared amplitudes. The authors prove the one-loop factorization and compute J^(1) using an axial-gauge analysis, revealing two-particle colour correlations and potential higher-order multiparton correlations. They further show that in two- and three-parton processes the colour algebra collapses to Casimir operators, yielding QED-like factorization and enabling NNLO calculations for e^+e^- → 2 jets and 3 jets. Overall, the work provides a comprehensive framework for infrared factorization at one loop and lays the groundwork for higher-loop generalizations and precise NNLO predictions.

Abstract

We study the soft limit of one-loop QCD amplitudes and we derive the process-independent factorization formula that controls the singular behaviour in this limit. This is obtained from the customary eikonal factorization formula valid at tree (classical) level by introducing a generalized soft-gluon current that embodies the quantum corrections. We compute the explicit expression of the soft-gluon current at one-loop order. It contains purely non-abelian correlations between the colour charges of each pair of hard-momentum partons in the matrix element. This leads to colour correlations between (two and) three hard partons in the matrix element squared. Exploiting colour conservation, we recover QED-like factorization for the square of the matrix elements with two and three hard partons.

The soft-gluon current at one-loop order

TL;DR

The paper derives a process-independent factorization formula for the soft limit of one-loop QCD amplitudes by introducing a generalized soft-gluon current that incorporates quantum corrections. It provides an explicit, gauge-invariant expression for the one-loop soft current J^(1) that exhibits purely non-abelian colour correlations and reduces, via color conservation, to a tree-level-like structure for two- and three-hard-parton squared amplitudes. The authors prove the one-loop factorization and compute J^(1) using an axial-gauge analysis, revealing two-particle colour correlations and potential higher-order multiparton correlations. They further show that in two- and three-parton processes the colour algebra collapses to Casimir operators, yielding QED-like factorization and enabling NNLO calculations for e^+e^- → 2 jets and 3 jets. Overall, the work provides a comprehensive framework for infrared factorization at one loop and lays the groundwork for higher-loop generalizations and precise NNLO predictions.

Abstract

We study the soft limit of one-loop QCD amplitudes and we derive the process-independent factorization formula that controls the singular behaviour in this limit. This is obtained from the customary eikonal factorization formula valid at tree (classical) level by introducing a generalized soft-gluon current that embodies the quantum corrections. We compute the explicit expression of the soft-gluon current at one-loop order. It contains purely non-abelian correlations between the colour charges of each pair of hard-momentum partons in the matrix element. This leads to colour correlations between (two and) three hard partons in the matrix element squared. Exploiting colour conservation, we recover QED-like factorization for the square of the matrix elements with two and three hard partons.

Paper Structure

This paper contains 7 sections, 58 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Soft-gluon factorization at tree level
  3. Soft-gluon factorization at one loop
  4. Proof of factorization and calculation of the one-loop current
  5. Proof of factorization
  6. Calculation of the one-loop current
  7. Processes with two and three partons and $e^+e^-$ $\rightarrow 3$ jets at NNLO

Figures (4)

  • Figure 1: Feynman diagrams that contribute to the soft behaviour of the one-loop amplitude. The shaded blob denotes the tree-level amplitude and the virtual gluon with soft momentum $k$ can either $(a)$ connect two different external legs $i$ and $j$ or $(b)$ be emitted and reabsorbed by the same leg.
  • Figure 2: Graphs that contribute to the one-loop soft current.
  • Figure 3: Feynman diagrams that depend on a single hard momentum.
  • Figure 4: Feynman diagrams that depend on two or three different hard momenta.