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The Two-loop Soft Anomalous Dimension Matrix and Resummation at Next-to-next-to Leading Pole

S. Mert Aybat, Lance J. Dixon, George Sterman

TL;DR

This work computes the two-loop soft anomalous-dimension matrix governing color exchange in massless 2→n QCD processes and shows it is proportional to the one-loop matrix via the cusp-related constant K, enabling NNLL resummation of infrared poles. By formulating amplitudes in jet-soft-hard factorization and using eikonal Wilson lines, the authors reproduce the NNLO single-pole structure and connect it to Catani’s H^(2) framework, including an explicit identity linking H^(2) to anomalous dimensions. The results, valid for arbitrary parton multiplicities, provide a practical route to threshold resummation and enhance understanding of soft-gluon dynamics in multi-jet production. This framework extends the Sudakov analysis to nontrivial color mixing and offers a path toward precise, color-resolved predictions at NNLO and beyond.

Abstract

We extend the resummation of dimensionally-regulated amplitudes to next-to-next-to-leading poles. This requires the calculation of two-loop anomalous dimension matrices for color mixing through soft gluon exchange. Remarkably, we find that they are proportional to the corresponding one-loop matrices. Using the color generator notation, we reproduce the two-loop single-pole quantities H^(2) introduced by Catani for quark and gluon elastic scattering. Our results also make possible threshold and a variety of other resummations at next-to-next-to leading logarithm. All of these considerations apply to 2 to n processes with massless external lines.

The Two-loop Soft Anomalous Dimension Matrix and Resummation at Next-to-next-to Leading Pole

TL;DR

This work computes the two-loop soft anomalous-dimension matrix governing color exchange in massless 2→n QCD processes and shows it is proportional to the one-loop matrix via the cusp-related constant K, enabling NNLL resummation of infrared poles. By formulating amplitudes in jet-soft-hard factorization and using eikonal Wilson lines, the authors reproduce the NNLO single-pole structure and connect it to Catani’s H^(2) framework, including an explicit identity linking H^(2) to anomalous dimensions. The results, valid for arbitrary parton multiplicities, provide a practical route to threshold resummation and enhance understanding of soft-gluon dynamics in multi-jet production. This framework extends the Sudakov analysis to nontrivial color mixing and offers a path toward precise, color-resolved predictions at NNLO and beyond.

Abstract

We extend the resummation of dimensionally-regulated amplitudes to next-to-next-to-leading poles. This requires the calculation of two-loop anomalous dimension matrices for color mixing through soft gluon exchange. Remarkably, we find that they are proportional to the corresponding one-loop matrices. Using the color generator notation, we reproduce the two-loop single-pole quantities H^(2) introduced by Catani for quark and gluon elastic scattering. Our results also make possible threshold and a variety of other resummations at next-to-next-to leading logarithm. All of these considerations apply to 2 to n processes with massless external lines.

Paper Structure

This paper contains 26 sections, 153 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Factorized amplitudes in dimensional regularization
  3. The jet functions and the Sudakov form factor
  4. The soft function
  5. Resumming the soft function
  6. The jet functions to two loops
  7. Eikonal amplitudes at one and two loops
  8. $2 \rightarrow 2$ eikonal diagrams at one loop
  9. $2 \rightarrow 2$ eikonal diagrams at two loops
  10. The 2E diagrams and $\Gamma_i^{(2)}$
  11. Vanishing of three-gluon diagram with three eikonal lines
  12. Exponentiation of the remaining 3E diagrams
  13. Expansion of the soft function
  14. Iterative color-matrix form
  15. Generalizations
  16. ...and 11 more sections

Figures (4)

  • Figure 1: One-loop diagrams that contribute to $\Gamma_{S_{\rm f}}^{(1)}$
  • Figure 2: Color basis $\{c_1,c_2\}$ for four-quark process
  • Figure 3: Two-loop diagrams that contribute to $\Gamma_{W_{\rm f}}^{(2)}$
  • Figure 4: a-c) Pairs of 3E diagrams.