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Bootstrapping the QCD soft anomalous dimension

Øyvind Almelid, Claude Duhr, Einan Gardi, Andrew McLeod, Chris D. White

TL;DR

This work shows that the three-loop non-dipole part of the QCD soft anomalous dimension for massless partons can be obtained via a bootstrap approach, without full Feynman integral evaluation. By arguing that the cross-ratio dependent piece is expressible in terms of single-valued harmonic polylogarithms and imposing symmetry, Regge, and collinear constraints, the authors fix the remaining degrees of freedom up to an overall normalization. The resulting Δ_n^(3) matches the known result and demonstrates the power of a function-space bootstrap in non-planar perturbative gauge theory, suggesting a path toward simplified higher-loop calculations. The analysis also clarifies how SVHPLs and conformal cross ratios organize the non-dipole structure, with potential applications to massive cases and other observables.

Abstract

The soft anomalous dimension governs the infrared singularities of scattering amplitudes to all orders in perturbative quantum field theory, and is a crucial ingredient in both formal and phenomenological applications of non-abelian gauge theories. It has recently been computed at three-loop order for massless partons by explicit evaluation of all relevant Feynman diagrams. In this paper, we show how the same result can be obtained, up to an overall numerical factor, using a bootstrap procedure. We first give a geometrical argument for the fact that the result can be expressed in terms of single-valued harmonic polylogarithms. We then use symmetry considerations as well as known properties of scattering amplitudes in collinear and high-energy (Regge) limits to constrain an ansatz of basis functions. This is a highly non-trivial cross-check of the result, and our methods pave the way for greatly simplified higher-order calculations.

Bootstrapping the QCD soft anomalous dimension

TL;DR

This work shows that the three-loop non-dipole part of the QCD soft anomalous dimension for massless partons can be obtained via a bootstrap approach, without full Feynman integral evaluation. By arguing that the cross-ratio dependent piece is expressible in terms of single-valued harmonic polylogarithms and imposing symmetry, Regge, and collinear constraints, the authors fix the remaining degrees of freedom up to an overall normalization. The resulting Δ_n^(3) matches the known result and demonstrates the power of a function-space bootstrap in non-planar perturbative gauge theory, suggesting a path toward simplified higher-loop calculations. The analysis also clarifies how SVHPLs and conformal cross ratios organize the non-dipole structure, with potential applications to massive cases and other observables.

Abstract

The soft anomalous dimension governs the infrared singularities of scattering amplitudes to all orders in perturbative quantum field theory, and is a crucial ingredient in both formal and phenomenological applications of non-abelian gauge theories. It has recently been computed at three-loop order for massless partons by explicit evaluation of all relevant Feynman diagrams. In this paper, we show how the same result can be obtained, up to an overall numerical factor, using a bootstrap procedure. We first give a geometrical argument for the fact that the result can be expressed in terms of single-valued harmonic polylogarithms. We then use symmetry considerations as well as known properties of scattering amplitudes in collinear and high-energy (Regge) limits to constrain an ansatz of basis functions. This is a highly non-trivial cross-check of the result, and our methods pave the way for greatly simplified higher-order calculations.

Paper Structure

This paper contains 20 sections, 108 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The soft anomalous dimension
  3. Wilson lines and the Riemann sphere
  4. Ansatz for $\Delta_n^{(3)}$
  5. Colour structure of $\Delta_n^{(3)}$
  6. Symmetries
  7. Constraints from $\mathcal{N}=4$ Super Yang-Mills
  8. An ansatz for $\Delta_n^{(3)}$
  9. The Regge limit
  10. The Regge limit of $\Delta_4^{(3)}$
  11. Constraints from the Regge limit
  12. A. Analytic continuation to incoming Wilson lines.
  13. B. The momentum conserving limit.
  14. C. Constraints from the Regge limit.
  15. The collinear limit
  16. ...and 5 more sections

Figures (1)

  • Figure 1: Paths in the complex plane for the analytic continuation of the variables $z$ and $\bar{z}$ to the physical region where partons 1 and 2 are incoming.