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The two-loop MHV amplitudes in N=4 supersymmetric Yang-Mills theory

C. Vergu

TL;DR

The paper computes the even part of the planar two-loop MHV scattering amplitude in N=4 super Yang-Mills for an arbitrary number of external legs, expressing the result as a finite sum of conformal integrals with rational coefficients. Using a unitarity-based approach, the authors classify loop topologies (double boxes, kissing boxes, box-pentagons, double pentagons) and determine the conformal dressings by matching two-particle cuts, solving a linear system with random kinematics. They present explicit coefficient expressions for a comprehensive set of topologies across various massless/massive leg attachments, showing that the 2-loop even part can be assembled into a compact integral representation, while noting an undetermined μ-term and the parity-odd contribution remains to be computed. The work highlights structural features such as the dominance of conformal integrals, potential reorganizations into double-pentagon topologies, and connections to Wilson-loop results, informing future higher-point and higher-loop analyses in planar N=4 SYM.

Abstract

We compute the even part of the planar two-loop MHV amplitude in N=4 supersymmetric Yang-Mills theory, for an arbitrary number of external particles. The answer is expressed as a sum of conformal integrals.

The two-loop MHV amplitudes in N=4 supersymmetric Yang-Mills theory

TL;DR

The paper computes the even part of the planar two-loop MHV scattering amplitude in N=4 super Yang-Mills for an arbitrary number of external legs, expressing the result as a finite sum of conformal integrals with rational coefficients. Using a unitarity-based approach, the authors classify loop topologies (double boxes, kissing boxes, box-pentagons, double pentagons) and determine the conformal dressings by matching two-particle cuts, solving a linear system with random kinematics. They present explicit coefficient expressions for a comprehensive set of topologies across various massless/massive leg attachments, showing that the 2-loop even part can be assembled into a compact integral representation, while noting an undetermined μ-term and the parity-odd contribution remains to be computed. The work highlights structural features such as the dominance of conformal integrals, potential reorganizations into double-pentagon topologies, and connections to Wilson-loop results, informing future higher-point and higher-loop analyses in planar N=4 SYM.

Abstract

We compute the even part of the planar two-loop MHV amplitude in N=4 supersymmetric Yang-Mills theory, for an arbitrary number of external particles. The answer is expressed as a sum of conformal integrals.

Paper Structure

This paper contains 29 sections, 80 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Review
  3. Description of the computational procedure
  4. Results
  5. Results
  6. Double box topologies
  7. No legs attached
  8. One massless leg attached
  9. Two massless legs attached
  10. One massive leg attached
  11. One massless leg and one massive leg attached
  12. Two massive legs attached
  13. Extra double boxes
  14. Kissing double-box topologies
  15. Box-Pentagon topologies
  16. ...and 14 more sections

Figures (3)

  • Figure 1: The generic form of the cuts we computed. The subsets of particles $B$ and $D$ can be empty while the number of particles in the subsets $A$ and $C$ is at least two.
  • Figure 2: Two integrals with one massive leg, but with different integral coefficients. The six-point integral has coefficient zero as shown in Ref. Bern:2008ap while the seven-point integral has coefficient non-zero as shown in Ref. Vergu:2009zm.
  • Figure 3: The legs shown in this figure must be massless. Integrals where at least one of these legs is massive have coefficient zero and therefore don't appear in the MHV result.