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Three-Loop Leading Singularities and BDS Ansatz for Five Particles

Marcus Spradlin, Anastasia Volovich, Congkao Wen

TL;DR

The paper tackles the planar three-loop five-point amplitude in ${\cal N}=4$ SYM and tests the three-loop BDS ansatz using a combination of leading singularities and obstruction theory.It expresses the amplitude as a linear combination of 17 dual-conformal basis integrals, determining all coefficients analytically via residues on carefully chosen contours.Using obstructions, the authors extract the constants in the three-loop BDS relation, finding $C^{(3)} = 17.8241$ and fixing $a \approx 85.263$ and $b \approx 17.8241$, with $b$ matching the obstruction to $C^{(3)}$.The results reinforce the structure implied by dual conformal invariance for $n=5$ and demonstrate a powerful framework for high-loop amplitude calculations, while noting potential contributions from non-leading singularities and $\-3010

Abstract

We use the leading singularity technique to determine the planar three-loop five-particle amplitude in N=4 super Yang-Mills in terms of a simple basis of integrals. We analytically compute the integral coefficients for both the parity-even and the parity-odd parts of the amplitude. The parity-even part involves only dual conformally invariant integrals. Using the method of obstructions we numerically evaluate two previously unfixed coefficients which appear in the three-loop BDS ansatz.

Three-Loop Leading Singularities and BDS Ansatz for Five Particles

TL;DR

The paper tackles the planar three-loop five-point amplitude in ${\cal N}=4$ SYM and tests the three-loop BDS ansatz using a combination of leading singularities and obstruction theory.It expresses the amplitude as a linear combination of 17 dual-conformal basis integrals, determining all coefficients analytically via residues on carefully chosen contours.Using obstructions, the authors extract the constants in the three-loop BDS relation, finding $C^{(3)} = 17.8241$ and fixing $a \approx 85.263$ and $b \approx 17.8241$, with $b$ matching the obstruction to $C^{(3)}$.The results reinforce the structure implied by dual conformal invariance for $n=5$ and demonstrate a powerful framework for high-loop amplitude calculations, while noting potential contributions from non-leading singularities and $\-3010

Abstract

We use the leading singularity technique to determine the planar three-loop five-particle amplitude in N=4 super Yang-Mills in terms of a simple basis of integrals. We analytically compute the integral coefficients for both the parity-even and the parity-odd parts of the amplitude. The parity-even part involves only dual conformally invariant integrals. Using the method of obstructions we numerically evaluate two previously unfixed coefficients which appear in the three-loop BDS ansatz.

Paper Structure

This paper contains 11 sections, 36 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Outline of the Calculation
  3. Review of the Leading Singularity Method
  4. Integration Strategy: Collapse and Expand
  5. Choosing a Sufficient Set of Contours
  6. Example 1: Topology $L$
  7. Example 2: Topology $M$
  8. Example 3: Topology $R$
  9. The 3-loop 5-particle Amplitude
  10. The Three-Loop BDS Ansatz
  11. Summary

Figures (2)

  • Figure 1: The planar 3-loop 5-particle topologies associated to leading singularities. Each figure represents a sum over that subset of Feynman diagrams in which all of the indicated propagators are present. We label the external momenta clockwise with $k_1$ at the leg indicated with the arrow.
  • Figure 2: The 17 independent integrals appearing in the ansatz. Other integrals can be obtained by rotations or reflections. As in fig. \ref{['topologies']} we label the external momenta clockwise with $k_1$ at the position indicated by the arrow.