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Three-loop octagons and n-gons in maximally supersymmetric Yang-Mills theory

Simon Caron-Huot, Song He

TL;DR

The paper investigates scattering amplitudes in planar $\mathcal{N}=4$ SYM under $R^{1,1}$ kinematics, showing drastic simplifications and new structures that enable compact analytic octagon formulas up to three loops. By combining the $\bar{Q}$ equation with collinear-soft uplifting, the authors derive explicit one-loop NMHV and N${}^2$MHV octagons, upgrade them to higher points, and compute the three-loop MHV octagon; they also develop a symbol-integration framework to obtain explicit polylog expressions. A key outcome is that higher-point amplitudes can be constructed from the octagon through uplifting, with the octagon acting as a universal building block, and the results exhibit precise collinear, soft, and OPE-compatible behavior. The work highlights deep connections between integrability-based constraints, dual conformal symmetry, and the analytic structure of amplitudes in two-dimensional kinematics, offering a path to systematic higher-loop and higher-point predictions and potential links to OPE and strong-coupling data.

Abstract

We study the S-matrix of planar $\mathcal{N}=4$ supersymmetric Yang-Mills theory when external momenta are restricted to a two-dimensional subspace of Minkowski space. We find significant simplifications and new, interesting structures for tree and loop amplitudes in two-dimensional kinematics; in particular, the higher-point amplitudes we consider can be obtained from those with lowest-points by a collinear uplifting. Based on a compact formula for one-loop N${}^2$MHV amplitudes, we use an equation proposed previously to compute, for the first time, the complete two-loop NMHV and three-loop MHV octagons, which we conjecture to uplift to give the full $n$-point amplitudes up to simpler logarithmic terms or dilogarithmic terms.

Three-loop octagons and n-gons in maximally supersymmetric Yang-Mills theory

TL;DR

The paper investigates scattering amplitudes in planar SYM under kinematics, showing drastic simplifications and new structures that enable compact analytic octagon formulas up to three loops. By combining the equation with collinear-soft uplifting, the authors derive explicit one-loop NMHV and NMHV octagons, upgrade them to higher points, and compute the three-loop MHV octagon; they also develop a symbol-integration framework to obtain explicit polylog expressions. A key outcome is that higher-point amplitudes can be constructed from the octagon through uplifting, with the octagon acting as a universal building block, and the results exhibit precise collinear, soft, and OPE-compatible behavior. The work highlights deep connections between integrability-based constraints, dual conformal symmetry, and the analytic structure of amplitudes in two-dimensional kinematics, offering a path to systematic higher-loop and higher-point predictions and potential links to OPE and strong-coupling data.

Abstract

We study the S-matrix of planar supersymmetric Yang-Mills theory when external momenta are restricted to a two-dimensional subspace of Minkowski space. We find significant simplifications and new, interesting structures for tree and loop amplitudes in two-dimensional kinematics; in particular, the higher-point amplitudes we consider can be obtained from those with lowest-points by a collinear uplifting. Based on a compact formula for one-loop NMHV amplitudes, we use an equation proposed previously to compute, for the first time, the complete two-loop NMHV and three-loop MHV octagons, which we conjecture to uplift to give the full -point amplitudes up to simpler logarithmic terms or dilogarithmic terms.

Paper Structure

This paper contains 27 sections, 105 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. An invitation: all one-loop NMHV amplitudes from the octagon
  4. The 8-point NMHV amplitude
  5. Collinear-soft uplifting formalism
  6. The $n$-point one-loop NMHV amplitude from uplifting
  7. One-loop N${}^2$MHV amplitudes and uplifting
  8. Dual-conformal box-expansion and one-loop N${}^2$MHV octagon
  9. Uplifting and the N${}^2$MHV decagon
  10. The $\bar{Q}$ equation and the two-loop NMHV octagon
  11. Left-hand side of the $\bar{Q}$ equation
  12. Right-hand side of the $\bar{Q}$ equation
  13. Two-loop NMHV octagon
  14. Uplifting for two-loop NMHV
  15. Procedure for integrating symbols
  16. ...and 12 more sections

Figures (4)

  • Figure 1: All-loop $\bar{Q}$ equation for planar $\mathcal{N}=4$ S-matrix.
  • Figure 2: The octagon remainder function in the Euclidean region: (a) the normalized 3-loop function; (b) the difference between 3-loop and 2-loop normalized functions; (c) the difference between 2-loop and the strong coupling normalized functions.
  • Figure 3: Plot of $\bar{R}^{(2)}_{8,0}$ (blue), $\bar{R}^{(3)}_{8,0}$ (red) and $\bar{R}^\textrm{strong}_{8,0}$ (yellow) as a function of $\phi$, at $|m|=0.2$ (left) and $|m|=0.45$ (right).
  • Figure 4: Wilson loops and factorizations in two-dimensional kinematics: the zigzag null polygon is parametrized by the coordinates $(x^+, x^-)$ (rotated by 45 degrees). When $\langle \widehat{n{-}1},i\rangle=0$, the $x^-$ (horizontal) coordinates of side $i$ and side $n{-}1$ coincide, and the amplitude becomes the product of the two lower-point amplitudes, with a prefactor $f_{n{-}1,i}:=(n{-}1\,1\,i)( [n{-}2\,n\,i{+}1]-[n{-}2\,n\,i{-}1])$.