The Soft-Collinear Bootstrap: N=4 Yang-Mills Amplitudes at Six and Seven Loops

TL;DR

This paper demonstrates that four-point amplitudes in N=4 super-Yang-Mills theory can be uniquely reconstructed at high loop orders by enforcing softer infrared behavior, dual conformal invariance, and symmetry, matching known results up to five loops.

Contribution

It introduces a novel bootstrap approach based on infrared softness, dual conformal invariance, and symmetry to determine multi-loop integrands in N=4 SYM theory.

Findings

Explicit formulas for four-point integrands up to seven loops.

Enumeration of all dual conformally invariant integrands through seven loops.

Identification of 229 and 1873 contributing diagrams at six and seven loops, respectively.

Abstract

Infrared divergences in scattering amplitudes arise when a loop momentum $\ell$ becomes collinear with a massless external momentum $p$. In gauge theories, it is known that the L-loop logarithm of a planar amplitude has much softer infrared singularities than the L-loop amplitude itself. We argue that planar amplitudes in N=4 super-Yang-Mills theory enjoy softer than expected behavior as $\ell \parallel p$ already at the level of the integrand. Moreover, we conjecture that the four-point integrand can be uniquely determined, to any loop-order, by imposing the correct soft-behavior of the logarithm together with dual conformal invariance and dihedral symmetry. We use these simple criteria to determine explicit formulae for the four-point integrand through seven-loops, finding perfect agreement with previously known results through five-loops. As an input to this calculation we enumerate all four-point dual conformally invariant (DCI) integrands through seven-loops, an analysis which is aided by several graph-theoretic theorems we prove about general DCI integrands at arbitrary loop-order. The six- and seven-loop amplitudes receive non-zero contributions from 229 and 1873 individual DCI diagrams respectively.