A Three-Point Form Factor Through Five Loops

TL;DR

This paper computes the three-point form factor in planar $ 4$ super-Yang-Mills theory up to five loops, revealing new structures and providing insights into related QCD amplitudes and function spaces.

Contribution

It introduces new constraints and methods to bootstrap multi-loop form factors, expanding the understanding of function spaces relevant for high-loop calculations in supersymmetric and QCD theories.

Findings

New three, four, and five-loop results for the form factor.

Identification of a smaller function space with no independent $\

contribution

Abstract

We bootstrap the three-point form factor of the chiral part of the stress-tensor supermultiplet in planar $\mathcal{N}=4$ super-Yang-Mills theory, obtaining new results at three, four, and five loops. Our construction employs known conditions on the first, second, and final entries of the symbol, combined with new multiple-final-entry conditions, ``extended-Steinmann-like'' conditions, and near-collinear data from the recently-developed form factor operator product expansion. Our results are expected to give the maximally transcendental parts of the $gg\to Hg$ and $H\to ggg$ amplitudes in the heavy-top limit of QCD. At two loops, the extended-Steinmann-like space of functions we describe contains all transcendental functions required for four-point amplitudes with one massive and three massless external legs, and all massless internal lines, including processes such as $gg\to Hg$ and $\gamma^*\to q\bar{q}g$. We expect the extended-Steinmann-like space to contain these amplitudes at higher loops as well, although not to arbitrarily high loop order. We present evidence that the planar $\mathcal{N}=4$ three-point form factor can be placed in an even smaller space of functions, with no independent $\zeta$ values at weights two and three.