Recurrent Features of Amplitudes in Planar $\mathcal{N}=4$ Super Yang-Mills Theory

TL;DR

This paper analyzes high-loop perturbative data of a form factor in planar $ ext{N}=4$ super Yang-Mills theory, revealing patterns, relations, and recursive structures that advance understanding of scattering amplitudes at all loop orders.

Contribution

It uncovers new recursive relations and closed-form expressions for amplitude coefficients, enhancing the understanding of amplitude structures at arbitrary loop levels.

Findings

Identification of zero coefficient patterns

Derivation of closed-form coefficient expressions

Discovery of recursion relations linking loop orders

Abstract

The planar three-gluon form factor for the chiral stress tensor operator in planar maximally supersymmetric Yang-Mills theory is an analog of the Higgs-to-three-gluon scattering amplitude in QCD. The amplitude (symbol) bootstrap program has provided a wealth of high-loop perturbative data about this form factor, with results up to eight loops available. The symbol of the form factor at $L$ loops is given by words of length $2L$ in six letters with associated integer coefficients. In this paper, we analyze this data, describing patterns of zero coefficients and relations between coefficients. We find many sequences of words whose coefficients are given by closed-form expressions which we expect to be valid at any loop order. Moreover, motivated by our previous machine-learning analysis, we identify simple recursion relations that relate the coefficient of a word to the coefficients of particular lower-loop words. These results open an exciting door for understanding scattering amplitudes at all loop orders.