Amplitudes and Correlators to Ten Loops Using Simple, Graphical Bootstraps

TL;DR

The paper develops a graphical bootstrap approach for planar $\mathcal{N}=4$ SYM that determines the four-point correlator and its dual four-point amplitude up to ten loops by combining three rules (square/rung, triangle, pentagon) with the rung rule. Central to the method are $f$-graphs, which encode conformal weights and planarity, and the fully symmetric function $\mathcal{F}^{(\ell)}$ whose light-like limits yield all-loop amplitudes and higher-point information. The authors prove and illustrate the rules, show they fix most coefficients through high loop orders, and provide explicit ten-loop results and data, including insights into coefficient distributions and special graph topologies like anti-prisms. This graphical framework dramatically reduces computation time (e.g., inspecting ten-loop results in minutes) and supports extensions to eleven or twelve loops, as well as potentially broader applicability to other quantum field theories. Overall, the work demonstrates that simple, graph-based consistency relations can capture rich, all-loop structure in a highly symmetric theory, with implications for higher-point amplitudes and the study of amplitude/correlator dualities.

Abstract

We introduce two new graphical-level relations among possible contributions to the four-point correlation function and scattering amplitude in planar, maximally supersymmetric Yang-Mills theory. When combined with the rung rule, these prove powerful enough to fully determine both functions through ten loops. This then also yields the full five-point amplitude to eight loops and the parity-even part to nine loops. We derive these rules, illustrate their applications, compare their relative strengths for fixing coefficients, and survey some of the features of the previously unknown nine and ten loop expressions. Explicit formulae for amplitudes and correlators through ten loops are available at: http://goo.gl/JH0yEc.