The SAGEX Review on Scattering Amplitudes, Chapter 5: Analytic Bootstraps for Scattering Amplitudes and Beyond

TL;DR

This chapter reviews an analytic bootstrap method for constructing scattering amplitudes in quantum field theory, focusing on six- and seven-particle amplitudes in N=4 super Yang-Mills theory, emphasizing the role of cluster algebras and kinematic limits.

Contribution

It introduces a bootstrap approach exploiting analytic structures and cluster algebras to efficiently determine scattering amplitudes, including a detailed example at two loops.

Findings

Constructed six-particle amplitudes using bootstrap methods.

Identified the function space structure via cluster algebras.

Applied the method to compute the two-loop six-particle amplitude.

Abstract

One of the main challenges in obtaining predictions for collider experiments from perturbative quantum field theory, is the direct evaluation of the Feynman integrals it gives rise to. In this chapter, we review an alternative bootstrap method that instead efficiently constructs physical quantities by exploiting their analytic structure. We present in detail the setting where this method has been originally developed, six- and seven-particle amplitudes in the large-color limit of $\mathcal{N}=4$ super Yang-Mills theory. We discuss the class of functions these amplitudes belong to, and the strong clues mathematical objects known as cluster algebras provide for rendering this function space both finite and of relatively small dimension at each loop order. We then describe how to construct this function space, as well as how to locate the amplitude inside of it with the help of kinematic limits, and apply the general procedure to a concrete example: The determination of the two-loop correction to the first nontrivial six-particle amplitude. We also provide an overview of other areas where the realm of the bootstrap paradigm is expanding, including other scattering amplitudes, form factors and Feynman integrals, and point out the analytic properties of potentially wider applicability that it has revealed.