The SAGEX review on scattering amplitudes Chapter 5: Analytic bootstraps for scattering amplitudes and beyond

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 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.

Symbol alphabets from the Landau singular locus

“…Specifically, if A ⊂ R 2 is the set of vertices of a convex n-gon, then the secondary polytope Σ(A) is the n−th associahedron, which is indeed isomorphic to the cluster polytope of the A n−3 cluster algebra. Since Newt(E A ) ≃ Σ(A), our approach for extracting symbol alphabets from the principal A-determinant E A (G) offers promise for providing a first-principle derivation and extension of the intriguing clusteralgebraic structures observed in a wealth of different Feynman integrals [76][77][78], following similar observations in the context of scattering amplitudes in N = 4 super Yang-Mills theory [79,80], see also the recent review [81].…”

The SAGEX review on scattering amplitudes*

“…Applications within effective field theories that utilise all the different solving and simplification tools introduced in this section, as notably the post-Newtonian expansion of classical (non-linear) Einstein gravity, are given to five-loop order. [5] One of the main challenges in obtaining predictions for collider experiments from quantum field theory, is the evaluation of the Feynman integrals resulting from its traditional perturbative treatment. In this chapter, we review an alternative bootstrap method that bypasses this formidable task by constructing physical quantities from the knowledge of their expected analytic structure.…”