The SAGEX Review on Scattering Amplitudes, Chapter 1: Modern Fundamentals of Amplitudes

TL;DR

This chapter provides an accessible overview of modern scattering amplitude techniques, including spinor-helicity formalism, recursion relations, and symmetries, with examples in Yang-Mills and gravity theories.

Contribution

It introduces foundational amplitude methods and symmetries, making advanced topics accessible to readers with basic quantum field theory knowledge.

Findings

Explanation of spinor-helicity formalism

Overview of BCFW recursion and unitarity methods

Discussion of symmetries in $ ext{N}=4$ supersymmetric Yang-Mills

Abstract

This chapter introduces the foundational elements of scattering amplitudes. It is meant to be accessible to readers with only a basic understanding of quantum field theory. Topics covered include: the four-dimensional spinor-helicity formalism and the colour decomposition of Yang-Mills scattering amplitudes; the study of soft and collinear limits of Yang-Mills and gravity amplitudes; the BCFW recursion relation and generalised unitarity, also in the superamplitudes formalism of $\mathcal{N}{=}4$ supersymmetric Yang-Mills; an overview of standard and hidden symmetries of the $S$-matrix of $\mathcal{N}{=}4$ supersymmetric Yang-Mills, such as the conformal, dual conformal and Yangian symmetries; and a brief excursus on form factors of protected and non-protected operators in Yang-Mills theory. Several examples and explicit calculations are also provided.