The SAGEX review on scattering amplitudes Chapter 1: Modern fundamentals of amplitudes

This chapter introduces the foundational concepts and techniques for 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 N = 4 supersymmetric Yang-Mills; an overview of standard and hidden symmetries of the S-matrix of N = 4 super symmetric 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.