A Course in Amplitudes

TL;DR

This pedagogical work introduces on-shell scattering amplitudes in gauge theories, guiding readers from Dirac spinors to modern spinor-helicity techniques. It foregrounds two core methods—BCFW recursion for tree-level amplitudes and the unitarity cut method for loop corrections—demonstrating how amplitudes can be constructed from on-shell data and simple scalar integrals. The text emphasizes color decomposition, soft/collinear factorization, and supersymmetry Ward identities to reveal universal, efficient structures in gauge theories, including applications to gluon scattering and ${\cal N}=4$ SYM. Together, these methods offer a powerful framework for both analytic understanding and practical computation of amplitudes in the Standard Model and beyond.

Abstract

This a pedagogical introduction to scattering amplitudes in gauge theories. It proceeds from Dirac equation and Weyl fermions to the two pivot points of current developments: the recursion relations of Britto, Cachazo, Feng and Witten, and the unitarity cut method pioneered by Bern, Dixon, Dunbar and Kosower. In ten lectures, it covers the basic elements of on-shell methods.

Figures (10)

• Figure 1: Feynman diagrams and the corresponding group factors for four-gluon scattering
• Figure 2: Feynman diagrams contributing to $k\to 0$ soft singularity
• Figure 3: Feynman diagram singular in the collinear limit $p_m\parallel p_n$
• Figure 4: Factorization in $m$-particle channel.
• Figure 5: Contribution to the residues from an on-shell propagator. For a $z$-dependent momentum flow through the indicated internal propagator, particle $i$ has to be on the one side and particle $j$ has to be on the other side.