An Introduction to On-shell Recursion Relations

TL;DR

This paper introduces on-shell recursion relations for calculating tree-level amplitudes in gauge theories, covering foundational concepts, recent developments, and various applications in quantum field theory.

Contribution

It provides a comprehensive overview of on-shell recursion relations, including generalizations, new theoretical insights, and practical applications in different quantum field theories.

Findings

Derived recursion relations for off-shell currents

Extended recursion techniques to supersymmetric theories

Applied recursion relations to compute specific amplitude examples

Abstract

This article provides an introduction to on-shell recursion relations for calculations of tree-level amplitudes. Starting with the basics, such as spinor notations and color decompositions, we expose analytic properties of gauge-boson amplitudes, BCFW-deformations, the large $z$-behavior of amplitudes, and on-shell recursion relations of gluons. We discuss further developments of on-shell recursion relations, including generalization to other quantum field theories, supersymmetric theories in particular, recursion relations for off-shell currents, recursion relation with nonzero boundary contributions, bonus relations, relations for rational parts of one-loop amplitudes, recursion relations in 3D and a proof of CSW rules. Finally, we present samples of applications, including solutions of split helicity amplitudes and of N= 4 SYM theories, consequences of consistent conditions under recursion relation, Kleiss-Kuijf (KK) and Bern-Carrasco-Johansson (BCJ) relations for color-ordered gluon tree amplitudes, Kawai-Lewellen-Tye (KLT) relations.