An Introduction to On-shell Recursion Relations
Bo Feng, Mingxing Luo
TL;DR
The article provides a comprehensive introduction to on-shell recursion, starting from spinor-helicity and color decomposition to derive BCFW deformations and gluon recursion. It surveys broad generalizations (supersymmetry, off-shell currents, nonzero boundary contributions, loop rational parts, 3D theories, CSW rules) and illustrates powerful applications, including split-helicity amplitudes, tree-level N=4 SYM, KK/BCJ relations, and KLT relations. By emphasizing analyticity, factorization, and controlled complex deformations, the work highlights a unifying S-matrix perspective that yields deep constraints and efficient computational tools beyond traditional Feynman-diagram methods. The discussed framework significantly advances our understanding of gauge and gravity amplitudes, with implications for field theory, string-inspired relations, and higher-dimensional generalizations.
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.