An Introduction to On-shell Recursion Relations

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.

80 pages, 8 figures; references added; minor changes of text

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• Motivic Amplitudes and Cluster Coordinates
• Hepta-Cuts of Two-Loop Scattering Amplitudes
• The Rise of SMEFT On-shell Amplitudes
• Computation of Contour Integrals on ${\mathcal{M}}_{0,n}$
• A "Twistor String" Inspired Formula For Tree-Level Scattering Amplitudes in N=8 SUGRA
• Fundamental BCJ Relation in N=4 SYM From The Connected Formulation
• New structures in scattering amplitudes: a review
• On powercounting in perturbative quantum gravity theories through color-kinematic duality
• 2-, 3- and 4-Body Decays in the Constructive Standard Model
• On the refined counting of graphs on surfaces
• Determination of Boundary Contributions in Recursion Relation
• String theory in target space
• On Tensorial Spaces and BCFW Recursion Relations for Higher Spin Fields
• Color-Kinematics Duality for Sudakov Form Factor in Non-Supersymmetric Pure Yang-Mills Theory
• On Multi-step BCFW Recursion Relations
• $\Delta$-Algebra and Scattering Amplitudes
• Evaluating EYM amplitudes in four dimensions by refined graphic expansion
• General tree-level amplitudes by factorization limits
• Permutation Relations of Yangian Invariants, Unitarity Cuts, and Scattering Amplitudes
• Amplitudes for Astrophysicists: Known Knowns
• Boundary Operators of BCFW Recursion Relation
• Notes on Cubic and Quartic Light-Front Kinematics
• Nonplanar On-shell Diagrams and Leading Singularities of Scattering Amplitudes
• Exploring $HVV$ amplitudes with $CP$ violation by decomposition and on-shell scattering amplitude methods
• Factorizations for tree amplitudes in the double-cover framework: from gravity to other theories
• BCFW Deformation and Regge Limit
• Deconstructing Supersymmetric S-matrices in D <= 2 + 1
• Positivity, Grassmannian Geometry and Simplex-like Structures of Scattering Amplitudes
• Extended Projection Method for Massive Fermion
• Cachazo-Svrcek-Witten Rules for Tree-Level Gluonic Amplitudes Revisited
• On-shell recursion relations for generic integrands
• Coherent States and Particle Scattering in Loop Quantum Gravity
• On-shell Methods for Form Factors in N=4 SYM and Their Applications
• Brane Tilings as On-shell Diagrams
• Recursion in multiplet bases for tree-level MHV gluon amplitudes
• A First Step Towards Effectively Nonperturbative Scattering Amplitudes in the Perturbative Regime
• A Tale Of Two Amplitudes In High Energy Physics