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Contents

Idea

In quantum field theory scattering amplitudes are the probability amplitudes for processes of scattering of fundamental particles (or fundamental strings etc.) off each other. The collection of scattering amplitudes forms the S-matrix. In perturbative quantum field theory its contributions may be labeled by Feynman diagrams, whence it is also called the Feynman perturbation series.

Of particular interest are vacuum amplitudes which are scattering amplitudes “where nothing external scatters” hence for no incoming and no outgoing states. The 1-loop vacuum amplitudes are regularized traces over Feynman propagators. These are the incarnations of zeta functions, L-functions and eta functions in physics.

Examples

In Chern-Simons theory

The Feynman amplitudes of higher Chern-Simons theory, such as AKSZ sigma-models, regarded in their incarnation as Feynman amplitudes on compactified configuration spaces of points, serve to exhibit a graph complex-model for the de Rham complex of Fulton-MacPherson compactifications of configuration spaces of points by the construction recalled there. See the pointers at Chern-Simons theory here.

Of monopoles

See at moduli space of monopoles the section Scattering amplitudes of monopoles.

Related concepts

• scattering

• scattering matrix

• scattering cross section

• leptonic decay (of mesons), Dalitz decay

• perturbative quantum field theory

• n-point function, correlator

• Feynman diagram

• abstract scattering theory

• cluster decomposition

• renormalization

• on-shell recursion, planar limit

• form factor (QFT)

• KLT relations

• amplituhedron

• celestial amplitude

• string theory results applied elsewhere

• string scattering amplitude

• motives in physics

• graph complex, formality of the little n-disk operad

References

General

• Henriette Elvang, Yu-tin Huang, Scattering Amplitudes, Cambridge University Press (2015) [[arXiv:1308.1697](http://arxiv.org/abs/1308.1697), doi:10.1017/CBO9781107706620]

  “The purpose of this review is to bridge the gap between a standard course in quantum field theory and recent fascinating developments in the studies of on-shell scattering amplitudes.”

• Tomasz R. Taylor, A Course in Amplitudes (arXiv:1703.05670)

• Stephen J. Summers, Detlev Buchholz, Scattering in Relativistic Quantum Field Theory: Fundamental Concepts and Tools (arXiv:math-ph/0509047)

• Clifford Cheung, TASI Lectures on Scattering Amplitudes (arXiv:1708.03872)

• Simon Badger, Johannes Henn, Jan Plefka, Simone Zoia, Scattering Amplitudes in Quantum Field Theory [[arXiv:2306.05976](https://arxiv.org/abs/2306.05976)]

  “These lecture notes bridge a gap between introductory quantum field theory (QFT) courses and state-of-the-art research in scattering amplitudes”

Introduction and review:

A historical overview of the development of on-shell methods/analytic methods is in

• Henriette Elvang, Yu-tin Huang, Scattering Amplitudes, Cambridge University Press (2015) [[arXiv:1308.1697](http://arxiv.org/abs/1308.1697), doi:10.1017/CBO9781107706620]

  “The purpose of this review is to bridge the gap between a standard course in quantum field theory and recent fascinating developments in the studies of on-shell scattering amplitudes.”

• Tomasz R. Taylor, A Course in Amplitudes [[arXiv:1703.05670](https://arxiv.org/abs/1703.05670)]

• Stephen J. Summers, Detlev Buchholz, Scattering in Relativistic Quantum Field Theory: Fundamental Concepts and Tools [[arXiv:math-ph/0509047] (https://arxiv.org/abs/math-ph/0509047)]

• Clifford Cheung, TASI Lectures on Scattering Amplitudes [[arXiv:1708.03872](https://arxiv.org/abs/1708.03872)]

• Simon Badger, Johannes Henn, Jan Plefka, Simone Zoia, Scattering Amplitudes in Quantum Field Theory [[arXiv:2306.05976](https://arxiv.org/abs/2306.05976)]

  “These lecture notes bridge a gap between introductory quantum field theory (QFT) courses and state-of-the-art research in scattering amplitudes”

• Lance Dixon, Calculating Amplitudes, December 2013 (web)

For related reference on the S-matrix see there, such as

• Sebastian Mizera, Physics of the Analytic S-Matrix [[arXiv:2306.05395](https://arxiv.org/abs/2306.05395)]

A historical overview of the development of on-shell methods/analytic methods:

• Lance Dixon, Calculating Amplitudes, (December 2013) [[web](http://www.preposterousuniverse.com/blog/2013/10/03/guest-post-lance-dixon-on-calculating-amplitudes/)]

Annual conference series:

• Amplitudes 2019

  Amplitudes 2019

• Aplitudes 2020

• Amplitudes 2021

• Amplitudes 2022

• Amplitudes 2023

Analytic methods

• Ruth Britto, Loop amplitudes in gauge theories: modern analytic approaches (arXiv:1012.4493)

• Zvi Bern, Lance Dixon, David Kosower, On-Shell Methods in Perturbative QCD (arXiv:0704.2798)

• Joseph Polchinski, Matthew Strassler, Hard Scattering and Gauge/String Duality, Phys.Rev.Lett.88:031601,2002, (arXiv:hep-th/0109174)

See also at string theory results applied elsewhere and at motivic multiple zeta values.

In super Yang-Mills theory

In super Yang-Mills theory (and there in particular in the planar limit of N=4 D=4 super Yang-Mills theory) scattering amplitudes enjoy special symmetry properties, some of which can be used to extract information about scattering amplitudes in non-supersymmetric theories (see also at amplituhedron):

• Nima Arkani-Hamed, Jacob L. Bourjaily, Freddy Cachazo, Alexander B. Goncharov, Alexander Postnikov, Jaroslav Trnka, Scattering Amplitudes and the Positive Grassmannian (arXiv:1212.5605)

• Nima Arkani-Hamed, Jaroslav Trnka, The Amplituhedron, arxiv/1312.2007

Classification of massless scattering

Classification of possible long-range forces, hence of scattering processes of massless fields, by classification of suitably factorizing and decaying Poincaré-invariant S-matrices depending on particle spin, leading to uniqueness statements about Maxwell/photon-, Yang-Mills/gluon-, gravity/graviton- and supergravity/gravitino-interactions:

• Steven Weinberg, Feynman Rules for Any Spin. 2. Massless Particles, Phys. Rev. 134 (1964) B882 (doi:10.1103/PhysRev.134.B882)

• Steven Weinberg, Photons and Gravitons in $S$-Matrix Theory: Derivation of Charge Conservationand Equality of Gravitational and Inertial Mass, Phys. Rev. 135 (1964) B1049 (doi:10.1103/PhysRev.135.B1049)

• Steven Weinberg, Photons and Gravitons in Perturbation Theory: Derivation of Maxwell’s and Einstein’s Equations,” Phys. Rev. 138 (1965) B988 (doi:10.1103/PhysRev.138.B988)

• Paolo Benincasa, Freddy Cachazo, Consistency Conditions on the S-Matrix of Massless Particles (arXiv:0705.4305)

• David A. McGady, Laurentiu Rodina, Higher-spin massless S-matrices in four-dimensions, Phys. Rev. D 90, 084048 (2014) (arXiv:1311.2938, doi:10.1103/PhysRevD.90.084048)

Quick review:

• Daniel Baumann, What long-range forces are allowed?, 2019 (pdf)

In string theory and higher supergravity

• Wieland Staessens, Bert Vercnocke, Lectures on Scattering Amplitudes in String Theory (arXiv:1011.0456)

• Michael Green, Properties of low energy graviton scattering amplitudes, June 2010 (pdf)

For more see at string scattering amplitude.

Motivic structure

Motivic structures in scattering amplitudes (see at motives in physics) are discussed for instance in

• John Golden, Alexander B. Goncharov, Marcus Spradlin, Cristian Vergu, Anastasia Volovich, Motivic Amplitudes and Cluster Coordinates (arXiv:1305.1617)

See also the references at period.