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The probability amplitude for a process of scattering of fundamental particles or fundamental strings off each other. The collection of scattering amplitudes forms the S-matrix. In perturbation theory this is computed by the Feynman diagram perturbation series.
Of particular interest are vacuum amplitudes which are scattering amplitudes “where nothing externel scatters” hence for no incoming and no outgoing states. The 1-loop vacuum amplitudes are regularized traces over Feynman propagators/Dirac propagators. These are the incarnations of zeta functions, L-functions and eta functions in physics.
A good text is
intended to
bridge the gap between a standard course in quantum field theory and recent fascinating developments in the studies of on-shell scattering amplitudes.
A historical overview of the development of on-shell methods/analytic methods is in
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 (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
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 structures in scattering amplitudes (see at motives in physics) are discussed for instance in
See also the references at period.