Contents

# 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.

## References

### General

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

Annual conference series:

### Analytic methods

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):

### 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:

Quick review:

### 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)