Contents

# Contents

## Idea

The scattering amplitudes in Yang-Mills theory are traditionally expressed as sums of typically very many Feynman diagrams, which in turn are integrals over certain algebraic functions. At least in highly supersymmetric versions of Yang-Mills theory such as N=4 D=4 super Yang-Mills theory this can be expressed more efficiently (see Dixon 13 for a review of the general method and see at string theory results applied elsewhere) as fewer integrals of some integrand over some domain of certain convex polyhedra inside a positive Grassmanian (which is an infinite dimensional space related to the study of the phenomenon of total positivity). Therefore this has been called the amplituhedron (Arkani-Hamed & Trnka13).

In slightly more detail, the scattering amplitudes in N=4 D=4 super Yang-Mills theory can be given as functions of $4n$ real variables, the momenta of $n$ scattering particles, and at loop number $\ell$ they are given by $4 \ell$-fold integrals of algebraic functions of $4n+4 \ell$ real variables. (When $\ell=0$, then (ABCGPT 12, sections 16.4) claims that all the Galois conjugates? of the algebraic function occur symmetrically; so that one indeed has rational functions.)

This formulation is typically a drastic improvement of computational complexity for fixed $k$ (helicity) and fixed $\ell$ (loop number) but variable $n$. The Feynman diagram description is exponential in $n$, while the complexity of the amplituhedron description grows much more mildly.

In particular for $k = 0$ and $k = 1$ the resulting amplitude is just 0 and the amplituhedron description gives this immediately, but for $k = 1$ there are still many nontrivial Feynman integrals. That the sum of all these indeed vanish was maybe known earlier, though, due to a result by Parke and Taylor.

For $k = 2$ again there is the “Parke-Taylor formula?” efficiently expressing the amplitudes MHV amplitudes.

## References

A review of the history and background is in

For a good more technical general introduction of the background see

For scattering amplitudes via the “amplituhedron” the integrand is discussed in

and the integration domain in

Simple aspects of four particle scattering are treated in

Earlier lecture notes and announcements include

• Jaroslav Trnka, The amplituhedron, pdf

• Nima Arkani-Hamed, Grassmannian polytopes and scattering amplitudes, lecture at Perimeter Institute, video

Informed online discussion includes

• Logan Maingi on MathOverflow here

• David Speyer here

Journalistic coverage includes

• Natalie Wolchover, A Jewel at the Heart of Quantum Physics, Quanta Magazine, Sep 17, 2013 (hosted at Simons Foundation) html