nLab amplituhedron

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.

See also at motives in physics

Related concepts

• surfaceology

References

Review:

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

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

• Livia Ferro, Tomasz Lukowski, Amplituhedra, and Beyond, Topical Review invited by Journal of Physics A: Mathematical and Theoretical (arXiv:2007.04342)

• Shounak De, Dmitrii Pavlov, Marcus Spradlin, Anastasia Volovich: From Feynman diagrams to the amplituhedron: a gentle review [arXiv:2410.11757]

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

• Nima Arkani-Hamed, Jacob L. Bourjaily, Freddy Cachazo, Alexander B. Goncharov, Alexander Postnikov, Jaroslav Trnka, Scattering amplitudes and the positive Grassmannian, arxiv/1212.5605

and the integration domain in

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

Simple aspects of four particle scattering are treated in

• Nima Arkani-Hamed, Jaroslav Trnka, Into the Amplituhedron, arxiv/1312.7878

  .

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

See also

• Wikipedia: Amplituhedron
• Thomas Lam, Amplituhedron cells and Stanley symmetric functions, arxiv/1408.5531