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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
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)
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
Journalistic coverage includes
See also
Last revised on June 12, 2023 at 07:46:54. See the history of this page for a list of all contributions to it.