Scattering in three dimensions from rational maps

• Published in: The journal of high energy physics Vol. 2013; no. 10; pp. 1 - 20
• Main Authors: Cachazo, Freddy, He, Song, Yuan, Ellis Ye
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01-10-2013; Springer Nature B.V
• Subjects: Amplitudes; Classical and Quantum Gravitation; Elementary Particles; High energy physics; Infrared; Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Relativity Theory; Resultants; String Theory; Supersymmetry; Three dimensional; Wave functions; Field Theories in Lower Dimensions; Scattering Amplitudes; Extended Supersymmetry
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP10(2013)141

A
bstract
The complete tree-level S-matrix of four dimensional
super Yang-Mills and
supergravity has compact forms as integrals over the moduli space of certain rational maps. In this note we derive formulas for amplitudes in three dimensions by using the fact that when amplitudes are dressed with proper wave functions dimensional reduction becomes straightforward. This procedure leads to formulas in terms of rational maps for three dimensional maximally supersymmetric Yang-Mills and gravity theories. The integrand of the new formulas contains three basic structures: Parke-Taylor-like factors, Vandermonde determinants and resultants. Integrating out some of the Grassmann directions produces formulas for theories with less than maximal supersymmetry, which exposes yet a fourth kind of structure. Combining all four basic structures we start a search for consistent S-matrices in three dimensions. Very nicely, the most natural ones are those corresponding to ABJM and BLG theories. We also make a connection between the power of a resultant in the integrand, representations of the Poincaré group, infrared behavior and conformality of a theory. Extensions to other theories in three dimensions and to arbitrary dimensions are also discussed.