Scattering in three dimensions from rational maps

By F. Cachazo, Song He, E. Y. Yuan. 2013

142 citations recorded by the source corpus.

A bstractThe complete tree-level S-matrix of four dimensional $ \mathcal{N}=4 $ super Yang-Mills and $ \mathcal{N}=8 $ 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.

Subfields: Astronomy and Astrophysics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/c12f6c0778151a581cf7d2082b2d9b762de48656

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Astronomy and Astrophysics

1. N. Arkani-Hamed
2. Alexander Ochirov
3. F. Cachazo
4. Song He
5. Yuntao Bai
6. Thomas Lam
7. H. Johansson
8. E. Y. Yuan

In Nuclear and High Energy Physics

1. Alexander Ochirov
2. N. Arkani-Hamed
3. F. Cachazo
4. Song He
5. H. Johansson
6. Yuntao Bai
7. E. Y. Yuan
8. Thomas Lam

In Statistical and Nonlinear Physics

1. N. Arkani-Hamed
2. Alexander Ochirov
3. Yuntao Bai
4. F. Cachazo
5. Song He
6. H. Johansson
7. Thomas Lam
8. E. Y. Yuan

Citations in the corpus

• Color-kinematics duality for QCD amplitudes (2015, background): “The BCJ amplitude relations motivated the discovery of the scattering equations and associated elegant string-like formulas for gauge and gravity tree amplitudes [86–89].”
• Positive geometries and canonical forms (2017, methodology): “Our methods are not able to rigorously prove it, but a triangulation of this Grassmann polytope appears to be given by the images under Z of the four 4-dimensional positroid cells [2][3][4][5] : (∗1) = 0, [1][2][3][5] : (∗4) = 0, [1][2][3][4] : (∗5) = 0, [12][34][5] : (12) = (34) = 0 (7.”
• Scattering equations and matrices: from Einstein to Yang-Mills, DBI and NLSM (2014, background): “Evidence that this must be possible is that in particular dimensions such formulas exist [22, 40, 48].”
• Scattering equations and Kawai-Lewellen-Tye orthogonality (2013, background): “In [6] it was proven that for four dimensional kinematics the total number of solutions is (n − 3)!.”
• Scattering of massless particles: scalars, gluons and gravitons (2013, background): “One of the main ingredients of the formula are the scattering equations [6, 9] ∑”
• Scattering forms and the positive geometry of kinematics, color and the worldsheet (2017, background): “[28, 30]), an object whose importance has been highlighted in Nair’s observation [31] and Witten’s twistor string [4], and especially in the story of scattering equations and the CHY formulas for scattering amplitudes [5, 6, 15, 32].”
• Black Hole Formation and Classicalization in Ultra-Planckian 2 ! N Scattering (2014, unknown): “, N − 2, with each sector having ( N−3 k−2 ) solutions and N−2 ∑ k=2 ( N−3 k−2 ) = (N − 3)! [47].”