Scattering equations: from projective spaces to tropical grassmannians
Abstract
We introduce a natural generalization of the scattering equations, which connect the space of Mandelstam invariants to that of points on ℂℙ^{1}, to higherdimensional projective spaces ℂℙ^{ k  1}. The standard, k = 2 Mandelstam invariants, s _{ ab }, are generalized to completely symmetric tensors {s}_{a_1{a}_2\dots {a}_k} subject to a `massless' condition {s}_{a_1{a}_2\dots {a}_{k2}bb}=0 and to `momentum conservation'. The scattering equations are obtained by constructing a potential function and computing its critical points. We mainly concentrate on the k = 3 case: study solutions and define the generalization of biadjoint scalar amplitudes. We compute all `biadjoint amplitudes' for ( k, n) = (3 , 6) and find a direct connection to the tropical Grassmannian. This leads to the notion of k = 3 Feynman diagrams. We also find a concrete realization of the new kinematic spaces, which coincides with the spinorhelicity formalism for k = 2, and provides analytic solutions analogous to the MHV ones.
 Publication:

Journal of High Energy Physics
 Pub Date:
 June 2019
 DOI:
 10.1007/JHEP06(2019)039
 arXiv:
 arXiv:1903.08904
 Bibcode:
 2019JHEP...06..039C
 Keywords:

 Scattering Amplitudes;
 Differential and Algebraic Geometry;
 High Energy Physics  Theory;
 Mathematics  Combinatorics
 EPrint:
 27+7 pages. v2: typos corrected. Connection to trop G(3,7) added at end of section 4. Appendix with numerical seeds for all solutions to X(3,6) equations provided