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 ${\mathbb{CP}}^1$, to higher-dimensional projective spaces ${\mathbb{CP}}^{k-1}$. The standard, $k=2$ Mandelstam invariants, $s_{ab}$, are generalized to completely symmetric tensors ${\text{s}}_{a_1{a}_2\dots a_k}$ subject to a `massless' condition ${\text{s}}_{a_1{a}_2\cdots a_{k-2}bb}=0$ 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 spinor-helicity formalism for $k=2$, and provides analytic solutions analogous to the MHV ones.

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

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• Non-abelian $Z$-theory: Berends-Giele recursion for the ${\alpha }^{\prime }$-expansion of disk integrals
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Cited by in corpus (1)

• Singularities of eight- and nine-particle amplitudes from cluster algebras and tropical geometry