Abstract
We introduce a natural generalization of the scattering equations, which connect the space of Mandelstam invariants to that of points on ℂℙ1, to higher-dimensional projective spaces ℂℙk − 1. The standard, k = 2 Mandelstam invariants, sab, are generalized to completely symmetric tensors \( {\mathrm{s}}_{a_1{a}_2\dots {a}_k} \) subject to a ‘massless’ condition \( {\mathrm{s}}_{a_1{a}_2\dots {a}_{k-2}bb}=0 \) 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.
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Cachazo, F., Early, N., Guevara, A. et al. Scattering equations: from projective spaces to tropical grassmannians. J. High Energ. Phys. 2019, 39 (2019). https://doi.org/10.1007/JHEP06(2019)039
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DOI: https://doi.org/10.1007/JHEP06(2019)039