Scattering equations: from projective spaces to tropical grassmannians
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13130_2019_Article_10717.pdf
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Author(s)
Cachazo, Freddy
Early, Nick
Guevara, Alfredo
Mizera, Sebastian
Date Issued
June 11, 2019
Publisher
Springer Berlin Heidelberg
Citation
Journal of High Energy Physics. 2019 Jun 11;2019(6):39
Version
Final published version
Abstract
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
s
a
1
a
2
…
a
k
$$ {\mathrm{s}}_{a_1{a}_2\dots {a}_k} $$
subject to a ‘massless’ condition
s
a
1
a
2
…
a
k
−
2
b
b
=
0
$$ {\mathrm{s}}_{a_1{a}_2\dots {a}_{k-2}bb}=0 $$
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution
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DOI of Published Version
https://doi.org/10.1007/JHEP06(2019)039
