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.

arXiv comment27+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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Cited by in corpus (25)

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• Feynman Polytopes and the Tropical Geometry of UV and IR Divergences
• Two-loop Octagons, Algebraic Letters and $\bar{Q}$ Equations
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• Likelihood Equations and Scattering Amplitudes
• Singularities of eight- and nine-particle amplitudes from cluster algebras and tropical geometry
• Tropical fans, scattering equations and amplitudes
• Smoothly Splitting Amplitudes and Semi-Locality
• Truncated cluster algebras and Feynman integrals with algebraic letters
• Positive configuration space
• Symbol Alphabets from Tensor Diagrams
• Singular Solutions in Soft Limits
• Algebraic branch points at all loop orders from positive kinematics and wall crossing
• Notes on Biadjoint Amplitudes, $\mathrm{Trop}G(3,7)$ and $X(3,7)$ Scattering Equations
• Amplituhedra, and Beyond
• Notes on polytopes, amplitudes and boundary configurations for Grassmannian string integrals
• Generalized Planar Feynman Diagrams: Collections
• Planar Matrices and Arrays of Feynman Diagrams
• The SAGEX Review on Scattering Amplitudes, Chapter 5: Analytic Bootstraps for Scattering Amplitudes and Beyond
• Planar Matrices and Arrays of Feynman Diagrams: Poles for Higher $k$
• On polarized scattering equations for superamplitudes of 11D supergravity and ambitwistor superstring
• One-loop integrand from generalised scattering equations
• Double Soft Theorem for Generalised Biadjoint Scalar Amplitudes
• Computing characteristic polynomials of hyperplane arrangements with symmetries
• Polyhedral and Tropical Geometry of Flag Positroids