A Soft Theorem for the Tropical Grassmannian

Abstract

We study the soft limit of a recently proposed generalization of the biadjoint scalar amplitudes $m_n^{(k)}$, which have been conjectured to have a relation to the tropical Grassmannian $\text{Tr G}(k,n)$. Using the CHY formulation along with the Global Residue Theorem, we prove the soft factorization for $m_n^{(k)}$ amplitudes for arbitrary $k$ and $n$. We find that the soft factors are in direct correspondence to vertices of the associahedron ${\mathcal{A}}_{k-1}$, and hence take the form of $m_n^{(2)}$ amplitudes. This entails that all scattering amplitudes of the ordinary biadjoint scalar theory can be interpreted as an infinite family of soft factors. Additionally, Grassmannian duality reveals that generalized amplitudes $m_n^{(k)}$ with $k>2$ satisfy not only a soft theorem, but also a non-trivial "hard" theorem. We perform numerical checks of our theorems against previous results for $\text{Tr G}(4,7)$ and $\text{Tr G}(5,8)$, thereby providing strong evidence of their relation with the CHY formulation.

arXiv comment28 + 9 pages, 2 figures

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• One-loop integrand from generalised scattering equations
• Weighted blade arrangements and the positive tropical Grassmannian