Smoothly Splitting Amplitudes and Semi-Locality

Abstract

In this paper, we study a novel behavior developed by certain tree-level scalar scattering amplitudes, including the biadjoint, NLSM, and special Galileon, when a subset of kinematic invariants vanishes without producing a singularity. This behavior exhibits properties which we call $\text{smooth splitting}$ and $\text{semi-locality}$. The former means that an amplitude becomes the product of exactly three amputated Berends-Giele currents, while the latter means that any two currents share one external particle. We call these smooth splittings 3-splits. In fact, there are exactly $\left(\genfrac{}{}{0ex}{}{n}{3}\right)-n$ such 3-splits, one for each generic, interior triangle in a polygon; as they cannot be obtained from standard factorization, they are a new phenomenon in Quantum Field Theory. In fact, the resulting splitting is analogous to the one first seen in Cachazo-Early-Guevara-Mizera (CEGM) amplitudes which generalize standard cubic scalar amplitudes from their $\mathrm{Tr}G(2,n)$ formulation to $\mathrm{Tr}G(k,n)$, where $\mathrm{Tr}G(k,n)$ is the tropical Grassmannian. Along the way, we show how smooth splittings naturally lead to the discovery of mixed amplitudes in the NLSM and special Galileon theories and to novel BCFW-like recursion relations for NLSM amplitudes.

arXiv comment49 pages, 13 figures. V2. Corrected typos; added Section 6.7 on smooth 2-splits in k=3 CEGM generalized amplitudes

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