Smoothly Splitting Amplitudes and Semi-Locality

TL;DR

This paper uncovers a new phenomenon called smooth splitting in scalar scattering amplitudes, revealing novel factorization properties and leading to new amplitude relations in quantum field theory.

Contribution

It introduces the concept of smooth splitting and semi-locality in scalar amplitudes, a phenomenon not derived from standard factorization, and connects it to tropical Grassmannians and new recursion relations.

Findings

Discovery of smooth splitting (3-splits) in scalar amplitudes.

Identification of semi-locality sharing external particles.

Development of BCFW-like recursion relations for NLSM.

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 $\textit{smooth splitting}$ and $\textit{semi-locality}$. 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 $\binom{n}{3}-n$ 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 ${\rm Tr}\, G(2,n)$ formulation to ${\rm Tr}\, G(k,n)$, where ${\rm Tr}\, G(k,n)$ 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.