On universal splittings of tree-level particle and string scattering amplitudes

TL;DR

This paper investigates a universal splitting behavior of tree-level scattering amplitudes, showing how they factorize under specific kinematic conditions across various theories including string, gauge, and gravity theories.

Contribution

It systematically studies the splitting behavior of string and particle amplitudes, extending known factorization properties to a broad class of theories and integrands.

Findings

Amplitudes factorize into lower-point currents under 2-split kinematics.

Results apply to open- and closed-string amplitudes, CHY formulas, and multiple theories.

Provides new insights into soft theorems and factorization behaviors.

Abstract

In this paper, we study the newly discovered universal splitting behavior for tree-level scattering amplitudes of particles and strings~\cite{Cao:2024gln}: when a set of Mandelstam variables (and Lorentz products involving polarizations for gluons/gravitons) vanish, the $n$-point amplitude factorizes as the product of two lower-point currents with $n{+}3$ external legs in total. We refer to any such subspace of the kinematic space of $n$ massless momenta as "2-split kinematics", where the scattering potential for string amplitudes and the corresponding scattering equations for particle amplitudes nicely split into two parts. Based on these, we provide a systematic and detailed study of the splitting behavior for essentially all ingredients which appear as integrands for open- and closed-string amplitudes as well as Cachazo-He-Yuan (CHY) formulas, including Parke-Taylor factors, correlators in superstring and bosonic string theories, and CHY integrands for a variety of amplitudes of scalars, gluons and gravitons. These results then immediately lead to the splitting behavior of string and particle amplitudes in a wide range of theories, including bi-adjoint $\phi^3$ (with string extension known as $Z$ and $J$ integrals), non-linear sigma model, Dirac-Born-Infeld, the special Galileon, etc., as well as Yang-Mills and Einstein gravity (with bosonic and superstring extensions). Our results imply and extend some other factorization behavior of tree amplitudes considered recently, including smooth splittings~\cite{Cachazo:2021wsz} and factorizations near zeros~\cite{Arkani-Hamed:2023swr}, to all these theories. A special case of splitting also yields soft theorems for gluons/gravitons as well as analogous soft behavior for Goldstone particles near their Adler zeros.