Generalized $2$-split for higher-derivative YM and GR amplitudes at tree-level

TL;DR

This paper establishes that higher-derivative YM and GR tree amplitudes with special insertions do not factorize into a single current-current product, but universally decompose into a sum of multiple $2$-split contributions. By expanding $F^3$, $R^2$, and $R^3$ amplitudes into constituent theories such as ${\rm BAS}$, ${\rm YM}$, and ${\rm GR}$, and applying transmutation operators, the authors derive explicit multi-term factorization structures (involving currents $\mathcal{J}^X$, $\mathcal{J}^Y$, $\mathcal{J}^Z$) that reproduce the known hidden-zero behavior and align with open- and closed-string factorization patterns in the appropriate limits. The results generalize the standard $2$-split to a broader, unified framework and demonstrate consistency with string-theoretic expectations for open and closed strings, while revealing avenues for a more transparent understanding via CHY/BCFW formalisms. Although $R^3$ exhibits some discrepancies with the full closed-string corrections at $\mathcal{O}(\alpha'^2)$ due to missing sectors, the identified multi-term structure provides a robust scaffold for exploring high-derivative amplitudes, their soft limits, and their deeper connections to string theory and amplitude bootstrapping.

Abstract

We study the generalized $2$-split of higher-derivative amplitudes, including Yang-Mills (YM) and Gravity (GR) amplitudes with special insertions of higher-derivative vertices, by expanding them into ${\rm YM}\oplus{\rm BAS}$, ${\rm GR}\oplus{\rm YM}$, and ${\rm GR}\oplus{\rm YM}\oplus{\rm BAS}$ amplitude, respectively. By leveraging the established $2$-split properties of these constituent theories, we show that these higher-derivative amplitudes -- which also exhibit another newly discovered phenomenon called hidden zero -- do not factorize into a single product of two currents. Instead, their factorization universally appears as a sum of multiple $2$-split contributions.