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

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.