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 YM ⨁ BAS, GR ⨁ YM, and GR ⨁ YM ⨁ 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.