$2$-Split of Form Factors via BCFW Recursion Relation

Recently, \cite{Cao:2025hio} demonstrated the $2$-split for form factor under specific kinematic constraints. This factorization is analogous to that observed in scattering amplitudes. A key consequence of this structure is the presence of hidden zeros, where the form factors vanish on specific kinematic loci. We first establish these zeros and a new zero for the form factors of the composite operators $\mathcal{O}=\frac{1}{2}\text{Tr}((\mathrm{\partial }\phi {)}^2)+\text{Tr}({\phi }^3)$ and $\mathcal{O}=\text{Tr}(F^2)$, and then employ an inductive proof based on the BCFW recursion relation to prove the $2$-split factorization for any number of external particles.