$2$-Split of Form Factors via BCFW Recursion Relation
Liang Zhang
TL;DR
The work proves a $2$-split factorization for tree-level form factors of the operators $O_1=(1/2)Tr((partial phi)^2)+Tr(phi^3)$ and $O_2=Tr(F^2)$ by extending the on-shell BCFW framework to form factors. The proof hinges on identifying hidden zeros and an additional zero, and an inductive argument shows that under carefully chosen $2$-split kinematics the form factor factorizes into a product of a lower-point current and a lower-point form factor, with only four nonzero BCFW diagrams contributing. The results generalize the amplitude factorization structure to off-shell observables, strengthening the link between on-shell techniques and operator form factors. This approach opens prospects for applying similar factorization ideas to EFTs and massive theories and raises questions about bootstrapping form factors from their zeros.
Abstract
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 ${\cal O} =\frac{1}{2}\Tr((\partial φ)^2) + \Tr(φ^3)$ and ${\cal O} = \Tr(F^2)$, and then employ an inductive proof based on the BCFW recursion relation to prove the $2$-split factorization for any number of external particles.