Five-point partial waves, splitting constraints and hidden zeros

We study the partial-wave expansion of residues of five-point tree-amplitude involving identical scalar particles in the external legs. We check the construction using massive spinor-helicity building blocks and by matching to the tree-level five-point Veneziano amplitude at fixed mass levels. As an application, we express five-point splitting constraints - the reduction of the five-point amplitude to products of four-point amplitudes on special kinematic loci - as linear relations among the five-point partial-wave coefficients. At low mass levels these constraints, together with spin truncation, fix the full five-point partial-wave data in terms of the four-point coefficients and imply simple compatibility conditions; remarkably, imposing two independent splitting loci also forces the residue to vanish on their intersection, making the associated hidden zero manifest in partial-wave space. We also show that once both channels allow spin-2 exchange a genuine kernel can remain, indicating the need for additional higher-point input to achieve complete rigidity.