From Hidden Zeros to Exact Splittings and New Zeros

Abstract: Hidden zeros impose constraints on scattering amplitudes beyond their familiar factorization properties. We explore their structure in two related directions. We first promote hidden zeros to explicit splitting formulas that organize full amplitudes at generic kinematics. Ordered $\mathrm{Tr}(φ<sup>3)$ amplitudes, nonlinear sigma model (NLSM) amplitudes, and cosmological wavefunctions share the same defining exact splitting structure $$\mathcal A=\sum_αR_α\, {\mathcal J}<em>{α,L}{\mathcal J}</em>{α,R} {\mathcal J}<em>{α,\mathrm{out}}$$, adapted to any hidden zero. Each prefactor $R</em>α$ is a condition of the hidden zero locus, so this decomposition manifests the hidden vanishing term by term. The remaining factors are lower point amplitudes or currents. Using these representations, we can systematically find new zero loci beyond the usual ones. In particular, we obtain five new all-multiplicity zero families. We also find continuous families of zeros: the coefficients of their defining equations can vary continuously while the amplitude remains zero.