Endpoint anatomy of the baryon--meson sum rule in $b\to sν\barν$

Abstract

Baryon-meson sum rules enable cross-checks among exclusive mesonic and baryonic measurements. Unlike their heavy-to-heavy counterparts, analogous heavy-to-light relations lack an explicit heavy-quark-symmetry origin. We investigate their origin in the $b\to sν\barν$ system by studying the relevant helicity amplitudes at the two kinematic endpoints. Although the sum rule follows algebraically from the simple short-distance structure, its coefficient depends nontrivially on the hadronic dynamics. At maximum recoil, the coefficient for the daughter baryons ${\cal B}=Λ,Ξ$ is consistent with the approximate value $1/2$ within current form-factor uncertainties. At zero recoil, by contrast, a kinematic helicity projection enforces its exact vanishing. The fully integrated coefficients are numerically close to $1/4$. However, unlike the one-quarter coefficient in the heavy-to-heavy system, they are not local symmetry coefficients but integrated quantities shaped by the behavior between the two endpoints. This understanding may guide the construction of baryon-meson sum rules in more general heavy-to-light decays.