Constructing heavy-quark sum rule for $b \to c$ meson and baryon decays

TL;DR

This paper develops a spin-decomposition framework based on heavy-quark symmetry to derive fundamental heavy-quark sum rules for $b\\to c$ semileptonic decays among various hadrons. The key idea is that decay amplitudes factor into a universal quark-level part $M_{\\lambda_c,\\lambda_b}$ and a light-degree-of-freedom overlap, with Clebsch–Gordan orthogonality yielding sum rules on squared amplitudes that are independent of the specific final-state hadron and hold without phase-space integration. The authors reproduce the known $B\\to D^{(*)}$ and $\\Lambda_b\\to\\Lambda_c$ relations and extend the formalism to $\\Omega_b\\to\\Omega_c^{(*)}$ as well as transitions to excited states, introducing Isgur–Wise functions $\\xi(w),\\zeta(w),\\xi_1(w),\\xi_2(w),\\tau_{1/2}(w),\\tau_{3/2}(w)$. The framework provides a transparent, non-perturbative tool to test the Standard Model and potential new physics, with clear predictions for a range of channels and a path for future experimental validation and theoretical extensions.

Abstract

We study heavy-hadron semileptonic decays proceeding via $b \to c$ transition, such as $B \to D^{(*)}τ\barν_τ$ and $Λ_b \to Λ_cτ\barν_τ$. In the heavy-quark limit, where the heavy-quark symmetry holds, we provide a fundamental framework for heavy-quark sum rules among these decays based on the spin decomposition picture. The relation holds directly for the squared amplitudes without requiring phase-space integration. We then apply this relation to reproduce the sum rule among $B \to D^{(*)}τ\barν_τ$ and $Λ_b \to Λ_cτ\barν_τ$. Furthermore, we derive new sum rules for $Ω_b \to Ω_c^{(*)}$ transitions and those involving excited states, such as $B \to \{D_0^*,D_1^*\}$ and $B \to \{D_1,D_2^*\}$.