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

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 $B \to \{D_1,D_2^*\}$.

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