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

Abstract

We study heavy-hadron semileptonic decays proceeding via $b\to c$ transition, such as $B\to D^{(\ast )}\tau {\bar{\nu }}_{\tau }$ and ${\Lambda }_b\to {\Lambda }_c\tau {\bar{\nu }}_{\tau }$. 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 $B\to D^{(\ast )}\tau {\bar{\nu }}_{\tau }$ and ${\Lambda }_b\to {\Lambda }_c\tau {\bar{\nu }}_{\tau }$. Furthermore, we derive new sum rules for ${\Omega }_b\to {\Omega }_c^{(\ast )}$ transitions and those involving excited states, such as $B\to \{D_0^{\ast },D_1^{\ast }\}$ and $B\to \{D_1,D_2^{\ast }\}$.

19 pages, 2 tables; v2: matches published version