$b \to c$ semileptonic sum rule: SU(3)$_{\rm{F}}$ symmetry violation

Abstract

To clarify possible deviations in $b\to c\tau\overline\nu$ processes, the $b\to c$ semileptonic sum rule provides a valuable tool. This relation, derived based on heavy quark symmetry (HQS), offers a powerful consistency check among experimental results. In this work, we extend the previously proposed sum rule for $\{B\to D^{(*)} l\overline\nu,\,\Lambda_b\to \Lambda_c l\overline\nu\}$ to include $\{B_s\to D_s^{(*)} l\overline\nu,\,\Xi_b\to \Xi_c l\overline\nu\}$, thereby enabling more useful cross-checks. Although the relation is supported by HQS and SU(3) flavor symmetry, both symmetries are broken in reality, and the size of the violation needs to be quantified to assess the validity of the sum rule. While the violation is expected to be moderate based on chiral perturbation theory, we perform a numerical evaluation and compare it with future experimental sensitivities. We find that the violation remains smaller than the expected experimental uncertainty. Therefore another new physics agnostic and predictive sum rules are constructed to check the consistency.