$b\to c$ semileptonic sum rule: orbitally excited hadrons

We study semileptonic sum rules for $b\to c\tau \bar{\nu }$ transitions involving orbitally excited charm hadrons. Starting from the amplitude-level relation implied by the heavy quark symmetry, we construct sum rules relating these decays. We then examine deviations from the small-velocity limit. Our numerical analysis shows that the deviations generally increase once excited hadrons are involved, with tensor contributions often inducing sizable effects. At present, however, the relevant form factors are not yet sufficiently constrained. Further improvements in the hadronic inputs are essential for these sum rules to yield robust predictions for the corresponding lepton-universality ratios.