Exploring the $\tau$ polarization in $B\to X\tau\bar\nu$ along different axes

The $\tau$ polarization in semileptonic $B$ decays provides probes of new physics complementary to decay rate distributions of the three-body final state. Prior calculations for inclusive decays used a definition for the polarization axis that is different from the choice used in calculations (and the only measurement) for exclusive channels. To compare inclusive and exclusive predictions, we calculate the $\tau$ polarization in inclusive $B\to X\tau\bar\nu$ using the same choice as in the exclusive decays, and construct a sum rule relating the inclusive $\tau$ polarization to a weighted sum of exclusive decay polarizations. We use this relation, experimental data, and theoretical predictions for the decays to the lightest charm or up-type hadrons to make predictions for excited channels.