Predictions and Tests for Semi-leptonic Decays of Beauty Hadrons,
We make predictions and propose tests for the semi-leptonic decays $B\to D^{(*)} l \bar{\nu}_l$ and $\Lambda_b\to \Lambda_c l \bar{\nu}_l$, with $l$ = $e$, $\mu$ and $\tau$. Preliminarily, we study such processes according to the $\ell-s$ scheme, which helps interpreting the results of previous analyses. Then, we write the angular distributions of those decays in the helicity formalism and suggest for each decay some tests for distinguishing among the various Dirac operators which may explain the tensions with the standard model. In particular, we propose, for $l$ = $e$ and $\mu$, to study the fully differential distribution. In the cases of the $B\to D^*$ and $\Lambda_b$ decays, important physical information can be extracted even after integrating over the momentum of the charged lepton, which is quite suitable for $l$ = $\tau$. Especially, the decay angular distribution of polarized $\Lambda_b$ depends on five observables, which assume different values according to the various new-physics interactions; one of these observables is related to the longitudinal polarization of the $\tau-\bar{\nu}_\tau$ system, which can be compared with the $D^*$ longitudinal polarization in the $B\to D^*$ decay. Moreover, we analyze the $\tau$ anomaly in the branching ratios of the semi-leptonic decays; to this end, we make some assumptions, which have a remarkable predictive power and lead to results in agreement with previous theoretical calculations. Thanks to these assumptions, we combine our present analysis of the $\Lambda_b$ decay with the previous ones of the $B\to D^{(*)}$ decays, getting strong constraints on the parameters which characterize the different new-physics operators and an indication on the most likely one.