Impact of polarization observables and $ B_cto tau u$ on new physics explanations of the $bto c tau u$ anomaly

The combined analysis of the BaBar, Belle, and LHCb data on $Bto Dtau u$, $Bto D^*tau u$ and $B_cto J/Psitau u$ decay observables shows evidence of physics beyond the Standard Model (SM). In this article, we study all the one- and two-dimensional scenarios which can be generated by adding a single new particle to the SM. We put special emphasis on the model-discriminating power of $F_L(D^*)$ and of the $tau$ polarizations, and especially on the constraint from the branching fraction ${rm BR}(B_ctotau u)$. We critically review this constraint and do not support the aggressive limit of ${rm BR}(B_ctotau u)<10%$ used in some analyses. While the impact of $F_L(D^*)$ is currently still limited, the ${rm BR}(B_ctotau u)$ constraint has a significant impact: depending on whether one uses a limit of $60%$, $30%$ or $10%$, the pull for new physics (NP) in scalar operators changes drastically. More specifically, for a conservative $60%$ limit a scenario with scalar operators gives the best fit to data, while for an aggressive $10%$ limit this scenario is strongly disfavored and the best fit is obtained in a scenario in which only a left-handed vector operator is generated. We find a sum rule for the branching ratios of $Bto Dtau u$, $Bto D^*tau u$ and $Lambda_bto Lambda_ctau u$ which holds for any NP contribution to the Wilson coefficients. This sum rule entails an enhancement of ${rm BR}(Lambda_bto Lambda_ctau u)$ over its SM prediction by $(24pm 6)%$ for the current $mathcal{R}(D^{(*)})$ data.