$B$ decays proceeding via $bto cell u$ transitions with $ell=e$ or $mu$ are tree-level processes in the Standard Model. They are used to measure the CKM element $V_{cb}$, as such forming an important ingredient in the determination of e.g. the unitarity triangle; hence the question to which extent they can be affected by new physics contributions is important, specifically given the long-standing tension between $V_{cb}$ determinations from inclusive and exclusive decays and the significant hints for lepton flavour universality violation in $bto ctau u$ and $bto sellell$ decays. We perform a comprehensive model-independent analysis of new physics in $bto cell u$, considering all combinations of scalar, vector and tensor interactions occuring in single-mediator scenarios. We include for the first time differential distributions of $Bto D^*ell u$ angular observables for this purpose. We show that these are valuable in constraining non-standard interactions. Specifically, the zero-recoil endpoint of the $Bto Dell u$ spectrum is extremely sensitive to scalar currents, while the maximum-recoil endpoint of the $Bto D^*ell u$ spectrum with transversely polarized $D^*$ is extremely sensitive to tensor currents. We also quantify the room for $e$-$mu$ universality violation in $bto cell u$ transitions, predicted by some models suggested to solve the $bto ctau u$ anomalies, from a global fit to $Bto Dell u$ and $Bto D^*ell u$ for the first time. Specific new physics models, corresponding to all possible tree-level mediators, are also discussed. As a side effect, we present $V_{cb}$ determinations from exclusive $B$ decays, both with frequentist and Bayesian statistics, leading to compatible results. The entire numerical analysis is based on open source code, allowing it to be easily adapted once new data or new form factors become available.
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