$R_{K^{(*)}}$ and $R(D^{(*)})$ anomalies resolved with lepton mixing

In a recent paper arXiv:1706.08437, we have advanced a minimal resolution of some of the persistent anomalies in semileptonic $B$-decays. These include the neutral-current observables $R_K$ and $R_{K^*}$, as well as the charged-current observables $R(D)$ and $R(D^*)$. Recently, it has been observed that the semileptonic decays of the $B_c$ meson also hint at a similar type of anomaly. In this longer version, we discuss in detail why, if the anomalies are indeed there, it is a challenging task to explain the data consistently in terms of a simple and compelling new physics scenario. We find that the minimal scheme to achieve a reasonable fit involves the inclusion of just two (or, at worst, three with a possible symmetry relationship between their Wilson coefficients) new current-current operators, constructed in terms of the flavour eigenstates, augmented by a change of basis for the charged lepton fields. With only three unknown parameters, this class of models not only explain all the anomalies (including that in $B_c \to J/\psi \, \ell \nu$) to a satisfactory level but also predict some interesting signatures, like $B\to K\mu\tau$, $B_s\to \tau\tau$, or $B\to K$ plus missing energy, that can be observed at LHCb or Belle-II.