Tension in the inclusive versus exclusive determinations of $|V_{cb}|$: a possible role of new physics

We reconsider the possibility that the tension in the $|V_{cb}|$ determinations from inclusive and exclusive $B$ decay modes is due to a new physics effect. We modify the Standard Model effective Hamiltonian for semileptonic $b \to c$ transitions including a tensor operator with a lepton flavour dependent coupling $\epsilon_T^\ell$, and investigate separately the muon and electron modes. The interference term between SM and NP, proportional to the lepton mass, has different impact in the inclusive and exclusive $B$ modes to muon. Moreover, even when the lepton mass is small as for the electron, the NP effect is different in inclusive and exclusive $B$ channels. For both $\mu$ and $e$ we find a region of $\epsilon_T^{\mu,\,e}$ where the constraints from $B^- \to D^{(*)0} \ell^- {\bar \nu}_\ell$ and $B \to X_c \, \ell \, \bar \nu_\ell$ are satisfied for the same $|V_{cb}|$.