Combined explanations of B-physics anomalies: the sterile neutrino solution

In this paper we provide a combined explanation of charged- and neutral-current $B$-physics anomalies assuming the presence of a light sterile neutrino $N_R$ which contributes to the $B \to D^{(*)} \tau \nu$ processes. We focus in particular on two simplified models, where the mediator of the flavour anomalies is either a vector leptoquark $U_1^\mu \sim ({\bf 3}, {\bf 1}, 2/3)$ or a scalar leptoquark $S_1 \sim ({\bf \bar 3}, {\bf 1}, 1/3)$. We find that $U_1^\mu$ can successfully reproduce the required deviations from the Standard Model while being at the same time compatible with all other flavour and precision observables. The scalar leptoquark instead induces a tension between $B_s$ mixing and the neutral-current anomalies. For both states we present the limits and future projections from direct searches at the LHC finding that, while at present both models are perfectly allowed, all the parameter space will be tested with more luminosity. Finally, we study in detail the cosmological constraints on the sterile neutrino $N_R$ and the conditions under which it can be a candidate for dark matter.