Role of Tensor operators in $R_K$ and $R_{K^*}$

The recent LHCb measurement of $R_{K^*}$ in two $q^2$ bins, when combined with the earlier measurement of $R_K$, strongly suggests lepton flavour non-universal new physics in semi-leptonic $B$ meson decays. Motivated by these intriguing hints of new physics, several authors have considered vector, axial vector, scalar and pseudo scalar operators as possible explanations of these measurements. However, tensor operators have widely been neglected in this context. In this paper, we consider the effect of tensor operators in $R_K$ and $R_{K^*}$. We find that, unlike other local operators, tensor operators can comfortably produce both of $R_{K^*} ^{\rm low}$ and $R_{K^*} ^{\rm central}$ close to their experimental central values. However, a simultaneous explanation of $R_K$ is not possible with only Tensor operators, and other vector or axial vector operators are needed. In fact, we find that combination of vector and tensor operators can provide simultaneous explanations of all the anomalies comfortably at the $1 \sigma$ level, a scenario which is hard to achieve with only vector or axial vector operators. We also comment on the compatibility of the various new physics solutions with the measurements of the inclusive decay $B_d \to X_s \ell^+ \ell^-$.