Explaining the $R_K$ and $R_{K^*}$ anomalies

Recent LHCb results on $R_{K^*}$, the ratio of the branching fractions of $B \to K^* \mu^+ \mu^-$ to that of $B \to K^* e^+ e^-$, for the dilepton invariant mass bins $q^2 \equiv m_{\ell\ell}^2 = [0.045 - 1.1]$ GeV$^2$ and $[1.1 - 6]$ GeV$^2$ show approximately $2.5 \sigma$ deviations from the corresponding Standard Model prediction in each of the bins. This, when combined with the measurement of $R_K \, (q^2=[1-6]\, \rm GeV^2)$, a similar ratio for the decay to a pseudo scalar meson, highly suggests for lepton non-universal new physics in semi-leptonic $B$ meson decays. In this work, we perform a model independent analysis of these potential new physics signals and identify the operators that do the best job in satisfying all these measurements. We show that heavy new physics, giving rise to $q^2$ independent local 4-Fermi operators of scalar, pseudo-scalar, vector or axial-vector type, is unable to explain all the three measurements simultaneously, in particular $R_{K^*}$ in the bin [0.045 - 1.1], within their experimental $1\sigma$ regions. We point out the possibility to explain $R_{K^*}$ in the low bin by an additional light ($\lesssim 20 \, \rm MeV$) vector boson with appropriate coupling strengths to ($\bar b \, s$) and ($\bar e \, e$).