  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
($\bar e \, e$).
