  At present, the measurements of some observables in $B \to K^* \mu^+\mu^-$
and $B_s^0 \to \phi \mu^+ \mu^-$ decays, and of $R_{K^{(*)}} \equiv {\cal B}(B
\to K^{(*)} \mu^+ \mu^-)/{\cal B}(B \to K^{(*)} e^+ e^-)$, are in disagreement
with the predictions of the standard model. While most of these discrepancies
can be removed with the addition of new physics (NP) in $b \to s \mu^+ \mu^-$,
a difference of $>\sim 1.7 \sigma$ still remains in the measurement of
$R_{K^*}$ at small values of $q^2$, the dilepton invariant mass-squared. In the
context of a global fit, this is not a problem. However, it does raise the
question: if the true value of $R_{K^*}^{low}$ is near its measured value, what
is required to explain it? In this paper, we show that, if one includes NP in
$b \to s e^+ e^-$, one can generate values for $R_{K^*}^{low}$ that are within
$\sim 1\sigma$ of its measured value. Using a model-independent,
effective-field-theory approach, we construct many different possible NP
scenarios. We also examine specific models containing leptoquarks or a $Z'$
gauge boson. Here, additional constraints from lepton-flavour-violating
observables, $B_s^0$-${\bar B}_s^0$ mixing and neutrino trident production must
be taken into account, but we still find a number of viable NP scenarios. For
the various scenarios, we examine the predictions for $R_{K^{(*)}}$ in other
$q^2$ bins, as well as for the observable $Q_5 \equiv P^{\prime\mu\mu}_5
-P^{\prime ee}_5$.
