  Scalars that carry lepton number can help mediate would-be
lepton-number-violating processes, such as neutrinoless double $\beta$ decay or
lepton-scattering-mediated nucleon-antinucleon conversion. Here we show that
such new scalars can also solve the anomaly in precision determinations of the
fine-structure constant $\alpha$ from atom interferometry and from the
electron's anomalous magnetic moment, $a_e \equiv (g-2)_e/2$, by reducing
$|a_e|$. Study of the phenomenological constraints on these solutions favor a
doubly-charged scalar with mass below the GeV scale. Significant constraints
arise from the measurement of the parity-violating asymmetry in M{\o}ller
scattering, and we consider the implications of the next-generation MOLLER
experiment at Jefferson Laboratory and of an improved $a_e$ measurement.
