  The 4.2$\sigma$ discrepancy between the standard model prediction for the
muon anomalous magnetic moment $a_\mu$ and the experimental result is
accompanied by other anomalies. A crucial input for the prediction is the
hadronic vacuum polarization $a_\mu^{\rm HVP}$ inferred from $\sigma_{\rm had}
=\sigma(e^+e^- \to\,$hadrons) data. However, the two most accurate
determinations of $\sigma_{\rm had}$ from KLOE and BaBar disagree by almost
3$\,\sigma$. Additionally, the combined data-driven result disagrees with the
most precise lattice determination of $a_{\mu}^{\rm HVP}$ by $2.1\,\sigma$. We
show that all these discrepancies could be accounted for by a new boson
produced resonantly around the KLOE centre of mass energy and decaying promptly
yielding $e^+e^-$ and $\mu^+\mu^-$ pairs in the final states. This gives rise
to three different effects: (i) the additional $e^+e^-$ events will affect the
KLOE luminosity determination based on measurements of the Bhabha cross
section, and in turn the inferred value of $\sigma_{\rm had}$; (ii) the
additional $\mu^+\mu^-$ events will affect the determination of $\sigma_{\rm
had}$ via the (luminosity independent) measurement of the ratio of
$\pi^+\pi^-\gamma$ versus $\mu^+\mu^-\gamma$ events; (iii) loops involving the
new boson would contribute directly to the prediction for $a_\mu$. We discuss
in detail this possibility, and we present a simple model that can reconcile
the KLOE and BaBar results for $\sigma_{\rm had}$, the data-driven and the
lattice determinations of $a_\mu^{\rm HVP}$, the predicted and measured values
of $a_\mu$, while complying with all phenomenological constraints.
