  The QED hadronic vacuum polarization function plays an important role in the
determination of precision electroweak observables and of the anomalous
magnetic moment of the muon. These contributions have been computed from data,
by means of dispersion relations affecting the electron positron hadronic cross
sections, or by first principle lattice-QCD computations in the Standard Model.
Today there is a discrepancy between the two approaches for determining these
contributions, which affects the comparison of the measurement of the anomalous
magnetic moment of the muon with the theoretical predictions. In this article,
we revisit the idea that this discrepancy may be explained by the presence of a
new light gauge boson that couples to the first generation quark and leptons
and has a mass below the GeV scale. We discuss the requirements for its
consistency with observations and the phenomenological implications of such a
gauge extension.
