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The QED hadronic vacuum polarization function plays an important role in thedetermination of precision electroweak observables and of the anomalousmagnetic moment of the muon. These contributions have been computed from data,by means of dispersion relations affecting the electron positron hadronic crosssections, or by first principle lattice-QCD computations in the Standard Model.Today there is a discrepancy between the two approaches for determining thesecontributions, which affects the comparison of the measurement of the anomalousmagnetic 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 anew light gauge boson that couples to the first generation quark and leptonsand has a mass below the GeV scale. We discuss the requirements for itsconsistency with observations and the phenomenological implications of such agauge extension.

Resolving the muon $g-2$ tension through $Z'$-induced modifications to $\sigma_{\mathrm{had}}$