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The discrepancy between the Standard Model theory and experimentalmeasurement of the muon magnetic moment anomaly,$a_{\mu}=\left(g_{\mu}-2\right)/2$, is connected to precision electroweak (EW)predictions via their common dependence on hadronic vacuum polarizationeffects. The same data for the total $e^+e^- \rightarrow \text{hadrons}$ crosssection, $\sigma_{\rm had}(s)$, are used as input into dispersion relations toestimate the hadronic vacuum polarization contributions, $a_{\mu}^{\rmhad,\,VP}$, as well as the five-flavor hadronic contribution to the running QEDcoupling at the $Z$-pole, $\Delta\alpha_{\rm had}^{(5)}(M_{Z}^2)$, which entersnatural relations and global EW fits. The EW fit prediction of$\Delta\alpha_{\rm had}^{(5)}(M_{Z}^2) = 0.02722(41)$ agrees well with$\Delta\alpha_{\rm had}^{(5)}(M_{Z}^2) = 0.02761(11)$ obtained from thedispersion relation approach, but exhibits a smaller central value suggestiveof a larger discrepancy $\Delta a_{\mu}=a_{\mu}^{\rm exp} - a_{\mu}^{\rm SM}$than currently expected. Postulating that the $\Delta a_{\mu}$ difference maybe due to missing $\sigma_{\rm had}(s)$ contributions, implications for $M_W$,$\sin^2 \! \theta^{\rm lep}_{\rm eff}$ and $M_H$ obtained from global EW fitsare investigated. Shifts in $\sigma_{\rm had}(s)$ needed to bridge $\Deltaa_{\mu}$ are found to be excluded above $\sqrt{s} \gtrsim 0.7$ GeV at the95\%CL. Moreover, prospects for $\Delta a_{\mu}$ originating below that energyare deemed improbable given the required increases in the cross section. Suchhypothetical changes to the hadronic data are also found to affect otherrelated observables, such as the electron anomaly, $a_e^{\rm SM}$, the ratio$R_{e/\mu} = (m_\mu/m_e)^2 (a_{e}^{\rm had,\,LO\,VP}/a_{\mu}^{\rmhad,\,LO\,VP})$ and the running of the weak mixing angle at low energies,although the consequences of these are currently less constraining.

The muon $g$-2 and $\Delta \alpha$ connection