A measurement of additional radiation in $e^{+}e^{-} \to \mu\mu \gamma$ and $e^{+}e^{-} \to \pi\pi \gamma$ initial-state-radiation events is presented using the full $BaBar$ data sample.
For the first time results are presented at next-to- and next-to-next-to-leading order, with one and two additional photons, respectively, for radiation from the initial and final states.
The comparison with the predictions from \textsc{Phokhara} and \textsc{AfkQed} generators reveals discrepancies for the former in the one-photon rates and angular distributions.
While this disagreement has a negligible effect on the $e^{+}e^{-} \to \pi\pi (\gamma)$ cross section measured by $BaBar$, the impact on the KLOE and BESIII measurements is estimated and found to be indicative of significant systematic effects.
The findings shed a new light on the longstanding deviation among the muon $g-2$ measurement, the Standard Model prediction using the data-driven dispersive approach for calculation of the hadronic vacuum polarization~(HVP), and the comparison with lattice QCD calculations.