The Hadronic Contribution to the Muon $g-2$ from Hadron Production in Initial State Radiation Events at the $e^+ e^-$ Collider DA$\Phi$NE

A relevant reduction of the theoretical uncertainty on the muon anomalous magnetic moment, dominated by the error on the leading hadronic contribution $a^{\rm h}_\mu$, would come from a new precise measurement of the cross section of $e^+e^-$ annihilation into hadrons ($\sigma_h$) below 1~GeV. An experimental approach to the evaluation of the dispersion integral defining $a^{\rm h}_\mu$ is proposed here as an alternative to the conventional method based on the interpolation of different measurements of $\sigma_h$ performed with a center of mass energy scan. The wide occurrence of hadron production, at variable $q^2$, in the initial state radiation (ISR) events at the high luminosity $\phi-$factory DA$\Phi$NE suggests to redefine the dispersion integral in terms of the differential cross section for $e^+e^-\rightarrow hadrons + \gamma_{\rm ISR}$. The feasibility of such measurement, which can be performed without the need of dedicated run conditions and with the benefit of the full luminosity of the machine, is discussed. The required precision is shown to be easily achieved thanks to the resolutions and performances of the KLOE detector.