Atoms as electron accelerators for measuring the $e^+e^- \to\,$hadrons cross section

The hadronic vacuum polarization contribution to $(g-2)_\mu$ can be determined via dispersive methods from $e^+e^-\to\;$hadrons data. We propose a novel approach to measure the hadronic cross section $\sigma_{\mathrm{had}}$ as an alternative to the initial-state radiation and energy scan techniques, which relies on positron annihilation off atomic electrons of a high $Z$ target ($^{238}$U, $Z=92$). We show that by leveraging the relativistic electron velocities of the inner atomic shells, a high-intensity $12\,$GeV positron beam, such as the one foreseen at JLab, can allow to measure $\sigma_{\mathrm{had}}$ with high statistical accuracy from the two-pion threshold up to above $\sqrt{s} \sim 1\,$GeV.