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 ${\sigma }_{\mathrm{h}\mathrm{a}\mathrm{d}}$ 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 ${\sigma }_{\mathrm{h}\mathrm{a}\mathrm{d}}$ with high statistical accuracy from the two-pion threshold up to above $\sqrt{s}\sim 1$GeV.