Resonant production of dark photons in positron beam dump experiments

Positrons beam dump experiments have unique features to search for very narrow resonances coupled superweakly to $e^+ e^-$ pairs. Due to the continue loss of energy from soft photon bremsstrahlung, in the first few radiation lengths of the dump a positron beam can continuously scan for resonant production of new resonances via $e^+$ annihilation off an atomic $e^-$ in the target. In the case of a dark photon $A'$ kinetically mixed with the photon, this production mode is of first order in the electromagnetic coupling $\alpha$, and thus parametrically enhanced with respect to the $O(\alpha^2)$ $e^+e^- \to \gamma A'$ production mode and to the $O(\alpha^3)$ $A'$ bremsstrahlung in $e^--$nucleon scattering so far considered. If the lifetime is sufficiently long to allow the $A'$ to exit the dump, $A' \to e^+e^-$ decays could be easily detected and distinguished from backgrounds. We explore the foreseeable sensitivity of the Frascati PADME experiment in searching with this technique for the $17\,$MeV dark photon invoked to explain the $^8$Be anomaly in nuclear transitions.