Exclusive leptonium electroproduction

Purely leptonic bound states provide precision probes of QED. Positronium $(e^+e^-)$ and muonium $(μ^+e^-)$ have long been observed, whereas dimuonium $(μ^+μ^-)$ and tauonium $(τ^+τ^-)$ remain undiscovered. We study exclusive vector-leptonium electroproduction in $ep$ collisions within nonrelativistic QED. We include the Bethe--Heitler and double deeply virtual Compton scattering contributions and their interference, and calculate the NLO QCD hard-scattering kernels entering the dominant Compton form factor $\Hcal$ within collinear GPD factorization. The NLO QCD correction to the DDVCS contribution changes from a strong suppression at low photon virtuality to a sizable enhancement as the lower virtuality cut is raised, with the gluon channel providing the dominant contribution. Bethe--Heitler production dominates the exclusive rate, supporting dedicated dimuonium searches at the EIC and JLab, with larger samples expected at higher-energy electron--proton colliders. The much larger positronium samples provide a high-statistics environment for precision QED studies, whereas tauonium production remains strongly suppressed.