Binding energy for hydrogen-like atoms in the Nelson model without cutoffs

In the Nelson model particles interact through a scalar massless field. For hydrogen-like atoms there is a nucleus of infinite mass and charge Ze, Z > 0, fixed at the origin and an electron of mass m and charge e. This system forms a bound state with binding energy E_bin =me⁴Z²/8π² to leading order in e. We investigate the radiative corrections to the binding energy and prove upper and lower bounds which imply that E_bin=me⁴ Z²/8π² + c₀e⁶ O(e⁷ ln e) with explicit coefficient c₀ and independent of the ultraviolet cutoff. c₀ can be computed by perturbation theory, which however is only formal since for the Nelson Hamiltonian the smallest eigenvalue sits exactly at the bottom of the continuous spectrum.