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 Ebin=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 Ebin=me4Z2/8π2+c0e6+O(e7lne) 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.