  In this paper, we determine the exact expression for the hydrogen binding
energy in the Pauli-Fierz model up to the order $O(\alpha^5\log\alpha^{-1})$,
where $\alpha$ denotes the finestructure constant, and prove rigorous bounds on
the remainder term of the order $o(\alpha^5\log\alpha^{-1})$. As a consequence,
we prove that the binding energy is not a real analytic function of $\alpha$,
and verify the existence of logarithmic corrections to the expansion of the
ground state energy in powers of $\alpha$, as conjectured in the recent
literature.
