  We show that the correction to the gravitational binding energy for binary
black holes due to the tail effect resembles the Lamb shift in the Hydrogen
atom. In both cases a 'conservative' effect arises from interactions with
'radiation' modes, and moreover an explicit cancelation between near and far
zone divergences is at work. In addition, regularization scheme-dependence may
introduce ambiguity parameters. This is remediated, within an effective field
theory approach, by the implementation of the zero-bin subtraction. We
illustrate the procedure explicitly for the Lamb shift, by performing an
ambiguity-free derivation within the framework of non-relativistic
electrodynamics. We also derive the renormalization group equations from which
we reproduce Bethe logarithm (at order $\alpha_e^5 \log \alpha_e$), and
likewise the contribution to the gravitational potential from the tail effect
(proportional to $v^8 \log v$).
