Lamb shift and the gravitational binding energy for binary black holes

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 nonrelativistic electrodynamics. We also derive the renormalization group equations from which we reproduce Bethe logarithm (at order αₑ⁵logαₑ), and likewise the contribution to the gravitational potential from the tail effect (proportional to v⁸logv).