# The Lamb shift and the gravitational binding energy for binary black holes

## TL;DR

This paper draws an analogy between the Lamb shift in quantum electrodynamics and the correction to gravitational binding energy in binary black holes, highlighting the role of tail effects and regularization techniques.

## Contribution

It introduces an effective field theory approach to compute the tail effect correction in gravitational binding energy, paralleling the Lamb shift calculation, and addresses regularization ambiguities.

## Key findings

- Derived the gravitational tail effect correction analogous to the Lamb shift.
- Implemented zero-bin subtraction to remove regularization ambiguities.
- Reproduced Bethe logarithm and tail contribution to gravitational potential.

## Abstract

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 $v^8 \log v$).