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

TL;DR

Problem: precise accounting of tail-induced corrections to gravitational binding energy in binary black holes faces regularization ambiguities. Approach: formulate both QED (NRQED) and gravity (NRGR) calculations within an EFT framework, employing zero-bin subtraction to separate near/far zone physics and derive RG equations. Findings: obtain the Lamb-shift-like long-distance correction in electrodynamics, including Bethe logarithm, and reproduce a $v^8 \, \log v$ tail correction in gravity, with regulator-independent results. Significance: demonstrates a unified, ambiguity-free EFT method for long-distance quantum and classical corrections to bound-state dynamics, with potential impact on high-precision gravitational-wave templates.

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 $α_e^5 \log α_e$), and likewise the contribution to the gravitational potential from the tail effect (proportional to $v^8 \log v$).

Figures (6)

• Figure 1: One loop vertex correction in electrodynamics.
• Figure 2: The one-loop correction in \ref{['green']}. The double line represents the bound state, and the dots are the dipole-type coupling from \ref{['pnrqed']}. A similar diagram --albeit at the classical level-- appears in NRGR (see below).
• Figure 3: The EFT approach to the binary inspiral problem. See review for a thorough review.
• Figure 4: The tail contribution to the radiative quadrupole moment. Only the lines with an arrow propagate. The double-line represents the two-body system, treated as an external non-propagating source.
• Figure 5: The tail contribution to radiation-reaction. The $(+,-)$ labels are associated to the 'in-in' formalism, required to properly compute retardation effects. The wavy line is a radiation mode $p^\mu \sim \lambda_{\rm rad}^{-1}$, whereas the dashed line corresponds to a potential mode with ${\boldsymbol{q}} \sim \lambda_{\rm rad}^{-1}$. See nltail for more details.