Gravitational Sommerfeld Effects: Formalism, Renormalization, and Perturbation to $O(G^{10})$

TL;DR

This paper develops a systematic framework to compute the gravitational Sommerfeld factor for scalar perturbations in binary inspirals, incorporating tidal effects and achieving high-order analytical solutions.

Contribution

It introduces a new EFT-based method to analytically solve the Sommerfeld factor to $O(G^{10})$, including phase and magnitude, and establishes a renormalization group equation for radiative moments.

Findings

Derived a closed-form expression for the Sommerfeld factor in terms of the EFT connection matrix.

Proved the phase of the Sommerfeld factor matches elastic Compton scattering phase shift without tidal dissipation.

Solved the Sommerfeld factor analytically to $O(G^{10})$ for multiple partial waves.

Abstract

In the effective field theory (EFT) description of binary inspirals, the radiated gravitational waveform receives universal corrections from the curved background, the so-called ``tail effects'', that resum into the so-called ``Sommerfeld factor''. We develop a systematic framework for computing this gravitational Sommerfeld factor for scalar perturbations with the presence of tidal effects on the system. Using the worldline EFT, we recast the diagrammatic resummation as a solution to the $d$-dimensional wave equation with a localized source, and derive a closed-form expression for the Sommerfeld factor in terms of the EFT connection matrix. We prove that the phase of the Sommerfeld factor is exactly the same as elastic Compton scattering phase shift when there is no tidal dissipation. By combining the renormalization techniques in EFT with the Mano--Suzuki--Takasugi method in black hole perturbation theory, we analytically solve the Sommerfeld factor for both the magnitude and phase to $O(G^{10})$ for the $\ell = 0, 1, 2$ partial waves. We further establish a new renormalization group equation for the radiative multipole moments, whose exact solution yields an improved resummation of the waveform beyond the universal tail logarithms. These high-precision data and exact relations pave the way for future waveform resummation.