Radiation-Reaction and Angular Momentum Loss at $\mathcal{O}(G^4)$

TL;DR

This paper simplifies the calculation of angular momentum loss at fourth order in gravitational constant for two-body scattering by relating it to the tree-level waveform, enabling a closed-form expression that includes all fractional post-Newtonian corrections.

Contribution

It reveals that the odd-in-velocity contribution to angular momentum loss at  order can be computed from the tree-level waveform, reducing the loop order needed and allowing a resummation of fractional PN corrections.

Findings

Derived a closed-form expression for  angular momentum loss.

Reduced the loop order from three to two for calculations.

Resummed all fractional PN corrections starting at 1.5PN.

Abstract

We point out that the odd-in-velocity contribution to the $\mathcal{O}(G^4)$ radiated angular momentum for two-body scattering is determined by the radiation-reaction (RR) term in the one-loop waveform. This RR term is actually proportional to the tree-level waveform, and this reduces the calculation of the odd-in-velocity contribution to the $\mathcal{O}(G^4)$ angular momentum loss, $J_\text{2rad}$, to two loops, instead of three loops as one would expect by power counting. We exploit this simplification, which follows from unitarity, to obtain a closed-form expression for $J_\text{2rad}$ for generic velocities, which resums all fractional post-Newtonian (PN) corrections to the $\mathcal{O}(G^4)$ angular momentum loss starting at 1.5PN.