Mechanical angular momentum loss at $\mathcal{O}(G^4)$ from static and radiative gravitons

Abstract: We determine the loss of mechanical angular momentum $L^μ_{\rm loss}$ of two gravitationally scattering bodies at $\mathcal{O}(G^4)$ using worldline quantum field theory (WQFT). The gravitational memory effect, a phenomenon of the early- and late-time behaviour of the gravitational field, leads to static contributions to $L^μ_{\rm loss}$. We illuminate how these arise: the linear memory contributes through so-called static correlators, while the nonlinear memory vanishes at this order due to symmetry considerations. The computation of the remaining, dynamical part resembles that of the $\mathcal{O}(G^4)$ impulse. We generate its integrand using a reformulation of the WQFT Feynman rules that conserves four-momentum at every vertex and plainly exhibits the classical loops. The three-loop integrals are reduced to master integrals, whose differential equations and boundary matching in the post-Newtonian limit we describe in some detail. In contrast to conservative effects at this order, only polylogarithms appear. While individual contributions to the dynamical part are divergent, these cancel in $L^μ_{\rm loss}$. The quadratic-in-radiation part of our result reproduces earlier results, while the complete result agrees with existing post-Newtonian results.