Gravitational wave scattering at $\mathcal{O}(G^4)$: Murua construction and elliptics

We compute the amplitude for the scattering of a gravitational wave off of a spinless point particle at fourth order in Newton's constant, using the worldline quantum field theory formalism. A decomposition of our master integrals incorporating Murua coefficients allows us to entirely bypass the cut subtraction needed to convert the scattering amplitude into the Magnusian, the latter being desirable as it maps directly onto the scattering phase shift in partial wave space. This is then matched to the prediction from black hole perturbation theory, proving that point-particle worldline quantum field theory accurately describes Schwarzschild black holes up to $\mathcal{O}(G^4)$. Elliptic functions appear in momentum space for the first time for this process at this order.