Gravitational Wave Scattering in Spinless WQFT

We develop the computational framework for gravitational wave - black hole scattering in worldline quantum field theory (WQFT) without spin. Crucially, we prove on general grounds that, in the absence of dissipation, the exponential representation of the $S$-matrix maps -- through a partial-wave transformation -- directly onto the scattering phase shift from black hole perturbation theory (BHPT), indicating an exponentiation of the WQFT amplitude itself in partial-wave space. Computing explicitly, we reproduce the BHPT phase shift without spin up to $O(G^{3})$ from WQFT. While this result is expected, it lays the groundwork for higher-precision analyses involving non-minimal effects. Along the way, we outline our efficient diagram generation technique and include a pedagogical discussion on the computation of the required two-loop integrals.