Black Hole Response Theory and its Exact Shockwave Limit

We develop a black hole response formulation of worldline quantum field theory (WQFT) adapted to the gravitational self-force expansion. Integrating out the worldline and graviton fluctuations yields connected graviton response functions: the "zeroth response" one-point function encodes the exact background metric, the "first response" two-point function describes the scattering of a gravitational wave off the black hole, and higher-point functions provide effective vertices for a systematic gravitational self-force (SF) expansion. As a first application, we consider a massless primary source, corresponding to the Aichelburg-Sexl shockwave or an ultra-boosted black hole. We show that the one-point response resums to the exact shockwave metric and that the resummed probe vertices reproduce the exact geodesics in this background. We then compute the off-shell graviton two-point response exactly in the gravitational coupling $G$ by resumming the full post-Minkowskian series. For on-shell external gravitons this yields the exact transfer matrix for gravitational-wave scattering off the shockwave, including recoil. The PM expansion exponentiates to a compact form in which the Born term is dressed by an overall phase which includes the Weinberg phase. Our results provide the basic WQFT ingredients for future 1SF computations of observables such as the impulse and waveform in the ultra high-energy regime.