Dynamics of Spinning Binary at 2PM

We consider the covariant proposal for the gravitational Compton amplitude for a Kerr black hole. Employing the covariant three- and four-point Compton amplitudes, we assemble the classical one-loop integrand on the maximal cut at all orders in spin, utilizing the method of unitarity. Expanding in powers of spin, we evaluate the one-loop amplitude up to $\mathcal O(G^2 a^8)$. Supplemented with extra contact contributions derived from the far-zone data of the Teukolsky solutions, the one-loop amplitude is in agreement with results available in the literature. We showcase the classical eikonal in the aligned-spin case at $\mathcal O(G^2 a^7)$.