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 $\mathcal{O}\left(G^2{a}^8\right)$ \( \mathcal{O}\left({G}^2{a}^8\right) \) . 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 $\mathcal{O}\left(G^2{a}^7\right)$ \( \mathcal{O}\left({G}^2{a}^7\right) \) .