Dynamics of spinning binary at 2PM

A bstract 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 $$ \mathcal{O}\left({G}^2{a}^8\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>O</mml:mi> <mml:mfenced> <mml:mrow> <mml:msup> <mml:mi>G</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:msup> <mml:mi>a</mml:mi> <mml:mn>8</mml:mn> </mml:msup> </mml:mrow> </mml:mfenced> </mml:math> . 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 $$ \mathcal{O}\left({G}^2{a}^7\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>O</mml:mi> <mml:mfenced> <mml:mrow> <mml:msup> <mml:mi>G</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:msup> <mml:mi>a</mml:mi> <mml:mn>7</mml:mn> </mml:msup> </mml:mrow> </mml:mfenced> </mml:math> .