Conservative binary dynamics from gravitational tail emission processes

We reanalyze the far zone contribution to the two-body conservative dynamics arising from interaction between radiative and longitudinal modes, the latter sourced by mass and angular momentum, which in the mass case is known as tail process. We verify the expected correspondence between two loop self-energy amplitudes and the gluing of two classical (one leading order, one at one loop) emission amplitudes, with focus on the Ward identities. As part of our analysis, we originally compute emission and self-energy processes with the longitudinal mode sourced by angular momentum for generic electric and magnetic multipoles and we highlight the role of the contribution from source interaction with two gravitational fields.