Angular momentum tail contributions to compact binary dynamics

We derive the effective action governing the dynamics of a compact binary system when gravitational radiation is emitted by any mass or current multipole, scattered by the quasi-static field associated with the binary's angular momentum, and then reabsorbed. Among such angular momentum failed-tail processes, the ones involving multipole moments up to mass and current octupoles, which mix also with quadrupoles of opposite parity, contribute to the system dynamics at sixth post-Newtonian order; we display these terms explicitly as a particular case of our general derivation. Additionally, we derive the radiative multipole moments associated to arbitrary angular momentum failed-tails in emission processes.