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Tail contributions to gravitational conservative dynamics

We compute tail contributions to the conservative dynamics of a generic self-gravitating system, for every multipole order, of either electric and magnetic parity. Such contributions arise when gravitational radiation is backscattered by the static curvature generated by the source itself and reabsorbed. Tail processes are relevant for the study of compact binary dynamics starting from the fourth post-Newtonian order. Within this context, we resolve a discrepancy recently arised at fifth post-Newtonian order between Effective Field Theory and self-force results.