  We apply the classical double copy to the calculation of self-energy of
composite systems with multipolar coupling to gravitational field, obtaining
next-to-leading order results in the gravitational coupling $G_N$ by
generalizing color to kinematics replacement rules known in literature. When
applied to the multipolar description of the two-body system, the self-energy
diagrams studied in this work correspond to tail processes, whose physical
interpretation is of radiation being emitted by the non-relativistic source,
scattered by the curvature generated by the binary system and then re-absorbed
by the same source. These processes contribute to the conservative two-body
dynamics and the present work represents a decisive step towards the systematic
use of double copy within the multipolar post-Minkowskian expansion.
