  The mass-type quadrupole moment of inspiralling compact binaries (without
spins) is computed at the fourth post-Newtonian (4PN) approximation of general
relativity. The multipole moments are defined by matching between the field in
the exterior zone of the matter system and the PN field in the near zone,
following the multipolar-post-Minkowskian (MPM)-PN formalism. The matching
implies a specific regularization for handling infra-red (IR) divergences of
the multipole moments at infinity, based on the Hadamard finite part procedure.
On the other hand, the calculation entails ultra-violet (UV) divergences due to
the modelling of compact objects by delta-functions, that are treated with
dimensional regularization (DR). In future work we intend to systematically
study the IR divergences by means of dimensional regularization as well. Our
result constitutes an important step in the goal of obtaining the gravitational
wave templates of inspiralling compact binary systems with 4PN/4.5PN accuracy.
