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As part of the study of the two-body problem in Einstein's gravity, thefourth post-Newtonian order (4PN) of the two-body effective action is beingcomputed presently by both effective field theory (EFT) methods and others.Diagrams with 3 (or 4) classical loops appear to be a significant obstacle. Inthis paper we develop a method to compute such 3-loop diagrams and demonstrateit through a specific diagram. We reduce the classical diagrams throughshrinking the body worldlines to a form more familiar in Quantum Field Theory.A key ingredient in the evaluation is the Integration By Parts method forFeynman integrals.

Classical 3-loop 2-body diagrams