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Working within the post-Newtonian (PN) approximation to General Relativity,we use the effective field theory (EFT) framework to study the conservativedynamics of the two-body motion at fourth PN order, at fifth order in theNewton constant. This is one of the missing pieces preventing the computationof the full Lagrangian at fourth PN order using EFT methods. We exploit theanalogy between diagrams in the EFT gravitational theory and 2-point functionsin massless gauge theory, to address the calculation of 4-loop amplitudes bymeans of standard multi-loop diagrammatic techniques. For those terms which canbe directly compared, our result confirms the findings of previous studies,performed using different methods.

Effective field theory approach to the gravitational two-body dynamics, at fourth post-Newtonian order and quintic in the Newton constant