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