  We describe the formalism to compute gravitational-wave observables for
compact binaries via the effective field theory framework in combination with
modern tools from collider physics. We put particular emphasis on solving the
"multi-loop" integration problem via the methodology of differential equations
and expansion by regions. This allows us to "bootstrap" the two-body
relativistic dynamics in the Post-Minkowskian (PM) expansion from boundary data
evaluated in the near-static (soft) limit. We illustrate the procedure with the
derivation of the total spacetime impulse in the scattering of non-spinning
bodies to 4PM (three-loop) order, i.e. ${\cal O}(G^4)$, including conservative
and dissipative effects.
