  We compute the four-momentum radiated during the scattering of two spinless
bodies, at leading order in the Newton's contant $G$ and at all orders in the
velocities, using the Effective Field Theory worldline approach. Following
\cite{Mougiakakos:2021ckm}, we derive the conserved stress-energy tensor
linearly coupled to gravity generated by localized sources, at leading and
next-to-leading order in $G$, and from that the classical probability amplitude
of graviton emission. The total emitted momentum is obtained by phase-space
integration of the graviton momentum weighted by the modulo squared of the
radiation amplitude. We recast this as a two-loop integral that we solve using
techniques borrowed from particle physics, such as reverse unitarity, reduction
to master integrals by integration-by-parts identities and canonical
differential equations. The emitted momentum agrees with recent results
obtained by other methods. Our approach provides an alternative way of directly
computing radiated observables in the post-Minkowskian expansion without going
through the classical limit of scattering amplitudes.
