  We address the many-atom emission of a dilute cloud of two-level atoms
through a renormalized perturbation theory. An analytical solution for the
truncated coupled-dipole equations is derived, which contains an effective
spectrum associated to the initial conditions. Our solution is able to
distinguish precisely classical from semi-classical predictions for large
atomic ensembles. This manifests as a reduction of the cooperativity in the
radiated power for higher atomic excitation, in disagreement with the fully
classical prediction from linear optics. Moreover, the second-order cooperative
emission appears accurate over several single atom lifetimes and for
interacting regimes stronger than those permitted in conventional perturbation
theory. We can compute the semiclassical dynamics of hundred thousand of
interacting atoms with ordinary computational resources, which makes our
formalism particularly promising to probe the nonlinear dynamics of quantum
many-body systems that emerge from cumulant expansions.
