  Abstract We investigate the spectral shift in collective forward scattering
for a cold dense atomic cloud. The shift, sometimes called collective Lamb
shift, results from resonant dipole-dipole interaction mediated by real and
virtual photon exchange, forming many-body states displaying various super- and
subradiant spectral behavior. The scattering spectrum reflects the overall
radiative behavior from these states. However, it also averages out the
radiative details associated with a single collective state, causing ambiguity
in explaining the origin of the spectral shift and raising controversy on its
scaling property. We employ a Monte-Carlo simulation to study how the
collective states are occupied and contribute to emission. We thus distinguish
two kinds of collective shift that follow different scaling laws. One results
from dominant occupation of the near-resonant collective states. This shift is
usually small and insensitive to the density or the number of participating
atoms. The other comes from large spatial correlation of dipoles, associated
with the states of higher degree of emission. This corresponds to larger
collective shift that is approximately linearly dependent on the optical depth.
Our analysis provides not only a novel perspective for the spectral features in
collective scattering, but also a possible resolution to the controversy on the
scaling property that has been reported elsewhere because of different origins.
