  We investigate the scaling behavior of Nambu-Goldstone (NG) modes in the
ordered phase of the Vicsek model, introducing a phenomenological equation of
motion (EOM) incorporating a previously overlooked non-linear term. This term
arises from the interaction between velocity fields and density fluctuations,
leading to new scaling behaviors. We derive exact scaling exponents in two
dimensions, which reproduce the isotropic scaling behavior reported in a prior
numerical simulation.
