  Massive Nambu-Goldstone (mNG) bosons are quasiparticles whose gap is
determined exactly by symmetry. They appear whenever a symmetry is broken
spontaneously in the ground state of a quantum many-body system, and at the
same time explicitly by the system's chemical potential. In this paper, we
revisit mNG bosons and show that apart from their gap, symmetry also protects
their scattering amplitudes. Just like for ordinary gapless NG bosons, the
scattering amplitudes of mNG bosons vanish in the long-wavelength limit. Unlike
for gapless NG bosons, this statement holds for any scattering process
involving one or more external mNG states; there are no kinematic singularities
associated with the radiation of a soft mNG boson from an on-shell initial or
final state.
