  The Higgs boson in fermionic condensates with the BCS pairing interaction
describes the dynamics of the pairing amplitude. I show that the existence and
properties of this mode are sensitive to the energy dispersion of the
interaction. Specifically, when the pairing is suppressed at the Fermi level,
the Higgs mode may become unphysical (virtual) state or a resonance with finite
lifetime, depending on the details of interaction. Conversely, the Higgs mode
is discrete for the pairing interaction enhanced at the Fermi level. This work
illustrates conceptual difficulties associated with introducing collective
variables in the many-body pairing dynamics.
