  By considering a spin-$\frac{1}{2}$ degenerate Fermi gases in a ring cavity
where strong interaction between atoms and light gives rise to a superradiance,
we find the cavity dissipation could cause a severe broadening in some special
cases, breaking down the quasi-particle picture which was constantly assumed in
mean field theory studies. This broadening happens when the band gap resonant
with polariton excitation energy. Interestingly enough, this broadening is
highly spin selective depending on how the fermions are filled and the spectrum
becomes asymmetric due to dissipation. Further, a non-monotonous dependence of
the maximal broadening of the spectrum against cavity decay rate $\kappa$ is
found and the largest broadening emerges at $\kappa$ comparable to recoil
energy.
