  We study Cooper pairing in the Dirac composite fermion (CF) system. The
presence of the mass term in the Dirac CF description (which may simulate
Landau level mixing) i.e. breaking of particle-hole (PH) symmetry in this
system, is a necessary condition for the existence of a PH Pfaffian-like
topological state. In the scope of RPA approximation and hydrodynamic approach,
we find some signatures of pairing at finite frequencies. Motivated by this
insight, we extend our analysis to the case of a different but still Dirac
quasi-particle (CF) representation, appropriate in the presence of a mass term,
and discuss the likelihood of PH Pfaffian pairing and Pfaffian pairing in
general. On the basis of gauge field effects, we find for small Dirac mass
anti-Pfaffian or Pfaffian instability depending on the sign of mass, while for
large mass (Landau level mixing), irrespective of its sign, PH Pfaffian-like
instability.
