  In a recent proposal of the half-filled Landau level, the composite fermions
are taken to be Dirac particles and particle-hole symmetric. Cooper pairing of
these composite fermions in different angular momentum channels, $\ell$, can
give rise to different kinds of Pfaffian states. In addition to the well known
Moore-Read Pfaffian and anti-Pfaffian states, a new putative particle-hole
symmetric Pfaffian state, corresponding to the $s-$wave pairing channel, was
also proposed. However, the possible underlying pairing mechanism is not clear
at all. In this work we provide a specific pairing mechanism for realizing some
of these Pfaffian states. We show that there can be nonzero pairing in angular
momentum channels $|\ell|\ge 1$ depending on the magnitude of a coupling
constant. There is a quantum phase transition from the Dirac composite fermi
liquid state to Cooper pairing states in angular momentum channels $|\ell|\ge
1$ as the coupling constant is tuned across its critical point value.
Surprisingly the particle-hole symmetric $\ell=0$ channel pairing turns out to
be impossible irrespective of the size of the coupling constant.
