Pairing of particle-hole symmetric composite fermions in half-filled Landau level

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, l, 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 vertical bar l vertical bar >= 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 vertical bar l vertical bar >= 1 as the coupling constant is tuned across its critical point value. Surprisingly the particle-hole symmetric l = 0 channel pairing turns out to be impossible irrespective of the size of the coupling constant.