  A recent proposal argues that an alternate description of the half-filled
Landau level is a theory of massless Dirac fermions. We examine the possibility
of pairing of these Dirac fermions by numerically solving the {\em coupled}
Eliashberg equations unlike our previous calculation [Wang and Chakravarty,
Phys. Rev. B {\bf 94}, 165138 (2016)]. In addition, vertex corrections are
calculated to be zero from the Ward identity. We find that pairing is possible
in non-zero angular momentum channels; the only differences are minor numerical
shifts. As before, the pairing leads to the gapped Pfaffian and anti-Pfaffian
states. However, in our approximation scheme, pairing is not possible in the
putative particle-hole symmetric state for $\ell=0$ angular momentum. The
specific heat at low temperatures of a system of massless Dirac fermions
interacting with a transverse gauge field, expected to be relevant for the
half-filled Landau level, is calculated. Using the Luttinger formula, it is
found be $\propto T\ln T$ in the leading low temperature limit, due to the
exchange of transverse gauge bosons. The result agrees with the corresponding
one in the nonrelativistic composite fermion theory of Halperin, Lee and Read
of the half-filled Landau level.
