  The nu=1/2+1/2 quantum Hall bilayer has been previsously modeled using
Chern-Simons-RPA-Eliashberg (CSRPAE) theory to describe pairing between the two
layers. However, these approaches are troubled by a number of divergences and
ambiguities. By using a "modified" RPA approximation to account for mass
renormalization, we can work in a limit where the cyclotron frequency is taken
to infinity, effectively projecting to a single Landau level. This,
surprisingly, controls the important divergences and removes ambiguities found
in prior attempts at CSRPAE. Examining BCS pairing of composite fermions we
find that the angular momentum channel l=+1 dominates for all distances d
between layers and at all frequency scales. Examining BCS pairing of composite
fermion electrons in one layer with composite fermion holes in the opposite
layer, we find the l=0 pairing channel dominates for all d and all frequencies.
The strength of the pairing in these two different descriptions of the same
phase of matter is found to be almost identical. This agrees well with our
understanding that these are two different but dual descriptions of the same
phase of matter.
