  An effective field theory of composite Dirac fermions was proposed by Son
[Phys. Rev. X 5, 031027 (2015)] as a theory of the half-filled Landau level
with explicit particle-hole symmetry. We compute the electromagnetic response
of this Son-Dirac theory on the level of the random phase approximation (RPA),
where we pay particular attention to the effect of an additional
composite-fermion dipole term that is needed to restore Galilean invariance. We
find that once this dipole correction is taken into account, spurious interband
transitions and collective modes that are present in the response of the
unmodified theory either cancel or are strongly suppressed. We demonstrate that
this gives rise to a consistent theory of the half-filled Landau level valid at
all frequencies, at least to leading order in the momentum. In addition, the
dipole contribution modifies the Fermi-liquid response at small frequency and
momentum, which is a prediction of the Son-Dirac theory within the RPA that
distinguishes it from a separate description of the half-filled Landau level by
Halperin, Lee, and Read within the RPA.
