  We present a systematic derivation of the effective action for interacting
vortices in a non-relativistic two-dimensional superfluid described by the
Gross-Pitaevskii equation by integrating out longitudinal fluctuations of the
order parameter. There are no logarithmically divergent coefficients in the
equations of motion. Our analysis is valid in a dilute limit of vortices where
the intervortex spacing is large compared to the core size, and where number
fluctuations of atoms in vortex cores are suppressed. We analyze sound-induced
corrections to the dynamics of a vortex-antivortex pair and show that there is
no instability to annihilation, suggesting that sound-mediated interactions are
not strong enough to ruin an inverse energy cascade in two-dimensional
zero-temperature superfluid turbulence.
