  In a uniform ring-shaped one-dimensional superfluid, quantum fluctuations
that unwind the order parameter need to transfer momentum to quasiparticles
(phonons). We present a detailed calculation of the leading exponential factor
governing the rate of such phonon-assisted tunneling in a weakly-coupled Bose
gas at a low temperature $T$. We also estimate the preexponent. We find that
for small superfluid velocities the $T$-dependence of the rate is given mainly
by $\exp(-c_s P/ 2T)$, where $P$ is the momentum transfer, and $c_s$ is the
phonon speed. At low $T$, this represents a strong suppression of the rate,
compared to the non-uniform case. As a part of our calculation, we identify a
complex instanton, whose analytical continuation to suitable real-time segments
is real and describes formation and decay of coherent quasiparticle states with
nonzero total momenta.
