  We study homogeneous quenches in integrable quantum field theory where the
initial state contains zero-momentum particles. We demonstrate that the
two-particle pair amplitude necessarily has a singularity at the two-particle
threshold. Albeit the explicit discussion is carried out for special
(integrable) initial states, we argue that the singularity is inevitably
present and is a generic feature of homogeneous quenches involving the creation
of zero momentum particles. We also identify the singularity in quenches in the
Ising model across the quantum critical point, and compute it perturbatively in
phase quenches in the quantum sine-Gordon model which are potentially relevant
to experiments. We then construct the explicit time dependence of one-point
functions using a linked cluster expansion regulated by a finite volume
parameter. We find that the secular contribution normally linear in time is
modified by a $t\ln t$ term. We additionally encounter a novel type of secular
contribution which is shown to be related to parametric resonance. It is an
interesting open question to resum the new contributions and to establish their
consequences directly observable in experiments or numerical simulations.
