  The nature of quasiparticles in 2D quantum antiferromagnets at finite
temperature remains an open question despite decades of theoretical work. In
particular, it is not fully understood how long wavelength excitations
contribute to significant broadening of the experimentally observable spectrum.
Motivated by this problem, we consider the $XY$ model of easy-plane
antiferromagnets, and compute the dynamic structure factor by direct summation
of diagrams. In doing so, we find that non-interacting quasiparticles with
infinite lifetimes can still lead to a broad response. This forms the basis for
a new paradigm describing the interaction of experimental probes with a
physical system, where broadening is due neither to the lifetime, nor to the
emergence of fractional quasiparticles. Instead, strong fluctuations drive the
probe to absorb and radiate an infinite number of arbitrarily low energy
quasiparticles, leading us to draw parallels with the infrared catastrophe in
quantum electrodynamics.
