  Previously, we showed that, owing to effects arising from quantum
electrodynamics (QED), magnetohydrodynamic fast modes of sufficient strength
will break down to form electron-positron pairs while traversing the
magnetospheres of strongly magnetised neutron stars. The bulk of the energy of
the fast mode fuels the development of an electron-positron fireball. However,
a small, but potentially observable, fraction of the energy ($\sim 10^{33}$
ergs) can generate a non-thermal distribution of electrons and positrons far
from the star. In this paper, we examine the cooling and radiative output of
these particles. We also investigate the properties of non-thermal emission in
the absence of a fireball to understand the breakdown of fast modes that do not
yield an optically thick pair plasma. This quiescent, non-thermal radiation
associated with fast mode breakdown may account for the recently observed
non-thermal emission from several anomalous X-ray pulsars and soft-gamma
repeaters.
