  The coupling of an unparticle operator O_U to Standard Model particles opens
up the possibility of unparticle decays into standard model fields. We study
this issue by analyzing the pole structure (and spectral function) of the
unparticle propagator, corrected to account for one-loop polarization effects
from virtual SM particles. We find that the propagator of a scalar unparticle
(of scaling dimension 1 < d_U < 2) with a mass gap m_g develops an isolated
pole, m_p^2-i m_p Gamma_p, with m_p^2 < m_g^2 below the unparticle continuum
that extends above m_g (showing that the theory would be unstable without a
mass gap). If that pole lies below the threshold for decay into two standard
model particles the pole corresponds to a stable unparticle state (and its
width Gamma_p is zero). For m_p^2 above threshold the width is non zero and
related to the unparticle decay rate into Standard Model particles. This
picture is valid for any value of d_U in the considered range.
