  We derive general relationships between the number of complex poles of a
propagator and the sign of the spectral function originating from the branch
cut in the Minkowski region under some assumptions on the asymptotic behaviors
of the propagator. We apply this relation to the mass-deformed Yang-Mills model
with one-loop quantum corrections, which is identified with a low-energy
effective theory of the Yang-Mills theory, to show that the gluon propagator in
this model has a pair of complex conjugate poles or "tachyonic" poles of
multiplicity two, in accordance with the fact that the gluon field has a
negative spectral function, while the ghost propagator has at most one
"unphysical" pole. Finally, we discuss implications of these results for gluon
confinement and other non-perturbative aspects of the Yang-Mills theory.
