  We discuss the analytic continuation of the gluon propagator from the
Euclidean region to the complex squared-momentum plane towards the Minkowski
region from a viewpoint of gluon confinement. For this purpose, we investigate
the massive Yang-Mills model with one-loop quantum corrections, which is to be
identified with a low-energy effective theory of the Yang-Mills theory in the
sense that the confining decoupling solution for the Euclidean gluon and ghost
propagators of the Yang-Mills theory in the Landau gauge obtained by numerical
simulations on the lattice are reproduced with good accuracy from the massive
Yang-Mills model by taking into account one-loop quantum corrections. We show
that the gluon propagator in the massive Yang-Mills 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. These results are consistent with
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. Consequently, we give an analytical proof for violation of the
reflection positivity as a necessary condition for gluon confinement for any
choice of the parameters in the massive Yang-Mills model, including the
physical point. Moreover, the complex structure of the propagator enables us to
explain why the gluon propagator in the Euclidean region is well described by
the Gribov-Stingl form.
