  In order to understand the confining decoupling solution of the Yang-Mills
theory in the Landau gauge, we consider the massive Yang-Mills model which is
defined by just adding a gluon mass term to the Yang-Mills theory with the
Lorentz-covariant gauge fixing term and the associated Faddeev-Popov ghost
term. First of all, we show that massive Yang-Mills model is obtained as a
gauge-fixed version of the gauge-invariantly extended theory which is
identified with the gauge-scalar model with a single fixed-modulus scalar field
in the fundamental representation of the gauge group. This equivalence is
obtained through the gauge-independent description of the Brout-Englert-Higgs
mechanism proposed recently by one of the authors. Then, we reconfirm that the
Euclidean gluon and ghost propagators 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. Moreover,
we demonstrate in a numerical way that the Schwinger function calculated from
the gluon propagator in the Euclidean region exhibits violation of the
reflection positivity at the physical point of the parameters. In addition, we
perform the analytic continuation of the gluon propagator from the Euclidean
region to the complex momentum plane towards the Minkowski region. We give an
analytical proof that the reflection positivity is violated for any choice of
the parameters in the massive Yang-Mills model, due to the existence of a pair
of complex conjugate poles and the negativity of the spectral function for the
gluon propagator to one-loop order. 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.
