  In this talk we consider the mass-deformed Yang-Mills theory in the covariant
gauge which is obtained by just adding a gluon mass term to the Yang-Mills
theory with the covariant gauge fixing term and the associated Faddeev-Popov
ghost term. First, we reconfirm that the decoupling solution in the Landau
gauge Yang-Mills theory is well reproduced from the mass-deformed Yang-Mills
theory by taking into account loop corrections. Second, we show that the
mass-deformed Yang-Mills theory 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 as a
consequence of the gauge-independent Brout-Englert-Higgs mechanism proposed
recently by one of the authors. Third, we show that the reflection positivity
is violated for any value of the parameters in the mass-deformed Yang-Mills
theory to one-loop quantum corrections. Finally, we discuss the implications
for the existence of positivity violation/restoration crossover in light of the
Fradkin-Shenker continuity between Confinement-like and Higgs-like regions in a
single confinement phase in the gauge-scalar model on the lattice.
