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In this talk we consider the mass-deformed Yang-Mills theory in the covariantgauge which is obtained by just adding a gluon mass term to the Yang-Millstheory with the covariant gauge fixing term and the associated Faddeev-Popovghost term. First, we reconfirm that the decoupling solution in the Landaugauge Yang-Mills theory is well reproduced from the mass-deformed Yang-Millstheory by taking into account loop corrections. Second, we show that themass-deformed Yang-Mills theory is obtained as a gauge-fixed version of thegauge-invariantly extended theory which is identified with the gauge-scalarmodel with a single fixed-modulus scalar field in the fundamentalrepresentation of the gauge group. This equivalence is obtained as aconsequence of the gauge-independent Brout-Englert-Higgs mechanism proposedrecently by one of the authors. Third, we show that the reflection positivityis violated for any value of the parameters in the mass-deformed Yang-Millstheory to one-loop quantum corrections. Finally, we discuss the implicationsfor the existence of positivity violation/restoration crossover in light of theFradkin-Shenker continuity between Confinement-like and Higgs-like regions in asingle confinement phase in the gauge-scalar model on the lattice.

Mass-deformed Yang-Mills theory in the covariant gauge and its gauge-invariant extension through the gauge-independent BEH mechanism