Study of the gauge invariant, nonlocal mass operator $\mathrm{Tr}\int d^{4}x{F}_{\mu \nu }(D^2{)}^{-1}{F}_{\mu \nu }$ in Yang-Mills theories

The nonlocal mass operator $\mathrm{Tr}\int d^{4}x{F}_{\mu \nu }(D^2{)}^{-1}{F}_{\mu \nu }$ is considered in Yang-Mills theories in Euclidean space-time. It is shown that the operator $\mathrm{Tr}\int d^{4}x{F}_{\mu \nu }(D^2{)}^{-1}{F}_{\mu \nu }$ can be cast in local form through the introduction of a set of additional fields. A local and polynomial action is thus identified. Its multiplicative renormalizability is proven by means of the algebraic renormalization in the class of linear covariant gauges. The anomalous dimensions of the fields and of the mass operator are computed at one-loop order. A few remarks on the possible role of this operator for the issue of the gauge invariance of the dimension two condensates are outlined.