Study of the nonlocal gauge invariant mass operator $\mathrm{T}\mathrm{r}\int d^{4}x{F}_{\mu \nu }(D^2{)}^{-1}{F}_{\mu \nu }$ in the maximal Abelian gauge

The nonlocal gauge invariant mass operator $$\mathrm{T}\mathrm{r}$ \int d^{4}x F_{\mu\nu} (D^2)^{-1} F_{\mu\nu}$ is investigated in Yang-Mills theories in the maximal Abelian gauge. By means of the introduction of auxiliary fields a local action is achieved, enabling us to use the algebraic renormalization in order to prove the renormalizability of the resulting local model to all orders of perturbation theory.