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

The nonlocal gauge invariant mass operator $\mathrm{Tr} \int d^{4}x F_{\mu\nu} (D^2)^{-1} F_{\mu\nu}$ 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.