Modified St$\ddot u$ckelberg Formalism: Free Massive Abelian 2-Form Theory in 4D

We demonstrate that the celebrated St$\ddot u$ckelberg formalism gets modified in the case of a massive four (3+1)-dimensional (4D) Abelian 2-form theory due to the presence of a self-duality discrete symmetry in the theory. The latter symmetry entails upon the modified 4D massive Abelian 2-form gauge theory to become a massive model of Hodge theory within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism where there is existence of a set of (anti-)co-BRST transformations corresponding to the usual nilpotent (anti-)BRST transformations. The latter exist in any arbitrary dimension of spacetime for the usual St$\ddot u$ckelberg-modified massive Abelian 2-form gauge theory. The modification in the St$\ddot u$ckelberg technique is backed by the precise mathematical arguments from the differential geometry where the exterior derivative and Hodge duality operator play the decisive roles. The modified version of the St$\ddot u$ckelberg technique remains invariant under the discrete duality transformations which also establish a precise and deep connection between the off-shell nilpotent (anti-)BRST and (anti-)co-BRST transformations. We have clarified a simple trick to get rid of the higher derivative terms in the appropriate Lagrangian densities so that our 4D theory can become consistent and renormalizable.