Antisymmetric tensor propagator with spontaneous Lorentz violation

In this work, we study the spontaneous Lorentz symmetry breaking due to an antisymmetric 2-tensor field in Minkowski spacetime. For a smooth quadratic potential, the spectrum of the theory exhibits massless and massive excitations. We show that the equations of motion for the free field obey some constraints which lead to the massive mode be non-propagating at leading order. Besides, there exists a massless mode in the theory which can be identified with the usual Kalb-Ramond field, carrying only one on-shell degree of freedom. The same conclusion holds when one analyses the pole structure of its Feynman propagator. A new complete set of spin-type operators is found, which was the requirement to evaluate the propagator of the Kalb-Ramond field modified by the presence of a nonzero vacuum expectation value responsible for the Lorentz violation.