  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.
