Derivative self-interactions for a massive vector field
  In this work we revisit the construction of theories for a massive vector
field with derivative self-interactions such that only the 3 desired
polarizations corresponding to a Proca field propagate. We start from the
decoupling limit by constructing healthy interactions containing second
derivatives of the Stueckelberg field with itself and also with the transverse
modes. The resulting interactions can then be straightforwardly generalized
beyond the decoupling limit. We then proceed to a systematic construction of
the interactions by using the Levi-Civita tensors. Both approaches lead to a
finite family of allowed derivative self-interactions for the Proca field. This
construction allows us to show that some higher order terms recently introduced
as new interactions trivialize in 4 dimensions by virtue of the Cayley-Hamilton
theorem. Moreover, we discuss how the resulting derivative interactions can be
written in a compact determinantal form, which can also be regarded as a
generalization of the Born-Infeld lagrangian for electromagnetism. Finally, we
generalize our results for a curved background and give the necessary
non-minimal couplings guaranteeing that no additional polarizations propagate
even in the presence of gravity.
