Derivative self-interactions for a massive vector field

TL;DR

The paper develops a comprehensive construction of derivative self-interactions for a massive vector field (generalized Proca theories) that preserve exactly $3$ propagating polarizations. It employs a decoupling-limit approach for the Stueckelberg field and a Levi-Civita tensor–based systematic method to identify a finite interaction family, showing higher-order terms trivialize in $D=4$ via Cayley–Hamilton and expressing the result in a compact determinantal (Born–Infeld–like) form. The framework naturally extends to curved spacetimes with non-minimal couplings that maintain the correct dof, linking to Horndeski-like structures in the Stueckelberg limit. The work also clarifies dimensional dependence, noting that certain seventh-order terms appear only in dimensions $>4$, and it raises questions about the technical naturalness and quantum stability of the full vector theory.

Abstract

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.