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.
Highlights
The discovery of the cosmic acceleration of the universe triggered a plethora of attempts to unveil the physical mechanism behind it
We discuss how the resulting derivative interactions can be written in a compact determinantal form, which can be regarded as a generalization of the Born-Infeld lagrangian for electromagnetism
Since a gravitational theory based on a massless spin 2 particle needs to coincide with General Relativity (GR) at low energies, modifications of gravity on large distances inevitably leads to the introduction of additional degrees of freedom
Summary
Where we have used matrix notation and 1⁄2 stands for the identity matrix. We can recognize that M vanishes in 4 dimensions by virtue of the Cayley-Hamilton theorem applied on F μν and, the interaction is trivial in 4 dimensions, but it can be present in higher dimensions. Since we need the Levi-Civita symbol with 5 indices, this term identically vanishes in 4 dimensions This is nothing but an alternative way of writing the Cayley-Hamilton theorem. We shall end this Section by noticing that the structure of the interactions based on the antisymmetry of the Levi-Civita tensor allows a nice determinantal formulation of the generalized Proca interactions The existence of such a formulation is not surprising and it is in the same spirit as the interactions in massive gravity and scalar Galileon interactions, which can be compactly written in terms of a determinantal interaction. The first interesting vector Galileon interaction is encoded in the second order symmetric elementary polynomial e2.
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