On the 4D generalized Proca action for an Abelian vector field

We summarize previous results on the most general Proca theory in 4 dimensions containing only first-order derivatives in the vector field (second-order at most in the associated Stückelberg scalar) and having only three propagating degrees of freedom with dynamics controlled by second-order equations of motion. Discussing the Hessian condition used in previous works, we conjecture that, as in the scalar galileon case, the most complete action contains only a finite number of terms with second-order derivatives of the Stückelberg field describing the longitudinal mode, which is in agreement with the results of http://dx.doi.org/10.1088/1475-7516/2014/05/015 and http://dx.doi.org/10.1016/j.physletb.2016.04.017 and complements those of http://dx.doi.org/10.1088/1475-7516/2016/02/004. We also correct and complete the parity violating sector, obtaining an extra term on top of the arbitrary function of the field A_μ, the Faraday tensor F_μν and its Hodge dual F-tilde_μν.