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 St\"uckelberg 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
St\"uckelberg field describing the longitudinal mode, which is in agreement
with the results of JCAP 1405, 015 (2014) and Phys. Lett. B 757, 405 (2016) and
complements those of JCAP 1602, 004 (2016). We also correct and complete the
parity violating sector, obtaining an extra term on top of the arbitrary
function of the field $A_\mu$, the Faraday tensor $F_{\mu \nu}$ and its Hodge
dual $\tilde{F}_{\mu \nu}$.
