A no-go theorem for generalized vector Galileons on flat spacetime

We explore the possibility for generalized electromagnetism on flat spacetime. For a single copy of $U(1)$ gauge theory, we show that the Galileon-type generalization of electromagnetism is forbidden. Given that the equations of motion for the vector field are gauge invariant and Lorentz invariant, follow from an action and contain no more than second derivative on $A_\mu$, the equations of motion are at most linear with respect to second derivative of $A_\mu$.