How to determine local gauge invariance of action - massive complex field?

Question

attempted some preliminary computations, but was unable to determine the answer to this query. I am inquiring into the properties of an action of the form: $$ \int dx^4 \: \left(D_\mu \psi (D^\mu\psi)^* - m^2 \psi \psi^* + A^\mu A_\mu \right)$$ Under the Gauge transformation $\psi \to \psi e^{i \Lambda(x^\mu)}$, where $D_\mu = \partial_\mu - ie A_\mu$ and $A_\mu$ is the electromagnetic 4-potential and $\psi$ is complex scalar field. While it is well know that the first two terms are gauge invariant. I am not sure how to approach the third term. I figured I could gain some insight by considering the transformation $A_\mu \to A_\mu + \partial_\mu \phi$, this did not really provide me with any insight as I am not sure how to relate this to the original transformation, nor do I think that this is appropriate to prove/disprove the gauge invariance of the action.

Answer 1

Your Lagrangian does not look gauge invariant. You should check the source where you got it from. There was probably an error there. Instead of $A^\mu A_\mu$ there should be $F^{\mu\nu} F_{\mu \nu}$ there, maybe with a constant factor.

EDIT (11/2/2017): According to the OP's comment, it looks like there is no reference or rationale for the Lagrangian. It is obvious that the Lagrangian is not gauge invariant, as all its terms but $A^\mu A_\mu$ are gauge invariant and the term $A^\mu A_\mu$ is not: for example, if $A^\mu=0$, then $A^\mu A_\mu=0$ and generally $(A^\mu+\partial^\mu \phi) (A_\mu+\partial_\mu \phi)=\partial^\mu \phi\partial_\mu \phi\neq 0$.