# What forbids the existence of a $\lambda (A^\mu A_\mu)^2$ term in the Stueckelberg action?

In QFT, the Stueckelberg "trick" is often used to show how one can write a fully gauge invariant Lagrangian out of one that is not. For example, if we have

$\mathcal{L} = -\frac{1}{4}F^{\mu\nu}F_{\mu\nu} + m^2 A^\mu A_\mu$,

the gauge invariance becomes apparent when we rewrite the massive gauge boson in terms of a new vector field and a scalar field $\phi$: $A^\mu \rightarrow A^\mu + \frac{1}{m}\partial_\mu \phi$. Then, the Lagrangian is then invariant under $\delta \phi = -m \Lambda(x)$ and $\delta A_\mu = \partial_\mu \Lambda(x)$.

My question is this: typically, we do not ever see terms present in the above Lagrangian like $\lambda (A^\mu A_\mu)^2$. Moreover, it seems like we can always continue to add terms like $(A^\mu A_\mu)^4/m^2$, which clearly seems like a problem. If we regard the Stueckelberg theory as one in which the Higgs has been integrated out and we are only left with the massless Goldstone bosons $\phi$, terms like $\lambda (A^\mu A_\mu)^2$ should become very relevant at high energies by dimensional analysis. I would love some clarification on why they are never present in the Lagrangian.

The reason is that the Stueckelberg-Lagrangian is written down for a massive photon, a vector boson of the $U(1)$ gauge group. As photons are not self-interacting, interaction terms (i.e. $A^3$ or higher order) are not present.