I am trying to understand, from the QFT perspective, why photons are massless. Let's consider the following Lagrangian for a massive U(1) gauge boson
$\mathcal{L}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+m^2A^{\mu}A_{\mu}$,
which is the Proca Lagrangian. Usually, people say that this theory is not acceptable because the mass term breaks gauge invariance. I don't agree with this statement, since we could perform the Stueckelberg procedure and introduce a new scalar field via
$A_{\mu}\rightarrow A_{\mu}+\frac{1}{m}\partial_{\mu}\phi$
With this parametrisation, the Lagrangian is invariant under
$A_{\mu}\rightarrow A_{\mu}+\partial_{\mu}\Lambda\\ \phi\rightarrow \phi-m\Lambda$
If I am not mistaken, one could now choose the unitary gauge and set $\phi=0$, where the longitudinal dof decouples, and we recover the Proca Lagrangian. As far as I understand, the Proca Lagrangian is perfectly fine and although hidden, it preserves gauge invariance. Therefore, saying that the mass term breaks gauge invariance and that is why photons are massless is not convicing for me. So, why are photons massless?
