What are the Global gauge transformations of gauge bosons in Standard Model?
To elaborate: Initially, we consider the global $U(1)$ transformations of scalars ($\phi$) and fermions ($\psi$) as
And when the phase $\alpha$ depends on spacetime, $x$, then the transformation becomes local and these Dirac Lagrangian is not invariant under these gauge transformations as we end up with $\partial_\mu\alpha(x)$. At this point, we introduce a gauge field, $A_\mu(x)$, so as to make the complete Lagrangian invariant under these local gauge transformations, which we call QED or scalar QED after adding the gauge field's dynamic terms.
In Classical Field Theory such as Classical Electrodynamics, we simply have,
$$A_\mu' \rightarrow A_\mu - \partial_\mu\alpha,$$ in four vector form. Certainly, I suppose, we can't call this transformation local as it is classical(?) and $\alpha$ here is a subsidiary function which just depends on spacetime.
Now, in Quantum Field theory, as I described in the beginning, what are the global U(1) transformations of the gauge field $A_\mu(x)$ once we close the deal as QED or scalar QED? Is it just, $$A_\mu'\rightarrow A_\mu$$ as $\alpha$ doesn't depend on spacetime?