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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

$$\psi'\rightarrow e^{\alpha}\psi.$$

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?

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Yes, OP is right: A global $U(1)$ gauge transformation in E&M does not change the gauge field $A_{\mu}$ itself: $\delta A_{\mu}=0$. Only the matter fields transform.

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The literal answer to your question is simply "Yes.", but it seems to me that you are confused about several foundational aspects of gauge theory, so let me comment on three topics you mention off-handedly:

  1. "Making the global symmetry local" is for some reason a popular pedogogy, but it doesn't actually make any physical sense as a motivation. See this answer of mine or this questions and its answers for better motivations for gauge theory.

    You seem to think that quantum gauge theories are somehow completely different from classical gauge theories because the "from global to local" motivation doesn't work in classical electrodynamics. This is not the case, a quantum gauge theory is the quantization of a classical gauge theory like any other quantum field theory is the quantization of its corresponding classical field theory.

  2. The transformation $$ A_\mu \mapsto A'_\mu = A_\mu + \partial_\mu\alpha$$ is a local transformation. The definition of a local transformation in this case is simply that it is a transformation in which the transformation parameter ($\alpha$) depends on spacetime.

    (The transformation is also "classical". In fact, this transformation only makes sense for the classical field, not for the quantum operator-valued field, since $\alpha$ is not a dynamical field of the classical theory and therefore quantization does not turn it into an operator-valued field.)

  3. The global transformations of a $\mathrm{U}(1)$ gauge theory are those where $\alpha$ does not depend on spacetime, and indeed those are the transformations under which the gauge field is invariant.

    It is crucial to note that it is the latter and not the former property that generalizes to the notion of "global" transformations in non-Abelian gauge theories. The gauge symmetry is, in a very real sense, "unphysical" in that it encodes a redundancy in our mathematical description, not a property of the physical system, see e.g. this question and its answers or this answer of mine. Its global part - in the meaning of the transformations which leave the gauge field invariant - is a true property of the system, not of the description, for instance, it is the global symmetry, not the gauge symmetry, that is broken in the Higgs mechanicism, see this excellent answer by Dominik Else.

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