On page 690 in Peskin & Schroeder's book, the Higgs mechanism is discussed in the context of an Abelian Gauge theory. After the SSB, among the many terms that appear, there is a mass term for the electromagnetic field $A_{\mu}$ and a term bilinear in the electromagnetic field and the Goldstone's pseudo-boson $\phi_2$, that is $\sqrt{2}e\phi_0A_{\mu}\partial^{\mu}\phi_2$, with $\phi_0$ the VEV, cf. eq. (20.7).
Then is discussed that, if I interpret this term as a vertex in perturbation theory, it contributes to restore the transversality of the electromagnetic field propagator. This shows the deep relation between the third physical degree of freedom, that $A_{\mu}$ acquires post the mechanism, and the DOF given by the pseudo-boson. Probably more evident in unitary gauge with only physical DOF.
My question is if in this section Peskin and Schroeder are even trying to give me a hint to see that even at quantum level gauge invariance is preserved.
What I mean is, at the classical level the gauge invariance is obviously still present, since I have just done a change of variables, expanding around one particular vacuum among the many degenerate ones. But at the quantum level, I should show that the Ward identities are still valid even after the SSB. And since the decoupling of the nonphysical degrees of freedom of $A_{\mu}$ is ensured by the presence of the Ward Identity in QED, it seems there is a relation here. Basically, I think, they are showing that the constraints of the Ward identity, on the Green's function of the electromagnetic field, are still present given its transversality even after the SSB. Giving a hint that may the Ward identity still holds.
Is this right? If it is (I am absolutely not sure it is), it would be nice if the answerer could expand in a more precise and rigorous way what I have tried to state here, probably in a rather confused way; and show how do we formally check that indeed gauge invariance is still present, even at the quantum level.
Note: in the paragraph before I meant "hint" because, the fact that I recover the transversality of the Green's function, I guess doesn't prove anything about the validity of the Ward identity, since the implication works in the other direction (Ward identity $\to$ decoupling of nonphysical DOF, and not necessarily vice-versa).
