 Solution

Let $ϵ$ here be a constant infinitesimal transformation parameter, unrelated to the dimensional regularization parameter in question 1. The matter and Maxwell terms undergo an ordinary infinitesimal gauge transformation with $\alpha =ϵ\varphi$, and their variations cancel as before. The gauge fixing contribution isThe extra real scalar field kinetic term varies byAdding the two gives a total derivative:The last equality uses vanishing boundary contributions, for example sufficiently fast decay of the fields. No field equations were used. This gauge-scalar global symmetry of covariantly fixed QED leaves the action invariant even though the density need not be invariant pointwise; the scalar's variation supplies exactly the missing boundary-term cancellation.