= Gauge-scalar global symmetry of covariantly fixed QED
Adding a free real <scalar field> $\phi$ to covariantly gauge-fixed <QED> gives an infinitesimal global invariance $\delta A_\mu=\varepsilon\partial_\mu\phi$, $\delta\psi=-ie\varepsilon\phi\psi$, $\delta\phi=-\varepsilon(\partial\cdot A)/\xi$. The matter terms transform as an ordinary <gauge transformation>. The scalar kinetic variation combines with the <gauge fixing> variation into $\delta\mathcal L=-(\varepsilon/\xi)\partial_\mu[(\partial\cdot A)\partial^\mu\phi]$. Consequently the action is invariant under boundary conditions removing this total derivative, without using field equations.
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