Gauge-scalar global symmetry of covariantly fixed QED

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