Let's have massive spinor QED:

$$

\tag 1 L = -\frac{1}{4}F_{\mu \nu}F^{\mu \nu} + \mu^{2}A^{2} + \bar{\Psi}(i\gamma^{\mu}D_{\mu} - m)\Psi , \quad D_{\mu} \equiv \partial_{\mu} - ieA_{\mu}

$$

This theory doesn't have local gauge symmetry because of existence of mass term for $A$ field, but has global invariance under transformations $\Psi \to e^{i \alpha}\Psi$, thus in two cases (massive and massless QED) current $J_{\mu} = \bar{\Psi}\gamma_{\mu}\Psi$ is conserved. I want to establish the differences between massive and massless spinor QED.

1. It seems to me that the derivation generalized Ward identity,

$$

(l - k)_{\mu}\Gamma^{\mu}(l, k) = iS^{-1}(k) - iS^{-1}(l),

$$

isn't depend on the value of photon mass, because the derivation is started from calculation of divergence of correlator $\partial^{\mu}_{x}\langle \hat{T} \left( J_{\mu}(x)\Psi (y)\bar{\Psi}(z)\right)$, and $J_{\mu}$ is conserved and has equal commutation relations $[J_{\mu}(\mathbf x), \Psi (\mathbf y)]$ in both cases. Thus all charges in both theories are renormalized equivalently, $e \to \sqrt{Z_{3}}^{-1}e$ ($Z_{3}$ denotes $A$ field renormalization constant).

2. Identity $k^{i}_{\mu}M^{\mu}(p_{1},...,p_{n}, k_{1},..., k_{i},...,k_{m}) = 0$, where $k^{i}$ corresponds to external photon momentum and $M_{\mu}$ corresponds to the amplitude with $m$ external photonic lines without polarization vector $\epsilon^{\mu}(k_{i})$, $M =\epsilon^{\mu}(k_{i})M_{\mu} $, is also hold for both cases. Thus longitudinal polarizations don't take role as in- and out-states.

Thus, if I haven't missed something, the main difference between massive and massless QED is in the pole structure of propagators:

$$

D_{\mu \nu}^{\text{massive}}(p) \sim \frac{1}{p^{2} - \mu^{2}}, \quad D_{\mu \nu}^{\text{massless}}(p) \sim \frac{1}{p^{2}},

$$

and in corresponding nonrelativistic limits (Coulomb law and Yukawa law correspondingly).

Questions.

1. Does massive QED have problem with renormalizability due to the mass term? It seems to me that $(2)$ makes it renormalizable, but I'm not sure.

2. If the first question has negative answer, than suppose following idea: mass term of gauge field $A$ in $(1)$ could be obtained from theory with scalar field through Stueckelberg mechanism. This theory is renormalizable, but is completely equivalent to $(1)$ when we neglect the scalar field part. Thus the situation may be following: the theory $(1)$ is explicitly renormalizable if we able to use $R_{\epsilon}$ gauge.

3. Finally, if the first question has positive answer, are two theories completely equivalent up to the pole structure of the propagator.