I've found the following Lagrangian $$\mathcal{L}=i\bar{\psi}\gamma^\mu\left(\partial_\mu-ieA_\mu -ieA'_\mu\right)\psi-m\bar\psi\psi -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{4}F'_{\mu\nu}F'^{\mu\nu}.$$ Could I think of it as concealing the "standard" QED Lagrangian, i.e. $$\mathcal{L}=i\bar{\psi}\gamma^\mu\left(\partial_\mu-ieA_\mu\right)\psi-m\bar\psi\psi -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}?$$
I tried to make a trivial field redefinition, like $$A_\mu(x)\rightarrow A_\mu(x)+A'_\mu(x),$$ but I've found that the $F_{\mu\nu}F^{\mu\nu}\equiv F^2$ term transforms as $$F^2\rightarrow F^2+F'^2+F_{\mu\nu}F'^{\mu\nu}+F'_{\mu\nu}F^{\mu\nu}$$ under this field redefinition. Am I wrong or is it possible to trace this theory back to QED, at least for the $\psi$ field?