The action for a modified version of electrodynamics is given by

$$I=\int d^{4} x\left(-\frac{1}{4} F_{a b} F^{a b}-\frac{1}{2} m^{2} A_{a} A^{a}+\mu_{0} J^{a} A_{a}\right)$$

where $m$ is an arbitrary constant, $F_{a b}=\partial_{a} A_{b}-\partial_{b} A_{a}$ and $J^{a}$ is a conserved current.

(i) By varying $A_{a}$, derive the field equations analogous to Maxwell's equations by demanding that $\delta I=0$ for an arbitrary variation $\delta A_{a}$.

(ii) Show that $\partial_{a} A^{a}=0$.

(iii) Suppose that the current $J^{a}(x)$ is a function of position only. Show that

$$A^{a}(\mathbf{x})=\frac{\mu_{0}}{4 \pi} \int d^{3} x^{\prime} \frac{J^{a}\left(\mathbf{x}^{\prime}\right)}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|} e^{-m\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}$$