(i) Consider the action

$$S=-\frac{1}{4 \mu_{0} c} \int\left(F_{\mu \nu} F^{\mu \nu}+2 \lambda^{2} A_{\mu} A^{\mu}\right) d^{4} x+\frac{1}{c} \int A_{\mu} J^{\mu} d^{4} x$$

where $A_{\mu}(x)$ is a 4-vector potential, $F_{\mu \nu}=\partial_{\mu} A_{\nu}-\partial_{\nu} A_{\mu}$ is the field strength tensor, $J^{\mu}(x)$ is a conserved current, and $\lambda \geqslant 0$ is a constant. Derive the field equation

$$\partial_{\mu} F^{\mu \nu}-\lambda^{2} A^{\nu}=-\mu_{0} J^{\nu} .$$

For $\lambda=0$ the action $S$ describes standard electromagnetism. Show that in this case the theory is invariant under gauge transformations of the field $A_{\mu}(x)$, which you should define. Is the theory invariant under these same gauge transformations when $\lambda>0$ ?

Show that when $\lambda>0$ the field equation above implies

$$\partial_{\mu} \partial^{\mu} A^{\nu}-\lambda^{2} A^{\nu}=-\mu_{0} J^{\nu}$$

Under what circumstances does $(*)$ hold in the case $\lambda=0$ ?

(ii) Now suppose that $A_{\mu}(x)$ and $J_{\mu}(x)$ obeying $(*)$ reduce to static 3 -vectors $\mathbf{A}(\mathbf{x})$ and $\mathbf{J}(\mathbf{x})$ in some inertial frame. Show that there is a solution

$$\mathbf{A}(\mathbf{x})=-\mu_{0} \int G\left(\left|\mathbf{x}-\mathbf{x}^{\prime}\right|\right) \mathbf{J}\left(\mathbf{x}^{\prime}\right) d^{3} \mathbf{x}^{\prime}$$

for a suitable Green's function $G(R)$ with $G(R) \rightarrow 0$ as $R \rightarrow \infty$. Determine $G(R)$ for any $\lambda \geqslant 0$. [Hint: You may find it helpful to consider first the case $\lambda=0$ and then the case $\lambda>0$, using the result $\nabla^{2}\left(\frac{1}{R} f(R)\right)=\nabla^{2}\left(\frac{1}{R}\right) f(R)+\frac{1}{R} f^{\prime \prime}(R)$, where $\left.R=\left|\mathbf{x}-\mathbf{x}^{\prime}\right| .\right]$

If $\mathbf{J}(\mathbf{x})$ is zero outside some bounded region, comment on the effect of the value of $\lambda$ on the behaviour of $\mathbf{A}(\mathbf{x})$ when $|\mathbf{x}|$ is large. [No further detailed calculations are required.]