Quantum Field Theory-Gauge Transformations

[jameson2]

Homework Statement

Given the Lagrangian density $$L(\phi^{\mu})=-\frac{1}{2}(\partial_{\mu}\phi^{\nu})(\partial^{\mu}\phi_{\nu}) + \frac{1}{2}(\partial_{\mu}\phi^{\mu})^2+\frac{m^2}{2}(\phi^{\mu}\phi_{\mu})$$
and gauge transformation $$\phi^{\mu}\rightarrow \phi^{\mu} + \partial^{\mu}\alpha$$

(c) Introduce one extra real scalar field $$\sigma$$ and write some interacting Lagrangian $$L^{\prime}=L(\phi^{\mu})+L^2(\phi^{\mu},\sigma)$$ which is invariant under the gauge transformation and gives the original L for $$\sigma=0$$.

(d) Can we solve (c) with sigma that has a canonical kinetic term $$-\frac{1}{2}(\partial_{\mu}\sigma)^2$$

Homework Equations

The Attempt at a Solution

The first parts of the question show that $$\partial_{\mu}\phi^{\mu}=0$$ which simplifies the Lagrangian, and also that the initial Lagrangian is not invariant under the gauge transformation. I got those out.
But parts (c) and (d) seems like the kind of thing you either know or you don't, is there a way of working it out?

 

[Maybe_Memorie]

Bumping this because I would also like to know if (c) and (d) can be solved in a way that doesn't involve trial and error or just knowing the answer.