Gauge symmetry for massless Klein-Gordon field

[Plaetean]

Homework Statement

I need to gauge the symmetry:

$$\phi \rightarrow \phi + a(x)$$

for the Lagrangian:
$$L=\partial_\mu\phi\partial^\mu\phi$$

Homework Equations

The Attempt at a Solution

We did this in class for the Dirac equation with a phase transformation and I understood the method, but can't seem to figure out a way to apply the same approach to this field. All I have done so far is plug the transformation into the Lagrangian, but what I'm not sure about is what form the gauge field $$A_\mu$$ needs to take, or even where in the Lagrangian to put it! I have a load of QFT books too but the only mention of gauge invariance in them is very brief, not at all comprehensive, and is way further into the book than I am currently. Any guidance on where to start or where to read further would be really appreciated. Thanks as always!

 

[nrqed]

Homework Statement

I need to gauge the symmetry:

$$\phi \rightarrow \phi + a(x)$$

for the Lagrangian:
$$L=\partial_\mu\phi\partial^\mu\phi$$

Homework Equations

The Attempt at a Solution

We did this in class for the Dirac equation with a phase transformation and I understood the method, but can't seem to figure out a way to apply the same approach to this field. All I have done so far is plug the transformation into the Lagrangian, but what I'm not sure about is what form the gauge field $$A_\mu$$ needs to take, or even where in the Lagrangian to put it! I have a load of QFT books too but the only mention of gauge invariance in them is very brief, not at all comprehensive, and is way further into the book than I am currently. Any guidance on where to start or where to read further would be really appreciated. Thanks as always!

First: under that gauge transformation, how does the Lagrangian
$L=\partial_\mu\phi\partial^\mu\phi$ transform?

Next, you have to add a term that contains $A_\mu$. It will have to be Lorentz invariant and be of dimension 4. What form can it take? Then you will have to see how the gauge field transforms to make the total Lagrangian invariant.