What Is the Propagator of the Proca Lagrangian?

[Muoniex]

Homework Statement

I want to show that the propagator of Proca Lagrangian:

$$\mathcal{L}=-\frac{1}{4}F_{\mu \nu}F^{\mu \nu}+\frac{1}{2}M^2A_\mu A^\mu$$

Is given by:

$$\widetilde{D}_{\mu \nu}(k)=\frac{i}{k^2-M^2+i\epsilon}[-g_{\mu\nu}+\frac{k_\mu k_\nu}{M^2}]$$

Homework Equations

Remember that: $$F_{\mu \nu}=\partial_\mu A_\nu - \partial_\nu A_\mu$$

The Attempt at a Solution

I tried to use the Euler-Lagrange equation, and I obtained:

$$\partial_{\mu} (\partial^{\mu} A_{\nu} - \partial_{\nu} A^{\mu} ) + M^2 A^{\nu} = 0$$

I suppose I have to do a Fourier Transform in order to express that equation in terms of $$k^\mu$$
but I don't know how to do it. I don't even know if I have started the problem properly, or if there's another way.
Can anyone help me, please?

 

[MathematicalPhysicist]

Check it in Problem Book in QFT of Voja's the answers to problems 5.2b, 5.7,6.15.

This should answer your questions.

 

[Muoniex]

Check it in Problem Book in QFT of Voja's the answers to problems 5.2b, 5.7,6.15.

This should answer your questions.

Sorry for the late answer, but I wanted to check all the steps with calm.
The problems you told me helped a lot, thanks!