Feynman rules for Lagrangian with derivative Interaction

[silverwhale]

Homework Statement

The lagrangian is given by:
$$L = \frac{1}{2} \partial_{\mu} \phi \partial^{\mu} \phi + \frac{\alpha}{2} \phi \partial_{\mu} \phi \partial^{\mu} \phi$$

And the question is to find the feynman rules.

Homework Equations

The Attempt at a Solution

I started by using the generating functional with interaction terms method, but the calculation is huge and with it I get all the feynman graphs this Lagrangian can generate. But I am just interested in deriving the rules from the Lagrangian. How can I do that? I am clearly missing something, but what?

 

[fzero]

As a start, you should rewrite that term in the momentum picture by rewriting $\phi$ in terms of its Fourier transform. You should find something of the form $V(p_1,p_2,p_3) \phi(p_1)\phi(p_2)\phi(p_3)$, where $V$ should be appropriately symmetrized. You will be able to read off the vertex from this term.

 

[silverwhale]

I indeed got the vertex function.

It is: $$-i \alpha (p_1 p_2 + p_1 p_3 + p_2 p_3) \delta(p_1 + p_2 + p_3)$$.

Thank you.