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Feynman rules for Lagrangian with derivative Interaction

  1. Oct 30, 2015 #1
    1. The problem statement, all variables and given/known data
    The lagrangian is given by:
    [tex] L = \frac{1}{2} \partial_{\mu} \phi \partial^{\mu} \phi + \frac{\alpha}{2} \phi \partial_{\mu} \phi \partial^{\mu} \phi [/tex]

    And the question is to find the feynman rules.

    2. Relevant equations

    3. 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?
  2. jcsd
  3. Oct 30, 2015 #2


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    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.
  4. Oct 31, 2015 #3
    I indeed got the vertex function.

    It is: [tex] -i \alpha (p_1 p_2 + p_1 p_3 + p_2 p_3) \delta(p_1 + p_2 + p_3) [/tex].

    Thank you.
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