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Homework Help: Feynman Diagram for phi^4 theory (path integral)

  1. Feb 12, 2015 #1
    1. The problem statement, all variables and given/known data
    Hey guys!

    So basically in the question I'm given the action

    [itex]S=\int d^{d}x \left[ \frac{1}{2}\partial_{\mu}\phi\partial_{\nu}\phi\eta^{\mu\nu} - \frac{m^{2}}{2}\phi^{2} -\frac{\lambda}{4!}\phi^{4}\right][/itex].

    I have use the feynman rules to calculate the tree level diagram with 6 external momentum states [itex]k_{1},k_{2}\dots k_{6}[/itex]

    2. Relevant equations
    Not sure -- please ask if you need something as I have no idea what information is needed

    3. The attempt at a solution
    As far as I understand, I have to construct the Z function by isolating the vertex. Doing this, i get

    [itex]Z[J]=e^{-i\frac{\lambda}{4!}\int d^{d}x\left( i\frac{\delta}{\delta J(x)} \right)^{4}}\int D\phi e^{i\int d^{d}x \left[ \frac{1}{2}\partial_{\mu}\phi\partial_{\nu}\phi\eta^{\mu\nu} - \frac{m^{2}}{2}\phi^{2} -iJ(x)\right]}[/itex].

    Now you can further isolate the [itex]Z[J=\lambda=0][/itex] factor to get

    Z[J]=Z[0,0]e^{-i\frac{\lambda}{4!}\int d^{d}x\left( i\frac{\delta}{\delta J(x)} \right)^{4}} e^{-(i/2)\int\int d^{d}x\, d^{d}y J(x)D(x-y)J(y)}


    [itex]D(x-y)=\int \frac{d^{d}k}{(2\pi)^{d}}\frac{e^{ik\cdot(x-y)}}{k^{2}-m^{2}+i\epsilon}[/itex].

    Now I'm stuck. I dont know how many vertices I should consider, or even what to do with the 6 momenta. I dont know why this is all d-dimensional or how to deal with it.

    Help please :D thanks!
  2. jcsd
  3. Feb 12, 2015 #2


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    Education Advisor
    Gold Member

    You are going to have to look at the question again. Since you didn't post the question but only "as far as I understand" it's not really possible to help you very much.
  4. Feb 12, 2015 #3
    First of all, you probably studied φ3 theory (if not, do so by all means!), so it would be a very good idea to go back there and see how you can deduce the φ4 analog. (For example, vertices are related to the power of your functional derivative, which is related to the exponent of the interaction term –the one controlled by λ here– in the Lagrangian. In φ3 a vertex connects 3 lines...)
    To construct the tree-level diagrams, use Feynman rules: you must have some kind of table or detailed prescriptions in your textbook and you must have read it if you are given such a problem. Then again, use the analogy with φ3...
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