Feynman Diagram for phi^4 theory (path integral)

[Dixanadu]

Homework Statement

Hey guys!

So basically in the question I'm given the action

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

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

Homework Equations

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

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

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

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

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

where

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

Now I'm stuck. I don't know how many vertices I should consider, or even what to do with the 6 momenta. I don't know why this is all d-dimensional or how to deal with it.

Help please :D thanks!

 

[DEvens]

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.

 

[Goddar]

Hi.
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...