# Feynman Diagram for phi^4 theory (path integral)

## 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 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.

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