- #1
Dixanadu
- 254
- 2
Homework Statement
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]
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
[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
[itex]
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],
where
[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 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!