How to Count Wick Contractions in Phi^3 Theory?

[Dixanadu]

Hi guys,

So basically I'm trying to find the time ordered product:

$T(:\phi^{3}(x)::\phi^{3}(y):)$

for a phi^{3} theory and I'm really having trouble counting the Wick contractions. I've counted the first order contractions which I found to be 9. Second order I got 12 but I don't know if its right, and 3rd order is hopeless. So here's where I'm at so far:

$T(:\phi^{3}(x)::\phi^{3}(y):)\\ =:\phi^{3}(x)\phi^{3}(y):+9i\Delta_{F}(x-y):\phi^{2}(x)\phi^{2}(y):+12(i\Delta_{F}(x-y))^{2}:\phi(x)\phi(y): + A(i\Delta_{F}(x-y))^{3}$

and I don't know if the factor of 12 is right, or how to find A. I need help!

Thanks guys!

 

[Dixanadu]

Okay I think i got A = 6 but no idea lol...:\