# Help with Wick's theorem in a $\phi^4$ QFT

QFT noob here. I am currently working out the momentum space two-point function for a $$\phi^4$$ qft in Euclidean space time, and considering the $$\lambda^1$$ order contribution, I am encountering a correlation function of the form

$$\Big\langle\phi(x_1)\phi(x_2)\phi^4(y_1)\Big\rangle$$

I wish to expand it in terms of simpler green's function to then work out the Feynman diagrams, but I am a bit unsure as to how to approach the expansion of this. I need to use Wick's theorem, yes? I am not sure why, but the $$\phi^4$$ I feel is tripping me up slightly. How should I approach this?

• this is just i.stack.imgur.com/zK0Ad.png Dec 17, 2021 at 4:50
• Yes, I think I just misunderstood the role of Feynman diagrams on correlation functions. So, essentially what I am gathering is that the correlation function I posted above is difficult and annoying to calculate normally, so we employ Feynman diagrams to cut the corners so to speak Dec 17, 2021 at 5:23
• That's indeed what Feynman diagrams are for -- they're just a tool for keeping track of all the different Wick's theorem contractions, no more and no less. The one you posted is simple enough that you should be able to do it without diagrams, so you should try doing it both ways and comparing your answers.
– Zack
Dec 17, 2021 at 6:02

Omitting conceptual details, I can speak that $$\phi(x_1)$$, ... are Gassian distributed. According to the mentioned theorem, I can simply write the following: $$\langle\phi(x_1)\phi(x_2)\phi(y_1)^4\rangle = \text{sum of all possible two-point averages},$$ where the right hand side is given by (correct me if I miss something): $$\langle\phi(x_1)\phi(x_2)\rangle\langle\phi(y_1)\phi(y_1)\rangle\langle\phi(y_1)\phi(y_1)\rangle+\langle\phi(x_1)\phi(y_1)\rangle\langle\phi(y_1)\phi(y_1)\rangle\langle\phi(y_1)\phi(x_2)\rangle.$$ Next, you know that in representation picture you have integration over space, so you should add integration over $$y_1$$. From free theory analysis, you know that $$D(x-y)=\langle\phi(x)\phi(y)\rangle$$ is the simple quantity, which is called free propagator. Finally, you two-point function becomes $$\int d^4y_1\,D(x_1-x_2)D(y_1-y_1)D(y_1-y_1)+\int d^4y_1\,D(x_1-y_1)D(y_1-y_1)D(y_1-x_2).$$