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?
 A: Roughly speaking, you can interprete Wick theorem for quantum fields as the generalization of Wick theorem for Gaussian random variable, which also called Isserlis' theorem.
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
$$\int d^4y_1\,D(x_1-x_2)D(y_1-y_1)D(y_1-y_1)+\int d^4y_1\,D(x_1-y_1)D(y_1-y_1)D(y_1-x_2).$$
The key point of using Feynman diagrams is do not to repeat this boring procedure of writing down all the possible Wick contractions. From the lagrangian you can immediately identify what is the bare vertex and what is the free theory propagator. It is enough to write down any possible correction for two-point correlator function
