Making Feynman Diagrams for a given process

Question

I do not have much experience in field theory so please do suggest a text or source where I can find detailed explanation for my question.

My question is, how to draw feynman diagram for a given process? Just start with tree level.

Say the process is q$\bar{q}$ $\rightarrow$ $\gamma \gamma$

What I know is that you can draw the feynman diagram only when you know the hamiltonian and get the expression for S matrix element. Is there a way to draw them only using Feynman Rules? I see Peskin, takes a process for example and directly draws its tree level diagram without giving the form of lagrangian or hamiltonian.

Also tell me about higher order diagrams at the end of the answer if possible, currently I just want to know about the tree level diagrams

Answer 1

First you define the in-asymptote as a quark and an antiquark with momenta $\mathbf{p}$ and $\mathbf{p}'$, $$|\psi_{\text{in}}\rangle = a^\dagger_\mathbf{p}b^\dagger_{\mathbf{p}'}|0\rangle$$ where $a^\dagger$ and $b^\dagger$ are fermion and antifermion creation operators. Then the out-asymptote are two photons with momenta $\mathbf{k}$ and $\mathbf{k}'$, $$|\psi_{\text{out}}\rangle = c^\dagger_\mathbf{k}c^\dagger_{\mathbf{k}'}|0\rangle\,.$$ The scattering operator can be decomposed as $S = \mathbb{1} + \mathrm{i}T$, where the identity is when there is effectively no scattering. The $T$-matrix expansion will give you all of the scattering processes. To calculate this, you will need Wick's theorem. This is very nicely explained in the book by Peskin and Schroeder in chapter 4.