Are QED Feynman diagrams readable in any order?

Question

I have somewhat of a basic question regarding QED Feynman diagrams. To expose my doubts let's take the Feynman diagram of the Compton scattering (at the second order) as an example:

With the solid lines we of course want to indicate electrons, and with the squiggly ones photons; we of course have two graphs associated with this second order interaction. Also note that the arrow of time is assumed to be oriented horizontally, from left to right.

We are of course interested in finding the scattering matrix element $\langle f | S^{(2)}| i\rangle$, and we can show that it is given by the following formula:

$$\langle f | S^{(2)}| i \rangle=(2\pi)^4\delta ^4 (p'+k'-p-k) \sqrt{\frac{m}{VE_p}}\sqrt{\frac{m}{VE_{p'}}}\sqrt{\frac{1}{2VE_k}}\sqrt{\frac{1}{2VE_{k'}}}(\mathcal{M}_a+\mathcal{M}_b)\tag{1}$$

Where $\mathcal{M}_a,\mathcal{M}_b$ are two terms, corresponding to the two possible second order processes. So far so good for me.

Now should come the fun part: we know that $\mathcal{M}_a,\mathcal{M}_b$ can be determined from the Feynman diagrams of the processes by applying the Feynman rules. In the picture above the Feynman rules have been partially explicated, but we also need to take into account the vertices and the propagator. Let's focus on the process $(a)$, we should get (I am following my textbook here):

$$\mathcal{M}_a=\epsilon _\mu (k')\epsilon _\nu (k)\bar{u}(p')(-ie\gamma ^\mu)\frac{i}{\hat{q}-m+i\varepsilon}(-ie\gamma ^\nu)u(p) \ \ \ , \ \ \ with: \ q=p+k \tag{2}$$

My question regards this expression. As we can see all the terms dictated in the Feynman rules linked above are present (but for some reason there is no complex conjugation on $\epsilon (k')$... I don't know why), so no problem here (more or less). My question is: is the order of the terms in $(2)$ important? I would be inclined to assume that the order does matter since there seems to be no way that all of the spinors and operators involved in $\mathcal{M _i}$ are commuting no matter the circumstances, but on the other hand I don't understand how to determine in which order to write down the terms based on the shape of the graph, and I cannot find a text that explains that clearly either.

I understand how to associate each part of the graph with the corresponding numerical factor using the Feynman rules, but I don't understand how to order them to compose the shape of $\mathcal{M}_i$. For example I would not be sure how to write down $\mathcal{M}_b$, but I am positive I would be able to list all the factors that compose it.

Answer 1

Cooking recipe:

Diagram (a):

1. Start with the outgoing electron (i.e. the electron in the final state with momentum $p^\prime$): $\, \bar{u}(p^\prime)$
2. Following the electron line backwards (i.e. opposite to the direction of the arrows) you arrive at the interaction vertex with Lorentz index $\mu$: $\, -ie \gamma^\mu$
3. Now comes the electron propagator with momentum $q=p+k$: $\, \frac{i}{q\! \! /-m+i \epsilon}=i \frac{q\! \! /+m}{q^2-m^2+i\epsilon}$
4. Interaction vertex with Lorentz index $\nu$: $\, -ie \gamma^\nu$
5. Finally you arrive at the incoming electron (i.e. electron in the initial state with momentum $p$): $\, u(p)$
6. Write down the polarization vector $\varepsilon_\nu(k)$ for the photon in the initial state
7. Write down the polarization vector $\varepsilon_\mu^\ast(k^\prime)$ for the photon in the final state

Your final result reads: $$\bar{u}(p^\prime) (-ie \gamma^\mu) \frac{i}{q\!\!\!/ -m+i\epsilon}(-ie \gamma^\nu)u(p)\varepsilon_\nu(k)\varepsilon_\mu^\ast(k^\prime).$$ The order of the factors originating from the fermion line is important! As a simple check, remember that $\bar{u}(p^\prime)$ is a $1 \times 4$ matrix in Dirac space, $\gamma^\mu$, $\frac{i}{q \!\!/-m+i\epsilon}$, $\gamma^\nu$ are $4\times 4$ matrices and $u(p)$ is a $4 \times 1$ matrix! The positions of $\epsilon_\nu(k)$ and $\epsilon_\mu^\ast(k^\prime)$ are irrelevant.

Diagram (b):

Proceeding analogously you find $$\bar{u}(p^\prime) (-ie \gamma^\nu) \frac{i}{p\! \! \!/-k^\prime \! \! \! \! \!/-m+i \epsilon}(-ie \gamma^\mu) u(p) \varepsilon_\nu(k) \varepsilon_\mu^\ast(k^\prime).$$