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So, I have a Feynman diagram as shown below

enter image description here

When I try to calculate its amplitude I get

$$\mathcal{M}=ig_e^2\gamma^\mu\epsilon_\mu\frac{\not p_1+m}{p_1^2-m^2}\frac{\not p_2+m}{p_2^2-m^2}\gamma^\nu\epsilon_\nu^*$$

Where $p_1$ and $p_2$ are momentum of electron and positron, $\epsilon_\mu$ is polarization vector of incoming photon, $\epsilon_\nu$ is polarization vector of outcoming photon, which must be same.

Which is clearly not a scalar rather than a $4\times 4$ matrix. What is my mistake?

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  • $\begingroup$ (a) It would help if you defined your notation, especially what $\epsilon_\mu$, $\epsilon_\nu$, $p_{1\mu}$, $p_{2\mu}$ are. (b) Among other things, you should have an integral over a momentum variable running in the loop. (In fact you should get a divergent answer). $\endgroup$
    – Andrew
    Commented Dec 2, 2022 at 19:00
  • $\begingroup$ You’re using the same index $\mu$ in three different contractions. $\endgroup$
    – Ghoster
    Commented Dec 2, 2022 at 19:01
  • $\begingroup$ @Ghoster by $\gamma^\mu p_\mu$ I mean Feynman slash notation. I dob't know how to do it in latex. so I wrote like this $\endgroup$ Commented Dec 2, 2022 at 19:02
  • $\begingroup$ $\not p$ is “\not p” $\endgroup$
    – Ghoster
    Commented Dec 2, 2022 at 19:03
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    $\begingroup$ In addition to integrating over the loop momentum, because the virtual particles can have any momentum, you should be tracing over the spinor indices, because the virtual particles can have any spin. $\endgroup$
    – Ghoster
    Commented Dec 2, 2022 at 19:04

1 Answer 1

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The vertices, as well as the propagators are matrices ($\sim\gamma^\mu$). We can make the matrix structure explicit by writing out the indices in the diagram

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where $S$ is the fermion propagator. With this we can see that the loop contribution gives you a trace over these matrix indices \begin{equation} \mathcal M\sim\varepsilon_\mu\varepsilon_\nu\mathrm{tr}\big[\,\gamma^{\mu} S\,\gamma^{\nu} S\,\big]\ . \end{equation}

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