Scalar electron-muon scattering

[Milsomonk]

Homework Statement

Consider the scattering of a scalar electron and a spin half muon, draw and label the feynman diagram for this process.
write down the invariant amplitude M and the spin average |M|^2, use the trace notation.

Homework Equations

Photon propagator
$$\dfrac {-i\eta_{\mu\nu}}{k^2}$$

The Attempt at a Solution

Ok so I have written down what I think the correct labelled feynman diagram is, apologies that its attached as a photo, I did draw it in latex but it doesn't want to compile on here. It's possible I havn't gone about this the correct way, I can't find any examples of scalar particles scattering with spin half particles so I just tried it the only way I could think of. I'm not sure how to write the spin average matrix in terms of traces, I know it should have something to do with the fact that are multiple spin states for the initial and final states of the muon (two) but I don't know how to rwite this down mathematically. Any guidance would be greatly appreciated :)

 

[NateD]

A:The Feynman diagram looks correct. The invariant amplitude is$$M~=~\bar{u}_f \,\epsilon^\mu \, \frac{-i \eta_{\mu\nu}}{k^2}\, \epsilon_\nu \, v_i$$where we have used the photon propagator and $\epsilon^\mu$ and $\epsilon_\nu$ are the polarization vectors of the incoming and outgoing photons.The spin-averaged square of the invariant amplitude is given by $$ |M|^2~=~\frac{1}{4}\sum\limits_{s_i,s_f} \left| \bar{u}_{s_f} \,\epsilon^\mu \, \frac{-i \eta_{\mu\nu}}{k^2}\, \epsilon_\nu \, v_{s_i} \right|^2 $$where $s_i$ and $s_f$ denote the initial and final spin states of the muon, respectively. Using the trace notation, $$|M|^2~=~\frac{1}{4}\sum\limits_{s_i,s_f} \text{Tr}\left(\bar{u}_{s_f} \,\epsilon^\mu \, \frac{-i \eta_{\mu\nu}}{k^2}\, \epsilon_\nu \, v_{s_i} \, \bar{v}_{s_i} \,\epsilon_\rho \, \frac{-i \eta_{\rho\sigma}}{k^2}\, \epsilon^\sigma \, u_{s_f}\right)$$$$ = \frac{1}{4} \, \text{Tr}\left( \frac{1}{k^4} \, \epsilon^\mu \, \eta_{\mu\nu} \,\epsilon_\nu \, \epsilon_\rho \, \eta_{\rho\sigma} \, \epsilon^\sigma \right).