Moller scattering
Definition: Moller scattering
Moller scattering is the process $e^-e^-\to e^-e^-$ in scalar QED.
To calculate the Cross sections we can calculate the matrix element $|\mathcal{M}|^2$
There are two possible diagrams. They are classified according to the momentum transfer in terms of the Mandelstam variables as t- and u-channel:
# t-channel:
The momentum transfer is $k^\mu=p^\mu_3 - p^\mu_1$, so that $k^2=t$
Inserting the photon propagator and the proper vertex factors gives:
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Note that because $k^\mu(p^\mu_1 + p^\mu_3) = p^2_3 - p^2_1 = m^2-m^2 = 0$ the $\xi$ dependence vanishes. This is expected and neccessary because of gauge invariance, no final results should depend on the choice of gauge $\xi!$
The final result is $$\mathcal{M}_t=e^2 \frac{(p^1_\mu + p^3_\mu)(p^\mu_2 + p^\mu_4)}{t}$$
# u-channel:
For the u-channel the momentum transfer is $k^\mu = p^\mu_4 - p^\mu_1$ giving $k^2 = u^2$
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Again the gauge dependence vanishes
The final result is $$\mathcal{M}_u = e^2\frac{(p^\mu_1 + p^\mu_4)(p^2_\mu + p^3_\mu)}{u}$$
From $\mathcal{M}=\mathcal{M}_t + \mathcal{M}u$ we get the final expression for the Cross sections by inserting and substituting the mandelstam variables (note $E{cm} = s$) $$\frac{d\sigma(e^-e^-\to e^-e^-)}{d\Omega} = \frac{e^4}{64\pi^2 s}\left[\frac{s-u}{t} + \frac{s-t}{u }\right]^2$$