# Møller scattering amplitudes

In order to compute the scattering cross section for Møller scattering, one needs the amplitudes for both the $t$- and the $u$-channel. Since the cross section is proportional to $|\mathcal{M}|^2$, the matrix element squared, and since there are two channels, we will get mixed terms. One of these terms is, after averaging over the possible spin states:

$$-\frac{e^4}{4 t u}\mathop{Tr}\left[ (p_3\!\!\!\!\!/+m)\gamma^\mu(p_1\!\!\!\!\!/+m)\gamma^\beta(p_4\!\!\!\!\!/+m)\gamma_\mu(p_2\!\!\!\!\!/+m)\gamma_\beta \right],$$ where $e$ is the electron charge, $m$ the electron mass, $p_1,p_3$ the momenta of the incoming electrons, $p_2,p_4$ the momenta of the outgoing ones and $\gamma$ the $\gamma$-matrices. $\mathop{Tr}$ stands for the trace of an operator. $t$ and $u$ are Mandelstam variables.

Now, one could simply expand the expression I have to take the trace of but this will lead to an exploding amount of terms. So, is there any smart way of calculating this trace without getting lost in cumbersome calculations?

Typically, one uses the fact that the trace of an odd number of gamma matrices vanishes in $d$ dimensions other than select odd $d$. This allows you to see, by inspection really, the contributing terms. It usually amounts to dropping some mass contributions, for example. In your case, there is more simplication to be made first by anti commuting the $\gamma^{\mu}$ and $\gamma^{\beta}$ so as to e.g rewrite the pair $\gamma^{\mu}\gamma_{\mu} = d1$. This then leaves you with a trace of at most four gamma matrices, which is relatively simple and whose evaluation is explicitly given in the literature.