quantum field theory - How to calculate the tree-level probability amplitude for the electron-positron to muon-antimuon process?

Consider the following process: $e^+ + e^- \rightarrow \mu^+ + \mu^-$. I'm trying to calculate the probability amplitude of such a process in leading order.

In leading order the amplitude is given by:

$$\mathcal{M} = ie^2 [\bar{v}(p_2,s_2)\gamma^{\mu}u(p_1,s_1)]\frac{1}{(p_1+p_2)^2}[\bar{u}(p_3,s_3)\gamma_{\mu}v(p_4,s_4)].$$

To get the full probability amplitude we need to sum over all spin states and square the amplitude. In the end the expression reduces to:

$$\sum_{ s} |\mathcal{M}|^2 = \frac{e^4}{s^2} \operatorname{Tr} [\gamma^{\mu}(\gamma^{\sigma}p_{1,\sigma}+m_1)\gamma^{\nu}(\gamma^{\alpha}p_{2,\alpha}-m_2)] \operatorname{Tr} [\gamma_{\mu}(\gamma^{\beta}p_{4,\beta} - m_4)\gamma_{\nu}(\gamma^{\delta}p_{3,\delta}+m_3)]$$

where $\gamma^{\mu}$ are the Dirac matrices, $p_{1,2,3,4}$ are the incoming momenta of the positron and electron (1 and 2) and outgoing momenta of the muons (3 and 4). The masses have the same labeling.

Using the Trace relations for the Dirac matrices I got so far with computing the traces:

$$\operatorname{Tr} [\gamma^{\mu}(\gamma^{\sigma}p_{1,\sigma}+m_1)\gamma^{\nu}(\gamma^{\alpha}p_{2,\alpha}-m_2)] \\= 4(g^{\mu \sigma}g^{\nu\alpha}p_{\sigma 1}p_{\alpha 2} - g^{\mu \nu}g^{\sigma \alpha}p_{\sigma 1}p_{\alpha 2} + g^{\mu \alpha}g^{\sigma \nu}p_{\sigma1}p_{\alpha2}- g^{\mu \nu} m_1 m_2)$$

(where I used that all traces with an odd number of gamma matrices are identically zero), and an analogous term for the second trace.

If someone could check if this is good so far, I'd appreciate it, as I'm quite a newbie with this notation. Also how do I proceed from here? I'm a bit confused when I want to multiply these metric tensors with each other, especially when I consider the product with the 2nd trace aswell. I'd be grateful if someone could maybe finish the calculation in babysteps.

Answer

actually you should see Peskin part 5.1 . The trick is calculating electron-positron part and muon-antimuon part separately and because muon mass is much more heavier than electron mass, you can neglect the electron and positron mass(briefly take mass = 0 ). It would make your calculations easier. Good luck.