# 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.

• On this notes is some info that may be useful to you. Also, are you interested in the $m_e \ll m_\mu$ limit which is often computed in introductory references or in the complete amplitude? – glS Jan 6 '15 at 17:08
• Thanks, I'll check those notes out for some complimentary information tomorrow :) I was interested in the complete amplitude at first, taking the small mass limit afterwards. – user17574 Jan 7 '15 at 20:10

Remembering that $g^{\mu \nu} P_{\mu} = P^{\nu}$ helped calculating the trace.