# Electrons and X-rays to Muon Pairs (EXMP)
## 4 Simulation Results for Muon Beams  
The significant values concerning the emitted particles for the analysed working points are reported in Table 2. The MPP events were generated by means of the Whizard [15; 16], and ran in such a way to take into account the incoming beam features. The muon beam transverse normalised emittance is also reported in Table 2 and the number of muons per second divided by the emittance, the crucial figure of merit, as suggested in [17], was calculated. The muon beam normalised transverse emittance value for the WP200 is 4.7 \(\pi\) nm rad and of 3.3 \(\pi\) nm rad for WP500: these values compare to the analytical prediction of Equation (6), giving 5.9 \(\pi\) nm rad and 4.6 \(\pi\) nm rad, respectively. The MPP features for the two working points are displayed in Figure 3.  
Concerning WP200, the outstanding value of the transverse normalised emittance combined with the number of muon pair per second, returns the value \(\mathcal{N}_{\mu^{\pm}}/\epsilon_{x_{\mu}}^{n}=2\times 1.2\times 10^{19}\) m\({}^{-1}\)s\({}^{-1}\). If we consider the muons around the energy distribution peak of 50-100 GeV corresponding to a 10% RMS relative energy spread, the 20% of the produced muons are selected (with a longitudinal emittance value of \(\sim\)4.5 mm). The above-mentioned coefficient corresponding to this selection is \(\mathcal{N}_{\mu^{\pm}}^{10\%}/\epsilon_{x_{\mu}}^{n}=2\times 2.3\times 10^{18}\) m\({}^{-1}\)s\({}^{-1}\), is comparable with the best option of the Gamma Factory [18; 19] combined with LEMMA [3], analysed in reference [17]. The corresponding value for WP500 is \(\mathcal{N}_{\mu^{\pm}}^{10\%}/\epsilon_{x_{\mu}}^{n}=2\times 16.3\times 10^{18}\) m\({}^{-1}\)s\({}^{-1}\). The top line of Figure 4 shows the energy spectrum of the ICS photons produced in the collision. As expected, in a high electron recoil regime, the energy peaks at around 200 GeV and 500 GeV for WP200 and WP500, respectively. The TPP, which is the most probable collateral reaction, involves \(2\times 4.8\times 10^{15}\) and \(2\times 2.4\times 10^{16}\) primary electrons, still a small fraction of the total (\(2\times 10^{18}\) per second). Moreover, the bottom line of Figure 4 shows that most of the primary electrons involved in TTP would face a negligible energy loss since the \(e^{+}/e^{-}\) pairs are mainly generated at a very low energy (see middle line). A useful characteristic of the EXMP layout shown in Figure 2 involves the possibility of accelerating muons in the same LINACs used to accelerate the two twin primary electron beams. This is possible thanks to the very small emittance of muons and the non-interacting muon-electron beams. Beam optics could rely on RF focusing, which would be effective both on muons and on electrons, and will be further analysed in future work. LINAC acceleration of muons would allow to bring them up rapidly to the TeV kinetic energy range requested by muon collider physics, just in a few passes (each EXMP pass is \(400-1000\) GeV energy gain) through the LINAC twin system, using proper muon recirculation arcs. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **EXMP WP200** & **EXMP WP500** \\ \hline \(E_{CM}\) (MeV) & & 346 \\ \(X\) & & \(4.7\times 10^{5}\) \\ \(\sigma_{tt^{\prime}}^{TPp}\) (mb) & & 19 \\ \(\sigma_{tt^{\prime}}^{\rm ICS}\) (ub) & & 14.7 \\ \(\sigma_{tt^{  
Figure 4: EXMP scheme: emitted ICS photon (column 1), TPP positron (column 2), and the initial electron beam after the TPP reaction (column 3) beam features for WP200 (dotted blue) and WP500 (solid red). Muon beam energy spectra (top) and angular distributions (bottom). Figure 3: EXMP scheme: MPP features for WP200 (dotted blue) and WP500 (solid red). Muon beam energy spectra and angular distributions (top), phase space distribution in x and normalised x, emittance (bottom).