# Electrons and X-rays to Muon Pairs (EXMP)
## 2 Theory  
Here, we analyse the most important aspects concerning the electron-photon beam collisions at the center of mass energy, around 350 MeV. The center of mass energy is given by \(E_{CM}=\sqrt{2E_{e}h\
u-2(\underline{p}_{e}\cdot\underline{k})+M_{e}^{2}}\) where \(E_{e}\), \(\underline{p}_{e}\) and \(h\
u,\ \underline{k}\) are electron and photon energies and momenta, respectively, and \(M_{e}=0.511\) MeV/c\({}^{2}\) is the electron mass (natural units are used i.e., c = 1). Assuming a head-on collision within the electron and the incident photon with energy \(h\
u<<E_{e}\) and supposing \(M_{e}^{2}<<E_{e}h\
u\), the Lorentz factor of the center of mass is  
\[\gamma_{CM}=\frac{E_{tot}^{LAB}}{E_{CM}}\simeq\frac{E_{e}+h\
u}{\sqrt{4E_{e} h\
u+M_{e}^{2}}}\simeq\frac{E_{e}}{2\sqrt{E_{e}h\
u}}=\frac{\sqrt{E_{e}}}{2 \sqrt{h\
u}}. \tag{1}\]The CM threshold energy for the muon pair production (MPP) is given by \(E_{CM}^{th}=M_{e}+2M_{\mu}\). If for example \(E_{e}=200\) GeV, the equality \(M_{e}+2M_{\mu}=\sqrt{4E_{e}h\
u+M_{e}^{2}}\) implies that \(h\
u^{th}=56.6\) keV. The photon energy in the electron rest frame is \(h\
u^{th^{\ast}}=2\gamma_{e}h\
u^{th}=44.3\) GeV, while the photon energy in the CM frame is \(h\
u^{th\ast}=2\gamma_{CM}h\
u^{th}=106.5\) MeV. Besides MPP, the other predominant reactions in the mentioned CM energy range are triplet pair production (TPP--\(e^{-}+\gamma\to e^{-}+e^{+}+e^{-}\)) and inverse Compton scattering (ICS - \(e^{-}+\gamma\to e^{-}+\gamma\)). The electron recoil parameter, defined as  
\[X=2h\
u^{\prime}/M_{e}=4\gamma_{e}h\
u/M_{e}, \tag{2}\]  
achieves very high values in the cases of our interest (in the order of \(10^{5}\)). We also notice the center of mass energy scales, such as the square root of X: \(E_{cm}\simeq M_{e}\sqrt{1+X}\). The TPP cross section, calculated as in reference [7], is  
\[\sigma_{tot}^{TPP}=\sigma_{Bethe-Heitler}^{TPP}-\sigma_{Borsellino}^{TPP}\]  
\[ \begin{cases}\sigma_{Bethe-Heitler}^{TPP}=\pi r_{e}^{2}[3.11\,\ln(X)-8.07]\\ \sigma_{Borsellino}^{TPP}=\frac{2\pi r_{e}^{2}}{X}\left[\frac{4}{3}\,\ln^{3}(X)- 3\,\ln^{2}(X)+6.84\,\ln(X)-21.51\right]\end{cases}  \tag{3}\]  
where \(\alpha\simeq 1/137\) is the fine-structure constant and \(r_{e}^{2}=0.079\) barn is the classical electron radius squared. In a similar way, the MPP cross section can be obtained from the one above by substituting \(r_{e}\) with \(r_{\mu}=r_{e}(M_{e}/M_{\mu})\) and \(X\) with \(X(M_{e}/M_{\mu})\). The ICS cross section is given by [8]  
\[\sigma_{tot}^{ICS}=2\pi r_{e}^{2}\frac{1}{X}\bigg{[}\bigg{(}1-\frac{4}{X}-\frac {8}{X^{2}}\bigg{)}\ln(1+X)+\frac{1}{2}+\frac{8}{X}-\frac{1}{2(1+X)^{2}}\bigg{]}. \tag{4}\]  
The total cross sections \(\sigma_{tot}\) as a function of \(E_{CM}\) are depicted in Figure 1.  
Figure 1: Total cross sections of predominant reactions in the electron–photon collision as a function of \(E_{cm}\) (MeV): muon pair production (MPP--\(e^{-}+\gamma\to e^{-}+\mu^{+}+\mu^{-}\)), triplet pair production (TPP--\(e^{-}+\gamma\to e^{-}+e^{+}+e^{-}\)), and inverse Compton scattering (ICS—\(e^{-}+\gamma\to e^{-}+\gamma\)). The luminosity of the \(e^{-}-\gamma\) collider is defined as  
\[\mathcal{L}=\frac{N_{e}\,N_{ph}\,r}{2\,\pi\,\sqrt{\sigma_{x_{e}}^{2}+\sigma_{y_{e }}^{2}}\sqrt{\sigma_{x_{ph}^{2}}+\sigma_{y_{ph}}^{2}}}\,[\text{cm}^{-2}\;\text{s} ^{-1}] \tag{5}\]  
with \(N_{e},N_{ph}\) being the number of electron and photons per bunch, \(r\) the repetition rate of the collisions, and \(\sigma_{x_{e}},\sigma_{y_{e}},\sigma_{y_{ph}}\) the transverse dimensions of the electron and the photon beams, respectively. The number of muon and electron pairs and ICS events per second can be obtained by multiplying the primary collider luminosity by the respective total cross sections. The transverse normalised emittance of the produced muon is determined by the intrinsic thermal contribution of the reaction and by the features of the incoming electron beam.