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The decay products for a muon are an electron, a muon neutrino, and a electron antineutrino. As the decay products for a neutron (electron, proton, neutrino) can combine together to form a neutron again, would it be possible to generate muons using electrons and neutrinos as long as they have sufficient energy?

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    $\begingroup$ Yes, trivially, but with very tiny chances of happening. $\endgroup$ Feb 23 at 2:57
  • $\begingroup$ It almost impossible collide two neutrinos $\endgroup$
    – Sancol.
    Feb 23 at 7:59
  • $\begingroup$ But you can collide one electron with one electron antineutrino and get one muon with one muon antineutrino $\endgroup$
    – Sancol.
    Feb 23 at 8:07
  • $\begingroup$ why would the chance of it happening be very low? $\endgroup$ Feb 24 at 0:11

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Yes, there are two possibilities for inverse muon decay: $\nu_\mu e^- \rightarrow \nu_e \mu^-$ and $\bar{\nu}_e e^- \rightarrow \bar{\nu}_\mu \mu^-$. The first process has been observed, but not yet the second.

The standard model $\nu_\mu e^- \rightarrow \nu_e \mu^-$ lowest-order cross-section for a muon neutrino - electron interaction with centre-of-mass energy squared $s$ is (for $M_W^2\gg s \gg m_\mu^2$):

$$ \sigma(\nu_\mu e^- \rightarrow \nu_e \mu^-) \approx \frac{G^2_F}{\pi} s $$

I believe this process was first observed by the CHARM experiment at CERN and later by the CCFR and other experiments at Fermilab. The subsequent CHARM II experiment observed $15758 \pm 324$ $\nu_\mu e^- \rightarrow \nu_e \mu^-$ events, in agreement with the expected number.

The cross-section for $\bar{\nu}_e e^- \rightarrow \bar{\nu}_\mu \mu^-$ is comparable (for $M_W^2\gg s \gg m_\mu^2$): $$ \sigma(\bar{\nu}_e e^- \rightarrow \bar{\nu}_\mu \mu^-) \approx \frac{G^2_F}{3\pi} s $$ but it has not yet been observed because it is almost impossible to make a pure enough high-energy anti-electron-neutrino beam. For electrons at rest, the minimum neutrino energy threshold for $\bar{\nu}_e e^- \rightarrow \bar{\nu}_\mu \mu^-$ is about $m_\mu^2/2E_\nu = 10.9$ GeV. Such high energy neutrinos are produced by smashing protons into targets which produce lots of charged pions and kaons whose decays mostly produce muon neutrinos and anti-neutrinos. For example, the NuTeV "anti-neutrino" beam was $98\%$ $\bar{\nu}_\mu$ and only $1.6\%$ $\bar{\nu}_e$, and although they observed 24 events consistent with $\bar{\nu} e^- \rightarrow \bar{\nu} \mu^-$, this number was consistent with backgrounds such as beam impurities and muon charge misidentification. These backgrounds were about $7$ times greater than the expected $\bar{\nu}_e e^- \rightarrow \bar{\nu}_\mu \mu^-$ signal.

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