High energy muons in extensive air showers

The production of very high energy muons inside an extensive air shower is observable at $\nu$ telescopes and sensitive to the composition of the primary cosmic ray. Here we discuss five different sources of these muons: pion and kaon decays; charmed hadron decays; rare decays of unflavored mesons; photon conversion into a muon pair; and photon conversion into a $J/\psi$ vector meson decaying into muons. We solve the cascade equations for a $10^{10.5}$ GeV proton primary and find that unflavored mesons and gamma conversions are the two main sources of $E\ge 10^{8.5}$ GeV muons, while charm decays dominate at $10^{5.5}\,{\rm GeV} 10^6$ GeV deposition within 500 g/cm$^2$, while only in 1 out of 330 showers it is above $10^7$ GeV. We also show that the production of high energy muons is very different in proton, iron or photon showers ({e.g., conversions $\gamma\to \mu^+ \mu^-$ are the main source of $E\ge 10^4$ GeV muons in photon showers). Finally, we use Monte Carlo simulations to discuss the validity of our results.