Muons and Neutrinos

Muons and neutrinos are the only particles generated in cosmic-ray showers that can penetrate the Earth surface.

Muon and Neutrino Production in the atmosphere

Muons and neutrinos are produced in the development of hadronic extensive air showers. The atmosphere is to be seen as a large calorimeter where interactions and decay compete. They are also produced in any beam dump experiment where a beam of protons or nuclei is accelerated against a target of matter or, such as in cosmic sources, a target of radiation.

We focus here on atmospheric neutrinos and muons. An extensive review is in T.K. Gaisser, 2019 paper and in his book.

The most important channels for muon and neutrino production are:

• Pion and Kaon production (so called `conventinal neutrinos'). Depending on the energy one or the other dominates.

  • Two body decays processes such as:

    • $\pi^{\pm} \rightarrow \mu^{\pm} + \nu_{\mu}({ \bar \nu_\mu}) \rm{\;\; (\sim 100\%)}$. This is the most important channle below 100 GeV and the kinematics of the decay is covered in Special Relativity and High-Energy Astrophysical Phenomena.

    • $K^{\pm} \rightarrow \mu^{\pm} + \nu_{\mu}({\bar \nu_\mu}) \rm{\;\; (\sim 63.5\%)}$. The dominance of one or the other channel is shown in the plot below. Most of the neutrinos seen by IceCube for energies above 100 GeV are from kaon decay (this can be seen in the plots of fractions below where kaon decay dominates over pion one above about 300 GeV.

  • Three body decay process $K^0_L \rightarrow \pi^{\pm} e^{\pm}\nu_e({\bar \nu_e}) \rm{\;\; (\sim 38.7\%)}$and $K^\pm \rightarrow \pi^0 e \nu_e$ .

• Above about 50 TeV `prompt neutrinos' begin to dominate over conventional ones. They come from the prompt decays of charmed mesons, such as

• At lower energies, muons decay into neutrinos:

$$\mu^{\pm} \rightarrow e^{\pm} + \nu_{e}({\bar \nu_e}) + {\bar \nu_{\mu}}(\nu_\mu)$$

From these reaction we then get the fluxes of atmospheric neutrinos, where we distinguish the subdominant $\nu_e$ with respect to $\nu_\mu$. The cross over of prompt neutrinos should happen beyond 50 TeV but it is not yet experimentally proven.

From the review paper by TK Gaisser, 2019.

Below we also show the fractional contributions to the neutrino and muon fluxes in the atmosphere from different parent particles, including prompt. In the Earth’s atmosphere the muon decay length becomes larger than its typical production height (about 15 km at the zenith) for $E_\mu \gtrsim 2.5 \rm \, GeV$ . When all muons decay $\frac{\nu_e+\bar{\nu}_e}{\nu_\mu+\bar{\nu}_{\mu}} \sim 0.5$. The $\nu_e/\nu_\mu$ ratio therefore quickly decreases with energy above a GeV or so, until at high energy the only source of $\nu_e$ is the small contributions from $K_L^0$.

From T.K. Gaisser, 2019

In the references above the muon flux in the atmosphere is calculated by solving the cascade equations. At energies where muon decay can be ignored ( $\gtrsim 10 \rm GeV$ ):

$$\frac{\rm{d}N_\mu}{\rm{d}E_\mu} \sim \frac{0.14 E_\mu^{-2.7}}{\rm cm^2 s sr GeV}\{ \frac{1}{1+\frac{1.11E_\mu \cos\theta}{115 \rm GeV}} + \frac{0.054}{1+\frac{ 1.11E_\mu\cos\theta}{850 \rm GeV}}\}$$

Where the two term in brakets are for the pion and kaon 2 body decays, and the global relevant dependence is $\sim E_\mu^{-3.7}/\cos\theta$ (the dependency of the flux follows $sec\theta$ ). The first term is basically the primary spectrum, which ignores the knee.

Muon energy losses

The Average rate of muon energy losses are described by:

$$-dE/dx = a(E) + b(E) E$$

where $a(E)\sim 0.002 \rm \, GeV g^{-1} cm^2$ is the ionisation energy loss and $b(E)$(nearly constant) takes into account of pair production, bremsstrahlung and photonuclear contributions. From the DPB, 2020.

From DPB and references therein. Contributions to the fractional energy loss by muons in Fe due to e+e− pair production, bremsstrahlung, and photonuclear interactions.