The most important secondary particles for the detection of neutrinos are the charged leptons l = e, µ, τwhich are produced in the CC interaction νl+N→ l+X. Depending on the kinematics of the interaction, these may carry a significant amount of the initial neutrino energy. It is important to understand how these leptons propagate in ice and how likely they are to either decay or interact on their path. The different behavior of each lepton in ice is the basis for flavor discrimination in IceCube.
Decay The electron is a stable particle and does not decay. Both the muon and the tau are unstable and have a lifetime of τµ = 2.2· 10−6s and ττ = 290.3· 10−15s, respectively [78]. The decay time in the lab frame depends on the energy of the lepton due to the time dilation described in the theory of special relativity. The corresponding propagation length before the lepton decays (called decay length) is λdec = cβγτ, where c is the speed of light, β = v/c is the speed of the lepton, γ = 1/√1 − β2 is the Lorentz factor, and τ is the lifetime of the lepton in the center-of-mass frame. In the energy range of IceCube, all secondary leptons effectively travel at the speed of light, thus approximating β ≃ 1. The decay length can then be written as λdec = (Eτ )/(mc), using E = γmc2 where E is the energy of the lepton and m is its mass. The decay length gives the distance after which the survival probability of a lepton decreases to 1/e ≃ 36.8%. This is based on the exponential decay law p(L) = exp(−L/λdec)which gives the survival probability for a lepton with the decay length λdec after the propagation length L. For example, a 1 TeV muon has a decay length of λdec, µ = 6242 km, whereas a tau of the same energy has a much smaller decay length of λdec, τ = 4.9 cm using mµ = 105.7 MeV/c2 and mτ= 1776.9 MeV/c2 for the muon and tau masses, respectively [78].
Interaction Charged leptons undergo constant interactions while propagating in the ice, which may cause them to change direction and/or lose energy. The energy loss profile dE/dX is a measure of how much energy dE the lepton loses while traversing the amount of ice dX, commonly given in units of g/cm2. Four processes are generally considered: continuous energy losses due to ionization as well as radiative losses due to bremsstrahlung, pair production, and photonuclear interactions. Ionization is caused by the collision of the traversing lepton with shell electrons of the target atoms. The corresponding energy loss is continuous and only scales logarithmically with energy. In the energy range of IceCube, it can be approximated as constant ⟨dE/dX⟩ion ≃ 2 MeV/(g/cm2) for leptons with βγ ≫ 1 [94]. Hence, in ice with a density of ρice ≃ 0.9 g/cm3, a lepton continuously loses ∼ 180 MeV/m along its propagation path. Neglecting radiative losses, it requires at least ∼ 180 GeV in order to traverse ∼ 1 km through the IceCube detector. At energies above 1 TeV that are relevant to the analysis presented in this thesis, radiative interactions dominate and energy losses due to ionization are negligible. Radiative energy losses occur randomly along the propagation path of the lepton. An important quantity to describe this process is the radiation length X0
1 X0 = 4α 3 ℏ2 c2m2 NA A ( Z2(ln(184.15Z−1/3) − f (Z))+ Z ln(1194Z−2/3) ) , (3.4)
which is defined as the average propagation length after which the lepton has lost 1/e of its initial energy due to radiative processes [78]. Here, α is the electromagnetic fine-structure constant, ℏ the reduced Planck constant, c the speed of light, and NA the Avogadro constant. The formula is valid for a lepton with mass m traversing matter with an atomic number Z and a mass number A with a polynomial function f (Z) as given in [78]. Bremsstrahlung is emitted when the traversing lepton undergoes Coulomb scattering and is deflected by either the shell electrons or the nucleus of the target atom. The energy loss due to bremsstrahlung is stochastic and scales linearly with energy. Its average is given by ⟨dE/dX⟩brems≃ −E/X0[94]. Pair production is a secondary process where a photon creates an electron-positron pair in the vicinity of the nuclear Coulomb field. A requirement is that its energy is above the production threshold of Eγ ≥ 2me. The energy loss due to pair production is also stochastic and scales linearly with energy. Above a few GeV, its average is given by ⟨dE/dX⟩pair ≃ −E/(97X0)[94]. Photonuclear interaction is a process where a high-energy photon interacts with an atomic nucleus and causes its disintegration into smaller fragments. The energy threshold is above the binding energy of the nucleus at a few MeV. Energy losses due to photonuclear interactions are ∼ 50% smaller than bremsstrahlung and pair production [195].
The processes described above are generally valid in the energy range of IceCube. However, at very high energies above 10 PeV, the LPM-effect, named after Lev Landau, Isaak Pomeranchuk, and Arkady Migdal [196–198], significantly reduces the cross sections for bremsstrahlung and pair production. Consequently, leptons may propagate much farther. At even higher energies above 100 EeV, energy losses are predominantly due to photonuclear interactions [199]. The radiation length as defined in Equation 3.4 depends on the mass of the traversing lepton via X0∼ m2. Consequently, it is very different for each lepton in the same medium. In ice it is X0 ≃ 36 cm for electrons, X0 ≃ 15 km for muons and X0 ≃ 4353 km for taus. It follows that electrons lose their entire energy on a very short path (a few meters) whereas muons can on average propagate much farther (a few kilometers) before losing energy. Although taus could in principle cover an even greater distance without interacting, they have a much shorter lifetime than muons and usually decay before they lose a significant amount of energy.