Mathematical Note on semi-analytical PDF
approach for high energy muon track
Shigeru Yoshida
March 1, 2006
A high energy muon track is associated with multiple cascades mainlyby the pair creation reaction but also bremsstrahlung and photo-nuclear interaction. Their cross sections calculated by JULIeT are shown in the left
panel of Figure 1. The pair creation cross section dominates over those of
the other channels. Its mean free path isNA
σe+e− ρiceA' 1.6 × 10−25 [cm2 ] × 0.9 [g/cm3 ] × 6.02 × 1023 /18∼ 3 × 10−3[cm−1 ]

λ−1e+e− ∼(1)

Thus, a muon sees its pair creation cascade almost every 3m along itstrack. This situation is reasonably described by the continuous energy loss(CEL) approximation. In this picture, the average energy loss per unit
length dE/dX is given by−dEdX≡ βE
NA= EA0Z

dyydσ(E)dy(2)

where y = 1 − Eµ /Eµ is the inelasticity of muons. The β term is shown in
the right panel of Figure 1. As you can see, β is almost energy independentin UHE/EHE regime except the photo-nuclear interaction, which simplifies
our calculation.

1µ−25µe+ e−µ−6µ Photonucl.

µ Photonucl.
µβ [g −1cm 2 ]σ [cm2 ]µ Bremss.
µ µ+ µ−−30µe+ e−µ Bremss.

−8µµµ+ µ−CC−10NC

CC106107108µ τ +τ −µNC10910101011−12Energy [GeV]10610710810910Energy [GeV]

Figure 1: Left panel : Total cross sections of muons for various interaction
channels as a function of energy. Right panel : The energy loss coefficient βof muons.
When β is constant, Eq. 2 can be easily resolved and we get the energy
profile asE(X) = E0 exp [−βX](3)where X [g/cm2 ] is the slant depth of a given track and E0 is initial energy at
X = 0. Because the mean free path of the pair creation (∼ [m]) is definitely
shorter than our detector resolution, we can assume a series of small cascadeis generated every ∆X ≤ λe+e− along the track with energy of∆E = E(X) − E(X + ∆X) ' E0 exp [−βX] β∆X(4)Therefore, a cascade energy relative to the primary (initial) energy,∆E= exp−βX β∆XE0(5)is energy independent, which implies that you can scale this energy of the“small” cascade to any given primary muon energy.
The analysis above allows us to build a PDF of a given muon track assuperpose of single cascade PDF at vertex every ∆X along track geometry
with energy given by Eq. 5. You can give E0 as whatever you want because
2101011

you can scale it to true muon energy anyway. E0 is just a normalization
factor of PDF which can be derived in minimizing the likelihood, in otherwords. An individual cascade PDF will be built by using the photonics tableor an alternative analytical function.
There are some issues. First, the CEL approximation should work perfectly for the pair creation, but not for the Bremsstrahlung and the photonuclear reaction, because their cross sections are two orders of magnitude
lower than that of the pair creation. Those two reactions are sources ofstochastic energy loss. But significant fraction of cascades generated by thesereactions deposits energies too small to be resolved. So you have to include
their contributions in this COE approximation. Only when the cascade energy happens pretty big, we are hopefully able to resolve it by the Sean’s
module as a 1st order approximation (Note that the CEL approximation
discussed here can be considered as “0th order”). Secondly, β of the photonuclear reaction is not energy independent, as shown in Figure 1. The crosssection itself is also rather uncertain. So its treatment is not so simple andleaves us significant systematic errors in reconstruction. We should keep thispossibility in our mind.

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