**Shigeru Yoshida**

A high energy muon track is associated with multiple cascades
mainly by 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
is

Thus, a muon sees its pair creation cascade almost every
along its track. This situation is reasonably described by
the continuous energy loss (CEL) approximation. In this picture,
the **average** energy loss per unit length is given by

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

When is constant, Eq. 2
can be easily resolved and we get the energy profile as

(4) |

Therefore, a cascade energy relative to the primary (initial) energy,

The analysis above allows us to build a PDF of a given muon track
as superpose of single cascade PDF at vertex every
along track geometry with energy given by Eq. 5.
You can give as whatever you want because you can scale it
to true muon energy anyway. is just a normalization factor
of PDF which can be derived in minimizing the likelihood, in other words.
An individual cascade PDF will be built by using the photonics table
or 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 photo-nuclear reaction, because their cross sections are two orders
of magnitude lower than that of the pair creation. Those two reactions
are sources of stochastic energy loss. But significant fraction of
cascades generated by these reactions 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, of the photo-nuclear reaction is
not energy independent, as shown in Figure 1.
The cross section itself is also rather uncertain. So its treatment
is not so simple and leaves us significant systematic errors in
reconstruction. We should keep this possibility in our mind.