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