content_image/321769.jpg content_image/321770.jpg content_image/321771.jpg
n ≡ n ( p$_{T}$ ). For the pure power law case, the partonic modification factor, R$_{Q}$ = dσ final Q /dσ initial Q , (prior to fragmentation) is greatly simplified. This enables us to perform the path length fluctuations numerically via 

<img src='content_image/321769.jpg'>

where 

<img src='content_image/321770.jpg'>

Both the mean and width of those fractional energy losses depend on the local path length. We emphasize that no externally specified a priori path length, L , appears in Eq. (4.4); the path lengths explore the whole geometry. Fig. 4.3 shows the broad distribution of lengths traversed by hard partons in approach I. 

In the second approach, we determine effective path lengths, L$_{Q}$ , for each parton flavor, Q , by varying fixed L predictions R II $_{Q}$( p$_{⊥}$, L ) and comparing to approach I; see Fig. 4.3. In approach II, R II $_{Q}$( p$_{⊥}$, L ) is calculated directly from Eq. (1) with P ( E$_{i}$ → E$_{f}$ ) in Eq. (2) replaced by the fixed L approximation 

<img src='content_image/321771.jpg'>

Here, jet quenching is performed via two independent branching processes in contrast to the additive convolution in Eq. (4.4). For small energy losses the two approaches are similar. They differ however in the case of long path lengths 

