98 Chapter 4: Fluctuations in Jet Tomography Figure 4.3: Distribution of path lengths (given by Eq. (4.3)) traversed by hard scatterers; the lengths, L ( ⃗ x$_{⊥}$, φ ) , are weighted by the probability of pro- duction and averaged over azimuth. An equivalent formulation of Eq. (4.4) is R I Q = ∫ dL 1 /N$_{bin}$dN$_{bin}$/dL ∫ dϵ (1 − ϵ ) $^{n}$P I $_{Q}$( ϵ ; L ) . Since this is a purely geometric quantity, it is the same for all jet varieties. Also displayed are the single, represen- tative pathlengths, L$_{Q}$ , used as input in approach II. Note the hierarchy of scales with glue requiring the shortest, then charm, light quarks, and bottom the longest effective pathlength. tions) adds a high computational cost to the extended theory specified by Eqs. (4.1,4.2). In this first study with the extended theory, we explore the relative order of magnitude of the competing effects by combining two simpler approaches. In approach I, we parameterize the heavy quark pQCD spectra by a simpler power law, Ed $^{3}$σ$_{Q}$ /d $^{3}$k ∝ 1 /p n +2 ⊥ , with a slowly varying logarithmic index