208 Appendix D: Formation Time and Brick Estimates Changing variables to y = L − ℓ ⇒ dy = − dℓ and defining ω = 3 κ 2 / $^{3}$ν , the remaining integral in Eq. (D.31) becomes − e ωτ 1 / 3 0 L ∫ dye − ωy 1 / 3 = e ωτ 1 / 3 0 L $^{{}$3 e − ωy 1 / 3 ω 3 $^{[}$2 + 2 ωy 1 / 3 + ω $^{2}$y2 / 3 ] } = e ωτ 1 / 3 0 L $^{{}$3 e − ω ( L − ℓ ) 1 / 3 ω 3 × $^{[}$2 + 2 ω ( L − ℓ ) 1 / 3 + ω $^{2}$( L − ℓ ) 2 / 3 ] } ℓ = L − τ$_{0}$ ℓ =0 = e ωτ 1 / 3 0 L $^{{}$3 e − ωτ 1 / 3 0 ω 3 $^{[}$2 + 2 ωτ 1 / 3 0 + ω $^{2}$τ 2 / 3 0 ] − 3 e − ωL 1 / 3 ω 3 $^{[}$2 + 2 ωL 1 / 3 + ω $^{2}$L2 / 3 ] } . (D.35) R$_{AA}$ ≃ 3 τ 1 / 3 0 ωL + τ$_{0}$ L = 3 τ 1 / 3 0 + ωτ$_{0}$ ωL . (D.36) ω is in fact so large that for any reasonable τ$_{0}$ ∼ . 2 fm, ωτ$_{0}$ ≫ 1. Therefore we arrive at, to leading order, Since ω ∝ n$_{Q}$/M$_{Q}$ , with the proportionality constant the same for charm and bottom quarks, we still find that the ratio of charm to bottom R$_{AA}$ goes as Note the essential importance of the plasma formation time, τ$_{0}$ , in Eq. (D.36) in order to reproduce the linear in M$_{c}$/M$_{b}$ behavior. The results of the full R cb $_{AA}$( p$_{T}$ ) ≃ n$_{b}$ n$_{c}$ M$_{c}$ M$_{b}$ . (D.37)