218 Appendix E: Standard pQCD Results We wish to transform coordinates Thus the Jacobian gives d $^{3}$k = k$_{⊥}$dkdk$_{⊥}$dφ = k$_{⊥}$dωdk$_{⊥}$dφ . Hence k$_{x}$ = k$_{⊥}$ cos( φ ) (E.38) k$_{y}$ = k$_{⊥}$ sin( φ ) (E.39) k$_{z}$ = $^{√}$k 2 − k 2 ⊥ ≃ k (E.40) d $^{3}$N (0) γ k$_{⊥}$dωdk$_{⊥}$dφ = Q $^{2}$α π $^{2}$ω k 2 (E.41) d $^{2}$N (0) γ k$_{⊥}$dωdk$_{⊥}$ = 2 Q $^{2}$α πω k 2 (E.42) d $^{2}$N (0) γ dωd k 2 = Q $^{2}$α πω k $_{2}$. (E.43) N (0) γ = Q $^{2}$α π log$^{(}$ ω$_{max}$ ω$_{min}$$^{)}$ log$^{(}$ k 2 max k 2 min ) . (E.44) Nicely, we recover the Sudakov double logarithm [251]: The idea is to reproduce Djordjevic’s calculation of zeroth order radiation in [305]. The diagram is the same as Fig. E.1 except that the quark has a mass E.2.4 Diagrammatic Calculation, Massive Photon and Massive Quark Monojet