217 Appendix E: Standard pQCD Results We will need the following dot products: With these in hand we have that p · P = [(1 − x ) E $^{+}$, k 2 (1 − x ) E $_{+}$, − k ] · [ M, M, 0 ] ≃ 1 2 ME + (E.29) p · k = 1 2$^{(}$ k 2 x + x k 2 ) + k 2 ≃ k 2 2 x (E.30) k · P = 1 2$^{(}$ MxE + + M k 2 xE + ) ≃ 1 2 MxE + (E.31) ∑ λ |M$_{β}$| 2 = ( Qe ) 2 { 2 ( 1 $_{2}$ME + ) ( k 2 2 x ) ( 1 $_{2}$MxE + ) − M 2 ( 1 $_{2}$MxE$^{+}$) 2 ) } (E.32) = 4( Qe ) 2 k 2 − 4( Qe ) 2 ( xE $^{+}$)2 (E.33) ≃ 4( Qe ) 2 k 2 , (E.34) d $^{3}$N(0) γ = d 3 ⃗ k (2 π ) $^{3}$2 ω ∑ λ |M$_{β}$| 2 (E.35) ⇒ d $^{3}$N (0) γ d 3 ⃗ k = ( Qe ) 2 4 π $^{3}$ω k 2 = Q $^{2}$α π $^{2}$ω k 2 (E.36) (E.37) where the final relation comes from k ≪ xE + due to eikonality, Eq. (E.10). So