219 Appendix E: Standard pQCD Results M$_{Q}$ and the photon has mass m$_{γ}$ . The relevant four-vectors are now P ≃ [ E $^{+}$, M 2 Q E $_{+}$, 0 ] (E.45) p = [(1 − x ) E $^{+}$, k 2 + M 2 Q (1 − x ) E $_{+}$, − k ] (E.46) k = [ xE $^{+}$, k 2 + m 2 γ xE + , k ] (E.47) ϵ = [0 , ϵ · k xE $_{+}$, ϵ ] . (E.48) i M = ¯ u ( p ) iQeγ $^{µ}$ϵ ∗ µ i ( / P + M ) P 2 − M 2 + iϵ M$_{0}$ ( P, P$_{0}$ ) u ( P$_{0}$ ) (E.49) ≃ i $^{2}$Qe 2 p · ϵ ∗ 2 p · k + m 2 γ + iϵ ¯ u ( P ) M$_{0}$ ( P, P$_{0}$ ) u ( P$_{0}$ ) , (E.50) The full matrix element is then where Peskin’s trick, Eq. (E.11), was used. The latter amplitude corresponds to factorized production and will be dropped. We need the following relations 2 p · k + m 2 γ = [ (1 − x )( k 2 + m 2 $_{γ}$) x + x ( k 2 + M 2 $_{Q}$) (1 − x ) ] +2 k 2 + m 2 γ (E.51) = k 2 + (1 − x ) $^{2}$m2 γ + x $^{2}$M2 Q x (1 − x ) + m 2 γ (E.52) = k 2 + (1 − x ) m 2 γ + x $^{2}$M2 Q x (1 − x ) (E.53) 2 p · ϵ = $^{[}$2(1 − x ) k · ϵ x + 2 k · ϵ$^{]}$ = 2 k · ϵ x . (E.54)