244 Appendix F: Heavy Quark Production Radiation Derivation external field “tickling” the charge, we find that In order to make contact with the classical limit we assume a soft photon: | ⃗ k | ≪ | ⃗ p ′ − ⃗ p | . Then we can take i M = − ie ¯ u ( p $^{′}$) ( M$_{0}$ ( p $^{′}$, p − k ) i ( / p − / k + m ) ( p − k ) 2 − m $_{2}$γ · ϵ $^{∗}$( k ) + γ · ϵ $^{∗}$( k ) i ( / p ′ + / k + m ) ( p ′ + k ) 2 − m $_{2}$M$_{0}$ ( p ′ + k, p )$^{)}$ u ( p ) . (F.82) M$_{0}$ ( p ′ + k, p ) ≈ M$_{0}$ ( p $^{′}$, p − k ) ≈ M$_{0}$ ( p $^{′}$, p ) , (F.83) i M = ¯ u ( p $^{′}$) $^{[}$M$_{0}$ ( p $^{′}$, p ) $^{]}$u ( p ) · [ e$^{(}$ p ′ · ϵ ∗ p ′ · k − p · ϵ ∗ p · k )] . (F.84) and we can ignore the / k in the numerators. Using Eq. (E.11) we have that Eq. (F.82) becomes Summing over polarization states, integrating over phase space, and taking the conditional probability yields For massless QED we have that in a diagram that satisfies the Ward iden- tity ∑ λ ϵ λ $_{µ}$ϵ ∗ λ ν = − g$_{µν}$ . Then we have that the energy radiated, to lowest order, 〈 N$_{γ}$ 〉 = ∫ d $^{3}$k (2 π ) 3 1 2 ω e 2 ∑ λ ∣ ∣ ϵ$_{λ}$ · ( p ′ k · p ′ − p k · p$^{)}$$^{∣}$ ∣ ∣ 2 . (F.85)