241 Appendix F: Heavy Quark Production Radiation Derivation and From Eq. (F.46) we find where we have used the vector identity ( ⃗ a × ⃗ b ) · ( ⃗ c × ⃗ d ) = ( ⃗ a · ⃗ c )( ⃗ b · ⃗ d ) − ( ⃗ a · ⃗ d )( ⃗ b · ⃗ c ), and Plugging these pieces into Eq. (F.21) we find that ⃗ E$_{k}$ · ⃗ E ∗ k = k 0 2 ⃗ A 2 k + ⃗ k 2 A 0 k 2 − 2 k $^{0}$A0 $_{k}$⃗ k · ⃗ A$_{k}$ . (F.72) ⃗ B$_{k}$ · ⃗ B$_{−}$$_{k}$ = $^{[}$i ( ⃗ k × ⃗ A$_{k}$ ) ][ − i ( ⃗ k × ⃗ A$_{−}$$_{k}$ ) ] = ⃗ k 2 ⃗ A$_{k}$ · ⃗ A$_{−}$$_{k}$ − ⃗ k · ⃗ A$_{k}$ ⃗ k · ⃗ A$_{−}$$_{k}$ (F.73) = ⃗ B ∗ k · ⃗ B ∗ − $_{k}$, (F.74) ⃗ B$_{k}$ · ⃗ B ∗ k = ⃗ k 2 ⃗ A 2 k − ( ⃗ k · ⃗ A$_{k}$ ) $^{2}$. (F.75) E$_{Proca}$ = 1 4 ∫ d $^{3}$k (2 π ) 3 $^{[}$⃗ k $^{2}$A0 $_{k}$A 0 − k − k 0 2 ⃗ A$_{k}$ · ⃗ A$_{−}$$_{k}$ + ⃗ k 2 ⃗ A$_{k}$ · ⃗ A$_{−}$$_{k}$ + ( ⃗ k · ⃗ A$_{k}$ )( ⃗ k · ⃗ A$_{−}$$_{k}$ ) + µ $^{2}$A0 $_{k}$A 0 − k + µ 2 ⃗ A$_{k}$ · ⃗ A$_{−}$$_{k}$ $^{]}$cos 2 k $^{0}$t + k 0 2 ⃗ A 2 k + ⃗ k 2 A 0 k 2 − 2 k $^{0}$A0 $_{k}$⃗ k · ⃗ A$_{k}$ + ⃗ k 2 ⃗ A 2 k − ( ⃗ k · ⃗ A$_{k}$ ) 2 + µ 2 A 0 k 2 + µ 2 ⃗ A 2 $_{k}$. (F.76) First let’s prove that the coefficient of the cos term is 0. It’s clear that the