249 Appendix F: Heavy Quark Production Radiation Derivation What we will do is to solve a problem more relevant to finding the zeroth order contribution than bremsstrahlung radiation: beta decay radiation (pair production). We now take as our current a (charge) pair production we have allowed ourselves the freedom to have both difference masses and different momenta for the two (opposite) charges. The Fourier transform of our current is The only difference classically from the bremsstrahlung result is that the ˜ j $^{µ}$( k ) = ∫ d $^{4}$xe ik · $^{x}$j$^{µ}$( x ) = e lim ϵ → 0 + ∫ ∞ 0 dτ [ p µ 1 m$_{1}$ e τ $^{(}$− ϵ + i k · p 1 m 1 ) − p µ 2 m$_{2}$ e τ $^{(}$− ϵ + i k · p 2 m 2 ) ] = ie$^{[}$ p$_{1}$ k · p$_{1}$ + iϵ − p$_{2}$ k · p$_{2}$ + iϵ$^{]}$ . (F.102) y µ 1 , $_{2}$( τ ) =      0 , τ < 0 p µ 1 , 2 $_{m$_{1}$$_{,}$$_{2}$}$τ, τ > 0 ; (F.101) j $^{µ}$( x ) = e$^{∫}$ dτ $^{[}$dy µ $_{1}$( τ ) dτ δ (4) $^{(}$( x − y$_{1}$ ( τ ) ) − dy µ $_{2}$( τ ) dτ δ (4) $^{(}$( x − y$_{2}$ ( τ ) ] , (F.100) where we have F.4 Pair Production F.4.1 Classical