248 Appendix F: Heavy Quark Production Radiation Derivation with the other momenta the same, is In the case of a massless field the QFT and classical results are necessarily the same; we have Everything is consistent with taking m$_{γ}$ → 0 from our previous result. Again we see the overall factor of (1 − x ) 2 out front. We’re tempted to simply take M$_{Q}$ → 0 to recover the completely massless limit. However in the massless limit we must actually solve a completely different problem: for a massless charge we cannot take it to be at rest for t < 0. Perhaps of anecdotal interest the totally massless with recoil bremsstrahlung result, using this is clearly not what we are interested in. is dE d $^{3}$k = C x 2 k $_{2}$; (F.99) k = [ xE $^{+}$, k 2 + m 2 γ xE + , k ] , (F.98) p ′ = [(1 − x ) E $^{+}$, k 2 + M 2 Q (1 − x ) E $_{+}$, − k ] (F.97) p = [ E $^{+}$, 0 , 0 ] (F.96) dE d $^{3}$k = C 4(1 − x ) $^{2}$k2 $^{(}$k 2 + x $^{2}$M 2 Q ) $_{2}$. (F.95) dE d $^{3}$k = C 4 $^{(}$(1 + x ) $^{2}$k2 + m 2 γ ) $^{(}$(1 + x ) $^{2}$k2 + m 2 γ + x $^{2}$M 2 Q ) $_{2}$. (F.94) F.3.3 Massive Charge and Massless Field with Recoil