237 Appendix F: Heavy Quark Production Radiation Derivation Then We immediately arrive at our final result: When we include an external current in the Proca Lagrangian we have The EOM are then Proca $^{∑}$ϵ$_{µ}$ ϵ ∗ $_{ν}$A$^{µ}$A ∗ ν = | A$_{1}$ | 2 + | A$_{2}$ | 2 = − | A$_{0}$ | 2 + | A$_{1}$ | 2 + | A$_{2}$ | 2 + | A$_{3}$ | 2 = − g$_{µν}$A $^{µ}$A ∗ $^{ν}$. (F.58) E$_{EM}$ = ∫ d $^{3}$k (2 π ) 3 e 2 2 ( − g$_{µν}$ )$^{(}$ p ′ µ k · p ′ − p µ k · p$^{) (}$ p ′ ν k · p ′ − p ν k · p$^{)}$ = ∫ d $^{3}$k (2 π ) 3 e 2 2 ( 2 p · p ′ ( k · p $^{′}$)( k · p ) − m 2 ( k · p $^{′}$) 2 − m 2 ( k · p ) 2 ) . (F.59) L$_{Proca}$ = − 1 4 F $^{µν}$F$_{µν}$ + µ 2 2 A $^{µ}$A$_{µ}$ − j$_{µ}$A $^{µ}$. (F.60) ∂$_{α}$F αβ + µ $^{2}$A β = j $^{β}$. (F.61)