Appendix E: Standard pQCD Results 216 Therefore Compute electron (mass m = 0) beta decay starting with Eq. (F.84), We will want to exploit the replacement $^{∑}$ϵ$_{µ}$ ϵ ∗ ν → − g$_{µν}$ so we will have to preserve the Ward identity (charge conservation), To do so we will have to keep the second term explicitly with P$_{M}$ ≃ [ M, M, 0 , 0] being the momentum of the “proton” after (as well as before) beta emission. Then (essentially Eq. (F.86)), ∑ λ |M| 2 = ( Qe ) $^{2}$( − g$_{µν}$ )$^{(}$ p µ k · p − P µ M k · P$_{M}$$^{)(}$ p ν k · p − P ν M k · P$_{M}$$^{)}$ = ( Qe ) 2 ( 2 p · P$_{M}$ ( k · p )( k · P$_{M}$ ) − M 2 ( k · P$_{M}$ ) 2 − m 2 ( k · p ) 2 ) (E.28) k$_{µ}$$^{(}$ p µ k · p − P µ k · P$^{)}$ = 0 . (E.27) i M = ¯ u ( p ) [ M$_{0}$ ( p, P )] u ( P ) ·$^{[}$ Qe$^{(}$ p · ϵ ∗ p · k − P · ϵ ∗ P · k$^{)]}$ . (E.26) ∑ λ |M$_{β}$| = 4( Qe ) 2 k 2 . (E.25) E.2.3 Soft Photon Approximation, Massless Dijet