where u and v are positive and negative energy solutions of Dirac equation. Momentum Polarization vector where q is reference momentum, reflecting the freedom of on-shell gauge transformation, k is the vector boson momentum. Useful properties: antisymmetry Fierz rearrangement charge conjugation eikonal identity in an N -point amplitude, momentum conservation imply 25 〈 ij 〉 ≡ 〈 i $^{−}$| j $^{+}$〉 = √ | s$_{ij}$ | e $^{iφ$_{ij}$}$, [ ij ] ≡ 〈 i $^{+}$| j $^{−}$〉 = | s$_{ij}$ | e − i ( φ$_{ij}$ + π $^{)}$. (A2) 〈 i $^{±}$| γ $^{µ}$| i $^{±}$〉 = 2 k$_{i}$. (A3) ϵ ± $_{µ}$( k, q ) = ± 〈 q $^{±}$| γ$_{µ}$ | k $^{∓}$〉 √$_{2}$ 〈 q $^{∓}$| k $^{±}$〉 , (A4) 〈 ij 〉 [ ji ] = s$_{ij}$ (A5) 〈 ij 〉 = −〈 ji 〉 , [ ij ] = − [ ji ] , 〈 ii 〉 = [ ii ] = 0 , (A6) 〈 i $^{+}$| γ $^{µ}$| j $^{+}$〉〈 k $^{+}$| γ$_{µ}$ | l $^{+}$〉 = 2[ ik ] 〈 lj 〉 , (A7) 〈 i $^{+}$| γ $^{µ}$| j $^{+}$〉 = 〈 j $^{−}$| γ $^{µ}$| i $^{−}$〉 , (A8) k − 1 ∑ i = j 〈 i, i + 1 〉 〈 iq 〉〈 q, i + 1 〉 = 〈 jk 〉 〈 jq 〉〈 qk 〉 . (A9) n ∑ i =1 ,i = k [ ji ] 〈 ik 〉 = 0 . (A10)