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Knecht and Urech's result for general n = N _ f:

diff1 = (Collect[endres1 - knechtres1 // NMExpand // Expand // CycleUTraces, _UTrace1] // UReduce // IndicesCleanup // Expand) /. {GLeft[LorentzIndex[μ2], LorentzIndex[μ1]][x] -> -GLeft[LorentzIndex[μ1], LorentzIndex[μ2]][x], GRight[LorentzIndex[μ2], LorentzIndex[μ1]][x] -> -GRight[LorentzIndex[μ1], LorentzIndex[μ2]][x]} /. NM[Adjoint[CovariantFieldDerivative[MM[x], x, LorentzIndex[μ2]]], GRight[LorentzIndex[μ1], LorentzIndex[μ2]][x], CovariantFieldDerivative[MM[x], x, LorentzIndex[μ1]]] -> -NM[Adjoint[CovariantFieldDerivative[MM[x], x, LorentzIndex[μ1]]], GRight[LorentzIndex[μ1], LorentzIndex[μ2]][x], CovariantFieldDerivative[MM[x], x, LorentzIndex[μ2]]]

1/24 < G _ (μ _ 1 μ _ 2)^L >^2 + 1/12 < G _ (μ _ 1 μ _ 2)^R > < G _ (μ _ 1 μ _ 2)^L > + 1/24 < G _ (μ _ 1 μ _ 2)^R >^2 - 1/6 γ _ (μ _ 1 μ _ 2)^2 < Q >^2

UTrace1[GRight[LorentzIndex[μ1], LorentzIndex[μ2]][x]]^2

< G _ (μ _ 1 μ _ 2)^R >^2

(UTrace1[GRight[LorentzIndex[μ1], LorentzIndex[μ2]][x]]^2 //. Select[$Substitutions, FreeQ[#, _ ? (! FreeQ[#, UChiralSpurion | UChiralSpurionRight | UChiralSpurionLeft, Heads -> True] &) :> _, Heads -> True] &] // NMExpand // Expand // CommutatorReduce) /. UTrace1[UMatrix[(UChiralSpurionRight | UChiralSpurionLeft)[]][x]] -> UTrace1[UMatrix[UChiralSpurion[]][x]] /. UTrace1[FieldDerivative[UMatrix[(UChiralSpurionRight | UChiralSpurionLeft)[]][x], x, LorentzIndex[_]]] -> 0

(∂ _ μ _ 1 γ^( ) _ μ _ 2^ó )^2 < Q >^2 + (∂ _ μ _ 2 γ^( ) _ μ _ 1^ó )^2 < Q >^2 - 2 ∂ _ μ _ 1 γ^( ) _ μ _ 2^ó  ∂ _ μ _ 2 γ^( ) _ μ _ 1^ó  < Q >^2

FieldStrengthTensor[LorentzIndex[μ1], QuantumField[Particle[Vector[1]], LorentzIndex[μ2]][x], x, Explicit -> True]^2 UTrace1[UMatrix[UChiralSpurion[]][x]]^2 // Expand

(∂ _ μ _ 1 γ^( ) _ μ _ 2^ó )^2 < Q >^2 + (∂ _ μ _ 2 γ^( ) _ μ _ 1^ó )^2 < Q >^2 - 2 ∂ _ μ _ 1 γ^( ) _ μ _ 2^ó  ∂ _ μ _ 2 γ^( ) _ μ _ 1^ó  < Q >^2

%% - % // Simplify

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Converted by Mathematica  (July 10, 2003)