Configuration:  "ChPT2"

•UGammaTrick

SetOptions[#, Explicit -> False] & /@ {MM, SMM, UChi, FieldStrengthTensorFull, FieldStrengthTensor, LeftComponent, RightComponent} ;

IsoDot[IsoVector[QuantumField[Particle[lr : (LeftComponent[0] | RightComponent[0])], r___]][x_], IsoVector[UMatrix[UGenerator[]]]] := QuantumField[Particle[lr], r][x]

UTrace1[FieldStrengthTensorFull[LorentzIndex[_], QuantumField[Particle[LeftComponent[0] | RightComponent[0]], LorentzIndex[_]][x_], _, ___]] := 0 ;

$UMatrices = Union[$UMatrices, {LeftComponent[0], RightComponent[0]}] ;

(7.14) and (7.20) from Gasser and Leutwyler (1985). Changed to the Left/Right and fieldstrength conventions of Donoghue et al.

gammaRule = FieldDerivative[UGamma[LorentzIndex[μ2]][x], x, LorentzIndex[μ1]] -> FieldDerivative[UGamma[LorentzIndex[μ1]][x], x, LorentzIndex[μ2]] + UCommutator[UGamma[LorentzIndex[μ2]][x], UGamma[LorentzIndex[μ1]][x]] - 1/4 UCommutator[USmall[LorentzIndex[μ2]][x], USmall[LorentzIndex[μ1]][x]] - 1/2 I NM[Adjoint[SMM[x]], FieldStrengthTensorFull[{μ2}, QuantumField[Particle[LeftComponent[0]], LorentzIndex[μ1]][x], x, I], SMM[x]] - 1/2 I NM[SMM[x], FieldStrengthTensorFull[{μ2}, QuantumField[Particle[RightComponent[0]], LorentzIndex[μ1]][x], x, I], Adjoint[SMM[x]]]

∂ _ μ _ 1(Γ _ μ _ 2) -> ∂ _ μ _ 2(Γ _ μ _ 1) - Γ _ μ _ 1 '6 Γ _ μ _ 2 + Γ _ μ _ 2 '6 Γ _ μ _ 1 + 1/4 (u _ μ _ 1 '6 u _ μ _ 2 - u _ μ _ 2 '6 u _ μ _ 1) - 1/2 i (öÆ^† '6 L^( ) _ (μ _ 2 μ _ 1) '6 öÆ) - 1/2 i (öÆ '6 R^( ) _ (μ _ 2 μ _ 1) '6 öÆ^†)

gammaRule[[1]] - gammaRule[[2]] // UGammaTrick // NMExpand // Simplify

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SetOptions[#, Explicit -> True] & /@ {FieldStrengthTensorFull, FieldStrengthTensor, CovariantFieldDerivative} ;

-gammaRule[[1]] + gammaRule[[2]] //. $Substitutions /. MM[x] -> NM[SMM[x], SMM[x]] // NMExpand // Expand // SUNReduce // UReduce // NMExpand // Expand // IndicesCleanup // CycleUTraces

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Converted by Mathematica  (July 10, 2003)