urechsu3res = Collect[3/32 * UTrace[NM[CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]], Adjoint[CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]]]]] * UTrace[NM[CovariantFieldDerivative[MM[x], x, LorentzIndex[ν]], Adjoint[CovariantFieldDerivative[MM[x], x, LorentzIndex[ν]]]]] + 3/16 * UTrace[NM[Adjoint[CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]]], CovariantFieldDerivative[MM[x], x, LorentzIndex[ν]]]]^2 + 1/8 * UTrace[NM[Adjoint[CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]]], CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]]]] * (UTrace[NM[Adjoint[MM[x]], UMatrix[UChi[]][x]]] + UTrace[NM[Adjoint[UMatrix[UChi[]][x]], MM[x]]]) + 3/8 * UTrace[NM[Adjoint[CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]]], CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]], NM[Adjoint[MM[x]], UMatrix[UChi[]][x]] + NM[Adjoint[UMatrix[UChi[]][x]], MM[x]]]] + 11/144 * (UTrace[NM[Adjoint[MM[x]], UMatrix[UChi[]][x]]] + UTrace[NM[Adjoint[UMatrix[UChi[]][x]], MM[x]]])^2 + 5/48 * UTrace[NM[Adjoint[UMatrix[UChi[]][x]], MM[x], Adjoint[UMatrix[UChi[]][x]], MM[x]] + NM[Adjoint[MM[x]], UMatrix[UChi[]][x], Adjoint[MM[x]], UMatrix[UChi[]][x]]] - I/4 * (UTrace[NM[GLeft[LorentzIndex[μ], LorentzIndex[ν]][x], Adjoint[CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]]], CovariantFieldDerivative[MM[x], x, LorentzIndex[ν]]] + NM[GRight[LorentzIndex[μ], LorentzIndex[ν]][x], CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]], Adjoint[CovariantFieldDerivative[MM[x], x, LorentzIndex[ν]]]]]) - 1/4 * UTrace[NM[GRight[LorentzIndex[μ], LorentzIndex[ν]][x], MM[x], GLeft[LorentzIndex[μ], LorentzIndex[ν]][x], Adjoint[MM[x]]]] - 1/8 * (UTrace[NM[GLeft[LorentzIndex[μ], LorentzIndex[ν]][x], GLeft[LorentzIndex[μ], LorentzIndex[ν]][x]]] + UTrace[NM[GRight[LorentzIndex[μ], LorentzIndex[ν]][x], GRight[LorentzIndex[μ], LorentzIndex[ν]][x]]]) + 5/24 * UTrace[NM[Adjoint[UMatrix[UChi[]][x]], UMatrix[UChi[]][x]]] + 3/8 * f^2 * UTrace[NM[Adjoint[CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]]], CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]]]] * UTrace[NM[UMatrix[UChiralSpurionRight[]][x], UMatrix[UChiralSpurionRight[]][x]] + NM[UMatrix[UChiralSpurionLeft[]][x], UMatrix[UChiralSpurionLeft[]][x]]] + c/f^2 * UTrace[NM[Adjoint[CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]]], CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]]]] * UTrace[NM[UMatrix[UChiralSpurionRight[]][x], MM[x], UMatrix[UChiralSpurionLeft[]][x], Adjoint[MM[x]]]] - 3/4 * f^2 * UTrace[NM[Adjoint[CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]]], UMatrix[UChiralSpurionRight[]][x], MM[x]]]^2 - 3/4 * f^2 * UTrace[NM[CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]], UMatrix[UChiralSpurionLeft[]][x], Adjoint[MM[x]]]]^2 + 2 c/f^2 * UTrace[NM[Adjoint[CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]]], UMatrix[UChiralSpurionRight[]][x], MM[x]]] * UTrace[NM[CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]], UMatrix[UChiralSpurionLeft[]][x], Adjoint[MM[x]]]] - 9/4 * f^2 * UTrace[NM[Adjoint[CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]]], CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]], UMatrix[UChiralSpurionLeft[]][x], UMatrix[UChiralSpurionLeft[]][x]] + NM[CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]], Adjoint[CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]]], UMatrix[UChiralSpurionRight[]][x], UMatrix[UChiralSpurionRight[]][x]]] + (-1/4 f^2 + 3/2 c/f^2) UTrace[NM[Adjoint[CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]]], CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]], UMatrix[UChiralSpurionLeft[]][x], Adjoint[MM[x]], UMatrix[UChiralSpurionRight[]][x], MM[x]] + NM[CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]], Adjoint[CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]]], UMatrix[UChiralSpurionRight[]][x], MM[x], UMatrix[UChiralSpurionLeft[]][x], Adjoint[MM[x]]]] + 1/2 * f^2 * UTrace[NM[Adjoint[CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]]], UMatrix[UChiralSpurionRight[]][x], CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]], UMatrix[UChiralSpurionLeft[]][x]]] + c/f^2 * UTrace[NM[Adjoint[MM[x]], UMatrix[UChi[]][x]] + NM[Adjoint[UMatrix[UChi[]][x]], MM[x]]] * UTrace[NM[UMatrix[UChiralSpurionRight[]][x], MM[x], UMatrix[UChiralSpurionLeft[]][x], Adjoint[MM[x]]]] - 1/4 * f^2 * UTrace[NM[NM[Adjoint[MM[x]], UMatrix[UChi[]][x]] + NM[Adjoint[UMatrix[UChi[]][x]], MM[x]], UMatrix[UChiralSpurionLeft[]][x], UMatrix[UChiralSpurionLeft[]][x]] + NM[NM[MM[x], Adjoint[UMatrix[UChi[]][x]]] + NM[UMatrix[UChi[]][x], Adjoint[MM[x]]], UMatrix[UChiralSpurionRight[]][x], UMatrix[UChiralSpurionRight[]][x]]] + (1/4 f^2 + 3/2 c/f^2) * UTrace[NM[NM[Adjoint[MM[x]], UMatrix[UChi[]][x]] + NM[Adjoint[UMatrix[UChi[]][x]], MM[x]], UMatrix[UChiralSpurionLeft[]][x], Adjoint[MM[x]], UMatrix[UChiralSpurionRight[]][x], MM[x]] + NM[NM[MM[x], Adjoint[UMatrix[UChi[]][x]]] + NM[UMatrix[UChi[]][x], Adjoint[MM[x]]], UMatrix[UChiralSpurionRight[]][x], MM[x], UMatrix[UChiralSpurionLeft[]][x], Adjoint[MM[x]]]] + 3/8 * f^2 * UTrace[NM[CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]], CQLeft[LorentzIndex[μ]][x], Adjoint[MM[x]], UMatrix[UChiralSpurionRight[]][x]] + NM[Adjoint[CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]]], CQRight[LorentzIndex[μ]][x], MM[x], UMatrix[UChiralSpurionLeft[]][x]]] + 3/8 * f^2 * UTrace[NM[Adjoint[CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]]], UMatrix[UChiralSpurionRight[]][x], MM[x], CQLeft[LorentzIndex[μ]][x]] + NM[CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]], UMatrix[UChiralSpurionLeft[]][x], Adjoint[MM[x]], CQRight[LorentzIndex[μ]][x]]] + 3/8 * f^2 * UTrace[NM[Adjoint[CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]]], UCommutator[CQRight[LorentzIndex[μ]][x], UMatrix[UChiralSpurionRight[]][x]], MM[x]] + NM[CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]], UCommutator[CQLeft[LorentzIndex[μ]][x], UMatrix[UChiralSpurionLeft[]][x]], Adjoint[MM[x]]]] + (3/2 * f^4 + 2 * c + 20 * c^2/f^4) * UTrace[NM[UMatrix[UChiralSpurionRight[]][x], MM[x], UMatrix[UChiralSpurionLeft[]][x], Adjoint[MM[x]]]]^2 - (6 * c + 24 * c^2/f^4) * UTrace[NM[UMatrix[UChiralSpurionRight[]][x], UMatrix[UChiralSpurionRight[]][x], MM[x], UMatrix[UChiralSpurionLeft[]][x], UMatrix[UChiralSpurionLeft[]][x], Adjoint[MM[x]]]] - 3/2 * f^4 * UTrace[NM[UMatrix[UChiralSpurionRight[]][x], UMatrix[UChiralSpurionRight[]][x]] + NM[UMatrix[UChiralSpurionLeft[]][x], UMatrix[UChiralSpurionLeft[]][x]]] * UTrace[NM[UMatrix[UChiralSpurionRight[]][x], MM[x], UMatrix[UChiralSpurionLeft[]][x], Adjoint[MM[x]]]] + 3/8 * f^4 (UTrace[NM[UMatrix[UChiralSpurionLeft[]][x], UMatrix[UChiralSpurionLeft[]][x]]]^2 + UTrace[NM[UMatrix[UChiralSpurionRight[]][x], UMatrix[UChiralSpurionRight[]][x]]]^2) + (3/4 * f^4 + c + 10 * c^2/f^4) * UTrace[NM[UMatrix[UChiralSpurionLeft[]][x], UMatrix[UChiralSpurionLeft[]][x]]] * UTrace[NM[UMatrix[UChiralSpurionRight[]][x], UMatrix[UChiralSpurionRight[]][x]]] /. {μ -> μ1, ν -> μ2, f -> DecayConstant[PhiMeson], z -> CouplingConstant[ChPTVirtualPhotons3[2]]/DecayConstant[PhiMeson]^4, c -> CouplingConstant[ChPTVirtualPhotons3[2]]} /. NM -> nm //. (* Transform back to Minkowski space *) nm[a__] :> (-1)^(Count[{a}, LorentzIndex[__], Infinity, Heads -> True]/2) * NM[a] // Expand // UReduce, _UTrace1]

3/8 < Q _ L '6 Q _ L >^2 (f _ ϕ^(ó  ))^4 + 3/8 < Q _ R '6 Q _ R >^2 (f _ ϕ^(ó  ))^4 - 3/2 < Q _ R '6 Q _ R > < ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L > (f _ ϕ^(ó  ))^4 + 3/4 < ÷s _ μ _ 1(÷„)^† '6 Q _ R '6 ÷„ >^2 (f _ ϕ^(ó  ))^2 + 3/4 < ÷„^† '6 ÷s _ μ _ 1(÷„) '6 Q _ L >^2 (f _ ϕ^(ó  ))^2 + 9/4 < ÷s _ μ _ 1(÷„)^† '6 ÷s _ μ _ 1(÷„) '6 Q _ L '6 Q _ L > (f _ ϕ^(ó  ))^2 - 3/8 < ÷s _ μ _ 1(÷„)^† '6 c _ μ _ 1^R Q _ R '6 ÷„ '6 Q _ L > (f _ ϕ^(ó  ))^2 + 3/8 < ÷s _ μ _ 1(÷„)^† '6 c _ μ _ 1^R Q _ R '6 Q _ R '6 ÷„ > (f _ ϕ^(ó  ))^2 - 1/2 < ÷s _ μ _ 1(÷„)^† '6 Q _ R '6 ÷s _ μ _ 1(÷„) '6 Q _ L > (f _ ϕ^(ó  ))^2 - 3/8 < ÷s _ μ _ 1(÷„)^† '6 Q _ R '6 ÷„ '6 c _ μ _ 1^L Q _ L > (f _ ϕ^(ó  ))^2 - 3/8 < ÷s _ μ _ 1(÷„)^† '6 Q _ R '6 c _ μ _ 1^R Q _ R '6 ÷„ > (f _ ϕ^(ó  ))^2 + 9/4 < ÷s _ μ _ 1(÷„)^† '6 Q _ R '6 Q _ R '6 ÷s _ μ _ 1(÷„) > (f _ ϕ^(ó  ))^2 + 3/8 < ÷„^† '6 ÷s _ μ _ 1(÷„) '6 c _ μ _ 1^L Q _ L '6 Q _ L > (f _ ϕ^(ó  ))^2 - 3/8 < ÷„^† '6 ÷s _ μ _ 1(÷„) '6 Q _ L '6 c _ μ _ 1^L Q _ L > (f _ ϕ^(ó  ))^2 - 3/8 < ÷„^† '6 c _ μ _ 1^R Q _ R '6 ÷s _ μ _ 1(÷„) '6 Q _ L > (f _ ϕ^(ó  ))^2 - 1/4 < ÷„^† '6 χ '6 Q _ L '6 Q _ L > (f _ ϕ^(ó  ))^2 - 3/8 < ÷„^† '6 Q _ R '6 ÷s _ μ _ 1(÷„) '6 c _ μ _ 1^L Q _ L > (f _ ϕ^(ó  ))^2 - 1/4 < ÷„^† '6 Q _ R '6 Q _ R '6 χ > (f _ ϕ^(ó  ))^2 - 1/4 < χ^† '6 ÷„ '6 Q _ L '6 Q _ L > (f _ ϕ^(ó  ))^2 - 1/4 < χ^† '6 Q _ R '6 Q _ R '6 ÷„ > (f _ ϕ^(ó  ))^2 + 3/16 < ÷s _ μ _ 2(÷„)^† '6 ÷s _ μ _ 1(÷„) >^2 + 11/144 < ÷„^† '6 χ >^2 + 11/144 < χ^† '6 ÷„ >^2 + (3/2 (f _ ϕ^(ó  ))^4 + 2 C^( ) + (20 (C^( ))^2)/(f _ ϕ^(ó  ))^4) < ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L >^2 + 5/24 < χ^† '6 χ > - 1/8 < G _ (μ _ 1 μ _ 2)^L '6 G _ (μ _ 1 μ _ 2)^L > - 1/8 < G _ (μ _ 1 μ _ 2)^R '6 G _ (μ _ 1 μ _ 2)^R > - 1/4 i < ÷s _ μ _ 1(÷„)^† '6 ÷s _ μ _ 2(÷„) '6 G _ (μ _ 1 μ _ 2)^L > - 1/4 i < ÷s _ μ _ 2(÷„)^† '6 G _ (μ _ 1 μ _ 2)^R '6 ÷s _ μ _ 1(÷„) > - 3/8 < ÷s _ μ _ 1(÷„)^† '6 ÷s _ μ _ 1(÷„) '6 ÷„^† '6 χ > - 3/8 < ÷s _ μ _ 1(÷„)^† '6 ÷s _ μ _ 1(÷„) '6 χ^† '6 ÷„ > - 1/4 < ÷„^† '6 G _ (μ _ 1 μ _ 2)^R '6 ÷„ '6 G _ (μ _ 1 μ _ 2)^L > + 5/48 < ÷„^† '6 χ '6 ÷„^† '6 χ > + < Q _ L '6 Q _ L > ((3/4 (f _ ϕ^(ó  ))^4 + C^( ) + (10 (C^( ))^2)/(f _ ϕ^(ó  ))^4) < Q _ R '6 Q _ R > - 3/2 (f _ ϕ^(ó  ))^4 < ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L >) + < ÷s _ μ _ 1(÷„)^† '6 ÷s _ μ _ 1(÷„) > (-3/8 < Q _ L '6 Q _ L > (f _ ϕ^(ó  ))^2 - 3/8 < Q _ R '6 Q _ R > (f _ ϕ^(ó  ))^2 + 3/32 < ÷s _ μ _ 2(÷„)^† '6 ÷s _ μ _ 2(÷„) > - 1/8 < ÷„^† '6 χ > - 1/8 < χ^† '6 ÷„ > - (C^( ) < ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L >)/(f _ ϕ^(ó  ))^2) + < ÷„^† '6 χ > (11/72 < χ^† '6 ÷„ > + (C^( ) < ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L >)/(f _ ϕ^(ó  ))^2) + (1/4 (f _ ϕ^(ó  ))^2 + (3 C^( ))/(2 (f _ ϕ^(ó  ))^2)) < ÷„^† '6 Q _ R '6 χ '6 Q _ L > + 5/48 < χ^† '6 ÷„ '6 χ^† '6 ÷„ > + (1/4 (f _ ϕ^(ó  ))^2 + (3 C^( ))/(2 (f _ ϕ^(ó  ))^2)) < χ^† '6 Q _ R '6 ÷„ '6 Q _ L > + (1/4 (f _ ϕ^(ó  ))^2 - (3 C^( ))/(2 (f _ ϕ^(ó  ))^2)) < ÷s _ μ _ 1(÷„)^† '6 ÷s _ μ _ 1(÷„) '6 Q _ L '6 ÷„^† '6 Q _ R '6 ÷„ > + (1/4 (f _ ϕ^(ó  ))^2 - (3 C^( ))/(2 (f _ ϕ^(ó  ))^2)) < ÷s _ μ _ 1(÷„)^† '6 Q _ R '6 ÷„ '6 Q _ L '6 ÷„^† '6 ÷s _ μ _ 1(÷„) > + (1/4 (f _ ϕ^(ó  ))^2 + (3 C^( ))/(2 (f _ ϕ^(ó  ))^2)) < ÷„^† '6 Q _ R '6 ÷„ '6 χ^† '6 ÷„ '6 Q _ L > + (-(24 (C^( ))^2)/(f _ ϕ^(ó  ))^4 - 6 C^( )) < ÷„^† '6 Q _ R '6 Q _ R '6 ÷„ '6 Q _ L '6 Q _ L > + (1/4 (f _ ϕ^(ó  ))^2 + (3 C^( ))/(2 (f _ ϕ^(ó  ))^2)) < ÷„ '6 Q _ L '6 ÷„^† '6 χ '6 ÷„^† '6 Q _ R > - (2 C^( ) < ÷s _ μ _ 1(÷„)^† '6 Q _ R '6 ÷„ > < ÷„^† '6 ÷s _ μ _ 1(÷„) '6 Q _ L >)/(f _ ϕ^(ó  ))^2 + (C^( ) < χ^† '6 ÷„ > < ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L >)/(f _ ϕ^(ó  ))^2

Converted by Mathematica  (July 10, 2003)