Knecht and Urech's result for general
:
Converted by
Mathematica
(July 10, 2003)
::![knechtres = n/48 * UTrace[NM[CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]], Adjoint[CovariantFieldDerivative[MM[x], x, LorentzIndex[ν]]], CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]], Adjoint[CovariantFieldDerivative[MM[x], x, LorentzIndex[ν]]]]] + n/24 * UTrace[NM[CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]], Adjoint[CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]]], CovariantFieldDerivative[MM[x], x, LorentzIndex[ν]], Adjoint[CovariantFieldDerivative[MM[x], x, LorentzIndex[ν]]]]] + 1/8 * UTrace[NM[Adjoint[CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]]], CovariantFieldDerivative[MM[x], x, LorentzIndex[ν]]]]^2 + 1/16 * 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]]]]] - I * n/12 * (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]]]]]) - n/12 * UTrace[NM[GRight[LorentzIndex[μ], LorentzIndex[ν]][x], MM[x], GLeft[LorentzIndex[μ], LorentzIndex[ν]][x], Adjoint[MM[x]]]] - n/24 (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]]]) + n/8 * UTrace[NM[Adjoint[CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]]], CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]], NM[Adjoint[UMatrix[UChi[]][x]], MM[x]] + NM[Adjoint[MM[x]], UMatrix[UChi[]][x]]]] + 1/8 * UTrace[NM[Adjoint[CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]]], CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]]]] * (UTrace[NM[Adjoint[UMatrix[UChi[]][x]], MM[x]]] + UTrace[NM[Adjoint[MM[x]], UMatrix[UChi[]][x]]]) + (n^2 - 4)/16/n * 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]]] + (n^2 + 2)/16/n^2 * (UTrace[NM[Adjoint[MM[x]], UMatrix[UChi[]][x]]] + UTrace[NM[Adjoint[UMatrix[UChi[]][x]], MM[x]]])^2 + (n^2 - 4)/8/n * UTrace[NM[Adjoint[UMatrix[UChi[]][x]], UMatrix[UChi[]][x]]] + 1/6 * FieldStrengthTensor[LorentzIndex[μ], QuantumField[Vector[1], LorentzIndex[ν]][x], x, Explicit -> False]^2 * UTrace[UMatrix[UChiralSpurion[]][x]]^2 - 3/4 * f^2 * (UTrace[NM[Adjoint[CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]]], UMatrix[UChiralSpurionRight[]][x], MM[x], 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]], CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]], UMatrix[UChiralSpurionLeft[]][x], Adjoint[MM[x]]]]) - 3/4 * f^2 * (UTrace[NM[Adjoint[CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]]], CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]], UMatrix[UChiralSpurionLeft[]][x], UMatrix[UChiralSpurionLeft[]][x]]] + UTrace[NM[CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]], Adjoint[CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]]], UMatrix[UChiralSpurionRight[]][x], UMatrix[UChiralSpurionRight[]][x]]]) - f^2 * (1/4 - n z/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]]] + UTrace[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]]]]) + z * f^2 * UTrace[NM[CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]], Adjoint[CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]]]]] * UTrace[NM[UMatrix[UChiralSpurionRight[]][x], MM[x], UMatrix[UChiralSpurionLeft[]][x], Adjoint[MM[x]]]] - z * f^2 * (UTrace[NM[Adjoint[CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]]], CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]], UMatrix[UChiralSpurionLeft[]][x] (* , Adjoint[MM[x]] *) (* Misprint !! - extra Adjoint[MM[x]] *)]] + UTrace[NM[CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]], Adjoint[CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]]], UMatrix[UChiralSpurionRight[]][x] (* , MM[x] *) (* Misprint !! - extra MM[x] *)]]) * UTrace[UMatrix[UChiralSpurion[]][x]] + f^2/2 * UTrace[NM[Adjoint[CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]]], UMatrix[UChiralSpurionRight[]][x], CovariantFieldDerivative[MM[x], x, LorentzIndex[μ]], UMatrix[UChiralSpurionLeft[]][x]]] + 2 * z * 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]]]] - f^2/4 * UTrace[NM[NM[MM[x], Adjoint[UMatrix[UChi[]][x]]] + NM[UMatrix[UChi[]][x], Adjoint[MM[x]]], UMatrix[UChiralSpurionRight[]][x], UMatrix[UChiralSpurionRight[]][x]] + NM[NM[Adjoint[MM[x]], UMatrix[UChi[]][x]] + NM[Adjoint[UMatrix[UChi[]][x]], MM[x]], UMatrix[UChiralSpurionLeft[]][x], UMatrix[UChiralSpurionLeft[]][x]]] + f^2 * (1/4 + n z/2) * UTrace[NM[Adjoint[UMatrix[UChi[]][x]], UMatrix[UChiralSpurionRight[]][x], MM[x], UMatrix[UChiralSpurionLeft[]][x]] + NM[UMatrix[UChi[]][x], UMatrix[UChiralSpurionLeft[]][x], Adjoint[MM[x]], UMatrix[UChiralSpurionRight[]][x]]] + f^2 * (1/4 + n z/2) * UTrace[NM[Adjoint[UMatrix[UChi[]][x]], MM[x], UMatrix[UChiralSpurionLeft[]][x], Adjoint[MM[x]], UMatrix[UChiralSpurionRight[]][x], MM[x]] + NM[UMatrix[UChi[]][x], Adjoint[MM[x]], UMatrix[UChiralSpurionRight[]][x], MM[x], UMatrix[UChiralSpurionLeft[]][x], Adjoint[MM[x]]]] - z * f^2 * UTrace[NM[NM[MM[x], Adjoint[UMatrix[UChi[]][x]]] + NM[UMatrix[UChi[]][x], Adjoint[MM[x]]], UMatrix[UChiralSpurionRight[]][x]] + NM[NM[Adjoint[UMatrix[UChi[]][x]], MM[x]] + NM[Adjoint[MM[x]], UMatrix[UChi[]][x]], UMatrix[UChiralSpurionLeft[]][x]]] UTrace[UMatrix[UChiralSpurion[]][x]] + z * f^2 * UTrace[NM[Adjoint[UMatrix[UChi[]][x]], MM[x]] + NM[Adjoint[MM[x]], UMatrix[UChi[]][x]]] * UTrace[NM[UMatrix[UChiralSpurionRight[]][x], MM[x], UMatrix[UChiralSpurionLeft[]][x], Adjoint[MM[x]]]] + f^2/4 * 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]]]] + f^2/4 * 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]]] + (* Forgotten by Knecht & Urech *) f^2/4 * 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]]] + (2 * z + 2 * n * z^2) * f^4 * UTrace[NM[UMatrix[UChiralSpurionRight[]][x], MM[x], UMatrix[UChiralSpurionLeft[]][x], Adjoint[MM[x]], UMatrix[UChiralSpurionRight[]][x], MM[x], UMatrix[UChiralSpurionLeft[]][x], Adjoint[MM[x]]]] - (2 * z - 2 * n * z^2) * (* Misprint !! - forgotten f^4 *) f^4 * UTrace[NM[UMatrix[UChiralSpurionRight[]][x], UMatrix[UChiralSpurionRight[]][x], MM[x], UMatrix[UChiralSpurionLeft[]][x], UMatrix[UChiralSpurionLeft[]][x], Adjoint[MM[x]]]] - 8 * z^2 f^4 UTrace[NM[UMatrix[UChiralSpurionLeft[]][x], UMatrix[UChiralSpurionLeft[]][x], Adjoint[MM[x]], UMatrix[UChiralSpurionRight[]][x], MM[x]] + NM[UMatrix[UChiralSpurionRight[]][x], UMatrix[UChiralSpurionRight[]][x], MM[x], UMatrix[UChiralSpurionLeft[]][x], Adjoint[MM[x]]]] * UTrace[UMatrix[UChiralSpurion[]][x]] + (3/2 + 8 * z^2) * f^4 * UTrace[NM[UMatrix[UChiralSpurionRight[]][x], MM[x], UMatrix[UChiralSpurionLeft[]][x], Adjoint[MM[x]]]]^2 - 3 * f^4/2 * 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 + z^2) * f^4 * UTrace[NM[UMatrix[UChiralSpurionRight[]][x], UMatrix[UChiralSpurionRight[]][x]] + NM[UMatrix[UChiralSpurionLeft[]][x], UMatrix[UChiralSpurionLeft[]][x]]]^2 - z^2 * f^4 * UTrace[NM[UMatrix[UChiralSpurionRight[]][x], UMatrix[UChiralSpurionRight[]][x]] - NM[UMatrix[UChiralSpurionLeft[]][x], UMatrix[UChiralSpurionLeft[]][x]]]^2](../HTMLFiles/index_299.gif)
![knechtres1 = (Collect[(knechtres /. {μ -> μ1, ν -> μ2, f -> DecayConstant[Pion], z -> CouplingConstant[ChPTVirtualPhotons2[2]]/DecayConstant[Pion]^4}) // 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]]] /. {(* UMatrix[(UChiralSpurionRight | UChiralSpurionLeft | UChiralSpurion)[]][_] -> 0, (CQLeft | CQRight)[_][_] -> 0, (GLeft | GRight)[__][_] -> 0 *)} // Expand](../HTMLFiles/index_301.gif)
