•Two-vertex amplitude of fourth order in the chiral expansion

IsoVector[QuantumField[Particle[AxialVector[0], ___], ___], ___][_] := 0 ; IsoVector[QuantumField[Particle[AxialVector[1], ___], ___], ___][_] := 0 ; QuantumField[Particle[AxialVector[1], ___], ___][_] := 0 ; IsoVector[QuantumField[Particle[Scalar[1 | 2], ___], ___], ___][_] := 0 ; QuantumField[Particle[Scalar[1 | 2], ___], ___][_] := 0 ; IsoVector[QuantumField[Particle[PseudoScalar[0], ___], ___], ___][_] := 0 ; QuantumField[Particle[PseudoScalar[0], ___], ___][_] := 0 ;

LoadLagrangian[ChPT2[4]] ;

Lagrangian[ChPT2[4]]

L _ 7^( ) ((< χ '6 ÷„^† > - < ÷„ '6 χ^† >) '6 (< χ '6 ÷„^† > - < ÷„ '6 χ^† >)) + L _ 6^( ) ((< ÷„ '6 χ^† > + < χ '6 ÷„^† >) '6 (< ÷„ '6 χ^† > + < χ '6 ÷„^† >)) + L _ 4^( ) (< ÷s _ μ(÷„) '6 ÷s _ μ(÷„)^† > '6 (< ÷„ '6 χ^† > + < χ '6 ÷„^† >)) + L _ 1^( ) (< ÷s _ μ(÷„) '6 ÷s _ μ(÷„)^† > '6 < ÷s _ ν(÷„) '6 ÷s _ ν(÷„)^† >) + L _ 2^( ) (< ÷s _ μ(÷„) '6 ÷s _ ν(÷„)^† > '6 < ÷s _ μ(÷„) '6 ÷s _ ν(÷„)^† >) + H _ 2^( ) < χ^† '6 χ > + H _ 1^( ) (< L _ (μ ν) '6 L _ (μ ν) > + < R _ (μ ν) '6 R _ (μ ν) >) + L _ 5^( ) < ÷s _ μ(÷„) '6 ÷s _ μ(÷„)^† '6 (÷„ '6 χ^† + χ '6 ÷„^†) > + i L _ 9^( ) (< L _ (μ ν) '6 ÷s _ μ(÷„) '6 ÷s _ ν(÷„)^† > + < R _ (μ ν) '6 ÷s _ μ(÷„)^† '6 ÷s _ ν(÷„) >) + L _ 8^( ) (< ÷„ '6 χ^† '6 ÷„ '6 χ^† > + < χ '6 ÷„^† '6 χ '6 ÷„^† >) + L _ 3^( ) < ÷s _ μ(÷„) '6 ÷s _ μ(÷„)^† '6 ÷s _ ν(÷„) '6 ÷s _ ν(÷„)^† > + L _ 10^( ) < L _ (μ ν) '6 ÷„ '6 R _ (μ ν) '6 ÷„^† >

lag = Lagrangian[ChPT2[4]] + Lagrangian[ChPTEM2[4]] /. {FST[LeftComponent[0], {μ}, {ν}] :> 2 FieldStrengthTensorFull[LorentzIndex[μ], UMatrix[UChiralSpurionLeft] QuantumField[Particle[LeftComponent[1]], LorentzIndex[ν]][x], x], FST[RightComponent[0], {μ}, {ν}] :> 2 FieldStrengthTensorFull[LorentzIndex[μ], UMatrix[UChiralSpurionRight] QuantumField[Particle[RightComponent[1]], LorentzIndex[ν]][x], x]}

k _ 13^( ) (< Q '6 Q > '6 < Q '6 Q >) (f _ π^(ó  ))^4 + k _ 9^( ) (< Q '6 Q > '6 < Q '6 ÷„ '6 Q '6 ÷„^† >) (f _ π^(ó  ))^4 + k _ 1^( ) (< Q '6 ÷„ '6 Q '6 ÷„^† > '6 < Q '6 ÷„ '6 Q '6 ÷„^† >) (f _ π^(ó  ))^4 + k _ 14^( ) ((< (÷„ '6 χ^† + χ '6 ÷„^†) '6 Q > + < (÷„^† '6 χ + χ^† '6 ÷„) '6 Q >) '6 < Q >) (f _ π^(ó  ))^2 + k _ 11^( ) (< Q '6 Q > '6 (< ÷„ '6 χ^† > + < χ '6 ÷„^† >)) (f _ π^(ó  ))^2 + k _ 10^( ) (< Q '6 Q > '6 < ÷s _ μ(÷„) '6 ÷s _ μ(÷„)^† >) (f _ π^(ó  ))^2 + k _ 4^( ) (< ÷„^† '6 ÷s _ μ(÷„) '6 Q > '6 < ÷s _ μ(÷„) '6 ÷„^† '6 Q >) (f _ π^(ó  ))^2 + k _ 3^( ) (< ÷„^† '6 ÷s _ μ(÷„) '6 Q > '6 < ÷„^† '6 ÷s _ μ(÷„) '6 Q > + < ÷s _ μ(÷„) '6 ÷„^† '6 Q > '6 < ÷s _ μ(÷„) '6 ÷„^† '6 Q >) (f _ π^(ó  ))^2 + k _ 7^( ) (< Q '6 ÷„ '6 Q '6 ÷„^† > '6 (< χ '6 ÷„^† > + < χ^† '6 ÷„ >)) (f _ π^(ó  ))^2 + k _ 2^( ) (< Q '6 ÷„ '6 Q '6 ÷„^† > '6 < ÷s _ μ(÷„) '6 ÷s _ μ(÷„)^† >) (f _ π^(ó  ))^2 + k _ 12^( ) (< ÷s _ μ(Q _ L) '6 ÷s _ μ(Q _ L) > + < ÷s _ μ(Q _ R) '6 ÷s _ μ(Q _ R) >) (f _ π^(ó  ))^2 + k _ 8^( ) < (÷„^† '6 χ - χ^† '6 ÷„) '6 (÷„^† '6 Q '6 ÷„ '6 Q - Q '6 ÷„^† '6 Q '6 ÷„) > (f _ π^(ó  ))^2 + k _ 5^( ) (< (Q '6 ÷s _ μ(Q) - ÷s _ μ(Q) '6 Q) '6 ÷„^† '6 ÷s _ μ(÷„) > - < (Q '6 ÷s _ μ(Q) - ÷s _ μ(Q) '6 Q) '6 ÷s _ μ(÷„) '6 ÷„^† >) (f _ π^(ó  ))^2 + k _ 6^( ) < ÷s _ μ(Q) '6 ÷„ '6 ÷s _ μ(Q) '6 ÷„^† > (f _ π^(ó  ))^2 + L _ 7^( ) ((< χ '6 ÷„^† > - < ÷„ '6 χ^† >) '6 (< χ '6 ÷„^† > - < ÷„ '6 χ^† >)) + L _ 6^( ) ((< ÷„ '6 χ^† > + < χ '6 ÷„^† >) '6 (< ÷„ '6 χ^† > + < χ '6 ÷„^† >)) + L _ 4^( ) (< ÷s _ μ(÷„) '6 ÷s _ μ(÷„)^† > '6 (< ÷„ '6 χ^† > + < χ '6 ÷„^† >)) + L _ 1^( ) (< ÷s _ μ(÷„) '6 ÷s _ μ(÷„)^† > '6 < ÷s _ ν(÷„) '6 ÷s _ ν(÷„)^† >) + L _ 2^( ) (< ÷s _ μ(÷„) '6 ÷s _ ν(÷„)^† > '6 < ÷s _ μ(÷„) '6 ÷s _ ν(÷„)^† >) + H _ 2^( ) < χ^† '6 χ > + k _ 15^( ) (γ^( ) _ (μ ν) '6 γ^( ) _ (μ ν)) < Q '6 Q > + H _ 1^( ) (< (i (Q _ L '6 γ^( ) _ μ '6 Q _ L '6 γ^( ) _ ν - Q _ L '6 γ^( ) _ ν '6 Q _ L '6 γ^( ) _ μ) + Q _ L ∂ _ μ γ^( ) _ ν^ó  - Q _ L ∂ _ ν γ^( ) _ μ^ó ) '6 (i (Q _ L '6 γ^( ) _ μ '6 Q _ L '6 γ^( ) _ ν - Q _ L '6 γ^( ) _ ν '6 Q _ L '6 γ^( ) _ μ) + Q _ L ∂ _ μ γ^( ) _ ν^ó  - Q _ L ∂ _ ν γ^( ) _ μ^ó ) > + < (i (Q _ R '6 γ^( ) _ μ '6 Q _ R '6 γ^( ) _ ν - Q _ R '6 γ^( ) _ ν '6 Q _ R '6 γ^( ) _ μ) + Q _ R ∂ _ μ γ^( ) _ ν^ó  - Q _ R ∂ _ ν γ^( ) _ μ^ó ) '6 (i (Q _ R '6 γ^( ) _ μ '6 Q _ R '6 γ^( ) _ ν - Q _ R '6 γ^( ) _ ν '6 Q _ R '6 γ^( ) _ μ) + Q _ R ∂ _ μ γ^( ) _ ν^ó  - Q _ R ∂ _ ν γ^( ) _ μ^ó ) >) + L _ 5^( ) < ÷s _ μ(÷„) '6 ÷s _ μ(÷„)^† '6 (÷„ '6 χ^† + χ '6 ÷„^†) > + i L _ 9^( ) (< (i (Q _ L '6 γ^( ) _ μ '6 Q _ L '6 γ^( ) _ ν - Q _ L '6 γ^( ) _ ν '6 Q _ L '6 γ^( ) _ μ) + Q _ L ∂ _ μ γ^( ) _ ν^ó  - Q _ L ∂ _ ν γ^( ) _ μ^ó ) '6 ÷s _ μ(÷„) '6 ÷s _ ν(÷„)^† > + < (i (Q _ R '6 γ^( ) _ μ '6 Q _ R '6 γ^( ) _ ν - Q _ R '6 γ^( ) _ ν '6 Q _ R '6 γ^( ) _ μ) + Q _ R ∂ _ μ γ^( ) _ ν^ó  - Q _ R ∂ _ ν γ^( ) _ μ^ó ) '6 ÷s _ μ(÷„)^† '6 ÷s _ ν(÷„) >) + L _ 8^( ) (< ÷„ '6 χ^† '6 ÷„ '6 χ^† > + < χ '6 ÷„^† '6 χ '6 ÷„^† >) + L _ 3^( ) < ÷s _ μ(÷„) '6 ÷s _ μ(÷„)^† '6 ÷s _ ν(÷„) '6 ÷s _ ν(÷„)^† > + L _ 10^( ) < (i (Q _ L '6 γ^( ) _ μ '6 Q _ L '6 γ^( ) _ ν - Q _ L '6 γ^( ) _ ν '6 Q _ L '6 γ^( ) _ μ) + Q _ L ∂ _ μ γ^( ) _ ν^ó  - Q _ L ∂ _ ν γ^( ) _ μ^ó ) '6 ÷„ '6 (i (Q _ R '6 γ^( ) _ μ '6 Q _ R '6 γ^( ) _ ν - Q _ R '6 γ^( ) _ ν '6 Q _ R '6 γ^( ) _ μ) + Q _ R ∂ _ μ γ^( ) _ ν^ó  - Q _ R ∂ _ ν γ^( ) _ μ^ó ) '6 ÷„^† >

ll = ArgumentsSupply[lag /. {UMatrix[UChiralSpurionRight] -> UQuarkChargeMatrix[RenormalizationState[0], DiagonalToU -> True], UMatrix[UChiralSpurionLeft] -> UQuarkChargeMatrix[RenormalizationState[0], DiagonalToU -> True], UMatrix[UChiralSpurion] -> UQuarkChargeMatrix[RenormalizationState[0], DiagonalToU -> True]}, x, RenormalizationState[0], DiagonalToU -> True, ExpansionOrder -> 0, DropOrder -> 0] ;

ArgumentsSupply :: argxpr : Warning : The argument x is already in the expression.

lle = ll // NMExpand // Expand ;

lle // Length

281

lll = (WriteString["stdout", "."] ; DiscardTerms[#, Retain -> {Particle[Photon, RenormalizationState[0]] -> 2}, Method -> Expand]) & /@ lle ;

.........................................................................................................................................................................................................................................................................................

llll = lll // CommutatorReduce // ExpandU // FullSimplify

5/9 (e^( ))^2 (L _ 10^( ) + 2 H _ 1^( ) + k _ 15^( )) (∂ _ μ γ^( ) _ ν^ó  - ∂ _ ν γ^( ) _ μ^ó )^2

fields = {QuantumField[Particle[Photon, RenormalizationState[0]], LorentzIndex[μ1]][p1], QuantumField[Particle[Photon, RenormalizationState[0]], LorentzIndex[μ2]][p2]}

{γ^( ) _ μ _ 1, γ^( ) _ μ _ 2}

amp4 = (-I FeynRule[llll, fields]) // Simplify

20/9 (e^( ))^2 (L _ 10^( ) + 2 H _ 1^( ) + k _ 15^( )) (p _ 2^μ _ 1 p _ 1^μ _ 2 - g^(μ _ 1 μ _ 2) p _ 1 · p _ 2)

ampf4 = Pair[LorentzIndex[μ1, D], Momentum[Polarization[p1, i], D]] (-Pair[LorentzIndex[μ2, D], Momentum[Polarization[p2, -i], D]] ) amp4 // Contract // Simplify

20/9 (e^( ))^2 (L _ 10^( ) + 2 H _ 1^( ) + k _ 15^( )) (p _ 1 · p _ 2 µ ( p _ 1 ) · µ^* ( p _ 2 ) - p _ 1 · µ^* ( p _ 2 ) p _ 2 · µ ( p _ 1 ))

Converted by Mathematica  (July 10, 2003)