•Two-vertex of second order in the chiral expansion

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

Lagrangian[ChPTEM2[2]]

1/4 (< χ '6 ÷„^† > + < χ^† '6 ÷„ > + < ÷s _ μ(÷„) '6 ÷s _ μ(÷„)^† >) (f _ π^(ó  ))^2 - 1/2 λ ∂ _ μ(γ^( ) _ μ) ∂ _ ν(γ^( ) _ ν) - 1/4 (γ^( ) _ (μ ν) '6 γ^( ) _ (μ ν)) + C^( ) < Q _ R '6 ÷„ '6 Q _ L '6 ÷„^† >

ll = ArgumentsSupply[Lagrangian[ChPTEM2[2]] /. {UMatrix[UChiralSpurionRight] -> UQuarkChargeMatrix[RenormalizationState[0], DiagonalToU -> True], UMatrix[UChiralSpurionLeft] -> UQuarkChargeMatrix[RenormalizationState[0], DiagonalToU -> True]}, x, RenormalizationState[0], ExpansionOrder -> 0, DropOrder -> 0, DiagonalToU -> True] ;

lll = DiscardTerms[ll, Retain -> {Particle[Photon , RenormalizationState[0]] -> 2}, Method -> Expand] // Simplify

1/4 (-(∂ _ μ γ^( ) _ ν^ó  '6 ∂ _ μ γ^( ) _ ν^ó ) + ∂ _ μ γ^( ) _ ν^ó  '6 ∂ _ ν γ^( ) _ μ^ó  + ∂ _ ν γ^( ) _ μ^ó  '6 ∂ _ μ γ^( ) _ ν^ó  - ∂ _ ν γ^( ) _ μ^ó  '6 ∂ _ ν γ^( ) _ μ^ó  - 2 λ ∂ _ μ γ^( ) _ μ^ó  ∂ _ ν γ^( ) _ ν^ó )

llll = lll // IndicesCleanup // CommutatorReduce // Simplify

1/2 (-(∂ _ ρ1 γ^( ) _ τ1^ó )^2 + ∂ _ τ1 γ^( ) _ τ2^ó  ∂ _ τ2 γ^( ) _ τ1^ó  - λ ∂ _ ω1 γ^( ) _ ω1^ó  ∂ _ ω2 γ^( ) _ ω2^ó )

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

{γ^( ) _ μ _ 1, γ^( ) _ μ _ 2}

amp2 = FeynRule[llll, fields] // Simplify

i (-p _ 2^μ _ 1 p _ 1^μ _ 2 + λ p _ 1^μ _ 1 p _ 2^μ _ 2 + g^(μ _ 1 μ _ 2) p _ 1 · p _ 2)

mfa = MomentaCollect[amp2 // Expand, ParticlesNumber -> 2, PerturbationOrder -> 2, ExtendedCollect -> False]

-i p _ 2^μ _ 1 p _ 1^μ _ 2 + i λ p _ 1^μ _ 1 p _ 2^μ _ 2 + i g^(μ _ 1 μ _ 2) p _ 1 · p _ 2

ampf2 = -I Pair[LorentzIndex[μ1, D], Momentum[Polarization[p1, i], D]] (-Pair[LorentzIndex[μ2, D], Momentum[Polarization[p2, -i], D]] ) amp2 // Contract // Simplify

p _ 1 · µ^* ( p _ 2 ) p _ 2 · µ ( p _ 1 ) - p _ 1 · p _ 2 µ ( p _ 1 ) · µ^* ( p _ 2 )

gencoup = GenericCoupling[mfa]

{p _ 1 _ μ _ 2 p _ 2 _ μ _ 1, p _ 1 _ μ _ 1 p _ 2 _ μ _ 2, g^(μ _ 1 μ _ 2) (p _ 1 ° p _ 2)}

classcoup = ClassesCoupling[mfa] // Together ; classcoup // StandardForm

{{-i}, {i $Gauge}, {i}}

$VeryVerbose = 2 ;

CheckF[gencoup, XName[PhiModel -> ChPTEM2, VertexFields -> {Vector[1][0], Vector[1][0]}, MomentaOrder -> 2, XFileName -> Automatic] <> ".Gen"] ;

Using file name D:\\Program Files\\Wolfram Research\\Mathematica\\4.1\\AddOns\\Applications\\HighEnergyPhysics\\Phi\\CouplingVectors\\ChPTEM2V10V10o2.Gen

File exists, loading

CheckF[classcoup, XName[PhiModel -> ChPTEM2, VertexFields -> {Vector[1][0], Vector[1][0]}, MomentaOrder -> 2, XFileName -> Automatic] <> ".Mod"] ;

Using file name D:\\Program Files\\Wolfram Research\\Mathematica\\4.1\\AddOns\\Applications\\HighEnergyPhysics\\Phi\\CouplingVectors\\ChPTEM2V10V10o2.Mod

File exists, loading

$VeryVerbose = 0 ;

Converted by Mathematica  (July 10, 2003)