•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 ;

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

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

1/(24 (f _ π^(ó  ))^2) (C^( ) (< Overscript[π^( ), ->] · Overscript[σ, ->] '6 Overscript[π^( ), ->] · Overscript[σ, ->] '6 σ^3 > - 2 < Overscript[π^( ), ->] · Overscript[σ, ->] '6 σ^3 '6 Overscript[π^( ), ->] · Overscript[σ, ->] > + < σ^3 '6 Overscript[π^( ), ->] · Overscript[σ, ->] '6 Overscript[π^( ), ->] · Overscript[σ, ->] > - 3 < σ^3 '6 Overscript[π^( ), ->] · Overscript[σ, ->] '6 Overscript[π^( ), ->] · Overscript[σ, ->] '6 σ^3 > + 6 < σ^3 '6 Overscript[π^( ), ->] · Overscript[σ, ->] '6 σ^3 '6 Overscript[π^( ), ->] · Overscript[σ, ->] > - 3 < σ^3 '6 σ^3 '6 Overscript[π^( ), ->] · Overscript[σ, ->] '6 Overscript[π^( ), ->] · Overscript[σ, ->] >) (e^( ))^2 + (f _ π^(ó  ))^2 (6 < ∂ _ μ(Overscript[π^( ), ->]) · Overscript[σ, ->] '6 ∂ _ μ(Overscript[π^( ), ->]) · Overscript[σ, ->] > - 6 (m _ π^(ó  ))^2 < Overscript[π^( ), ->] · Overscript[σ, ->] '6 Overscript[π^( ), ->] · Overscript[σ, ->] >))

llle = ExpandU[lll, CommutatorReduce -> True] // Simplify

(C^( ) ((Overscript[öõ(3), ->] · Overscript[π^( ), ->])^2 - 3 Overscript[öõ(3), ->] × Overscript[π^( ), ->] · Overscript[öõ(3), ->] × Overscript[π^( ), ->] - Overscript[π^( ), ->] · Overscript[π^( ), ->]) (e^( ))^2 + 2 (f _ π^(ó  ))^2 (∂ _ μ(Overscript[π^( ), ->]) · ∂ _ μ(Overscript[π^( ), ->]) - Overscript[π^( ), ->] · Overscript[π^( ), ->] (m _ π^(ó  ))^2))/(4 (f _ π^(ó  ))^2)

IsoIndicesCounter = 0 ;

llll = llle // IsoIndicesSupply // SUNReduce[#, FullReduce -> True] & // IndicesCleanup // CommutatorReduce[#, FullReduce -> True] & // Simplify

(C^( ) ((π^( )^3)^2 + π^( )^k1 (3 δ _ (3 k1)^(2) δ _ (3 k2)^(2) π^( )^k2 - 4 π^( )^k1)) (e^( ))^2 + 2 (f _ π^(ó  ))^2 ((∂ _ τ1 π^( ) _ ó ^k1)^2 - (m _ π^(ó  ))^2 (π^( )^k1)^2))/(4 (f _ π^(ó  ))^2)

fields = {QuantumField[Particle[PseudoScalar[2], RenormalizationState[0]], SUNIndex[i1]][p1], QuantumField[Particle[PseudoScalar[2], RenormalizationState[0]], SUNIndex[i2]][p2]}

{π^( )^i _ 1, π^( )^i _ 2}

$ConstantIsoIndices = Union[$ConstantIsoIndices, {i1, i2}] ;

amp2 = (-I FeynRule[llll, fields]) /. i2 -> i1 // SUNReduce // Simplify

(2 C^( ) (e^( ))^2 (δ _ (3 i _ 1)^(2) - 1) - (f _ π^(ó  ))^2 ((m _ π^(ó  ))^2 + p _ 1 · p _ 2))/(f _ π^(ó  ))^2

The pion mass has to be modified:

massrule[I1_] = ParticleMass[PseudoScalar[2], SUNIndex[I1], RenormalizationState[0]]^2 -> ParticleMass[PseudoScalar[2], RenormalizationState[0]]^2 - CouplingConstant[QED[1], RenormalizationState[0]]^2/DecayConstant[PseudoScalar[2], RenormalizationState[0]]^2 CouplingConstant[ChPTEM2[2], RenormalizationState[0]] 2 (SU2Delta[3, SUNIndex[I1]] - 1)

(m _ π^(I _ 1 ))^2 -> (m _ π^(ó  ))^2 - (2 C^( ) (e^( ))^2 (δ _ (3 I _ 1)^(2) - 1))/(f _ π^(ó  ))^2

Converted by Mathematica  (July 10, 2003)