•ππγ

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

Lagrangian[ChPTPhoton2[2]]

1/4 (f _ π^(ó  ))^2 (< ÷„ '6 χ^† > + < χ '6 ÷„^† > + < ÷s _ μ(÷„) '6 ÷s _ μ(÷„)^† >)

The expanded lagrangian:

ll = ArgumentsSupply[Lagrangian[ChPTPhoton2[2]], x, RenormalizationState[0], ExpansionOrder -> 2, DropOrder -> 2, DiagonalToU -> True] ;

Redundant terms are discarded:

lll = DiscardTerms[ll, Retain -> {Particle[Pion , RenormalizationState[0]] -> 2, Particle[Photon , RenormalizationState[0]] -> 1}, CommutatorReduce -> True, Method -> Coefficient] // Simplify

1/8 i e^( ) (< Overscript[π^( ), ->] · Overscript[σ, ->] '6 σ^3 '6 ∂ _ μ(Overscript[π^( ), ->]) · Overscript[σ, ->] > + < ∂ _ μ(Overscript[π^( ), ->]) · Overscript[σ, ->] '6 Overscript[π^( ), ->] · Overscript[σ, ->] '6 σ^3 > - < ∂ _ μ(Overscript[π^( ), ->]) · Overscript[σ, ->] '6 σ^3 '6 Overscript[π^( ), ->] · Overscript[σ, ->] > - < σ^3 '6 Overscript[π^( ), ->] · Overscript[σ, ->] '6 ∂ _ μ(Overscript[π^( ), ->]) · Overscript[σ, ->] >) γ^( ) _ μ

Matrices are traced:

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

1/2 e^( ) (Overscript[öõ(3), ->] × Overscript[π^( ), ->] · ∂ _ μ(Overscript[π^( ), ->]) - Overscript[öõ(3), ->] × ∂ _ μ(Overscript[π^( ), ->]) · Overscript[π^( ), ->]) γ^( ) _ μ

Indices are supplied:

$IsoIndicesCounter = 0 ;

llll = llle // IsoIndicesSupply // SUNReduce // IndicesCleanup // Simplify

1/2 e^( ) f _ (3 k1 k2)^(2) γ^( ) _ τ1 (π^( )^k1 ∂ _ τ1 π^( ) _ ó ^k2 - π^( )^k2 ∂ _ τ1 π^( ) _ ó ^k1)

Calculation of the Feynman rule:

fields = {QuantumField[Particle[Pion, RenormalizationState[0]], SUNIndex[I1]][p1], QuantumField[Particle[Pion, RenormalizationState[0]], SUNIndex[I2]][p2], QuantumField[Particle[Photon, RenormalizationState[0]], LorentzIndex[μ3]][p3]}

{π^( )^I _ 1, π^( )^I _ 2, γ^( ) _ μ _ 3}

melsimplified = Simplify[SUNReduce[FeynRule[llll, fields]]]

e^( ) (p _ 2^μ _ 3 - p _ 1^μ _ 3) f _ (3 I _ 1 I _ 2)^(2)

Converted by Mathematica  (July 10, 2003)