•ππγγ

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

1/16 (e^( ))^2 (-< Overscript[π^( ), ->] · Overscript[σ, ->] '6 σ^3 '6 Overscript[π^( ), ->] · Overscript[σ, ->] '6 σ^3 > + < Overscript[π^( ), ->] · Overscript[σ, ->] '6 σ^3 '6 σ^3 '6 Overscript[π^( ), ->] · Overscript[σ, ->] > + < σ^3 '6 Overscript[π^( ), ->] · Overscript[σ, ->] '6 Overscript[π^( ), ->] · Overscript[σ, ->] '6 σ^3 > - < σ^3 '6 Overscript[π^( ), ->] · Overscript[σ, ->] '6 σ^3 '6 Overscript[π^( ), ->] · Overscript[σ, ->] >) γ^( ) _ μ^2

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

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

$IsoIndicesCounter = 0 ;

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

1/2 (e^( ))^2 f _ (3 k1 k3)^(2) f _ (3 k2 k3)^(2) π^( )^k1 π^( )^k2 γ^( ) _ ρ1^2

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], QuantumField[Particle[Photon, RenormalizationState[0]], LorentzIndex[μ4]][p4]}

{π^( )^I _ 1, π^( )^I _ 2, γ^( ) _ μ _ 3, γ^( ) _ μ _ 4}

melsimplified = Simplify[SUNReduce[FeynRule[llll, fields], FullReduce -> True]]

-2 i (e^( ))^2 g^(μ _ 3 μ _ 4) (δ _ (3 I _ 1)^(2) δ _ (3 I _ 2)^(2) - δ _ (I _ 1 I _ 2)^(2))

Converted by Mathematica  (July 10, 2003)