•Aϕϕϕϕϕ

IsoVector[QuantumField[Particle[Vector[0], ___], ___], ___][_] := 0 ; <br /> IsoVector[QuantumField[Particle[PseudoScalar[0], ___], ___], ___][_] := 0 ;

Lagrangian[ChPT3[2]]

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

ll = ArgumentsSupply[Lagrangian[ChPT3[2]], x, RenormalizationState[0], ExpansionOrder -> 5, DropOrder -> 5, DiagonalToU -> True] // CycleUTraces ;

lll = DiscardTerms[ll, Retain -> {Particle[PhiMeson , RenormalizationState[0]] -> 5, Particle[AxialVector[0] , RenormalizationState[0]] -> 1}, Method -> Expand] // CycleUTraces // Simplify

-1/(240 (f _ ϕ^(ó  ))^3) (< ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[A^( ) _ μ, ->] · Overscript[σ, ->] > - 4 < ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[A^( ) _ μ, ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] > + 6 < ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[A^( ) _ μ, ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] > - 4 < ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[A^( ) _ μ, ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] > + < ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[σ, ->] '6 Overscript[A^( ) _ μ, ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] >)

Expand[lll] // Length

5

llle = (WriteString["stdout", "."] ; ExpandU[#, CommutatorReduce -> True]) & /@ Expand[lll] ;

.....

Expand[llle] // Length

86

llll = (WriteString["stdout", "."] ; $IsoIndicesCounter = 0 ; # // IsoIndicesSupply // SUNReduce[#, FullReduce -> True] & // CommutatorReduce[#, FullReduce -> True] & // Simplify) & /@ Expand[llle] ;

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

fields = {QuantumField[Particle[AxialVector[0], RenormalizationState[0]], LorentzIndex[μ1], SUNIndex[I1]][p1], QuantumField[Particle[PhiMeson, RenormalizationState[0]], SUNIndex[I2]][p2], QuantumField[Particle[PhiMeson, RenormalizationState[0]], SUNIndex[I3]][p3], QuantumField[Particle[PhiMeson, RenormalizationState[0]], SUNIndex[I4]][p4], QuantumField[Particle[PhiMeson, RenormalizationState[0]], SUNIndex[I5]][p5], QuantumField[Particle[PhiMeson, RenormalizationState[0]], SUNIndex[I6]][p6]}

{A^( ) _ μ _ 1^I _ 1, ϕ^( )^I _ 2, ϕ^( )^I _ 3, ϕ^( )^I _ 4, ϕ^( )^I _ 5, ϕ^( )^I _ 6}

llll // LeafCount

7970

llll // Expand // Length

86

melsimplified = I * (WriteString["stdout", "."] ; FunctionalD[PhiToFC[#] /. Dot -> Times, fields]) & /@ Expand[llll] ;

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

melsimplified // LeafCount

285236

melsimplified // Length

85

A test:

Table[Print["Starting ", ii] ; (WriteString["stdout", "."] ; (# /. {I1 -> 7, I2 -> 7, I3 -> 3, I4 -> 3, I5 -> ii, I6 -> ii} /. {p2 -> -p3 - p4, p5 -> q, p6 -> q} // Contract // SUNReduce[#, FullReduce -> True] &) /. q -> 0 // ScalarProductExpand // Simplify) & /@ melsimplified, {ii, 1, 1 (* 8 *)}]

Starting 1

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

{(p _ 3^μ _ 1 + p _ 4^μ _ 1)/(12 (f _ ϕ^(ó  ))^3)}

Converted by Mathematica  (July 10, 2003)