•4ϕAS

The evaluated leading order Lagrangian:

ll = ArgumentsSupply[Lagrangian[ChPTWS3[2]], x, RenormalizationState[0], ExpansionOrder -> 4, DropOrder -> 4, DiagonalToU -> True] ;

Redundant terms are discarded:

lll = DiscardTerms[ll, Retain -> {Particle[PhiMeson , RenormalizationState[0]] -> 4, Particle[AxialVector[0] , RenormalizationState[0]] -> 1, Particle[Scalar[1], RenormalizationState[0]] -> 1}, CommutatorReduce -> True, Method -> Expand] // Simplify

-1/(24 (f _ ϕ^(ó  ))^4) (i c _ 2^( ) (< σ^6 '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[σ, ->] '6 Overscript[A^( ) _ μ, ->] · Overscript[σ, ->] > + 2 < σ^6 '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[A^( ) _ μ, ->] · Overscript[σ, ->] '6 ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[σ, ->] > - < σ^6 '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[A^( ) _ μ, ->] · Overscript[σ, ->] > - 2 < σ^6 '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[σ, ->] '6 Overscript[A^( ) _ μ, ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] > - 3 < σ^6 '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, ->] > - 3 < σ^6 '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[A^( ) _ μ, ->] · Overscript[σ, ->] '6 ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] > + 2 < σ^6 '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, ->] > - 2 < σ^6 '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[A^( ) _ μ, ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] > + 3 < σ^6 '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[σ, ->] '6 Overscript[A^( ) _ μ, ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] > + 2 < σ^6 '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[A^( ) _ μ, ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[σ, ->] > + 2 < σ^6 '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, ->] > + 2 < σ^6 '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[A^( ) _ μ, ->] · Overscript[σ, ->] '6 ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] > - 2 < σ^6 '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, ->] > + 3 < σ^6 '6 ∂ _ 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, ->] > - 2 < σ^6 '6 ∂ _ 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, ->] > - 2 < σ^6 '6 Overscript[A^( ) _ μ, ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] > + < σ^6 '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, ->] > - < σ^6 '6 Overscript[A^( ) _ μ, ->] · Overscript[σ, ->] '6 ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] >) s^( ))

Generator matrices are traced:

llle = ExpandU[ExpandU[lll, CommutatorReduce -> True], CommutatorReduce -> True] ;

Indices are supplied:

$IsoIndicesCounter = 0 ;

llll = llle // IsoIndicesSupply // IndicesCleanup ;

Calculation of the Feynman rule:

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[Scalar[1], RenormalizationState[0]]][p6]}

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

lal = Expand[llll // PhiToFC] ;

melsimplified = ((WriteString["stdout", "."] ; Simplify[I * SUNReduce[SUNReduce[SUNReduce[FunctionalD[#, fields]]]]]) & /@ lal) ;

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

melsimplified1 = melsimplified /. DOT -> Times // Contract ;

melsimplified1[[1]]

-(4 i c _ 2^( ) p _ 3^μ _ 1 d _ (6 I _ 3 I _ 5)^(3) d _ (I _ 1 I _ 2 I _ 4)^(3))/(9 (f _ ϕ^(ó  ))^4)

melsimplified1[[2]]

(2 i c _ 2^( ) p _ 4^μ _ 1 d _ (6 I _ 3 I _ 5)^(3) d _ (I _ 1 I _ 2 I _ 4)^(3))/(3 (f _ ϕ^(ó  ))^4)

melsimplified2 = Collect[melsimplified1, {_DecayConstant, _CouplingConstant, _Pair, _SU3Delta}] ;

melsimplified2 // LeafCount

94999

WriteString["stdout", "Total length: ", Length[melsimplified2]] ; melsimplified3 = (WriteString["stdout", " leafs: ", LeafCount[#]] ; Collect[SUNReduce[SUNReduce[SUNReduce[SUNReduce[SUNReduce[#]]]]], {_DecayConstant, _CouplingConstant, _SU3Delta}]) & /@ Take[melsimplified2, {1, -1}] ;

Another check that two different evaluations with specific components give the same result:

(SUNReduce[#, Explicit -> True, HoldSums -> False] & /@ (melsimplified2 /. {I1 -> 7, I2 -> 3, I3 -> 3, I4 -> 3, I5 -> 3} // Expand)) // Simplify

-(c _ 2^( ) (p _ 2^μ _ 1 + p _ 3^μ _ 1 + p _ 4^μ _ 1 + p _ 5^μ _ 1))/(f _ ϕ^(ó  ))^4

(SUNReduce /@ SUNReduce /@ SUNReduce /@ SUNReduce /@ SUNReduce /@ SUNReduce /@ SUNReduce /@ SUNReduce /@ SUNReduce /@ (melsimplified2 /. {I1 -> 7, I2 -> 3, I3 -> 3, I4 -> 3, I5 -> 3} // Expand)) // Simplify

-(c _ 2^( ) (p _ 2^μ _ 1 + p _ 3^μ _ 1 + p _ 4^μ _ 1 + p _ 5^μ _ 1))/(f _ ϕ^(ó  ))^4

Converted by Mathematica  (July 10, 2003)