•2ϕAS

The evaluated leading order Lagrangian:

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

Redundant terms are discarded:

lll = DiscardTerms[ll, Retain -> {Particle[PhiMeson , RenormalizationState[0]] -> 2, Particle[AxialVector[0] , RenormalizationState[0]] -> 1, Particle[Scalar[1], RenormalizationState[0]] -> 1}, CommutatorReduce -> True, Method -> Expand] // Simplify

1/(2 (f _ ϕ^(ó  ))^2) (i c _ 2^( ) (< σ^6 '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[σ, ->] '6 Overscript[A^( ) _ μ, ->] · Overscript[σ, ->] > + < σ^6 '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[A^( ) _ μ, ->] · Overscript[σ, ->] '6 ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[σ, ->] > - < σ^6 '6 ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[σ, ->] '6 Overscript[A^( ) _ μ, ->] · 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, ->] >) 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[Scalar[1], RenormalizationState[0]]][p4]}

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

lal = Expand[llll] ;

melsimplified = If[Head[lal] == Plus, Plus @@ (IndicesCleanup[SUNReduce[FeynRule[#, fields], FullReduce -> True]] & /@ (List @@ lal))] ;

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

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

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

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

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

Converted by Mathematica  (July 10, 2003)