•ϕA

We don't need vectors.

IsoVector[QuantumField[Particle[Vector[0], ___], ___], ___][_] := 0 ;

The evaluated leading order Lagrangian:

ll = ArgumentsSupply[Lagrangian[ChPTW3[2]], x, RenormalizationState[0], ExpansionOrder -> 1, DropOrder -> 1, DiagonalToU -> True] ;

Redundant terms are discarded:

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

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

Generator matrices are traced:

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

-(c _ 2^( ) (2 i Overscript[öõ(6), ->] × ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[A^( ) _ μ, ->] + 2 i Overscript[öõ(6), ->] × Overscript[A^( ) _ μ, ->] · ∂ _ μ(Overscript[ϕ^( ), ->]) + 2 Overscript[öõ(6), ->] ⊗ ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[A^( ) _ μ, ->] + 2 Overscript[öõ(6), ->] ⊗ Overscript[A^( ) _ μ, ->] · ∂ _ μ(Overscript[ϕ^( ), ->])))/f _ ϕ^(ó  )

Indices are supplied:

$IsoIndicesCounter = 0 ;

llll = llle // IsoIndicesSupply // IndicesCleanup

-(2 c _ 2^( ) d _ (k1 k2 k3) δ _ (6 k3) ∂ _ τ1 ϕ^( ) _ ó ^k2 A^( ) _ τ1^k1)/f _ ϕ^(ó  ) - (2 i c _ 2^( ) δ _ (6 k3) f _ (k1 k2 k3) ∂ _ τ1 ϕ^( ) _ ó ^k2 A^( ) _ τ1^k1)/f _ ϕ^(ó  ) - (2 c _ 2^( ) d _ (k1 k2 k3) δ _ (6 k3) ∂ _ τ1 ϕ^( ) _ ó ^k1 A^( ) _ τ1^k2)/f _ ϕ^(ó  ) - (2 i c _ 2^( ) δ _ (6 k3) f _ (k1 k2 k3) ∂ _ τ1 ϕ^( ) _ ó ^k1 A^( ) _ τ1^k2)/f _ ϕ^(ó  )

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]}

{A^( ) _ μ _ 1^I _ 1, ϕ^( )^I _ 2}

lal = Expand[llll] ;

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

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

Converted by Mathematica  (July 10, 2003)