•Tree contribution of second order in the chiral expansion

IsoVector[QuantumField[Particle[Vector[0], ___], ___], ___][_] := 0 ; IsoVector[QuantumField[Particle[Scalar[1 | 2], ___], ___], ___][_] := 0 ; QuantumField[Particle[Scalar[1 | 2], ___], ___][_] := 0 ; QuantumField[Particle[PseudoScalar[0], ___], ___][_] := 0 ;

ll = ArgumentsSupply[Lagrangian[ChPT3[2]], x, RenormalizationState[0], ExpansionOrder -> 1, DropOrder -> 1, DiagonalToU -> True, QuarkToMesonMasses -> True] ;

Redundant terms are discarded:

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

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

Matrices are traced:

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

-f _ ϕ^(ó  ) ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[A^( ) _ μ, ->]

Indices are supplied:

$IsoIndicesCounter = 0 ;

llll = llle // IsoIndicesSupply

-f _ ϕ^(ó  ) (∂ _ μ ϕ^( ) _ ó ^i _ 1 '6 A^( ) _ μ^i _ 1)

Calculation of the amplitude:

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}

amp2 = -I Simplify[SUNReduce[FeynRule[llll, fields]]] /. I2 -> I1

i f _ ϕ^(ó  ) p _ 2^μ _ 1

Converted by Mathematica  (July 10, 2003)