•Tree contribution of second order in the chiral expansion

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

A lagrangian which is just ChPTW3[2] times a scalar field:

ll = ArgumentsSupply[Lagrangian[ChPTW3[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}, CommutatorReduce -> True, Method -> Expand] // Simplify

-1/(3 f _ ϕ^(ó  )) (i c _ 5^( ) (2 3^(1/2) (< σ^6 '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 σ^8 > - < σ^6 '6 σ^8 '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] >) (m _ π^(ó  ))^2 + (m _ K^+^(ó  ))^2 (3 < σ^6 '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 σ^3 > - 3^(1/2) < σ^6 '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 σ^8 > - 3 < σ^6 '6 σ^3 '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] > + 3^(1/2) < σ^6 '6 σ^8 '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] >) + (m _ K^0^(ó  ))^2 (-3 < σ^6 '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 σ^3 > - 3^(1/2) < σ^6 '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 σ^8 > + 3 < σ^6 '6 σ^3 '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] > + 3^(1/2) < σ^6 '6 σ^8 '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] >)))

Generator matrices are traced:

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

-1/(3 f _ ϕ^(ó  )) (i c _ 5^( ) (4 3^(1/2) (-i Overscript[öõ(6), ->] × Overscript[öõ(8), ->] · Overscript[ϕ^( ), ->] + i Overscript[öõ(6), ->] × Overscript[ϕ^( ), ->] · Overscript[öõ(8), ->] - Overscript[öõ(6), ->] ⊗ Overscript[öõ(8), ->] · Overscript[ϕ^( ), ->] + Overscript[öõ(6), ->] ⊗ Overscript[ϕ^( ), ->] · Overscript[öõ(8), ->]) (m _ π^(ó  ))^2 + (-6 i Overscript[öõ(6), ->] × Overscript[öõ(3), ->] · Overscript[ϕ^( ), ->] - 6 Overscript[öõ(6), ->] ⊗ Overscript[öõ(3), ->] · Overscript[ϕ^( ), ->] + 2 3^(1/2) (i Overscript[öõ(6), ->] × Overscript[öõ(8), ->] · Overscript[ϕ^( ), ->] + Overscript[öõ(6), ->] ⊗ Overscript[öõ(8), ->] · Overscript[ϕ^( ), ->]) + 6 (i Overscript[öõ(6), ->] × Overscript[ϕ^( ), ->] · Overscript[öõ(3), ->] + Overscript[öõ(6), ->] ⊗ Overscript[ϕ^( ), ->] · Overscript[öõ(3), ->]) - 2 3^(1/2) (i Overscript[öõ(6), ->] × Overscript[ϕ^( ), ->] · Overscript[öõ(8), ->] + Overscript[öõ(6), ->] ⊗ Overscript[ϕ^( ), ->] · Overscript[öõ(8), ->])) (m _ K^+^(ó  ))^2 + (-6 i Overscript[öõ(6), ->] × Overscript[ϕ^( ), ->] · Overscript[öõ(3), ->] + 6 (i Overscript[öõ(6), ->] × Overscript[öõ(3), ->] · Overscript[ϕ^( ), ->] + Overscript[öõ(6), ->] ⊗ Overscript[öõ(3), ->] · Overscript[ϕ^( ), ->]) + 2 3^(1/2) (i Overscript[öõ(6), ->] × Overscript[öõ(8), ->] · Overscript[ϕ^( ), ->] + Overscript[öõ(6), ->] ⊗ Overscript[öõ(8), ->] · Overscript[ϕ^( ), ->]) - 6 Overscript[öõ(6), ->] ⊗ Overscript[ϕ^( ), ->] · Overscript[öõ(3), ->] - 2 3^(1/2) (i Overscript[öõ(6), ->] × Overscript[ϕ^( ), ->] · Overscript[öõ(8), ->] + Overscript[öõ(6), ->] ⊗ Overscript[ϕ^( ), ->] · Overscript[öõ(8), ->])) (m _ K^0^(ó  ))^2))

Indices are supplied:

$IsoIndicesCounter = 0 ;

llll = llle // IsoIndicesSupply // SUNReduce[#, FullReduce -> True] & // IndicesCleanup // CommutatorReduce // Simplify

(4 c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^+^(ó  ))^2) ϕ^( )^7)/f _ ϕ^(ó  )

Calculation of the Feynman rule:

fields = {QuantumField[Particle[PhiMeson, RenormalizationState[0]], SUNIndex[I1]][p1]}

{ϕ^( )^I _ 1}

melsimplified = Simplify[SUNReduce[FeynRule[llll, fields], FullReduce -> True]]

(4 i c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^+^(ó  ))^2) δ _ (7 I _ 1)^(3))/f _ ϕ^(ó  )

amp2 = -I melsimplified

(4 c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^+^(ó  ))^2) δ _ (7 I _ 1)^(3))/f _ ϕ^(ó  )

Converted by Mathematica  (July 10, 2003)