•ϕ

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

The leading order Lagrangian in raw form:

Lagrangian[ChPTW3[2]]

c _ 5^( ) (< σ^6 '6 ÷„^† '6 χ > + < σ^6 '6 χ^† '6 ÷„ >) + c _ 2^( ) < σ^6 '6 ÷s _ μ(÷„)^† '6 ÷s _ μ(÷„) >

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}, 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[ExpandU[lll, CommutatorReduce -> True], 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 // IndicesCleanup // Simplify

1/(3 f _ ϕ^(ó  )) (2 c _ 5^( ) (-2 i 3^(1/2) (d _ (k1 k2 k3) (δ _ (6 k3) δ _ (8 k2) - δ _ (6 k2) δ _ (8 k3)) - 2 i δ _ (6 k2) δ _ (8 k3) f _ (k1 k2 k3)) (m _ π^(ó  ))^2 + (m _ K^0^(ó  ))^2 (2 (3^(1/2) δ _ (6 k2) δ _ (8 k3) - 3 δ _ (3 k2) δ _ (6 k3)) f _ (k1 k2 k3) - i d _ (k1 k2 k3) (3 δ _ (3 k3) δ _ (6 k2) - 3 δ _ (3 k2) δ _ (6 k3) + 3^(1/2) (δ _ (6 k2) δ _ (8 k3) - δ _ (6 k3) δ _ (8 k2)))) + (m _ K^+^(ó  ))^2 (i d _ (k1 k2 k3) (3 δ _ (3 k3) δ _ (6 k2) - 3 δ _ (3 k2) δ _ (6 k3) + 3^(1/2) (δ _ (6 k3) δ _ (8 k2) - δ _ (6 k2) δ _ (8 k3))) + 2 (3 δ _ (3 k2) δ _ (6 k3) + 3^(1/2) δ _ (6 k2) δ _ (8 k3)) f _ (k1 k2 k3))) ϕ^( )^k1)

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

lll // WriteOutIsoVectors // WriteOutUMatrices // Simplify

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

llll // Expand // SUNReduce[#, Explicit -> True, HoldSums -> False, FullReduce -> True, CommutatorReduce -> True] & // 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]] // IndicesCleanup // Simplify

(4 i c _ 5^( ) (-2 3^(1/2) f _ (6 8 I _ 1)^(3) (m _ π^(ó  ))^2 + (m _ K^0^(ó  ))^2 (3^(1/2) f _ (6 8 I _ 1)^(3) - 3 f _ (3 6 I _ 1)^(3)) + (m _ K^+^(ó  ))^2 (3 f _ (3 6 I _ 1)^(3) + 3^(1/2) f _ (6 8 I _ 1)^(3))))/(3 f _ ϕ^(ó  ))

Converted by Mathematica  (July 10, 2003)