•ϕϕA

The evaluated leading order Lagrangian:

ll = ArgumentsSupply[Lagrangian[ChPTW3[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}, CommutatorReduce -> True, Method -> Coefficient] // 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, ->] >))

Generator matrices are traced:

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

-1/(3 (f _ ϕ^(ó  ))^2) (c _ 2^( ) (3 Overscript[öõ(6), ->] ⊗ Overscript[ϕ^( ), ->] × ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[A^( ) _ μ, ->] + 3 Overscript[öõ(6), ->] ⊗ Overscript[ϕ^( ), ->] × Overscript[A^( ) _ μ, ->] · ∂ _ μ(Overscript[ϕ^( ), ->]) - 3 Overscript[öõ(6), ->] ⊗ ∂ _ μ(Overscript[ϕ^( ), ->]) × Overscript[A^( ) _ μ, ->] · Overscript[ϕ^( ), ->] - 3 Overscript[öõ(6), ->] ⊗ Overscript[A^( ) _ μ, ->] × ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[ϕ^( ), ->] + 3 i Overscript[öõ(6), ->] × Overscript[ϕ^( ), ->] · ∂ _ μ(Overscript[ϕ^( ), ->]) × Overscript[A^( ) _ μ, ->] + 3 i Overscript[öõ(6), ->] × Overscript[ϕ^( ), ->] · Overscript[A^( ) _ μ, ->] × ∂ _ μ(Overscript[ϕ^( ), ->]) - 3 i Overscript[öõ(6), ->] × ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[A^( ) _ μ, ->] × Overscript[ϕ^( ), ->] + 3 i Overscript[öõ(6), ->] × Overscript[A^( ) _ μ, ->] · Overscript[ϕ^( ), ->] × ∂ _ μ(Overscript[ϕ^( ), ->]) + 3 Overscript[öõ(6), ->] × Overscript[ϕ^( ), ->] ⊗ ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[A^( ) _ μ, ->] + 3 Overscript[öõ(6), ->] × Overscript[ϕ^( ), ->] ⊗ Overscript[A^( ) _ μ, ->] · ∂ _ μ(Overscript[ϕ^( ), ->]) - 3 Overscript[öõ(6), ->] × ∂ _ μ(Overscript[ϕ^( ), ->]) ⊗ Overscript[A^( ) _ μ, ->] · Overscript[ϕ^( ), ->] - 3 Overscript[öõ(6), ->] × Overscript[A^( ) _ μ, ->] ⊗ ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[ϕ^( ), ->] - 3 i Overscript[öõ(6), ->] ⊗ Overscript[ϕ^( ), ->] ⊗ ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[A^( ) _ μ, ->] - 3 i Overscript[öõ(6), ->] ⊗ Overscript[ϕ^( ), ->] ⊗ Overscript[A^( ) _ μ, ->] · ∂ _ μ(Overscript[ϕ^( ), ->]) + 3 i Overscript[öõ(6), ->] ⊗ ∂ _ μ(Overscript[ϕ^( ), ->]) ⊗ Overscript[A^( ) _ μ, ->] · Overscript[ϕ^( ), ->] + 3 i Overscript[öõ(6), ->] ⊗ Overscript[A^( ) _ μ, ->] ⊗ ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[ϕ^( ), ->] + 2 i Overscript[öõ(6), ->] · Overscript[A^( ) _ μ, ->] Overscript[ϕ^( ), ->] · ∂ _ μ(Overscript[ϕ^( ), ->]) + 2 i Overscript[öõ(6), ->] · ∂ _ μ(Overscript[ϕ^( ), ->]) Overscript[ϕ^( ), ->] · Overscript[A^( ) _ μ, ->] - 4 i Overscript[öõ(6), ->] · Overscript[ϕ^( ), ->] ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[A^( ) _ μ, ->]))

Indices are supplied:

$IsoIndicesCounter = 0 ;

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

1/(3 (f _ ϕ^(ó  ))^2) (i c _ 2^( ) (3 (f _ (k1 k4 k5) (δ _ (6 k4) f _ (k2 k3 k5) (ϕ^( )^k2 ∂ _ τ1 ϕ^( ) _ ó ^k3 A^( ) _ τ1^k1 - ϕ^( )^k3 ∂ _ τ1 ϕ^( ) _ ó ^k1 A^( ) _ τ1^k2 + ϕ^( )^k1 (∂ _ τ1 ϕ^( ) _ ó ^k3 A^( ) _ τ1^k2 + ∂ _ τ1 ϕ^( ) _ ó ^k2 A^( ) _ τ1^k3)) - i d _ (k2 k3 k4) δ _ (6 k5) (ϕ^( )^k3 (∂ _ τ1 ϕ^( ) _ ó ^k2 A^( ) _ τ1^k1 + ∂ _ τ1 ϕ^( ) _ ó ^k1 A^( ) _ τ1^k2) - ϕ^( )^k1 (∂ _ τ1 ϕ^( ) _ ó ^k3 A^( ) _ τ1^k2 + ∂ _ τ1 ϕ^( ) _ ó ^k2 A^( ) _ τ1^k3))) - d _ (k1 k4 k5) δ _ (6 k5) (d _ (k2 k3 k4) + i f _ (k2 k3 k4)) (ϕ^( )^k3 (∂ _ τ1 ϕ^( ) _ ó ^k2 A^( ) _ τ1^k1 + ∂ _ τ1 ϕ^( ) _ ó ^k1 A^( ) _ τ1^k2) - ϕ^( )^k1 (∂ _ τ1 ϕ^( ) _ ó ^k3 A^( ) _ τ1^k2 + ∂ _ τ1 ϕ^( ) _ ó ^k2 A^( ) _ τ1^k3))) - 2 δ _ (6 k1) (ϕ^( )^k2 (∂ _ τ1 ϕ^( ) _ ó ^k2 A^( ) _ τ1^k1 + ∂ _ τ1 ϕ^( ) _ ó ^k1 A^( ) _ τ1^k2) - 2 ϕ^( )^k1 ∂ _ τ1 ϕ^( ) _ ó ^k2 A^( ) _ τ1^k2)))

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

(llll /. {SU3Delta -> SUNDelta, SU3D -> SUND, SU3F -> SUNF} // Expand // SUNReduce[#, Explicit -> True, HoldSums -> False] &) /. {QuantumField[___, Particle[PhiMeson, RenormalizationState[0]], ExplicitSUNIndex[1 | 2 | 4 | 5 | 6 | 7 | 8]][_] -> 0, QuantumField[Particle[AxialVector[0], RenormalizationState[0]], LorentzIndex[μ], ExplicitSUNIndex[1 | 2 | 3 | 4 | 5 | 8]][_] -> 0} // Simplify

(c _ 2^( ) ϕ^( )^3 ∂ _ τ1 ϕ^( ) _ ó ^3 A^( ) _ τ1^7)/(f _ ϕ^(ó  ))^2

((lll // WriteOutIsoVectors // WriteOutUMatrices) /. {QuantumField[___, Particle[PhiMeson, RenormalizationState[0]], ExplicitSUNIndex[1 | 2 | 4 | 5 | 6 | 7 | 8]][_] -> 0, QuantumField[Particle[AxialVector[0], RenormalizationState[0]], LorentzIndex[μ], ExplicitSUNIndex[1 | 2 | 3 | 4 | 5 | 8]][_] -> 0}) /. NM -> Times // Simplify

(c _ 2^( ) ϕ^( )^3 ∂ _ μ ϕ^( ) _ ó ^3 A^( ) _ μ^7)/(f _ ϕ^(ó  ))^2

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]}

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

melsimplified = Simplify[SUNReduce[FeynRule[llll, fields]]] // IndicesCleanup // Simplify

-1/(3 (f _ ϕ^(ó  ))^2) (i c _ 2^( ) (p _ 2^μ _ 1 (-6 d _ (6 I _ 3 k1)^(3) d _ (I _ 1 I _ 2 k1)^(3) - 6 i f _ (6 I _ 3 k1)^(3) d _ (I _ 1 I _ 2 k1)^(3) + 3 d _ (6 I _ 1 k1)^(3) d _ (I _ 2 I _ 3 k1)^(3) - 4 δ _ (6 I _ 3)^(3) δ _ (I _ 1 I _ 2)^(3) + 2 δ _ (6 I _ 2)^(3) δ _ (I _ 1 I _ 3)^(3) + 2 δ _ (6 I _ 1)^(3) δ _ (I _ 2 I _ 3)^(3) + 3 i d _ (I _ 2 I _ 3 k1)^(3) f _ (6 I _ 1 k1)^(3) + 3 i d _ (I _ 1 I _ 3 k1)^(3) f _ (6 I _ 2 k1)^(3) + 3 d _ (6 I _ 2 k1)^(3) (d _ (I _ 1 I _ 3 k1)^(3) + i f _ (I _ 1 I _ 3 k1)^(3)) - 3 f _ (6 I _ 2 k1)^(3) f _ (I _ 1 I _ 3 k1)^(3) + 3 i d _ (6 I _ 1 k1)^(3) f _ (I _ 2 I _ 3 k1)^(3) - 3 f _ (6 I _ 1 k1)^(3) f _ (I _ 2 I _ 3 k1)^(3)) + p _ 3^μ _ 1 (-6 d _ (6 I _ 2 k1)^(3) d _ (I _ 1 I _ 3 k1)^(3) - 6 i f _ (6 I _ 2 k1)^(3) d _ (I _ 1 I _ 3 k1)^(3) + 3 d _ (6 I _ 1 k1)^(3) d _ (I _ 2 I _ 3 k1)^(3) + 2 δ _ (6 I _ 3)^(3) δ _ (I _ 1 I _ 2)^(3) - 4 δ _ (6 I _ 2)^(3) δ _ (I _ 1 I _ 3)^(3) + 2 δ _ (6 I _ 1)^(3) δ _ (I _ 2 I _ 3)^(3) + 3 i d _ (I _ 2 I _ 3 k1)^(3) f _ (6 I _ 1 k1)^(3) + 3 i d _ (I _ 1 I _ 2 k1)^(3) f _ (6 I _ 3 k1)^(3) + 3 d _ (6 I _ 3 k1)^(3) (d _ (I _ 1 I _ 2 k1)^(3) + i f _ (I _ 1 I _ 2 k1)^(3)) - 3 f _ (6 I _ 3 k1)^(3) f _ (I _ 1 I _ 2 k1)^(3) - 3 i d _ (6 I _ 1 k1)^(3) f _ (I _ 2 I _ 3 k1)^(3) + 3 f _ (6 I _ 1 k1)^(3) f _ (I _ 2 I _ 3 k1)^(3))))

A check:

Table[melsimplified /. {I1 -> 7, I2 -> i, I3 -> j} // SUNReduce[#, Explicit -> True, HoldSums -> False] &, {i, 8}, {j, 8}]

( μ μ ) c p 1 c p 1 2 2 2 3 ------------ + ------------ ó  2 ó  2 (f ) (f ) ϕ ϕ 0 0 0 0 0 0 0 μ μ c p 1 c p 1 2 2 2 3 ------------ + ------------ ó  2 ó  2 (f ) (f ) 0 ϕ ϕ 0 0 0 0 0 0 μ μ μ μ μ c p 1 c p 1 Sqrt[3] c p 1 Sqrt[3] c p 1 c p 1 2 2 2 2 2 2 3 2 3 3 ------------ + ------------ --------------------- + -------------------- + -------------------- ó  2 ó  2 ó  2 ó  2 ó  2 (f ) (f ) (f ) 4 (f ) 4 Sqrt[3] (f ) 0 0 ϕ ϕ 0 0 0 0 ϕ ϕ ϕ μ μ c p 1 c p 1 2 2 2 3 ------------- - ------------ ó  2 ó  2 (f ) (f ) 0 0 0 ϕ ϕ 0 0 0 0 μ μ c p 1 c p 1 2 2 2 3 ------------- - ------------ ó  2 ó  2 (f ) (f ) 0 0 0 0 ϕ ϕ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 μ μ μ μ μ Sqrt[3] c p 1 c p 1 Sqrt[3] c p 1 c p 1 c p 1 2 2 2 2 2 2 2 3 2 3 -------------------- + -------------------- - -------------------- ------------- - ------------ ó  2 ó  2 ó  2 ó  2 ó  2 4 (f ) 4 Sqrt[3] (f ) (f ) (f ) (f ) 0 0 ϕ ϕ ϕ 0 0 0 0 ϕ ϕ

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

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

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

melsimplified /. {I1 -> 3, I2 -> 3, I3 -> 7} // SUNReduce // SUNReduce

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

Converted by Mathematica  (July 10, 2003)