•Pϕϕ

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[PseudoScalar[0] , RenormalizationState[0]] -> 1}, CommutatorReduce -> True, Method -> Expand] // Simplify

(i c _ 5^( ) !, _ 0^( ) (< σ^6 '6 Overscript[p^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] > - < σ^6 '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[p^( ), ->] · Overscript[σ, ->] >))/(f _ ϕ^(ó  ))^2

Generator matrices are traced:

llle = lll // ExpandU // CommutatorReduce

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

Indices are supplied:

$IsoIndicesCounter = 0 ;

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

1/(3 (f _ ϕ^(ó  ))^2) (2 c _ 5^( ) !, _ 0^( ) (2 i p^( )^6 (ϕ^( )^k1)^2 + ϕ^( )^k2 (3 ((d _ (6 k1 k5)^(3) + i f _ (6 k1 k5)^(3)) f _ (k2 k3 k5)^(3) p^( )^k3 ϕ^( )^k1 - i d _ (6 k1 k4)^(3) d _ (k2 k3 k4)^(3) (p^( )^k3 ϕ^( )^k1 - p^( )^k1 ϕ^( )^k3) + d _ (k2 k3 k4)^(3) f _ (6 k1 k4)^(3) (p^( )^k3 ϕ^( )^k1 - p^( )^k1 ϕ^( )^k3)) - 2 i p^( )^k2 ϕ^( )^6)))

Calculation of the Feynman rule:

fields = {QuantumField[Particle[PseudoScalar[0], RenormalizationState[0]], SUNIndex[I1]][p1], QuantumField[Particle[PhiMeson, RenormalizationState[0]], SUNIndex[I2]][p2], QuantumField[Particle[PhiMeson, RenormalizationState[0]], SUNIndex[I3]][p3]}

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

melsimplified = FeynRule[llll, fields] // SUNReduce[#, FullReduce -> True] & // IndicesCleanup // Simplify

1/(3 (f _ ϕ^(ó  ))^2) (2 c _ 5^( ) !, _ 0^( ) (-6 d _ (6 I _ 1 k1)^(3) d _ (I _ 2 I _ 3 k1)^(3) - 6 i f _ (6 I _ 1 k1)^(3) d _ (I _ 2 I _ 3 k1)^(3) + 2 δ _ (6 I _ 3)^(3) δ _ (I _ 1 I _ 2)^(3) + 2 δ _ (6 I _ 2)^(3) δ _ (I _ 1 I _ 3)^(3) - 4 δ _ (6 I _ 1)^(3) δ _ (I _ 2 I _ 3)^(3) + 3 i d _ (I _ 1 I _ 3 k1)^(3) f _ (6 I _ 2 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 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)))

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

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

-(4 i c _ 5^( ) !, _ 0^( ))/(f _ ϕ^(ó  ))^2

(SUNReduce[#, FullReduce -> True] & /@ (melsimplified /. {I1 -> 7, I2 -> 3, I3 -> 3} // Expand)) // Simplify

-(4 i c _ 5^( ) !, _ 0^( ))/(f _ ϕ^(ó  ))^2

Converted by Mathematica  (July 10, 2003)