•ϕϕ

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}, CommutatorReduce -> True, Method -> Expand] // Simplify

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

Generator matrices are traced:

llle = ExpandU[ExpandU[lll, CommutatorReduce -> True], CommutatorReduce -> True] ;

Indices are supplied:

$IsoIndicesCounter = 0 ;

llll = llle // IsoIndicesSupply // IndicesCleanup ;

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

((lll // WriteOutIsoVectors // WriteOutUMatrices) /. {QuantumField[___, Particle[PhiMeson, RenormalizationState[0]], ExplicitSUNIndex[1 | 2 | 3 | 4 | 5]][_] -> 0}) // CommutatorReduce[#, FullReduce -> True] & // Simplify

(2 (c _ 5^( ) (m _ K^0^(ó  ))^2 ϕ^( )^6 ϕ^( )^8 - c _ 2^( ) ∂ _ μ ϕ^( ) _ ó ^6 ∂ _ μ ϕ^( ) _ ó ^8))/(3^(1/2) (f _ ϕ^(ó  ))^2)

(SUNReduce[#, Explicit -> True, HoldSums -> False] & /@ (llll // Expand)) /. {QuantumField[___, Particle[PhiMeson, RenormalizationState[0]], ExplicitSUNIndex[1 | 2 | 3 | 4 | 5]][_] -> 0} // CommutatorReduce[#, FullReduce -> True] & // Simplify

(2 (c _ 5^( ) (m _ K^0^(ó  ))^2 ϕ^( )^6 ϕ^( )^8 - c _ 2^( ) ∂ _ τ1 ϕ^( ) _ ó ^6 ∂ _ τ1 ϕ^( ) _ ó ^8))/(3^(1/2) (f _ ϕ^(ó  ))^2)

Calculation of the Feynman rule:

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

{ϕ^( )^I _ 1, ϕ^( )^I _ 2}

lal = Expand[llll] ;

melsimplified = If[Head[lal] == Plus, Plus @@ (IndicesCleanup[SUNReduce[FeynRule[#, fields], CommutatorReduce -> True, FullReduce -> True]] & /@ (List @@ lal)), lal] ;

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

(SUNReduce[#, Explicit -> True, HoldSums -> False] & /@ (melsimplified /. {I1 -> 6, I2 -> 3} // Expand)) // Simplify

(2 i (c _ 5^( ) (m _ K^0^(ó  ))^2 + c _ 2^( ) p _ 1 · p _ 2))/(f _ ϕ^(ó  ))^2

(SUNReduce /@ SUNReduce /@ SUNReduce /@ SUNReduce /@ (melsimplified /. {I1 -> 6, I2 -> 3} // Expand)) // Simplify

(2 i (c _ 5^( ) (m _ K^0^(ó  ))^2 + c _ 2^( ) p _ 1 · p _ 2))/(f _ ϕ^(ó  ))^2

Converted by Mathematica  (July 10, 2003)