•ϕϕϕ

The evaluated leading order Lagrangian:

ll = ArgumentsSupply[Lagrangian[ChPTW3[2]], x, RenormalizationState[0], ExpansionOrder -> 3, DropOrder -> 3, DiagonalToU -> True] ;

Redundant terms are discarded:

lll = DiscardTerms[ll, Retain -> {Particle[PhiMeson , RenormalizationState[0]] -> 3}, CommutatorReduce -> True, Method -> Expand] // Simplify

-1/(18 (f _ ϕ^(ó  ))^3) (i (9 c _ 2^( ) (< σ^6 '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[σ, ->] '6 ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[σ, ->] > - < σ^6 '6 ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[σ, ->] '6 ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] >) + c _ 5^( ) (2 3^(1/2) (< σ^6 '6 σ^8 '6 Overscript[ϕ^( ), ->] · 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[ϕ^( ), ->] · Overscript[σ, ->] '6 σ^8 >) (m _ π^(ó  ))^2 + (m _ K^0^(ó  ))^2 (3 < σ^6 '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '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 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 σ^8 > - 3 < σ^6 '6 σ^3 '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '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, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] >) + (m _ K^+^(ó  ))^2 (-3 < σ^6 '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '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 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 σ^8 > + 3 < σ^6 '6 σ^3 '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '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, ->] '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 // CommutatorReduce[#, FullReduce -> True] & ;

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}) /. NM -> Times // Expand

-(2 c _ 5^( ) (m _ π^(ó  ))^2 (ϕ^( )^7)^3)/(3 (f _ ϕ^(ó  ))^3) + (2 c _ 5^( ) (m _ K^+^(ó  ))^2 (ϕ^( )^7)^3)/(3 (f _ ϕ^(ó  ))^3) - (2 c _ 5^( ) (m _ π^(ó  ))^2 (ϕ^( )^6)^2 ϕ^( )^7)/(3 (f _ ϕ^(ó  ))^3) + (2 c _ 5^( ) (m _ K^+^(ó  ))^2 (ϕ^( )^6)^2 ϕ^( )^7)/(3 (f _ ϕ^(ó  ))^3) - (2 c _ 5^( ) (m _ π^(ó  ))^2 (ϕ^( )^8)^2 ϕ^( )^7)/(3 (f _ ϕ^(ó  ))^3) + (2 c _ 5^( ) (m _ K^+^(ó  ))^2 (ϕ^( )^8)^2 ϕ^( )^7)/(3 (f _ ϕ^(ó  ))^3) - (c _ 2^( ) (∂ _ μ ϕ^( ) _ ó ^8)^2 ϕ^( )^7)/(f _ ϕ^(ó  ))^3 + (c _ 2^( ) ϕ^( )^8 ∂ _ μ ϕ^( ) _ ó ^7 ∂ _ μ ϕ^( ) _ ó ^8)/(f _ ϕ^(ó  ))^3

((lll // WriteOutIsoVectors // WriteOutUMatrices) /. {QuantumField[___, Particle[PhiMeson, RenormalizationState[0]], ExplicitSUNIndex[1 | 2 | 3 | 4 | 5]][_] -> 0}) /. NM -> Times // Expand

-(2 (m _ π^(ó  ))^2 c _ 5^( ) (ϕ^( )^7)^3)/(3 (f _ ϕ^(ó  ))^3) + (2 (m _ K^+^(ó  ))^2 c _ 5^( ) (ϕ^( )^7)^3)/(3 (f _ ϕ^(ó  ))^3) - (2 (m _ π^(ó  ))^2 c _ 5^( ) (ϕ^( )^6)^2 ϕ^( )^7)/(3 (f _ ϕ^(ó  ))^3) + (2 (m _ K^+^(ó  ))^2 c _ 5^( ) (ϕ^( )^6)^2 ϕ^( )^7)/(3 (f _ ϕ^(ó  ))^3) - (2 (m _ π^(ó  ))^2 c _ 5^( ) (ϕ^( )^8)^2 ϕ^( )^7)/(3 (f _ ϕ^(ó  ))^3) + (2 (m _ K^+^(ó  ))^2 c _ 5^( ) (ϕ^( )^8)^2 ϕ^( )^7)/(3 (f _ ϕ^(ó  ))^3) - (c _ 2^( ) (∂ _ μ ϕ^( ) _ ó ^8)^2 ϕ^( )^7)/(f _ ϕ^(ó  ))^3 + (c _ 2^( ) ϕ^( )^8 ∂ _ μ ϕ^( ) _ ó ^7 ∂ _ μ ϕ^( ) _ ó ^8)/(f _ ϕ^(ó  ))^3

(llll // Expand) // Length

88

(WriteString["stdout", "."] ; SUNReduce[#, Explicit -> True, HoldSums -> True] /. USumHeld -> USum /. {QuantumField[___, Particle[PhiMeson, RenormalizationState[0]], ExplicitSUNIndex[1 | 2 | 3 | 4 | 5]][_] -> 0} /. NM -> Times // Expand) & /@ (llll // Expand)

........................................................................................

-(2 c _ 5^( ) (m _ π^(ó  ))^2 (ϕ^( )^7)^3)/(3 (f _ ϕ^(ó  ))^3) + (2 c _ 5^( ) (m _ K^+^(ó  ))^2 (ϕ^( )^7)^3)/(3 (f _ ϕ^(ó  ))^3) - (2 c _ 5^( ) (m _ π^(ó  ))^2 (ϕ^( )^6)^2 ϕ^( )^7)/(3 (f _ ϕ^(ó  ))^3) + (2 c _ 5^( ) (m _ K^+^(ó  ))^2 (ϕ^( )^6)^2 ϕ^( )^7)/(3 (f _ ϕ^(ó  ))^3) - (2 c _ 5^( ) (m _ π^(ó  ))^2 (ϕ^( )^8)^2 ϕ^( )^7)/(3 (f _ ϕ^(ó  ))^3) + (2 c _ 5^( ) (m _ K^+^(ó  ))^2 (ϕ^( )^8)^2 ϕ^( )^7)/(3 (f _ ϕ^(ó  ))^3) - (c _ 2^( ) (∂ _ τ1 ϕ^( ) _ ó ^8)^2 ϕ^( )^7)/(f _ ϕ^(ó  ))^3 + (c _ 2^( ) ϕ^( )^8 ∂ _ τ1 ϕ^( ) _ ó ^7 ∂ _ τ1 ϕ^( ) _ ó ^8)/(f _ ϕ^(ó  ))^3

Calculation of the Feynman rule:

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

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

lal = Expand[llll] ;

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

Another 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

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

(SUNReduce /@ SUNReduce /@ SUNReduce /@ SUNReduce /@ (melsimplified /. {I1 -> 7, I2 -> 3, I3 -> 3} // Expand)) // Simplify

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

Converted by Mathematica  (July 10, 2003)