•ϕϕϕ

lag = Lagrangian[ChPTW3[4]] /. CouplingConstant[ChPTW3[4], _ ? ((# > 15) &), ___] :> 0

1/(f _ ϕ^(ó  ))^2 (c _ 2^( ) (N _ 13^( ) (< Δ '6 χ _ - > '6 < χ _ - >) + N _ 11^( ) (< Δ '6 χ _ + > '6 < χ _ + >) + N _ 7^( ) (< Δ '6 χ _ + > '6 < u _ μ '6 u _ μ >) + N _ 6^( ) (< Δ '6 u _ μ > '6 < χ _ + '6 u _ μ >) + N _ 8^( ) (< Δ '6 u _ μ '6 u _ μ > '6 < χ _ + >) + N _ 12^( ) < Δ '6 χ _ - '6 χ _ - > + N _ 10^( ) < Δ '6 χ _ + '6 χ _ + > + i N _ 15^( ) < Δ '6 u _ μ '6 f _ + _ (μ ν) '6 u _ ν > + N _ 9^( ) (< Δ '6 χ _ - '6 u _ μ '6 u _ μ > - < Δ '6 u _ μ '6 u _ μ '6 χ _ - >) + N _ 5^( ) (< Δ '6 χ _ + '6 u _ μ '6 u _ μ > + < Δ '6 u _ μ '6 u _ μ '6 χ _ + >) + i N _ 14^( ) (< Δ '6 f _ + _ (μ ν) '6 u _ μ '6 u _ ν > + < Δ '6 u _ μ '6 u _ ν '6 f _ + _ (μ ν) >)))

First, UNMSplit is used to expand NM products of U matrices into meson fields:

lag // Expand // Length

14

llu = CheckF[(WriteString["stdout", "."] ; UNMSplit[#, x, DropOrder -> 3]) & /@ Expand[lag], "SU3Weak3mesonllu"] ;

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

lluu = NMExpand[llu] // Simplify ;

Expand[lluu] // Length

376

Redundant terms are discarded:

lld = (WriteString["stdout", "."] ; DiscardTerms[#, Retain -> {Particle[PhiMeson ] -> 3}, CommutatorReduce -> True, Method -> Coefficient]) & /@ Expand[lluu] ;

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

Remaining 'raw' quantites are given arguments:

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

ArgumentsSupply :: argxpr : Warning : The argument x is already in the expression.

ll // Length

56

DeclareUScalar[QuantumField[aa___, Particle[bb__], cc___][x_]] ;

DeclareUScalar[UTrace1] ;

ll[[1]] // ExpandAll // NMExpand // Simplify

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

lll = (WriteString["stdout", "."] ; Simplify[NMExpand[ExpandAll[#]]]) & /@ ll ;

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

lll // Length

56

lll // LeafCount

57197

Expand[lll] // Length

1074

lal = (WriteString["stdout", "."] ; $IsoIndicesCounter = 0 ; CommutatorReduce[IsoIndicesSupply[#], FullReduce -> True]) & /@ Expand[lll] ;

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

Cases[lal, UTrace1[__ ? ((! FreeQ[{#}, QuantumField, Infinity, Heads -> True]) &)], Infinity, Heads -> True] // Length

0

lala = Simplify /@ Collect[lal, _UTrace1] ;

DeclareNonCommutative[UMatrix[a__]] ;

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}

lala // Length

97

res = ((WriteString["stdout", "."] ; Simplify[I * SUNReduce[FunctionalD[PhiToFC[#], fields], FullReduce -> True] // Contract]) & /@ lala) ;

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

res // Length

97

res // LeafCount

52700

A few checks:

test = res /. {I1 -> 7, I2 -> 3, I3 -> 3} // WriteOutUMatrices ;

subpar = Table[(ParticleMass[PhiMeson, SUNIndex[i], RenormalizationState[0]] -> ParticleMass[Select[$IsoSpinProjectionRules, (! FreeQ[#, {i}] &)][[1]][[1]], RenormalizationState[0]]), {i, 8}]

{m _ ϕ^(1 ) -> m _ π^+^(ó  ), m _ ϕ^(2 ) -> m _ π^+^(ó  ), m _ ϕ^(3 ) -> m _ π^0^(ó  ), m _ ϕ^(4 ) -> m _ K^+^(ó  ), m _ ϕ^(5 ) -> m _ K^+^(ó  ), m _ ϕ^(6 ) -> m _ K^0^(ó  ), m _ ϕ^(7 ) -> m _ K^0^(ó  ), m _ ϕ^(8 ) -> m _ η^(ó  )}

udrules = {PionPlus -> Pion, PionZero -> Pion, KaonPlus -> Kaon, KaonZero -> Kaon}

{π^+ -> π, π^0 -> π, K^+ -> K, K^0 -> K}

KamborToBijnens = {cc[ChPTW3[4], 5] -> Ε _ 10 - Ε _ 11, cc[ChPTW3[4], 6] -> Ε _ 11 + 2 Ε _ 12, cc[ChPTW3[4], 7] -> 1/2 Ε _ 11 + Ε _ 13, cc[ChPTW3[4], 8] -> Ε _ 11, cc[ChPTW3[4], 9] -> Ε _ 15, cc[ChPTW3[4], 10] -> Ε _ 1 - Ε _ 5, cc[ChPTW3[4], 11] -> Ε _ 2, cc[ChPTW3[4], 12] -> Ε _ 3 - Ε _ 5, cc[ChPTW3[4], 13] -> -Ε _ 4, cc[ChPTW3[4], 36] -> Ε _ 5} /. cc[a__] :> HoldPattern[CouplingConstant[a]]

{HoldPattern[N _ 5^( )] -> Ε _ 10 - Ε _ 11, HoldPattern[N _ 6^( )] -> Ε _ 11 + 2 Ε _ 12, HoldPattern[N _ 7^( )] -> Ε _ 11/2 + Ε _ 13, HoldPattern[N _ 8^( )] -> Ε _ 11, HoldPattern[N _ 9^( )] -> Ε _ 15, HoldPattern[N _ 10^( )] -> Ε _ 1 - Ε _ 5, HoldPattern[N _ 11^( )] -> Ε _ 2, HoldPattern[N _ 12^( )] -> Ε _ 3 - Ε _ 5, HoldPattern[N _ 13^( )] -> -Ε _ 4, HoldPattern[N _ 36^( )] -> Ε _ 5}

onshellamp = I * Collect[Cancel[(test/I /. subpar /. udrules /. p1 -> -p2 - p3 // MomentumExpand // ExpandScalarProduct) /. Pair[Momentum[p2, ___], Momentum[p3, ___]] -> (ParticleMass[Kaon, RenormalizationState[0]]^2 - Pair[Momentum[p2], Momentum[p2]] - Pair[Momentum[p3], Momentum[p3]])/2 /. {Pair[Momentum[p2, ___], Momentum[p2, ___], ___] -> ParticleMass[Pion, RenormalizationState[0]]^2, Pair[Momentum[p3, ___], Momentum[p3, ___], ___] -> ParticleMass[Pion, RenormalizationState[0]]^2}], {_DecayConstant, _ParticleMass}] // Simplify // (FullSimplify /@ #) & // FullSimplify

1/(3 (f _ ϕ^(ó  ))^5) (4 i c _ 2^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) ((6 N _ 5^( ) + 12 N _ 7^( ) + 3 N _ 8^( ) - 6 N _ 10^( ) - 14 N _ 11^( ) - 6 N _ 12^( )) (m _ π^(ó  ))^2 + (3 N _ 5^( ) - 6 N _ 7^( ) + 6 N _ 8^( ) + 3 N _ 9^( ) - 4 (N _ 10^( ) + N _ 11^( ))) (m _ K^(ó  ))^2))

onshellamp /. KamborToBijnens // Simplify

-1/(3 (f _ ϕ^(ó  ))^5) (4 i c _ 2^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) ((6 Ε _ 1 + 14 Ε _ 2 + 6 Ε _ 3 - 12 Ε _ 5 - 6 Ε _ 10 - 3 Ε _ 11 - 12 Ε _ 13) (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 (4 Ε _ 1 + 4 Ε _ 2 - 4 Ε _ 5 - 3 Ε _ 10 + 6 Ε _ 13 - 3 Ε _ 15)))

UndeclareUScalar[QuantumField[aa___, Particle[bb__], cc___][x_]] ; UndeclareUScalar[UTrace1] ; UnDeclareNonCommutative[UMatrix[a__]] ;

Converted by Mathematica  (July 10, 2003)