•ϕAS

IsoVector[QuantumField[___, Particle[PseudoScalar[0], ___], ___], ___][_] := 0 ; <br /> QuantumField[___, Particle[PseudoScalar[0], ___], ___][_] := 0 ; <br /> IsoVector[QuantumField[___, Particle[Scalar[2], ___], ___], ___][_] := 0 ; <br /> QuantumField[___, Particle[Scalar[2], ___], ___][_] := 0 ; <br /> IsoVector[QuantumField[___, Particle[Vector[0], ___], ___], ___][_] := 0 ; <br /> QuantumField[___, Particle[Vector[0], ___], ___][_] := 0 ;

The evaluated  next to leading order Lagrangian:

lag = Lagrangian[ChPTW3[4]] /. CouplingConstant[ChPTW3[4], _ ? ((# < 14 || # === 36) &), ___] :> 0

1/(f _ ϕ^(ó  ))^2 (c _ 2^( ) (N _ 31^( ) (< Δ '6 u _ μ > '6 < ϵ^(μ ν ρ σ) '6 f _ - _ (ρ σ) '6 u _ ν >) + N _ 30^( ) (< Δ '6 u _ μ > '6 < ϵ^(μ ν ρ σ) '6 f _ + _ (ρ σ) '6 u _ ν >) + i N _ 28^( ) ϵ^(μ ν ρ σ) (< Δ '6 u _ μ > '6 < u _ ν '6 u _ ρ '6 u _ σ >) + i N _ 24^( ) (< Overscript[∇,^] _ μ Δ '6 u _ μ > '6 < χ _ - >) + i N _ 35^( ) < Δ '6 (f _ + _ (μ ν) '6 ϵ^(μ ν ρ σ) '6 f _ - _ (ρ σ) - ϵ^(μ ν ρ σ) '6 f _ - _ (ρ σ) '6 f _ + _ (μ ν)) > + N _ 34^( ) < Δ '6 ((ϵ^(μ ν ρ σ) f _ - _ (ρ σ) + ϵ^(μ ν ρ σ) f _ + _ (ρ σ)) '6 u _ μ '6 u _ ν - u _ μ '6 u _ ν '6 (ϵ^(μ ν ρ σ) f _ - _ (ρ σ) + ϵ^(μ ν ρ σ) f _ + _ (ρ σ))) > + N _ 29^( ) < Δ '6 (ϵ^(μ ν ρ σ) '6 (f _ + _ (ρ σ) - f _ - _ (ρ σ)) '6 u _ μ '6 u _ ν - u _ μ '6 u _ ν '6 ϵ^(μ ν ρ σ) '6 (f _ + _ (ρ σ) - f _ - _ (ρ σ))) > + N _ 27^( ) (-< f _ - _ (μ ν) '6 f _ + _ (μ ν) > - < f _ + _ (μ ν) '6 f _ - _ (μ ν) > + 2 < f _ + _ (μ ν) '6 f _ + _ (μ ν) >) + N _ 22^( ) < Overscript[∇,^] _ μ Δ '6 Overscript[∇,^] _ μ χ _ + > + i N _ 18^( ) (< Δ '6 f _ + _ (μ ν) '6 f _ + _ (μ ν) > - < Δ '6 f _ - _ (μ ν) '6 f _ - _ (μ ν) >) + N _ 37^( ) < Δ '6 (f _ - _ (μ ν) + f _ + _ (μ ν)) '6 (f _ - _ (μ ν) + f _ + _ (μ ν)) > + i N _ 32^( ) < Overscript[∇,^] _ μ Δ '6 Δ '6 (ϵ^(μ ν ρ σ) '6 f _ - _ (ρ σ) '6 u _ ν - u _ ν '6 ϵ^(μ ν ρ σ) '6 f _ - _ (ρ σ)) > + i N _ 32^( ) < Overscript[∇,^] _ μ Δ '6 Δ '6 (ϵ^(μ ν ρ σ) '6 f _ + _ (ρ σ) '6 u _ ν - u _ ν '6 ϵ^(μ ν ρ σ) '6 f _ + _ (ρ σ)) > + i N _ 23^( ) (< Overscript[∇,^] _ μ Δ '6 χ _ - '6 u _ μ > + < Overscript[∇,^] _ μ Δ '6 u _ μ '6 χ _ - >) + i N _ 21^( ) (< Overscript[∇,^] _ μ Δ '6 χ _ + '6 u _ μ > - < Overscript[∇,^] _ μ Δ '6 u _ μ '6 χ _ + >) + N _ 26^( ) (< Overscript[∇,^] _ μ Δ '6 f _ - _ (μ ν) '6 u _ ν > + < Overscript[∇,^] _ μ Δ '6 u _ ν '6 f _ - _ (μ ν) >) + N _ 25^( ) (< Overscript[∇,^] _ μ Δ '6 f _ + _ (μ ν) '6 u _ ν > + < Overscript[∇,^] _ μ Δ '6 u _ ν '6 f _ + _ (μ ν) >) + N _ 20^( ) (< Overscript[∇,^] _ μ Δ '6 ω^(μ ν) '6 u _ ν > + < Overscript[∇,^] _ μ Δ '6 u _ ν '6 ω^(μ ν) >) + i N _ 17^( ) < Δ '6 u _ μ '6 f _ - _ (μ ν) '6 u _ ν > + i N _ 15^( ) < Δ '6 u _ μ '6 f _ + _ (μ ν) '6 u _ ν > + i N _ 16^( ) (< Δ '6 f _ - _ (μ ν) '6 u _ μ '6 u _ ν > + < Δ '6 u _ μ '6 u _ ν '6 f _ - _ (μ ν) >) + 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) >) + i N _ 19^( ) (< Overscript[∇,^] _ μ Δ '6 u _ μ '6 u _ ν '6 u _ ν > - < Overscript[∇,^] _ μ Δ '6 u _ ν '6 u _ ν '6 u _ μ >)))

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

llu = CheckF[(WriteString["stdout", "."] ; UNMSplit[#, x, DropOrder -> 1]) & /@ Expand[lag], "SU3Weak1mesonSAllu"] ;

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

lux = Expand[NMExpand[llu]] ;

'raw' quantites are given arguments:

lla = ArgumentsSupply[lux, x, RenormalizationState[0], ExpansionOrder -> 2, DropOrder -> 2, DiagonalToU -> True] ;

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

Redundant terms are discarded:

lla // Length

247

ll = (WriteString["stdout", "."] ; DiscardTerms[#, Retain -> {Particle[AxialVector[0], RenormalizationState[0]] -> 1, Particle[PhiMeson, RenormalizationState[0]] -> 1, Particle[Scalar[1], RenormalizationState[0]] -> 1}, CommutatorReduce -> True, Method -> Expand]) & /@ lla ;

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

Expand[ll] // Length

108

DeclareUScalar[QuantumField[aa___, Particle[bb__], cc___][x_]] ; <br /> DeclareUScalar[UTrace1] ;

lll = Simplify[NMExpand[ExpandAll[#]]] & /@ ll ;

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

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

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

0

Cases[lal /. _UTrace1 -> 1, _Umatrix, Infinity, Heads -> True] // Length

0

DeclareNonCommutative[UMatrix[a__]] ;

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[Scalar[1], RenormalizationState[0]]][p3]}

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

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

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

res // Length

106

res // LeafCount

5258

resu = Collect[res, {_DecayConstant, _UTrace1}] ;

resu // LeafCount

3122

resul = Collect[resu // Contract, HoldPattern[Plus[__ ? ((! FreeQ[{##}, Momentum | ParticleMass, Infinity, Heads -> True]) &)]]] ;

resul // LeafCount

1236

resul // Length

26

resul[[1]]

(2 c _ 2^( ) N _ 24^( ) p _ 3^μ _ 1 (m _ K^+^(ó  ))^2 < σ^6 σ^I _ 1 > < σ^3 σ^I _ 2 >)/(f _ ϕ^(ó  ))^3

UndeclareUScalar[QuantumField[aa___, Particle[bb__], cc___][x_]] ; <br /> UndeclareUScalar[UTrace1] ; <br /> UnDeclareNonCommutative[UMatrix[a__]] ;

Converted by Mathematica  (July 10, 2003)