•2 ϕ S2

The next to leading order Lagrangian in raw form:

lag = Lagrangian[ChPTW3[4]] /. CouplingConstant[ChPTW3[4], _ ? ((# > 13) &), ___] :> 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 _ 4^( ) (< Δ '6 u _ μ > '6 < u _ μ '6 u _ ν '6 u _ ν >) + N _ 8^( ) (< Δ '6 u _ μ '6 u _ μ > '6 < χ _ + >) + N _ 3^( ) (< Δ '6 u _ μ '6 u _ ν > '6 < u _ μ '6 u _ ν >) + N _ 12^( ) < Δ '6 χ _ - '6 χ _ - > + N _ 10^( ) < Δ '6 χ _ + '6 χ _ + > + 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 _ + >) + N _ 1^( ) < Δ '6 u _ μ '6 u _ μ '6 u _ ν '6 u _ ν > + N _ 2^( ) < Δ '6 u _ μ '6 u _ ν '6 u _ ν '6 u _ μ >))

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

llu = (WriteString["stdout", "."] ; UNMSplit[#, x, DropOrder -> 2]) & /@ Expand[lag] ;

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

lluu = NMExpand[llu] ;

Expand[lluu][[1]]

(c _ 2^( ) N _ 7^( ) ℵ^2 (< σ^6 '6 χ > '6 < ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[σ, ->] '6 ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[σ, ->] >))/(f _ ϕ^(ó  ))^4

Remaining 'raw' quantites are given arguments:

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

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

Expand[ll] // Length

28

ll // LeafCount

11411

Redundant terms are discarded:

lldd = CheckF[(WriteString["stdout", "."] ; Expand[NMExpand[#]]) & /@ Expand[ll], "SU3Weak2mesonS2lldd"] ;

lldd // Length

1055

lldd[[-2]]

(c _ 2^( ) N _ 10^( ) (m _ K^0^(ó  ))^2 !, _ 0^( ) < σ^6 '6 s^( )^0 '6 σ^8 '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] >)/(3^(1/2) (f _ ϕ^(ó  ))^4)

lld = CheckF[(WriteString["stdout", "."] ; DiscardTerms[#, Retain -> {Particle[PhiMeson, RenormalizationState[0]] -> 2, Particle[Scalar[2], RenormalizationState[0] ] -> 1}, CommutatorReduce -> True, Method -> Coefficient]) & /@ Select[lldd, (! FreeQ[#, Scalar[2], Infinity, Heads -> True]) &], "SU3Weak2mesonS2lld"] ;

lld // Length

335

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

DeclareUScalar[UTrace1] ;

lld[[1]] // ExpandAll // NMExpand // Simplify

-(4 c _ 2^( ) N _ 13^( ) (m _ π^(ó  ))^2 !, _ 0^( ) < Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[s^( ), ->] · Overscript[σ, ->] > < σ^6 '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] >)/(3 (f _ ϕ^(ó  ))^4)

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

lal = lll // IsoIndicesSupply ;

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[PhiMeson[0], RenormalizationState[0]], SUNIndex[I1]][p1], QuantumField[Particle[PhiMeson, RenormalizationState[0]], SUNIndex[I2]][p2], QuantumField[Particle[Scalar[2], RenormalizationState[0]], SUNIndex[I3]][p3]}

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

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

res // Length

335

res // LeafCount

24836

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

resu // LeafCount

13381

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

resul // LeafCount

3740

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

Converted by Mathematica  (July 10, 2003)