•Pϕ

IsoVector[QuantumField[___, Particle[Scalar[1], ___], ___], ___][_] := 0 ; <br /> QuantumField[___, Particle[Scalar[1], ___], ___][_] := 0 ; IsoVector[QuantumField[___, Particle[Scalar[2], ___], ___], ___][_] := 0 ; <br /> QuantumField[___, Particle[Scalar[2], ___], ___][_] := 0 ;

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 -> 1]) & /@ Expand[lag] ;

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

lluu = NMExpand[llu] ;

Expand[lluu][[1]]

-(i c _ 2^( ) N _ 11^( ) ℵ (< σ^6 '6 χ > '6 < χ '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] >))/(2 (f _ ϕ^(ó  ))^3)

Remaining 'raw' quantites are given arguments:

ll = ArgumentsSupply[lluu, x, RenormalizationState[0], ExpansionOrder -> 1, DropOrder -> 1, DiagonalToU -> True] ;

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

Expand[ll] // Length

60

Redundant terms are discarded:

lldd = CheckF[(WriteString["stdout", "."] ; Expand[NMExpand[#]]) & /@ Expand[ll], "SU3Weak1mesonPlldd"] ;

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

lldd[[1]]

(8 c _ 2^( ) N _ 13^( ) (< σ^6 '6 Overscript[p^( ), ->] · Overscript[σ, ->] > '6 < Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 σ^8 >) (m _ π^(ó  ))^2 !, _ 0^( ))/(3^(1/2) (f _ ϕ^(ó  ))^3)

lld = CheckF[(WriteString["stdout", "."] ; DiscardTerms[#, Retain -> {Particle[PseudoScalar[0], RenormalizationState[0]] -> 1, Particle[PhiMeson, RenormalizationState[0] ] -> 1}, CommutatorReduce -> True, Method -> Coefficient]) & /@ lldd, "SU3Weak1mesonPlld"] ;

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

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

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

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

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

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

lal = 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[PseudoScalar[0], RenormalizationState[0]], SUNIndex[I1]][p1], QuantumField[Particle[PhiMeson, RenormalizationState[0]], SUNIndex[I2]][p2]}

{p^( )^I _ 1, ϕ^( )^I _ 2}

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

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

res // Length

120

res // LeafCount

5909

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

resu // LeafCount

3663

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

resul // LeafCount

2166

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

Converted by Mathematica  (July 10, 2003)