•ϕPS

(* IsoVector[QuantumField[___, Particle[AxialVector[0], ___], ___], ___][_] := 0 ; QuantumField[___, Particle[AxialVector[0], ___], ___][_] := 0 ; *) IsoVector[QuantumField[___, Particle[Vector[0], ___], ___], ___][_] := 0 ; QuantumField[___, Particle[Vector[0], ___], ___][_] := 0 ; (* IsoVector[QuantumField[___, Particle[LeftComponent[0], ___], ___], ___][_] := 0 ; QuantumField[___, Particle[LeftComponent[0], ___], ___][_] := 0 ; IsoVector[QuantumField[___, Particle[RightComponent[0], ___], ___], ___][_] := 0 ; QuantumField[___, Particle[RightComponent[0], ___], ___][_] := 0 ; *)

The next to leading order Lagrangian in raw form:

lag = Lagrangian[ChPTW3[4]] /. CouplingConstant[ChPTW3[4], _ ? ((! MatchQ[#, 22 | 23]) &), ___] :> 0

(c _ 2^( ) (N _ 22^( ) < Overscript[∇,^] _ μ Δ '6 Overscript[∇,^] _ μ χ _ + > + i N _ 23^( ) (< Overscript[∇,^] _ μ Δ '6 χ _ - '6 u _ μ > + < Overscript[∇,^] _ μ Δ '6 u _ μ '6 χ _ - >)))/(f _ ϕ^(ó  ))^2

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

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

...

lux = Expand[NMExpand[llu]] ;

'raw' quantites are given arguments:

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

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

Redundant terms are discarded:

lla // Length

13

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

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

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

DeclareUScalar[UTrace1] ;

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

lal = IsoIndicesSupply[Expand[lll]]

(4 c _ 2^( ) N _ 22^( ) !, _ 0^( ) < σ^6 '6 σ^i _ 1 > ∂ _ μ s^( ) _ ó ^ó  ϕ^( )^i _ 1 ∂ _ μ p^( ) _ ó ^0)/(f _ ϕ^(ó  ))^3 + (2 c _ 2^( ) N _ 22^( ) !, _ 0^( ) < σ^6 '6 σ^i _ 1 '6 σ^i _ 2 > ∂ _ μ s^( ) _ ó ^ó  ϕ^( )^i _ 2 ∂ _ μ p^( ) _ ó ^i _ 1)/(f _ ϕ^(ó  ))^3 + (2 c _ 2^( ) N _ 22^( ) !, _ 0^( ) < σ^6 '6 σ^i _ 1 '6 σ^i _ 2 > ∂ _ μ s^( ) _ ó ^ó  ϕ^( )^i _ 1 ∂ _ μ p^( ) _ ó ^i _ 2)/(f _ ϕ^(ó  ))^3 + (4 c _ 2^( ) N _ 22^( ) !, _ 0^( ) < σ^6 '6 σ^i _ 1 > ∂ _ μ s^( ) _ ó ^ó  p^( )^0 ∂ _ μ ϕ^( ) _ ó ^i _ 1)/(f _ ϕ^(ó  ))^3 + (8 c _ 2^( ) N _ 23^( ) !, _ 0^( ) < σ^6 '6 σ^i _ 1 > ∂ _ μ s^( ) _ ó ^ó  p^( )^0 ∂ _ μ ϕ^( ) _ ó ^i _ 1)/(f _ ϕ^(ó  ))^3 + (2 c _ 2^( ) N _ 22^( ) !, _ 0^( ) < σ^6 '6 σ^i _ 1 '6 σ^i _ 2 > ∂ _ μ s^( ) _ ó ^ó  p^( )^i _ 2 ∂ _ μ ϕ^( ) _ ó ^i _ 1)/(f _ ϕ^(ó  ))^3 + (4 c _ 2^( ) N _ 23^( ) !, _ 0^( ) < σ^6 '6 σ^i _ 1 '6 σ^i _ 2 > ∂ _ μ s^( ) _ ó ^ó  p^( )^i _ 2 ∂ _ μ ϕ^( ) _ ó ^i _ 1)/(f _ ϕ^(ó  ))^3 + (2 c _ 2^( ) N _ 22^( ) !, _ 0^( ) < σ^6 '6 σ^i _ 1 '6 σ^i _ 2 > ∂ _ μ s^( ) _ ó ^ó  p^( )^i _ 1 ∂ _ μ ϕ^( ) _ ó ^i _ 2)/(f _ ϕ^(ó  ))^3 + (4 c _ 2^( ) N _ 23^( ) !, _ 0^( ) < σ^6 '6 σ^i _ 1 '6 σ^i _ 2 > ∂ _ μ s^( ) _ ó ^ó  p^( )^i _ 1 ∂ _ μ ϕ^( ) _ ó ^i _ 2)/(f _ ϕ^(ó  ))^3

DeclareNonCommutative[UMatrix[a__]] ;

Calculation of the Feynman rule with:

fields = {QuantumField[Particle[PseudoScalar[0], RenormalizationState[0]], SUNIndex[I1]][p1], QuantumField[Particle[PhiMeson, RenormalizationState[0]], SUNIndex[I2]][p2], QuantumField[Particle[Scalar[1], RenormalizationState[0]]][p3]}

{p^( )^I _ 1, ϕ^( )^I _ 2, s^( )}

res = (SUNReduce[FeynRule[#, fields], FullReduce -> True]) & /@ lal // Simplify ;

resul = res // Simplify

-(2 i c _ 2^( ) (2 N _ 23^( ) p _ 2 · p _ 3 + N _ 22^( ) (p _ 1 · p _ 3 + p _ 2 · p _ 3)) !, _ 0^( ) (2 δ _ (0 I _ 1)^(3) < σ^6 . σ^I _ 2 > + < σ^6 . σ^I _ 1 . σ^I _ 2 > + < σ^6 . σ^I _ 2 . σ^I _ 1 >))/(f _ ϕ^(ó  ))^3

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

Converted by Mathematica  (July 10, 2003)