•1 ϕ S2 S

The next to leading order Lagrangian in raw form:

lag = Lagrangian[ChPTW3[4]] /. CouplingConstant[ChPTW3[4], _ ? ((# < 14) &), ___] :> 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 _ - >) + N _ 36^( ) < Δ '6 (-(χ _ - '6 χ _ -) - χ _ - '6 χ _ + + χ _ + '6 χ _ - + χ _ + '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 = (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

21

ll = (WriteString["stdout", "."] ; DiscardTerms[#, Retain -> {Particle[PhiMeson, RenormalizationState[0]] -> 1, Particle[Scalar[1], RenormalizationState[0]] -> 1, Particle[Scalar[2], RenormalizationState[0]] -> 1}, CommutatorReduce -> True, Method -> Expand]) & /@ lla ;

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

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

DeclareUScalar[UTrace1] ;

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

$IsoIndicesCounter = 0 ;

lal = IsoIndicesSupply[Expand[lll]]

(4 i c _ 2^( ) N _ 21^( ) !, _ 0^( ) < σ^6 '6 σ^i _ 1 '6 σ^i _ 2 > ∂ _ μ s^( ) _ ó ^ó  s^( )^i _ 2 ∂ _ μ ϕ^( ) _ ó ^i _ 1)/(f _ ϕ^(ó  ))^3 - (4 i c _ 2^( ) N _ 21^( ) !, _ 0^( ) < σ^6 '6 σ^i _ 1 '6 σ^i _ 2 > ∂ _ μ s^( ) _ ó ^ó  s^( )^i _ 1 ∂ _ μ ϕ^( ) _ ó ^i _ 2)/(f _ ϕ^(ó  ))^3 - (2 i c _ 2^( ) N _ 22^( ) !, _ 0^( ) < σ^6 '6 σ^i _ 1 > ∂ _ μ s^( ) _ ó ^ó  ϕ^( )^i _ 1 ∂ _ μ s^( ) _ ó ^0)/(f _ ϕ^(ó  ))^3 + (2 i c _ 2^( ) N _ 22^( ) !, _ 0^( ) < σ^i _ 1 '6 σ^6 > ∂ _ μ s^( ) _ ó ^ó  ϕ^( )^i _ 1 ∂ _ μ s^( ) _ ó ^0)/(f _ ϕ^(ó  ))^3 - (2 i c _ 2^( ) N _ 22^( ) !, _ 0^( ) < σ^6 '6 σ^i _ 1 '6 σ^i _ 2 > ∂ _ μ s^( ) _ ó ^ó  ϕ^( )^i _ 1 ∂ _ μ s^( ) _ ó ^i _ 2)/(f _ ϕ^(ó  ))^3 + (2 i c _ 2^( ) N _ 22^( ) !, _ 0^( ) < σ^i _ 1 '6 σ^6 '6 σ^i _ 2 > ∂ _ μ s^( ) _ ó ^ó  ϕ^( )^i _ 1 ∂ _ μ s^( ) _ ó ^i _ 2)/(f _ ϕ^(ó  ))^3

DeclareNonCommutative[UMatrix[a__]] ;

Calculation of the Feynman rule with:

fields = {QuantumField[Particle[PhiMeson, RenormalizationState[0]], SUNIndex[I1]][p1], QuantumField[Particle[Scalar[1], RenormalizationState[0]]][p2], QuantumField[Particle[Scalar[2], RenormalizationState[0]], SUNIndex[I3]][p3]}

{ϕ^( )^I _ 1, s^( ), s^( )^I _ 3}

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

1/(f _ ϕ^(ó  ))^3 (2 c _ 2^( ) !, _ 0^( ) (2 N _ 21^( ) p _ 1 · p _ 2 (< σ^6 . σ^I _ 1 . σ^I _ 3 > - < σ^6 . σ^I _ 3 . σ^I _ 1 >) + N _ 22^( ) p _ 2 · p _ 3 (δ _ (0 I _ 3)^(3) (< σ^I _ 1 . σ^6 > - < σ^6 . σ^I _ 1 >) - < σ^6 . σ^I _ 1 . σ^I _ 3 > + < σ^I _ 1 . σ^6 . σ^I _ 3 >)))

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

Converted by Mathematica  (July 10, 2003)