•ϕS

IsoVector[QuantumField[___, Particle[AxialVector[0], ___], ___], ___][_] := 0 ; <br /> QuantumField[___, Particle[AxialVector[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^( ) (i N _ 24^( ) (< Overscript[∇,^] _ μ Δ '6 u _ μ > '6 < χ _ - >) + 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 _ - >) + i N _ 21^( ) (< Overscript[∇,^] _ μ Δ '6 χ _ + '6 u _ μ > - < Overscript[∇,^] _ μ Δ '6 u _ μ '6 χ _ + >) + 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, RenormalizationState[0], DropOrder -> 1]) & /@ Expand[lag] ;

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

lluu = NMExpand[llu] ;

Expand[lluu][[1]]

-(i c _ 2^( ) N _ 24^( ) ℵ (< ∂ _ μ(s^( )) '6 σ^6 '6 ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[σ, ->] > '6 < χ >))/((f _ ϕ^(ó  ))^2 f _ ϕ^(ó  ))

Redundant terms are discarded:

lld = DiscardTerms[#, Retain -> {Particle[PhiMeson , RenormalizationState[0]] -> 1}, CommutatorReduce -> True, Method -> Coefficient] & /@ Expand[lluu] ;

Remaining 'raw' quantites are given arguments:

ll = ArgumentsSupply[lld, x, RenormalizationState[0], ExpansionOrder -> 3, DropOrder -> 3, DiagonalToU -> True] ;

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

DeclareUScalar[QuantumField[aa___, Particle[bb__], cc___][x_]] ; DeclareUScalar[UTrace1] ;

ll[[1]] // ExpandAll // NMExpand // Simplify

1/(3 (f _ ϕ^(ó  ))^3) (2 i c _ 2^( ) N _ 21^( ) ((< σ^6 '6 ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[σ, ->] > + 2 3^(1/2) < σ^6 '6 ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[σ, ->] '6 σ^8 >) (m _ π^(ó  ))^2 + (m _ K^0^(ó  ))^2 (< σ^6 '6 ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[σ, ->] > - 3 < σ^6 '6 ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[σ, ->] '6 σ^3 > - 3^(1/2) < σ^6 '6 ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[σ, ->] '6 σ^8 >) + (m _ K^+^(ó  ))^2 (< σ^6 '6 ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[σ, ->] > + 3 < σ^6 '6 ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[σ, ->] '6 σ^3 > - 3^(1/2) < σ^6 '6 ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[σ, ->] '6 σ^8 >)) ∂ _ μ s^( ) _ ó ^ó )

Expand[ll] // Length

2

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

..

Expand[lll] // Length

10

lal = IsoIndicesSupply[Expand[lll]] ;

lal[[1]]

-(2 i c _ 2^( ) N _ 21^( ) (m _ K^+^(ó  ))^2 < σ^6 '6 σ^3 '6 σ^i _ 4 > ∂ _ μ s^( ) _ ó ^ó  ∂ _ μ ϕ^( ) _ ó ^i _ 4)/(f _ ϕ^(ó  ))^3

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

0

DeclareNonCommutative[UMatrix[a__]] ;

Calculation of the Feynman rule:

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

{ϕ^( )^I _ 1, s^( )}

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

..........

res = res // Simplify

1/(3 (f _ ϕ^(ó  ))^3) (2 c _ 2^( ) N _ 21^( ) p _ 1 · p _ 2 (2 3^(1/2) (< σ^6 . σ^I _ 1 . σ^8 > - < σ^6 . σ^8 . σ^I _ 1 >) (m _ π^(ó  ))^2 + (m _ K^0^(ó  ))^2 (3 < σ^6 . σ^3 . σ^I _ 1 > + 3^(1/2) < σ^6 . σ^8 . σ^I _ 1 > - 3 < σ^6 . σ^I _ 1 . σ^3 > - 3^(1/2) < σ^6 . σ^I _ 1 . σ^8 >) + (m _ K^+^(ó  ))^2 (-3 < σ^6 . σ^3 . σ^I _ 1 > + 3^(1/2) < σ^6 . σ^8 . σ^I _ 1 > + 3 < σ^6 . σ^I _ 1 . σ^3 > - 3^(1/2) < σ^6 . σ^I _ 1 . σ^8 >)))

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

Converted by Mathematica  (July 10, 2003)