•ϕA

IsoVector[QuantumField[___, Particle[PseudoScalar[0], ___], ___], ___][_] := 0 ; QuantumField[___, Particle[PseudoScalar[0], ___], ___][_] := 0 ; IsoVector[QuantumField[Particle[Vector[0], ___], ___], ___][_] := 0 ; IsoVector[QuantumField[___, Particle[Scalar[2], ___], ___]][_] := 0 ; QuantumField[___, Particle[Scalar[2], ___], ___, (ExplicitSUNIndex | SUNIndex)[_], ___][_] := 0 ; <br /> QuantumField[Particle[LeftComponent[0], ___], LorentzIndex[μ], ExplicitSUNIndex[0]] := 0 ; <br /> FieldDerivative[UMatrix[UGenerator[_]], {__}] := 0 ; <br /> FieldDerivative[_ ? NumericQ, {__}] := 0 ;

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 _ 8^( ) (< Δ '6 u _ μ '6 u _ μ > '6 < χ _ + >) + 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 _ + >)))

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

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

...........

Expand[llu] // Length

135

Redundant terms are discarded:

lld = (WriteString["stdout", "."] ; DiscardTerms[#, Retain -> {Particle[PhiMeson ] -> 1, Particle[AxialVector[0]] -> 1}, CommutatorReduce -> True, Method -> Coefficient]) & /@ Expand[llu] ;

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

lld // Length

28

lld[[1]]

-(c _ 2^( ) N _ 7^( ) < ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[σ, ->] '6 Overscript[A^( ) _ μ, ->] · Overscript[σ, ->] > < σ^6 '6 χ >)/(f _ ϕ^(ó  ))^3

Remaining 'raw' quantites are put on arguments:

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

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

Generator matrices are traced:

llle = ExpandU[ll, CommutatorReduce -> True] ;

Indices are supplied:

$IsoIndicesCounter = 0 ;

Expand[llle] // Length

146

llll = (WriteString["stdout", "."] ; # // IsoIndicesSupply // SUNReduce // SUNReduce // IndicesCleanup // NMExpand // CommutatorReduce[#, FullReduce -> True] & // Simplify) & /@ Expand[llle] ;

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

llll[[1]]

-(8 c _ 2^( ) N _ 5^( ) (m _ K^+^(ó  ))^2 ∂ _ τ1 ϕ^( ) _ ó ^6 A^( ) _ τ1^3)/(3 (f _ ϕ^(ó  ))^3)

Calculation of the Feynman rule:

fields = {QuantumField[Particle[AxialVector[0], RenormalizationState[0]], LorentzIndex[μ1], SUNIndex[I1]][p1], QuantumField[Particle[PhiMeson, RenormalizationState[0]], SUNIndex[I2]][p2]}

{A^( ) _ μ _ 1^I _ 1, ϕ^( )^I _ 2}

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

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

melsimplified[[1]]

-(8 c _ 2^( ) N _ 5^( ) p _ 2^μ _ 1 (m _ π^(ó  ))^2 d _ (6 I _ 1 I _ 2)^(3))/(3 (f _ ϕ^(ó  ))^3)

A check that two different evaluations with specific components give the same result:

melsimplified /. {I1 -> 6, I2 -> 3} // SUNReduce[#, Explicit -> True, HoldSums -> False] & // Simplify

(4 c _ 2^( ) p _ 2^μ _ 1 (N _ 8^( ) ((m _ π^(ó  ))^2 + (m _ K^+^(ó  ))^2 + (m _ K^0^(ó  ))^2) + 2 (N _ 5^( ) (m _ K^0^(ó  ))^2 + N _ 6^( ) ((m _ K^0^(ó  ))^2 - (m _ K^+^(ó  ))^2))))/(f _ ϕ^(ó  ))^3

melsimplified /. {I1 -> 6, I2 -> 3} // SUNReduce // SUNReduce // SUNReduce // SUNReduce // SUNReduce // SUNReduce // Simplify

(4 c _ 2^( ) p _ 2^μ _ 1 (N _ 8^( ) ((m _ π^(ó  ))^2 + (m _ K^+^(ó  ))^2 + (m _ K^0^(ó  ))^2) + 2 (N _ 5^( ) (m _ K^0^(ó  ))^2 + N _ 6^( ) ((m _ K^0^(ó  ))^2 - (m _ K^+^(ó  ))^2))))/(f _ ϕ^(ó  ))^3

Converted by Mathematica  (July 10, 2003)