•ϕϕϕA

The evaluated leading order Lagrangian:

ll = ArgumentsSupply[Lagrangian[ChPTW3[2]], x, RenormalizationState[0], ExpansionOrder -> 3, DropOrder -> 3, DiagonalToU -> True] ;

Redundant terms are discarded:

$VeryVerbose = 2 ;

lll = DiscardTerms[ll, Retain -> {Particle[PhiMeson , RenormalizationState[0]] -> 3, Particle[AxialVector[0] , RenormalizationState[0]] -> 1}, CommutatorReduce -> True, Method -> Coefficient] // Simplify ;

Using Method->Coefficient

Putting on dummy factors

Expanding NM products

Expand DOT products

DotExpand |

Expanding

Finding the coefficient

Applying CommutatorReduce

Generator matrices are traced:

llle = ExpandU[lll, CommutatorReduce -> True] // Simplify ;

The gauge group is SU( 3 ); the dimension of the representation is 3

Expanding the NM products

Applying expansion rules

Applying CommutatorReduce

Indices are supplied:

$IsoIndicesCounter = 0 ;

llll = llle // IsoIndicesSupply // IndicesCleanup ;

Starting with number 0

Recursively resolving iso-vector products

Non-contracted indices will not be numerated

Using ExtendedCleanup->True

Renaming product functions and protecting constants

Applying renaming rules

Applying renaming rules

Applying renaming rules

Applying renaming rules

Applying renaming rules

Applying renaming rules

Applying renaming rules

Renaming product functions and protecting constants

Applying renaming rules

Putting back product function names and constants

Applying CommutatorReduce

Applying SortIndices

Combinations |

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], QuantumField[Particle[PhiMeson, RenormalizationState[0]], SUNIndex[I3]][p3], QuantumField[Particle[PhiMeson, RenormalizationState[0]], SUNIndex[I4]][p4]}

{A^( ) _ μ _ 1^I _ 1, ϕ^( )^I _ 2, ϕ^( )^I _ 3, ϕ^( )^I _ 4}

$VeryVerbose = 0 ;

lal = Expand[llll] ;

lal // Length

111

$ConstantIsoIndices = {I1, I2, I3, I4} ;

melsimplified = CheckF[I * (If[Head[lal] == Plus, Plus @@ ((WriteString["stdout", "."] ; IndicesCleanup[Contract[FunctionalD[PhiToFC[#], fields]] // SUNReduce // SUNReduce // SUNReduce // SUNReduce // SUNReduce // SUNReduce]) & /@ (List @@ lal))]), "A3PiWeakMel"] ;

melsimplified /. {I1 -> 6, I2 -> 3, I3 -> 3, I4 -> 3} // SUNReduce // SUNReduce // SUNReduce // SUNReduce // Simplify

-(c _ 2^( ) (p _ 2^μ _ 1 + p _ 3^μ _ 1 + p _ 4^μ _ 1))/(f _ ϕ^(ó  ))^3

Converted by Mathematica  (July 10, 2003)