•2πS

The leading order lagrangian in raw form:

Lagrangian[ChPT2[2]]

1/4 (f _ π^(ó  ))^2 (< ÷„ '6 χ^† > + < χ '6 ÷„^† > + < ÷s _ μ(÷„) '6 ÷s _ μ(÷„)^† >)

The evaluated lagrangian:

ll = ArgumentsSupply[Lagrangian[ChPT2[2]], x, RenormalizationState[0], ExpansionOrder -> 2, DropOrder -> 2, DiagonalToU -> True] ;

Redundant terms are discarded:

lll = DiscardTerms[ll, Retain -> {Particle[Pion , RenormalizationState[0]] -> 2, Particle[Scalar[2] , RenormalizationState[0]] -> 1}, Method -> Coefficient] // CycleUTraces // Simplify

-1/2 !, _ 0^( ) (< Overscript[s^( ), ->] · Overscript[σ, ->] '6 Overscript[π^( ), ->] · Overscript[σ, ->] '6 Overscript[π^( ), ->] · Overscript[σ, ->] > + < s^( )^0 '6 Overscript[π^( ), ->] · Overscript[σ, ->] '6 Overscript[π^( ), ->] · Overscript[σ, ->] >)

Matrices are traced:

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

-(Overscript[π^( ), ->] · Overscript[π^( ), ->]) !, _ 0^( ) s^( )^0

Indices are supplied:

$IsoIndicesCounter = 0 ;

llll = llle // IsoIndicesSupply // CommutatorReduce[#, FullReduce -> True] &

-!, _ 0^( ) (π^( )^i _ 1)^2 s^( )^0

Calculation of the Feynman rule:

fields = {QuantumField[Particle[Pion, RenormalizationState[0]], SUNIndex[I1]][p1], QuantumField[Particle[Pion, RenormalizationState[0]], SUNIndex[I2]][p2], QuantumField[Particle[Scalar[2], RenormalizationState[0]], SUNIndex[I3]][p3]}

{π^( )^I _ 1, π^( )^I _ 2, s^( )^I _ 3}

melsimplified = Simplify[SUNReduce[FeynRule[llll, fields]]]

-2 i !, _ 0^( ) δ _ (0 I _ 3)^(2) δ _ (I _ 1 I _ 2)^(2)

Converted by Mathematica  (July 10, 2003)