•Pϕ

The unevaluated lowest order Lagrangian:

Lagrangian[ChPT3[2]]

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

The evaluated lowest order lagrangian:

ll = ArgumentsSupply[Lagrangian[ChPT3[2]], x, RenormalizationState[0], ExpansionOrder -> 1, DropOrder -> 1] ;

Redundant terms are discarded:

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

1/2 f _ ϕ^(ó  ) !, _ 0^( ) (< Overscript[p^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] > + < Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[p^( ), ->] · Overscript[σ, ->] >)

Matrices are traced:

llle = ExpandU[lll, CommutatorReduce -> True]

2 f _ ϕ^(ó  ) Overscript[p^( ), ->] · Overscript[ϕ^( ), ->] !, _ 0^( )

Indices are supplied:

$IsoIndicesCounter = 0 ;

llll = llle // IsoIndicesSupply

2 f _ ϕ^(ó  ) (p^( )^i _ 1 '6 ϕ^( )^i _ 1) !, _ 0^( )

Calculation of the Feynman rule:

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

{p^( )^I _ 1, ϕ^( )^I _ 2}

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

2 i f _ ϕ^(ó  ) !, _ 0^( ) δ _ (I _ 1 I _ 2)^(3)

Converted by Mathematica  (July 10, 2003)