•Pϕ

The evaluated leading order Lagrangian:

ll = ArgumentsSupply[Lagrangian[ChPTW3[2]], x, RenormalizationState[0], ExpansionOrder -> 1, DropOrder -> 1, DiagonalToU -> True] ;

Redundant terms are discarded:

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

(2 c _ 5^( ) !, _ 0^( ) (< σ^6 '6 Overscript[p^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] > + < σ^6 '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[p^( ), ->] · Overscript[σ, ->] > + 2 < σ^6 '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] > p^( )^0))/f _ ϕ^(ó  )

Generator matrices are traced:

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

Indices are supplied:

$IsoIndicesCounter = 0 ;

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

(4 c _ 5^( ) !, _ 0^( ) (2 p^( )^0 ϕ^( )^6 + (d _ (6 k1 k2)^(3) + i f _ (6 k1 k2)^(3)) (p^( )^k2 ϕ^( )^k1 + p^( )^k1 ϕ^( )^k2)))/f _ ϕ^(ó  )

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}

lal = Expand[lala] ;

melsimplified = FeynRule[llll, fields] // SUNReduce // IndicesCleanup

(8 i c _ 5^( ) !, _ 0^( ) d _ (6 I _ 1 I _ 2)^(3))/f _ ϕ^(ó  ) + (8 i c _ 5^( ) !, _ 0^( ) δ _ (0 I _ 1)^(3) δ _ (6 I _ 2)^(3))/f _ ϕ^(ó  )

Converted by Mathematica  (July 10, 2003)