•Two-vertex to fourth order in the energy

IsoVector[QuantumField[Particle[AxialVector[0], ___], ___], ___][_] := 0 ; IsoVector[QuantumField[Particle[Vector[0], ___], ___], ___][_] := 0 ; IsoVector[QuantumField[Particle[Scalar[1 | 2], ___], ___], ___][_] := 0 ; QuantumField[Particle[Scalar[1 | 2], ___], ___][_] := 0 ;

lag = Lagrangian[ChPT2[4]] /. CouplingConstant[ChPT2[4], 1 | 2 | 3 | 7 | 9 | 10 | 11 | 12, ___][___] :> 0

L _ 6^( ) ((< ÷„ '6 χ^† > + < χ '6 ÷„^† >) '6 (< ÷„ '6 χ^† > + < χ '6 ÷„^† >)) + L _ 4^( ) (< ÷s _ μ(÷„) '6 ÷s _ μ(÷„)^† > '6 (< ÷„ '6 χ^† > + < χ '6 ÷„^† >)) + L _ 5^( ) < ÷s _ μ(÷„) '6 ÷s _ μ(÷„)^† '6 (÷„ '6 χ^† + χ '6 ÷„^†) > + L _ 8^( ) (< ÷„ '6 χ^† '6 ÷„ '6 χ^† > + < χ '6 ÷„^† '6 χ '6 ÷„^† >)

ll = ArgumentsSupply[<br /> lag, x, RenormalizationState[0], ExpansionOrder -> 1, DropOrder -> 1, DiagonalToU -> True] ;

lll = DiscardTerms[ll, Retain -> {Particle[PseudoScalar[0] , RenormalizationState[0]] -> 1, Particle[PseudoScalar[2] , RenormalizationState[0]] -> 1}, CommutatorReduce -> True, Method -> Expand] // Simplify

(8 (2 L _ 6^( ) + L _ 8^( )) (m _ π^(ó  ))^2 !, _ 0^( ) (< Overscript[p^( ), ->] · Overscript[σ, ->] '6 Overscript[π^( ), ->] · Overscript[σ, ->] > + < Overscript[π^( ), ->] · Overscript[σ, ->] '6 Overscript[p^( ), ->] · Overscript[σ, ->] >))/f _ π^(ó  )

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

$IsoIndicesCounter = 0 ;

llll = llle // IsoIndicesSupply // SUNReduce // IndicesCleanup // CommutatorReduce // Simplify

(32 (2 L _ 6^( ) + L _ 8^( )) (m _ π^(ó  ))^2 !, _ 0^( ) p^( )^k1 π^( )^k1)/f _ π^(ó  )

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

{p^( )^I _ 1, π^( )^I _ 2}

mel = FeynRule[llll, fields] // Simplify // SUNReduce // Simplify

(32 i (2 L _ 6^( ) + L _ 8^( )) (m _ π^(ó  ))^2 !, _ 0^( ) δ _ (I _ 1 I _ 2)^(2))/f _ π^(ó  )

amp4 = -I mel ;

Converted by Mathematica  (July 10, 2003)