•Two-vertex to second order in the energy

Lagrangian[ChPT2[2]]

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

ll = ArgumentsSupply[Lagrangian[ChPT2[2]], x, RenormalizationState[0], ExpansionOrder -> 1, DropOrder -> 1, DiagonalToU -> True]

1/4 (f _ π^(ó  ))^2 (< ((i ℵ ∂ _ μ(Overscript[π^( ), ->]) · Overscript[σ, ->])/f _ π^(ó  ) + 1/2 i ((Overscript[A^( ) _ μ, ->] · Overscript[σ, ->] + Overscript[V^( ) _ μ, ->] · Overscript[σ, ->]) '6 ((i ℵ Overscript[π^( ), ->] · Overscript[σ, ->])/f _ π^(ó  ) + ÷¬öé)) - 1/2 i (((i ℵ Overscript[π^( ), ->] · Overscript[σ, ->])/f _ π^(ó  ) + ÷¬öé) '6 (Overscript[V^( ) _ μ, ->] · Overscript[σ, ->] - Overscript[A^( ) _ μ, ->] · Overscript[σ, ->]))) '6 (-(i ℵ ∂ _ μ(Overscript[π^( ), ->]) · Overscript[σ, ->])/f _ π^(ó  ) + 1/2 i ((Overscript[V^( ) _ μ, ->] · Overscript[σ, ->] - Overscript[A^( ) _ μ, ->] · Overscript[σ, ->]) '6 (÷¬öé - (i ℵ Overscript[π^( ), ->] · Overscript[σ, ->])/f _ π^(ó  ))) - 1/2 i ((÷¬öé - (i ℵ Overscript[π^( ), ->] · Overscript[σ, ->])/f _ π^(ó  )) '6 (Overscript[A^( ) _ μ, ->] · Overscript[σ, ->] + Overscript[V^( ) _ μ, ->] · Overscript[σ, ->]))) > + 2 !, _ 0^( ) < ((i ℵ Overscript[π^( ), ->] · Overscript[σ, ->])/f _ π^(ó  ) + ÷¬öé) '6 (((÷¬öé/2 - σ^3/2) (m _ π^(ó  ))^2)/(2 !, _ 0^( )) + ((÷¬öé/2 + σ^3/2) (m _ π^(ó  ))^2)/(2 !, _ 0^( )) + Overscript[s^( ), ->] · Overscript[σ, ->] - i (Overscript[p^( ), ->] · Overscript[σ, ->] + ÷¬öé p^( )^0) + ÷¬öé s^( )^0) > + 2 !, _ 0^( ) < (((÷¬öé/2 - σ^3/2) (m _ π^(ó  ))^2)/(2 !, _ 0^( )) + ((÷¬öé/2 + σ^3/2) (m _ π^(ó  ))^2)/(2 !, _ 0^( )) + Overscript[s^( ), ->] · Overscript[σ, ->] + i (Overscript[p^( ), ->] · Overscript[σ, ->] + ÷¬öé p^( )^0) + ÷¬öé s^( )^0) '6 (÷¬öé - (i ℵ Overscript[π^( ), ->] · Overscript[σ, ->])/f _ π^(ó  )) >)

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

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

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

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

$IsoIndicesCounter = 0 ;

llll = llle // IsoIndicesSupply

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

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

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

melsimplified = -I mel ;

Converted by Mathematica  (July 10, 2003)