•Two-vertex of second order in the chiral expansion

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 ; QuantumField[Particle[PseudoScalar[0], ___], ___][_] := 0 ;

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

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

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

1/4 (< ∂ _ μ(Overscript[π^( ), ->]) · Overscript[σ, ->] '6 ∂ _ μ(Overscript[π^( ), ->]) · Overscript[σ, ->] > - (m _ π^(ó  ))^2 < Overscript[π^( ), ->] · Overscript[σ, ->] '6 Overscript[π^( ), ->] · Overscript[σ, ->] >)

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

1/2 (∂ _ μ(Overscript[π^( ), ->]) · ∂ _ μ(Overscript[π^( ), ->]) - Overscript[π^( ), ->] · Overscript[π^( ), ->] (m _ π^(ó  ))^2)

llll = llle // IsoIndicesSupply // IndicesCleanup // CommutatorReduce

1/2 (∂ _ τ1 π^( ) _ ó ^k1)^2 - 1/2 (m _ π^(ó  ))^2 (π^( )^k1)^2

fields = FieldsSet[QuantumField[Particle[Pion, RenormalizationState[0]]], ParticlesNumber -> 2]

{π^( )^I _ 1, π^( )^I _ 2}

amp2 = (-I FeynRule[llll, fields]) /. {SUNDelta -> SU2Delta, I1 -> 1, I2 -> 1} // Simplify

-(m _ π^(ó  ))^2 - p _ 1 · p _ 2

Converted by Mathematica  (July 10, 2003)