•Tree contribution of second order in the chiral expansion

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

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

1/24 (-2 (< Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] > + 3^(1/2) (< Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 σ^8 > + < σ^8 '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] >)) (m _ π^(ó  ))^2 - 2 (m _ K^0^(ó  ))^2 < Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] > + 6 < ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[σ, ->] '6 ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[σ, ->] > + 3 (m _ K^0^(ó  ))^2 < Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 σ^3 > + 3^(1/2) (m _ K^0^(ó  ))^2 < Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 σ^8 > + 3 (m _ K^0^(ó  ))^2 < σ^3 '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] > + 3^(1/2) (m _ K^0^(ó  ))^2 < σ^8 '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] > + (m _ K^+^(ó  ))^2 (-2 < Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] > - 3 < Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 σ^3 > + 3^(1/2) < Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 σ^8 > - 3 < σ^3 '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] > + 3^(1/2) < σ^8 '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] >))

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

1/12 (-2 3^(1/2) Overscript[öõ(8), ->] · Overscript[ϕ^( ), ->] ⊗ Overscript[ϕ^( ), ->] (m _ π^(ó  ))^2 - 2 Overscript[ϕ^( ), ->] · Overscript[ϕ^( ), ->] (m _ π^(ó  ))^2 - 3 Overscript[öõ(3), ->] ⊗ Overscript[ϕ^( ), ->] · Overscript[ϕ^( ), ->] (m _ K^+^(ó  ))^2 - 3 Overscript[öõ(3), ->] · Overscript[ϕ^( ), ->] ⊗ Overscript[ϕ^( ), ->] (m _ K^+^(ó  ))^2 + 3^(1/2) Overscript[öõ(8), ->] · Overscript[ϕ^( ), ->] ⊗ Overscript[ϕ^( ), ->] (m _ K^+^(ó  ))^2 - 2 Overscript[ϕ^( ), ->] · Overscript[ϕ^( ), ->] (m _ K^+^(ó  ))^2 + 3 Overscript[öõ(3), ->] ⊗ Overscript[ϕ^( ), ->] · Overscript[ϕ^( ), ->] (m _ K^0^(ó  ))^2 + 3 Overscript[öõ(3), ->] · Overscript[ϕ^( ), ->] ⊗ Overscript[ϕ^( ), ->] (m _ K^0^(ó  ))^2 + 3^(1/2) Overscript[öõ(8), ->] · Overscript[ϕ^( ), ->] ⊗ Overscript[ϕ^( ), ->] (m _ K^0^(ó  ))^2 - 2 Overscript[ϕ^( ), ->] · Overscript[ϕ^( ), ->] (m _ K^0^(ó  ))^2 + 6 ∂ _ μ(Overscript[ϕ^( ), ->]) · ∂ _ μ(Overscript[ϕ^( ), ->]) + 3^(1/2) Overscript[öõ(8), ->] ⊗ Overscript[ϕ^( ), ->] · Overscript[ϕ^( ), ->] (-2 (m _ π^(ó  ))^2 + (m _ K^+^(ó  ))^2 + (m _ K^0^(ó  ))^2))

IsoIndicesCounter = 0 ;

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

1/6 (-ϕ^( )^k1 (ϕ^( )^k1 + 2 3^(1/2) d _ (8 k1 k2)^(3) ϕ^( )^k2) (m _ π^(ó  ))^2 - (m _ K^0^(ó  ))^2 (ϕ^( )^k1)^2 + 3 (∂ _ τ1 ϕ^( ) _ ó ^k1)^2 + 3 (m _ K^0^(ó  ))^2 d _ (3 k1 k2)^(3) ϕ^( )^k1 ϕ^( )^k2 + 3^(1/2) (m _ K^0^(ó  ))^2 d _ (8 k1 k2)^(3) ϕ^( )^k1 ϕ^( )^k2 - (m _ K^+^(ó  ))^2 ϕ^( )^k1 (ϕ^( )^k1 + (3 d _ (3 k1 k2)^(3) - 3^(1/2) d _ (8 k1 k2)^(3)) ϕ^( )^k2))

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

{ϕ^( )^I _ 1, ϕ^( )^I _ 2}

amp2f[i1_, i2_] = (-I Simplify[SUNReduce[FeynRule[llll, fields]]] // IndicesCleanup) /. {I1 -> i1, I2 -> i2} // Simplify

1/3 (-2 3^(1/2) d _ (8 i _ 1 i _ 2)^(3) (m _ π^(ó  ))^2 - δ _ (i _ 1 i _ 2)^(3) (m _ π^(ó  ))^2 + (m _ K^0^(ó  ))^2 (3 d _ (3 i _ 1 i _ 2)^(3) + 3^(1/2) d _ (8 i _ 1 i _ 2)^(3) - δ _ (i _ 1 i _ 2)^(3)) - 3 p _ 1 · p _ 2 δ _ (i _ 1 i _ 2)^(3) - (m _ K^+^(ó  ))^2 (3 d _ (3 i _ 1 i _ 2)^(3) - 3^(1/2) d _ (8 i _ 1 i _ 2)^(3) + δ _ (i _ 1 i _ 2)^(3)))

amp2Pion = amp2f[1, 1] /. udrules // Simplify

-(m _ π^(ó  ))^2 - p _ 1 · p _ 2

amp2Kaon = amp2f[4, 4] /. udrules // Simplify

-(m _ K^(ó  ))^2 - p _ 1 · p _ 2

amp2Eta = amp2f[8, 8] /. FromK0Rules /. udrules // Simplify

-(m _ η^(ó  ))^2 - p _ 1 · p _ 2

amp2 = AmplitudeProjection[amp2f, Channel -> {{PionZero} -> {PionZero}}] // Simplify

-(m _ π^(ó  ))^2 - p _ 1 · p _ 2

Converted by Mathematica  (July 10, 2003)