•Two-vertex of fourth order in the chiral expansion

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 ÷„^† >)

lld = ArgumentsSupply[lag, x, RenormalizationState[0], ExpansionOrder -> 2, DropOrder -> 2] ;

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

(2 (2 L _ 4^( ) + L _ 5^( )) (m _ π^(ó  ))^2 < ∂ _ μ(Overscript[π^( ), ->]) · Overscript[σ, ->] '6 ∂ _ μ(Overscript[π^( ), ->]) · Overscript[σ, ->] > - 4 (2 L _ 6^( ) + L _ 8^( )) (m _ π^(ó  ))^4 < Overscript[π^( ), ->] · Overscript[σ, ->] '6 Overscript[π^( ), ->] · Overscript[σ, ->] >)/(f _ π^(ó  ))^2

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

(4 (2 L _ 4^( ) + L _ 5^( )) ∂ _ μ(Overscript[π^( ), ->]) · ∂ _ μ(Overscript[π^( ), ->]) (m _ π^(ó  ))^2 - 8 (2 L _ 6^( ) + L _ 8^( )) Overscript[π^( ), ->] · Overscript[π^( ), ->] (m _ π^(ó  ))^4)/(f _ π^(ó  ))^2

IsoIndicesCounter = 0 ;

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

(4 (2 L _ 4^( ) + L _ 5^( )) (m _ π^(ó  ))^2 (∂ _ τ1 π^( ) _ ó ^k1)^2 - 8 (2 L _ 6^( ) + L _ 8^( )) (m _ π^(ó  ))^4 (π^( )^k1)^2)/(f _ π^(ó  ))^2

amp4 = (-I FeynRule[llll, FieldsSet[QuantumField[Particle[Pion, RenormalizationState[0]]], ParticlesNumber -> 2]]) /. {SUNDelta -> SU2Delta, I1 -> 1, I2 -> 1} // FullSimplify

(-16 (2 L _ 6^( ) + L _ 8^( )) (m _ π^(ó  ))^4 - 8 (2 L _ 4^( ) + L _ 5^( )) p _ 1 · p _ 2 (m _ π^(ó  ))^2)/(f _ π^(ó  ))^2

Converted by Mathematica  (July 10, 2003)