•Expansion of the lagrangian


We use an alternative notation for the lowest order lagrangian (in minkowski
space the minus of euclidean space goes),

lag = 1/4 DecayConstant[PhiMeson]^2 (UTrace[NM[USmall[LorentzIndex[μ1]][x], USmall[LorentzIndex[μ1]][x]] + UChiPlus[x]])

1/4 (f _ ϕ^(ó  ))^2 (< u _ μ _ 1 '6 u _ μ _ 1 > + < χ _ + >)

Which is easily seen to agree with the usual one,

lag /. $Substitutions // UReduce[#, SMMToMM -> True] & // Simplify

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

Lagrangian[ChPT3[2]]

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


Perturbation around the solution of the equation of motion, keeping only the
squared terms.

lag0 = lag // UPerturb[#, ExpansionOrder -> {0, 2}] & // NMExpand // Expand // DiscardTerms[#, Retain -> {Particle[UPerturbation] -> 2}] & // CycleUTraces // Simplify

1/4 (2 < ∂ _ μ _ 1(Overscript[ξ^( ), ->]) · Overscript[σ, ->] '6 ∂ _ μ _ 1(Overscript[ξ^( ), ->]) · Overscript[σ, ->] > + 4 < Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 ∂ _ μ _ 1(Overscript[ξ^( ), ->]) · Overscript[σ, ->] '6 Γ _ μ _ 1 > - 4 < Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Γ _ μ _ 1 '6 ∂ _ μ _ 1(Overscript[ξ^( ), ->]) · Overscript[σ, ->] > - < χ _ + '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] > - 4 < Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Γ _ μ _ 1 '6 Γ _ μ _ 1 > - < Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 u _ μ _ 1 '6 u _ μ _ 1 > + 4 < Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Γ _ μ _ 1 '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Γ _ μ _ 1 > + < Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 u _ μ _ 1 '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 u _ μ _ 1 >)

Writing out with indices.

sss = -lag0 // IsoIndicesSupply // IndicesCleanup // Expand // CycleUTraces // Simplify

1/4 (< χ _ + '6 σ^k _ 1 '6 σ^k _ 2 > ξ^( )^k _ 1 ξ^( )^k _ 2 + 4 < σ^k _ 1 '6 σ^k _ 2 '6 Γ _ ρ _ 1 '6 Γ _ ρ _ 1 > ξ^( )^k _ 1 ξ^( )^k _ 2 + < σ^k _ 1 '6 σ^k _ 2 '6 u _ ρ _ 1 '6 u _ ρ _ 1 > ξ^( )^k _ 1 ξ^( )^k _ 2 - 4 < σ^k _ 1 '6 Γ _ ρ _ 1 '6 σ^k _ 2 '6 Γ _ ρ _ 1 > ξ^( )^k _ 1 ξ^( )^k _ 2 - < σ^k _ 1 '6 u _ ρ _ 1 '6 σ^k _ 2 '6 u _ ρ _ 1 > ξ^( )^k _ 1 ξ^( )^k _ 2 - 4 < σ^k _ 1 '6 σ^k _ 2 '6 Γ _ ρ _ 1 > ξ^( )^k _ 1 ∂ _ ρ _ 1 ξ^( ) _ ó ^k _ 2 + 4 < σ^k _ 1 '6 Γ _ ρ _ 1 '6 σ^k _ 2 > ξ^( )^k _ 1 ∂ _ ρ _ 1 ξ^( ) _ ó ^k _ 2 - 2 < σ^k _ 1 '6 σ^k _ 2 > ∂ _ ρ _ 1 ξ^( ) _ ó ^k _ 1 ∂ _ ρ _ 1 ξ^( ) _ ó ^k _ 2)

Converted by Mathematica  (July 10, 2003)