•Expansion of the lagrangian


We use an alternative notation for the lowest order lagrangian in euclidean
space,

lag = 1/4 DecayConstant[Pion]^2 (UTrace[NM[USmall[LorentzIndex[μ1]][x], USmall[LorentzIndex[μ1]][x]] - UChiPlus[x]]) + 1/2 $Gauge FieldDerivative[QuantumField[Particle[Photon], {μ1}][x], x, {μ1}] FieldDerivative[QuantumField[Particle[Photon], {μ2}][x], x, {μ2}] + 1/4 NM[FieldStrengthTensor[{μ1}, QuantumField[Particle[Photon], {μ2}][x], x], FieldStrengthTensor[{μ1}, QuantumField[Particle[Photon], {μ2}][x], x]] - CouplingConstant[ChPTVirtualPhotons2[2]] (UTrace[NM[HRight[x], HRight[x]]] - UTrace[NM[HLeft[x], HLeft[x]]])/4

1/4 (< u _ μ _ 1 '6 u _ μ _ 1 > - < χ _ + >) (f _ π^(ó  ))^2 + 1/4 (γ^( ) _ (μ _ 1 μ _ 2) '6 γ^( ) _ (μ _ 1 μ _ 2)) - 1/4 C^( ) (< H _ R '6 H _ R > - < H _ L '6 H _ L >) + 1/2 λ ∂ _ μ _ 1 γ^( ) _ μ _ 1^ó  ∂ _ μ _ 2 γ^( ) _ μ _ 2^ó 


Which is easily seen to agree with the usual one (when transformed to
euclidean space),

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

1/4 ((< ÷s _ μ _ 1(÷„)^† '6 ÷s _ μ _ 1(÷„) > - < ÷„^† '6 χ > - < χ^† '6 ÷„ >) (f _ π^(ó  ))^2 + γ^( ) _ (μ _ 1 μ _ 2) '6 γ^( ) _ (μ _ 1 μ _ 2) - 4 C^( ) < ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L > + 2 λ ∂ _ μ _ 1 γ^( ) _ μ _ 1^ó  ∂ _ μ _ 2 γ^( ) _ μ _ 2^ó )

Lagrangian[ChPTVirtualPhotons2[2]] // Expand // Simplify

1/4 (< χ '6 ÷„^† > (f _ π^(ó  ))^2 + < χ^† '6 ÷„ > (f _ π^(ó  ))^2 + < ÷s _ μ(÷„) '6 ÷s _ μ(÷„)^† > (f _ π^(ó  ))^2 - 2 λ ∂ _ μ(γ^( ) _ μ) ∂ _ ν(γ^( ) _ ν) - γ^( ) _ (μ ν) '6 γ^( ) _ (μ ν) + 4 C^( ) < Q _ R '6 ÷„ '6 Q _ L '6 ÷„^† >)


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

SetOptions[FieldStrengthTensor, Explicit -> True] ;

lag0 = lag /. {QuantumField[pd___, Particle[Photon], LorentzIndex[li_]][x_] -> QuantumField[pd, Particle[Photon], {li}][x] + 2^(1/2) QuantumField[pd, Particle[UPerturbation], {li}][x]} // UPerturb[#, ExpansionOrder -> {0, 2}] & // DiscardTerms[#, Retain -> {Particle[UPerturbation] -> 2}, Method -> Coefficient] & // CycleUTraces // CommutatorReduce

1/2 (f _ π^(ó  ))^2 < H _ L '6 H _ L > ξ^( ) _ μ _ 1^2 - f _ π^(ó  ) < H _ L '6 ∂ _ μ _ 1(Overscript[ξ^( ), ->]) · Overscript[σ, ->] > ξ^( ) _ μ _ 1 + f _ π^(ó  ) < H _ L '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Γ _ μ _ 1 > ξ^( ) _ μ _ 1 - f _ π^(ó  ) < H _ L '6 Γ _ μ _ 1 '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] > ξ^( ) _ μ _ 1 + 1/2 i f _ π^(ó  ) < H _ R '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 u _ μ _ 1 > ξ^( ) _ μ _ 1 - 1/2 i f _ π^(ó  ) < H _ R '6 u _ μ _ 1 '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] > ξ^( ) _ μ _ 1 + 1/2 (∂ _ μ _ 1 ξ^( ) _ μ _ 2^ó )^2 + 1/2 (∂ _ μ _ 2 ξ^( ) _ μ _ 1^ó )^2 + 1/2 < ∂ _ μ _ 1(Overscript[ξ^( ), ->]) · Overscript[σ, ->] '6 ∂ _ μ _ 1(Overscript[ξ^( ), ->]) · Overscript[σ, ->] > + < Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 ∂ _ μ _ 1(Overscript[ξ^( ), ->]) · Overscript[σ, ->] '6 Γ _ μ _ 1 > - < Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Γ _ μ _ 1 '6 ∂ _ μ _ 1(Overscript[ξ^( ), ->]) · Overscript[σ, ->] > + 1/4 < χ _ + '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] > - (C^( ) < H _ L '6 H _ L '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] >)/(2 (f _ π^(ó  ))^2) + (C^( ) < H _ L '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 H _ L '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] >)/(2 (f _ π^(ó  ))^2) + (C^( ) < H _ R '6 H _ R '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] >)/(2 (f _ π^(ó  ))^2) - (C^( ) < H _ R '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 H _ R '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] >)/(2 (f _ π^(ó  ))^2) - < Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Γ _ μ _ 1 '6 Γ _ μ _ 1 > - 1/4 < Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 u _ μ _ 1 '6 u _ μ _ 1 > + < Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Γ _ μ _ 1 '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Γ _ μ _ 1 > + 1/4 < Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 u _ μ _ 1 '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 u _ μ _ 1 > - ∂ _ μ _ 1 ξ^( ) _ μ _ 2^ó  ∂ _ μ _ 2 ξ^( ) _ μ _ 1^ó  + λ ∂ _ μ _ 1 ξ^( ) _ μ _ 1^ó  ∂ _ μ _ 2 ξ^( ) _ μ _ 2^ó 

Cases[lag0 // IsoIndicesSupply // IndicesCleanup // Expand // CycleUTraces // Expand, _ * QuantumField[Particle[UPerturbation], LorentzIndex[__]][_] QuantumField[PartialD[LorentzIndex[__]], Particle[UPerturbation], SUNIndex[__]][_], Infinity]

{-f _ π^(ó  ) < H _ L '6 σ^k _ 1 > ξ^( ) _ μ _ 1 ∂ _ μ _ 1 ξ^( ) _ ó ^k _ 1}

% /. a : (_ * QuantumField[Particle[UPerturbation], LorentzIndex[__]][_] QuantumField[PartialD[LorentzIndex[__]], Particle[UPerturbation], SUNIndex[__]][_]) :> (a + SurfaceReduce[a, DifferenceOrder -> 1])/2

{1/2 (f _ π^(ó  ) ξ^( )^k _ 1 (< ∂ _ μ _ 1(H _ L) '6 σ^k _ 1 > ξ^( ) _ μ _ 1 + < H _ L '6 σ^k _ 1 > ∂ _ μ _ 1 ξ^( ) _ μ _ 1^ó ) - f _ π^(ó  ) < H _ L '6 σ^k _ 1 > ξ^( ) _ μ _ 1 ∂ _ μ _ 1 ξ^( ) _ ó ^k _ 1)}

SetOptions[CovariantNabla, Explicit -> False] ;

sss = (lag0 // IsoIndicesSupply // IndicesCleanup // Expand // CycleUTraces // Expand) /. a : (_ * QuantumField[Particle[UPerturbation], LorentzIndex[__]][_] QuantumField[PartialD[LorentzIndex[__]], Particle[UPerturbation], SUNIndex[__]][_]) :> (a + SurfaceReduce[a, DifferenceOrder -> 1])/2 // HLeftRightTrick // NMExpand // Expand // CycleUTraces

1/2 (f _ π^(ó  ))^2 < H _ L '6 H _ L > ξ^( ) _ μ _ 1^2 + f _ π^(ó  ) < ∇ _ μ _ 1(H _ L) '6 σ^k _ 1 > ξ^( )^k _ 1 ξ^( ) _ μ _ 1 - 1/2 f _ π^(ó  ) < ∂ _ μ _ 1(H _ L) '6 σ^k _ 1 > ξ^( )^k _ 1 ξ^( ) _ μ _ 1 + 1/2 i f _ π^(ó  ) < H _ R '6 σ^k _ 1 '6 u _ μ _ 1 > ξ^( )^k _ 1 ξ^( ) _ μ _ 1 - 1/2 i f _ π^(ó  ) < H _ R '6 u _ μ _ 1 '6 σ^k _ 1 > ξ^( )^k _ 1 ξ^( ) _ μ _ 1 - 1/2 f _ π^(ó  ) < H _ L '6 σ^k _ 1 > ∂ _ μ _ 1 ξ^( ) _ ó ^k _ 1 ξ^( ) _ μ _ 1 + (∂ _ μ _ 1 ξ^( ) _ μ _ 1^ó )^2 + 1/4 < χ _ + '6 σ^k _ 1 '6 σ^k _ 2 > ξ^( )^k _ 1 ξ^( )^k _ 2 + (C^( ) < σ^k _ 1 '6 H _ L '6 σ^k _ 2 '6 H _ L > ξ^( )^k _ 1 ξ^( )^k _ 2)/(2 (f _ π^(ó  ))^2) - (C^( ) < σ^k _ 1 '6 H _ R '6 σ^k _ 2 '6 H _ R > ξ^( )^k _ 1 ξ^( )^k _ 2)/(2 (f _ π^(ó  ))^2) - (C^( ) < σ^k _ 1 '6 σ^k _ 2 '6 H _ L '6 H _ L > ξ^( )^k _ 1 ξ^( )^k _ 2)/(2 (f _ π^(ó  ))^2) + (C^( ) < σ^k _ 1 '6 σ^k _ 2 '6 H _ R '6 H _ R > ξ^( )^k _ 1 ξ^( )^k _ 2)/(2 (f _ π^(ó  ))^2) - < σ^k _ 1 '6 σ^k _ 2 '6 Γ _ μ _ 1 '6 Γ _ μ _ 1 > ξ^( )^k _ 1 ξ^( )^k _ 2 - 1/4 < σ^k _ 1 '6 σ^k _ 2 '6 u _ μ _ 1 '6 u _ μ _ 1 > ξ^( )^k _ 1 ξ^( )^k _ 2 + < σ^k _ 1 '6 Γ _ μ _ 1 '6 σ^k _ 2 '6 Γ _ μ _ 1 > ξ^( )^k _ 1 ξ^( )^k _ 2 + 1/4 < σ^k _ 1 '6 u _ μ _ 1 '6 σ^k _ 2 '6 u _ μ _ 1 > ξ^( )^k _ 1 ξ^( )^k _ 2 + 1/2 f _ π^(ó  ) < H _ L '6 σ^k _ 1 > ξ^( )^k _ 1 ∂ _ μ _ 1 ξ^( ) _ μ _ 1^ó  + < σ^k _ 1 '6 σ^k _ 2 '6 Γ _ μ _ 1 > ξ^( )^k _ 1 ∂ _ μ _ 1 ξ^( ) _ ó ^k _ 2 - < σ^k _ 1 '6 Γ _ μ _ 1 '6 σ^k _ 2 > ξ^( )^k _ 1 ∂ _ μ _ 1 ξ^( ) _ ó ^k _ 2 + 1/2 < σ^k _ 1 '6 σ^k _ 2 > ∂ _ μ _ 1 ξ^( ) _ ó ^k _ 1 ∂ _ μ _ 1 ξ^( ) _ ó ^k _ 2 - ∂ _ μ _ 1 ξ^( ) _ μ _ 2^ó  ∂ _ μ _ 2 ξ^( ) _ μ _ 1^ó  + λ ∂ _ μ _ 1 ξ^( ) _ μ _ 1^ó  ∂ _ μ _ 2 ξ^( ) _ μ _ 2^ó 

Converted by Mathematica  (July 10, 2003)