•Expansion of u _ μ

do = 1 ;

test = I * NM[Adjoint[uExpRight[x, ExpansionOrder -> do]], CDr[NM[uExpRight[x, ExpansionOrder -> do], uExpLeftAdj[x, ExpansionOrder -> do]], x, {μ}, Explicit -> True], Adjoint[uExpLeftAdj[x, ExpansionOrder -> do]]] /. {QuantumField[pd___, Particle[Photon], LorentzIndex[li_]][x_] -> QuantumField[pd, Particle[Photon], {li}][x] + 2^(1/2) QuantumField[Particle[UPerturbation], {li}][x]} // NMExpand // Expand // UReduce ;

test1 = DiscardTerms[test, Retain -> {Particle[UPerturbation] -> do}]

-(2^(1/2) ∂ _ μ(Overscript[ξ^( ), ->]) · Overscript[σ, ->])/f - (öÆ^† '6 ∂ _ μ(öÆ) '6 Overscript[ξ^( ), ->] · Overscript[σ, ->])/(2^(1/2) f) + (∂ _ μ(öÆ) '6 öÆ^† '6 Overscript[ξ^( ), ->] · Overscript[σ, ->])/(2^(1/2) f) + (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 öÆ^† '6 ∂ _ μ(öÆ))/(2^(1/2) f) - (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 ∂ _ μ(öÆ) '6 öÆ^†)/(2^(1/2) f) + 2^(1/2) (öÆ^† '6 Q _ R '6 ξ^( ) _ μ '6 öÆ) - 2^(1/2) (öÆ '6 Q _ L '6 ξ^( ) _ μ '6 öÆ^†) + (i (öÆ^† '6 Q _ R '6 γ^( ) _ μ '6 öÆ '6 Overscript[ξ^( ), ->] · Overscript[σ, ->]))/(2^(1/2) f) - (i (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 öÆ^† '6 Q _ R '6 γ^( ) _ μ '6 öÆ))/(2^(1/2) f) - (i (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 öÆ '6 Q _ L '6 γ^( ) _ μ '6 öÆ^†))/(2^(1/2) f) + (i (öÆ '6 Q _ L '6 γ^( ) _ μ '6 öÆ^† '6 Overscript[ξ^( ), ->] · Overscript[σ, ->]))/(2^(1/2) f)

tes = -I * NM[uExpLeftAdj[x, ExpansionOrder -> do], Adjoint[CDr[NM[uExpRight[x, ExpansionOrder -> do], uExpLeftAdj[x, ExpansionOrder -> do]], x, {μ}, Explicit -> True]], uExpRight[x, ExpansionOrder -> do]] /. {QuantumField[pd___, Particle[Photon], LorentzIndex[li_]][x_] -> QuantumField[pd, Particle[Photon], {li}][x] + 2^(1/2) QuantumField[Particle[UPerturbation], {li}][x]} // NMExpand // Expand // UReduce ;

tes1 = DiscardTerms[tes, Retain -> {Particle[UPerturbation] -> do}]

-(2^(1/2) ∂ _ μ(Overscript[ξ^( ), ->]) · Overscript[σ, ->])/f + (∂ _ μ(öÆ)^† '6 öÆ '6 Overscript[ξ^( ), ->] · Overscript[σ, ->])/(2^(1/2) f) - (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 ∂ _ μ(öÆ)^† '6 öÆ)/(2^(1/2) f) + (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 öÆ '6 ∂ _ μ(öÆ)^†)/(2^(1/2) f) - (öÆ '6 ∂ _ μ(öÆ)^† '6 Overscript[ξ^( ), ->] · Overscript[σ, ->])/(2^(1/2) f) + 2^(1/2) (öÆ^† '6 ξ^( ) _ μ '6 Q _ R^† '6 öÆ) - 2^(1/2) (öÆ '6 ξ^( ) _ μ '6 Q _ L^† '6 öÆ^†) + (i (öÆ^† '6 γ^( ) _ μ '6 Q _ R^† '6 öÆ '6 Overscript[ξ^( ), ->] · Overscript[σ, ->]))/(2^(1/2) f) - (i (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 öÆ^† '6 γ^( ) _ μ '6 Q _ R^† '6 öÆ))/(2^(1/2) f) - (i (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 öÆ '6 γ^( ) _ μ '6 Q _ L^† '6 öÆ^†))/(2^(1/2) f) + (i (öÆ '6 γ^( ) _ μ '6 Q _ L^† '6 öÆ^† '6 Overscript[ξ^( ), ->] · Overscript[σ, ->]))/(2^(1/2) f)

test1 - tes1 /. Adjoint[q : (QuantumField[Particle[RightComponent[0] | LeftComponent[0]], LorentzIndex[μ]][x] | UMatrix[UChiralSpurionLeft[] | UChiralSpurionRight[]][x])] :> q // CommutatorReduce // UReduce

0

-2^(1/2)/DecayConstant[PhiMeson] CovariantNabla[xi[x], x, {μ}, Explicit -> False] + QuantumField[Particle[UPerturbation], LorentzIndex[μ]][x] HLeft[x]

H _ L ξ^( ) _ μ - (2^(1/2) ∇ _ μ(Overscript[ξ^( ), ->] · Overscript[σ, ->]))/f

ref1 = -2^(1/2)/DecayConstant[PhiMeson] CovariantNabla[xi[x], x, {μ}] + 2^(1/2) QuantumField[Particle[UPerturbation], LorentzIndex[μ]][x] HLeft[x] //. $Substitutions // NMExpand // Expand // UReduce

-(2^(1/2) ∂ _ μ(Overscript[ξ^( ), ->]) · Overscript[σ, ->])/f - (öÆ^† '6 ∂ _ μ(öÆ) '6 Overscript[ξ^( ), ->] · Overscript[σ, ->])/(2^(1/2) f) + (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 öÆ^† '6 ∂ _ μ(öÆ))/(2^(1/2) f) + (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 öÆ '6 ∂ _ μ(öÆ)^†)/(2^(1/2) f) - (öÆ '6 ∂ _ μ(öÆ)^† '6 Overscript[ξ^( ), ->] · Overscript[σ, ->])/(2^(1/2) f) + (i (öÆ^† '6 Q _ R '6 γ^( ) _ μ '6 öÆ '6 Overscript[ξ^( ), ->] · Overscript[σ, ->]))/(2^(1/2) f) - (i (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 öÆ^† '6 Q _ R '6 γ^( ) _ μ '6 öÆ))/(2^(1/2) f) - (i (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 öÆ '6 Q _ L '6 γ^( ) _ μ '6 öÆ^†))/(2^(1/2) f) + (i (öÆ '6 Q _ L '6 γ^( ) _ μ '6 öÆ^† '6 Overscript[ξ^( ), ->] · Overscript[σ, ->]))/(2^(1/2) f) + 2^(1/2) (öÆ^† '6 Q _ R '6 öÆ) ξ^( ) _ μ - 2^(1/2) (öÆ '6 Q _ L '6 öÆ^†) ξ^( ) _ μ

ref1 - test1 // CommutatorReduce // UReduce

0

do = 2 ;

tes = -I * NM[uExpLeftAdj[x, ExpansionOrder -> do], Adjoint[CDr[NM[uExpRight[x, ExpansionOrder -> do], uExpLeftAdj[x, ExpansionOrder -> do]], x, {μ}, Explicit -> True]], uExpRight[x, ExpansionOrder -> do]] /. {QuantumField[pd___, Particle[Photon], LorentzIndex[li_]][x_] -> QuantumField[pd, Particle[Photon], {li}][x] + 2^(1/2) QuantumField[Particle[UPerturbation], {li}][x]} // NMExpand // Expand // UReduce ;

tes2 = DiscardTerms[tes, Retain -> {Particle[UPerturbation] -> do}]

(i (∂ _ μ(öÆ)^† '6 öÆ '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->]))/(4 f^2) - (i (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 ∂ _ μ(öÆ)^† '6 öÆ '6 Overscript[ξ^( ), ->] · Overscript[σ, ->]))/(2 f^2) + (i (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 ∂ _ μ(öÆ)^† '6 öÆ))/(4 f^2) + (i (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 öÆ '6 ∂ _ μ(öÆ)^†))/(4 f^2) - (i (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 öÆ '6 ∂ _ μ(öÆ)^† '6 Overscript[ξ^( ), ->] · Overscript[σ, ->]))/(2 f^2) + (i (öÆ '6 ∂ _ μ(öÆ)^† '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->]))/(4 f^2) + (i (öÆ^† '6 ξ^( ) _ μ '6 Q _ R^† '6 öÆ '6 Overscript[ξ^( ), ->] · Overscript[σ, ->]))/f - (i (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 öÆ^† '6 ξ^( ) _ μ '6 Q _ R^† '6 öÆ))/f - (i (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 öÆ '6 ξ^( ) _ μ '6 Q _ L^† '6 öÆ^†))/f + (i (öÆ '6 ξ^( ) _ μ '6 Q _ L^† '6 öÆ^† '6 Overscript[ξ^( ), ->] · Overscript[σ, ->]))/f - (öÆ^† '6 γ^( ) _ μ '6 Q _ R^† '6 öÆ '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->])/(4 f^2) + (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 öÆ^† '6 γ^( ) _ μ '6 Q _ R^† '6 öÆ '6 Overscript[ξ^( ), ->] · Overscript[σ, ->])/(2 f^2) - (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 öÆ^† '6 γ^( ) _ μ '6 Q _ R^† '6 öÆ)/(4 f^2) + (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 öÆ '6 γ^( ) _ μ '6 Q _ L^† '6 öÆ^†)/(4 f^2) - (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 öÆ '6 γ^( ) _ μ '6 Q _ L^† '6 öÆ^† '6 Overscript[ξ^( ), ->] · Overscript[σ, ->])/(2 f^2) + (öÆ '6 γ^( ) _ μ '6 Q _ L^† '6 öÆ^† '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->])/(4 f^2)

test = I * NM[Adjoint[uExpRight[x, ExpansionOrder -> do]], CDr[NM[uExpRight[x, ExpansionOrder -> do], uExpLeftAdj[x, ExpansionOrder -> do]], x, {μ}, Explicit -> True], Adjoint[uExpLeftAdj[x, ExpansionOrder -> do]]] /. {QuantumField[pd___, Particle[Photon], LorentzIndex[li_]][x_] -> QuantumField[pd, Particle[Photon], {li}][x] + 2^(1/2) QuantumField[Particle[UPerturbation], {li}][x]} // NMExpand // Expand // UReduce ;

test2 = DiscardTerms[test, Retain -> {Particle[UPerturbation] -> do}] // UReduce

-(i (öÆ^† '6 ∂ _ μ(öÆ) '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->]))/(4 f^2) - (i (∂ _ μ(öÆ) '6 öÆ^† '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->]))/(4 f^2) + (i (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 öÆ^† '6 ∂ _ μ(öÆ) '6 Overscript[ξ^( ), ->] · Overscript[σ, ->]))/(2 f^2) + (i (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 ∂ _ μ(öÆ) '6 öÆ^† '6 Overscript[ξ^( ), ->] · Overscript[σ, ->]))/(2 f^2) - (i (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 öÆ^† '6 ∂ _ μ(öÆ)))/(4 f^2) - (i (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 ∂ _ μ(öÆ) '6 öÆ^†))/(4 f^2) + (i (öÆ^† '6 Q _ R '6 ξ^( ) _ μ '6 öÆ '6 Overscript[ξ^( ), ->] · Overscript[σ, ->]))/f - (i (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 öÆ^† '6 Q _ R '6 ξ^( ) _ μ '6 öÆ))/f - (i (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 öÆ '6 Q _ L '6 ξ^( ) _ μ '6 öÆ^†))/f + (i (öÆ '6 Q _ L '6 ξ^( ) _ μ '6 öÆ^† '6 Overscript[ξ^( ), ->] · Overscript[σ, ->]))/f - (öÆ^† '6 Q _ R '6 γ^( ) _ μ '6 öÆ '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->])/(4 f^2) + (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 öÆ^† '6 Q _ R '6 γ^( ) _ μ '6 öÆ '6 Overscript[ξ^( ), ->] · Overscript[σ, ->])/(2 f^2) - (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 öÆ^† '6 Q _ R '6 γ^( ) _ μ '6 öÆ)/(4 f^2) + (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 öÆ '6 Q _ L '6 γ^( ) _ μ '6 öÆ^†)/(4 f^2) - (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 öÆ '6 Q _ L '6 γ^( ) _ μ '6 öÆ^† '6 Overscript[ξ^( ), ->] · Overscript[σ, ->])/(2 f^2) + (öÆ '6 Q _ L '6 γ^( ) _ μ '6 öÆ^† '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->])/(4 f^2)

test2 - tes2 /. Adjoint[q : (QuantumField[Particle[RightComponent[0] | LeftComponent[0]], LorentzIndex[μ]][x] | UMatrix[UChiralSpurionLeft[] | UChiralSpurionRight[]][x])] :> q // CommutatorReduce // UReduce // CommutatorReduce // UReduce

0

SetOptions[CovariantFieldDerivative, Explicit -> True] ;

ref2 = 1/4/DecayConstant[PhiMeson]^2 UCommutator[xi[x], UCommutator[USmall[μ][x], xi[x]]] - I/DecayConstant[PhiMeson] QuantumField[Particle[UPerturbation], LorentzIndex[μ]][x] UCommutator[xi[x], HRight[x]] /. $Substitutions /. MM[x] -> NM[SMM[x], SMM[x]] // NMExpand // Expand // UReduce

-(i (öÆ^† '6 ∂ _ μ(öÆ) '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->]))/(4 f^2) - (i (∂ _ μ(öÆ) '6 öÆ^† '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->]))/(4 f^2) + (i (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 öÆ^† '6 ∂ _ μ(öÆ) '6 Overscript[ξ^( ), ->] · Overscript[σ, ->]))/(2 f^2) + (i (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 ∂ _ μ(öÆ) '6 öÆ^† '6 Overscript[ξ^( ), ->] · Overscript[σ, ->]))/(2 f^2) - (i (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 öÆ^† '6 ∂ _ μ(öÆ)))/(4 f^2) - (i (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 ∂ _ μ(öÆ) '6 öÆ^†))/(4 f^2) - (öÆ^† '6 Q _ R '6 γ^( ) _ μ '6 öÆ '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->])/(4 f^2) + (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 öÆ^† '6 Q _ R '6 γ^( ) _ μ '6 öÆ '6 Overscript[ξ^( ), ->] · Overscript[σ, ->])/(2 f^2) - (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 öÆ^† '6 Q _ R '6 γ^( ) _ μ '6 öÆ)/(4 f^2) + (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 öÆ '6 Q _ L '6 γ^( ) _ μ '6 öÆ^†)/(4 f^2) - (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 öÆ '6 Q _ L '6 γ^( ) _ μ '6 öÆ^† '6 Overscript[ξ^( ), ->] · Overscript[σ, ->])/(2 f^2) + (öÆ '6 Q _ L '6 γ^( ) _ μ '6 öÆ^† '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->])/(4 f^2) + (i (öÆ^† '6 Q _ R '6 öÆ '6 Overscript[ξ^( ), ->] · Overscript[σ, ->]) ξ^( ) _ μ)/f - (i (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 öÆ^† '6 Q _ R '6 öÆ) ξ^( ) _ μ)/f - (i (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 öÆ '6 Q _ L '6 öÆ^†) ξ^( ) _ μ)/f + (i (öÆ '6 Q _ L '6 öÆ^† '6 Overscript[ξ^( ), ->] · Overscript[σ, ->]) ξ^( ) _ μ)/f

ref2 - test2 // CommutatorReduce // UReduce

0

SetOptions[CovariantNabla, Explicit -> False] ;

Coeff[USmall][0][li_, x_] = USmall[li][x]

u _ li

Coeff[USmall][1][li_, x_] = -Sqrt[2]/DecayConstant[PhiMeson] CovariantNabla[xi[x], x, {li}] + Sqrt[2] QuantumField[Particle[UPerturbation], LorentzIndex[li]][x] HLeft[x]

2^(1/2) H _ L ξ^( ) _ li - (2^(1/2) ∇ _ li(Overscript[ξ^( ), ->] · Overscript[σ, ->]))/f

Coeff[USmall][2][li_, x_] = 1/4/DecayConstant[PhiMeson]^2 UCommutator[xi[x], UCommutator[USmall[li][x], xi[x]]] - I/DecayConstant[PhiMeson] QuantumField[Particle[UPerturbation], LorentzIndex[li]][x] UCommutator[xi[x], HRight[x]]

(Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 (u _ li '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] - Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 u _ li) - (u _ li '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] - Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 u _ li) '6 Overscript[ξ^( ), ->] · Overscript[σ, ->])/(4 f^2) - (i (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 H _ R - H _ R '6 Overscript[ξ^( ), ->] · Overscript[σ, ->]) ξ^( ) _ li)/f

UPerturb :: nocoeff = "Warning: Yor are requesting expanding in UPerturbation to order `1`. Only up to order 2 is implemented in terms of USmall and CovariantNabla.  (If you have the energy, please do work out the expansion and send it to feyncalc@feyncalc.org)" ;

Coeff[USmall][do_ ? ((# > 2) &)][li_, x_] := Coeff[USmall][do][li, x] = (Message[UPerturb :: nocoeff, do] ; DiscardTerms[I * NM[Adjoint[uExpRight[x, ExpansionOrder -> do]], CDr[NM[uExpRight[x, ExpansionOrder -> do], uExpLeftAdj[x, ExpansionOrder -> do]], x, {li}, Explicit -> True], Adjoint[uExpLeftAdj[x, ExpansionOrder -> do]]] /. {QuantumField[pd___, Particle[Photon], LorentzIndex[l_]][xx_] -> QuantumField[pd, Particle[Photon], {l}][xx] + Sqrt[2] QuantumField[Particle[UPerturbation], {l}][xx]} // NMExpand // Expand, Retain -> {Particle[UPerturbation] -> do}] // UReduce) ;

Coeff[USmall][3][μ, x]

Converted by Mathematica  (July 10, 2003)