•Expansion of f +

do = 1

1

UFPlus[μ, ν][x]

f _ + _ (μ ν)

DiscardTerms[NM[uExpLeftAdj[x, ExpansionOrder -> do], FieldStrengthTensorFull[{μ}, UGeneratorMatrixIsoDot[QuantumField[Particle[LeftComponent[0]], {ν}][x]], x, -I], Adjoint[uExpLeftAdj[x, ExpansionOrder -> do]]] + NM[Adjoint[uExpRight[x, ExpansionOrder -> do]], FieldStrengthTensorFull[{μ}, UGeneratorMatrixIsoDot[QuantumField[Particle[RightComponent[0]], {ν}][x]], x, -I], uExpRight[x, ExpansionOrder -> do]] /. lrRule // NMExpand // Expand, Retain -> {Particle[UPerturbation] -> do}]

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

I/Sqrt[2]/DecayConstant[PhiMeson] UCommutator[xi[x], UFMinus[μ, ν][x]] /. $Substitutions /. lrRule // NMExpand // Expand

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

% - %%

0

do = 2

2

DiscardTerms[NM[uExpLeftAdj[x, ExpansionOrder -> do], FieldStrengthTensorFull[{μ}, UGeneratorMatrixIsoDot[QuantumField[Particle[LeftComponent[0]], {ν}][x]], x, -I], Adjoint[uExpLeftAdj[x, ExpansionOrder -> do]]] + NM[Adjoint[uExpRight[x, ExpansionOrder -> do]], FieldStrengthTensorFull[{μ}, UGeneratorMatrixIsoDot[QuantumField[Particle[RightComponent[0]], {ν}][x]], x, -I], uExpRight[x, ExpansionOrder -> do]] /. lrRule // NMExpand // Expand, Retain -> {Particle[UPerturbation] -> do}]

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

-1/4/DecayConstant[PhiMeson]^2 UCommutator[xi[x], UCommutator[xi[x], UFPlus[μ, ν][x]]] /. $Substitutions /. lrRule // NMExpand // Expand

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

% - %%

0

Coeff[UFPlus][0][li1_, li2_, x_] = UFPlus[li1, li2][x]

f _ + _ (li1 li2)

Coeff[UFPlus][1][li1_, li2_, x_] = I/Sqrt[2]/DecayConstant[PhiMeson] UCommutator[xi[x], UFMinus[li1, li2][x]]

(i (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 f _ - _ (li1 li2) - f _ - _ (li1 li2) '6 Overscript[ξ^( ), ->] · Overscript[σ, ->]))/(2^(1/2) f)

Coeff[UFPlus][2][li1_, li2_, x_] = -1/4/DecayConstant[PhiMeson]^2 UCommutator[xi[x], UCommutator[xi[x], UFPlus[li1, li2][x]]]

-(Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 f _ + _ (li1 li2) - f _ + _ (li1 li2) '6 Overscript[ξ^( ), ->] · Overscript[σ, ->]) - (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 f _ + _ (li1 li2) - f _ + _ (li1 li2) '6 Overscript[ξ^( ), ->] · Overscript[σ, ->]) '6 Overscript[ξ^( ), ->] · Overscript[σ, ->])/(4 f^2)

Coeff[UFPlus][do_ ? ((# > 2) &)][li1_, li2_, x_] := Coeff[UChiPlus][do][li1, li2, x] = (Message[UPerturb :: nocoeff, do] ; DiscardTerms[NM[uExpLeftAdj[x, ExpansionOrder -> do], FieldStrengthTensorFull[{li1}, UGeneratorMatrixIsoDot[QuantumField[Particle[LeftComponent[0]], {li2}][x]], x, -I], Adjoint[uExpLeftAdj[x, ExpansionOrder -> do]]] + NM[Adjoint[uExpRight[x, ExpansionOrder -> do]], FieldStrengthTensorFull[{li1}, UGeneratorMatrixIsoDot[QuantumField[Particle[RightComponent[0]], {li2}][x]], x, -I], uExpRight[x, ExpansionOrder -> do]] // NMExpand // Expand, Retain -> {Particle[UPerturbation] -> do}] // UReduce) ;

Coeff[UFPlus][4][μ, ν, x]

UPerturb :: nocoeff : Warning: Yor are requesting expanding in UPerturbation to order 4 . 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)

(öÆ^† '6 R^( ) _ (μ ν) '6 öÆ '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->])/(96 f^4) - (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 öÆ^† '6 R^( ) _ (μ ν) '6 öÆ '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->])/(24 f^4) + (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 öÆ^† '6 R^( ) _ (μ ν) '6 öÆ '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->])/(16 f^4) - (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 öÆ^† '6 R^( ) _ (μ ν) '6 öÆ '6 Overscript[ξ^( ), ->] · Overscript[σ, ->])/(24 f^4) + (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 öÆ^† '6 R^( ) _ (μ ν) '6 öÆ)/(96 f^4) + (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 öÆ '6 L^( ) _ (μ ν) '6 öÆ^†)/(96 f^4) - (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 öÆ '6 L^( ) _ (μ ν) '6 öÆ^† '6 Overscript[ξ^( ), ->] · Overscript[σ, ->])/(24 f^4) + (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 öÆ '6 L^( ) _ (μ ν) '6 öÆ^† '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->])/(16 f^4) - (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 öÆ '6 L^( ) _ (μ ν) '6 öÆ^† '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->])/(24 f^4) + (öÆ '6 L^( ) _ (μ ν) '6 öÆ^† '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->])/(96 f^4)

Converted by Mathematica  (July 10, 2003)