•Expansion of <u _ μ u _ ν >

SetOptions[CovariantNabla, Explicit -> True] ;

USmallSqExp[li1_, li2_, x_] := NM[CovariantNabla[xi[x], x, {li1}], CovariantNabla[xi[x], x, {li2}]] + 1/4 NM[UCommutator[USmall[li1][x], xi[x]], UCommutator[USmall[li2][x], xi[x]]] // NMExpand // Simplify ;

USmallSqExp[li1_, x_] := NM[CovariantNabla[xi[x], x, {li1}], CovariantNabla[xi[x], x, {li1}]] + 1/4 NM[UCommutator[USmall[li1][x], xi[x]], UCommutator[USmall[li1][x], xi[x]]] // NMExpand // Simplify ;

USmallSqExp[μ, x] /. $Substitutions /. MM[x_] -> NM[SMM[x], SMM[x]] // NMExpand // Expand // UReduce[#, FullReduce -> False] &

SetOptions[CovariantNabla, Explicit -> False] ;

USmallSqExp[μ, ν, x] // UTrace // CycleUTraces // Expand

Tr _ 1(∇ _ μ(Overscript[ξ^( ), ->] · Overscript[σ, ->]) '6 ∇ _ ν(Overscript[ξ^( ), ->] · Overscript[σ, ->])) - 1/4 Tr _ 1(u _ μ '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 u _ ν) + 1/2 Tr _ 1(u _ μ '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 u _ ν '6 Overscript[ξ^( ), ->] · Overscript[σ, ->]) - 1/4 Tr _ 1(u _ μ '6 u _ ν '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->])

DecayConstant[PhiMeson]^2/2 * DiscardTerms[UPerturb[NM[USmall[μ][x], USmall[ν][x]], ExpansionOrder -> {0, 2}] // NMExpand // Expand, Retain -> {Particle[PseudoScalar[12]] -> 2}, Method -> Coefficient] // UTrace // CycleUTraces // Expand

Tr _ 1(∇ _ μ(Overscript[ξ^( ), ->] · Overscript[σ, ->]) '6 ∇ _ ν(Overscript[ξ^( ), ->] · Overscript[σ, ->])) - 1/4 Tr _ 1(u _ μ '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 u _ ν) + 1/2 Tr _ 1(u _ μ '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 u _ ν '6 Overscript[ξ^( ), ->] · Overscript[σ, ->]) - 1/4 Tr _ 1(u _ μ '6 u _ ν '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->])

% - %%

0

USmallSqExp[μ, x] // Expand

∇ _ μ(Overscript[ξ^( ), ->] · Overscript[σ, ->]) '6 ∇ _ μ(Overscript[ξ^( ), ->] · Overscript[σ, ->]) + 1/4 (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 u _ μ '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 u _ μ) - 1/4 (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 u _ μ '6 u _ μ '6 Overscript[ξ^( ), ->] · Overscript[σ, ->]) - 1/4 (u _ μ '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 u _ μ) + 1/4 (u _ μ '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 u _ μ '6 Overscript[ξ^( ), ->] · Overscript[σ, ->])

DecayConstant[PhiMeson]^2/2 * DiscardTerms[UPerturb[NM[USmall[μ][x], USmall[μ][x]], ExpansionOrder -> {0, 2}] // NMExpand // Expand, Retain -> {Particle[PseudoScalar[12]] -> 2}, Method -> Coefficient] // Expand

∇ _ μ(Overscript[ξ^( ), ->] · Overscript[σ, ->]) '6 ∇ _ μ(Overscript[ξ^( ), ->] · Overscript[σ, ->]) - 1/8 (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 u _ μ '6 u _ μ) + 1/4 (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 u _ μ '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 u _ μ) - 1/4 (u _ μ '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 u _ μ) + 1/4 (u _ μ '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 u _ μ '6 Overscript[ξ^( ), ->] · Overscript[σ, ->]) - 1/8 (u _ μ '6 u _ μ '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->])

% - %%

-1/8 (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 u _ μ '6 u _ μ) + 1/4 (Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 u _ μ '6 u _ μ '6 Overscript[ξ^( ), ->] · Overscript[σ, ->]) - 1/8 (u _ μ '6 u _ μ '6 Overscript[ξ^( ), ->] · Overscript[σ, ->] '6 Overscript[ξ^( ), ->] · Overscript[σ, ->])

Converted by Mathematica  (July 10, 2003)