•The identity ∇ _ μ(H _ (R, L)) = H _ (μ ±)+i/2[u _ μ,H _ (L, R)],              H _ (μ ±):= u^†c _ R^μQ _ Ru ± u c _ L^μQ _ Lu^†

The equation can be modified to obtain an equation of motion term which contains no covariant derivatives (partial integration):

(H _ (μ -)+i/2[u _ μ,H _ R]) H _ R u _ μ+ H _ L(H _ (μ +)+i/2[u _ μ,H _ L])u _ μ- (H _ (μ +)+i/2[u _ μ,H _ L])H _ L- H _ R(H _ (μ -)+i/2[u _ μ,H _ R]) =

(∇ _ μH _ L H _ R u _ μ + H _ L∇ _ μH _ R u _ μ)- (∇ _ μH _ R H _ L u _ μ + H _ R∇ _ μH _ L u _ μ)= -H _ LH _ R∇ _ μ u _ μ + H _ RH _ L∇ _ μ u _ μ=

-[H _ L, H _ R]{i/2(χ -+(2 C)/f _ π^2[H _ L,H _ R] - 1/n < χ _ - >)}.

CayleyHamiltonRules[{{HRight[x], HLeft[x], CovariantNabla[USmall[μ][x], {μ}]}}] /. UTrace1[(HLeft | USmall[_])[_] | CovariantNabla[USmall[_][_], {_}]] -> 0

{< ∂ _ μ(u _ μ) '6 H _ R '6 H _ L > -> < H _ R > < ∂ _ μ(u _ μ) '6 H _ L > + < ∂ _ μ(u _ μ) > < H _ L '6 H _ R > - < ∂ _ μ(u _ μ) '6 H _ L '6 H _ R > + < H _ R > < H _ L '6 Γ _ μ '6 u _ μ > - < H _ R > < H _ L '6 u _ μ '6 Γ _ μ > - < H _ L '6 H _ R '6 Γ _ μ '6 u _ μ > + < H _ L '6 H _ R '6 u _ μ '6 Γ _ μ > - < H _ L '6 Γ _ μ '6 u _ μ '6 H _ R > + < H _ L '6 u _ μ '6 Γ _ μ '6 H _ R >}

hminus[μ_][x_] = NM[Adjoint[SMM[x]], CQRight[LorentzIndex[μ]][x], SMM[x]] - NM[SMM[x], CQLeft[LorentzIndex[μ]][x], Adjoint[SMM[x]]]

öÆ^† '6 c _ μ^R Q _ R '6 öÆ - öÆ '6 c _ μ^L Q _ L '6 öÆ^†

hplus[μ_][x_] = NM[Adjoint[SMM[x]], CQRight[LorentzIndex[μ]][x], SMM[x]] + NM[SMM[x], CQLeft[LorentzIndex[μ]][x], Adjoint[SMM[x]]]

öÆ^† '6 c _ μ^R Q _ R '6 öÆ + öÆ '6 c _ μ^L Q _ L '6 öÆ^†

SetOptions[CovariantFieldDerivative, Explicit -> False] ;

rh0 = UTrace[NM[hminus[μ][x] + I/2 UCommutator[USmall[μ][x], HRight[x]], HRight[x], USmall[μ][x]] + NM[HLeft[x], hplus[μ][x] + I/2 UCommutator[USmall[μ][x], HLeft[x]], USmall[μ][x]] - NM[hplus[μ][x] + I/2 UCommutator[USmall[μ][x], HLeft[x]], HLeft[x], USmall[μ][x]] - NM[HRight[x], hminus[μ][x] + I/2 UCommutator[USmall[μ][x], HRight[x]], USmall[μ][x]]] // NMExpand // Expand

-1/2 i < H _ L '6 H _ L '6 u _ μ '6 u _ μ > + i < H _ L '6 u _ μ '6 H _ L '6 u _ μ > + 1/2 i < H _ R '6 H _ R '6 u _ μ '6 u _ μ > - i < H _ R '6 u _ μ '6 H _ R '6 u _ μ > - 1/2 i < u _ μ '6 H _ L '6 H _ L '6 u _ μ > + 1/2 i < u _ μ '6 H _ R '6 H _ R '6 u _ μ > - < öÆ^† '6 c _ μ^R Q _ R '6 öÆ '6 H _ L '6 u _ μ > + < öÆ^† '6 c _ μ^R Q _ R '6 öÆ '6 H _ R '6 u _ μ > + < H _ L '6 öÆ^† '6 c _ μ^R Q _ R '6 öÆ '6 u _ μ > + < H _ L '6 öÆ '6 c _ μ^L Q _ L '6 öÆ^† '6 u _ μ > - < H _ R '6 öÆ^† '6 c _ μ^R Q _ R '6 öÆ '6 u _ μ > + < H _ R '6 öÆ '6 c _ μ^L Q _ L '6 öÆ^† '6 u _ μ > - < öÆ '6 c _ μ^L Q _ L '6 öÆ^† '6 H _ L '6 u _ μ > - < öÆ '6 c _ μ^L Q _ L '6 öÆ^† '6 H _ R '6 u _ μ >

rh1 = CycleUTraces[rh0] /. {CQRight -> cQRight, CQLeft -> cQLeft} /. $Substitutions /. {cQRight -> CQRight, cQLeft -> CQLeft} // NMExpand // Expand

2 i < öÆ^† '6 öÆ^† '6 ÷s _ μ(÷„) '6 öÆ^† '6 öÆ^† '6 Q _ R '6 öÆ '6 öÆ '6 c _ μ^L Q _ L > - 2 i < öÆ^† '6 öÆ^† '6 Q _ R '6 öÆ '6 öÆ^† '6 ÷s _ μ(÷„) '6 öÆ^† '6 öÆ '6 c _ μ^L Q _ L > - 2 i < öÆ^† '6 c _ μ^R Q _ R '6 öÆ '6 öÆ^† '6 ÷s _ μ(÷„) '6 öÆ^† '6 öÆ '6 Q _ L '6 öÆ^† > + 2 i < öÆ^† '6 c _ μ^R Q _ R '6 öÆ '6 öÆ '6 Q _ L '6 öÆ^† '6 öÆ^† '6 ÷s _ μ(÷„) '6 öÆ^† > + 2 i < öÆ^† '6 Q _ R '6 öÆ '6 öÆ^† '6 ÷s _ μ(÷„) '6 öÆ^† '6 öÆ '6 Q _ L '6 öÆ^† '6 öÆ^† '6 ÷s _ μ(÷„) '6 öÆ^† > - 2 i < öÆ^† '6 Q _ R '6 öÆ '6 öÆ '6 Q _ L '6 öÆ^† '6 öÆ^† '6 ÷s _ μ(÷„) '6 öÆ^† '6 öÆ^† '6 ÷s _ μ(÷„) '6 öÆ^† > + 2 i < öÆ '6 Q _ L '6 öÆ^† '6 öÆ^† '6 ÷s _ μ(÷„) '6 öÆ^† '6 öÆ^† '6 Q _ R '6 öÆ '6 öÆ^† '6 ÷s _ μ(÷„) '6 öÆ^† > - 2 i < öÆ '6 Q _ L '6 öÆ^† '6 öÆ^† '6 Q _ R '6 öÆ '6 öÆ^† '6 ÷s _ μ(÷„) '6 öÆ^† '6 öÆ^† '6 ÷s _ μ(÷„) '6 öÆ^† >

rha = rh1 // UReduce[#, SMMToMM -> True] &

-2 i < ÷s _ μ(÷„)^† '6 c _ μ^R Q _ R '6 ÷„ '6 Q _ L > - 4 i < ÷s _ μ(÷„)^† '6 Q _ R '6 ÷s _ μ(÷„) '6 Q _ L > - 2 i < ÷s _ μ(÷„)^† '6 Q _ R '6 ÷„ '6 c _ μ^L Q _ L > - 2 i < ÷„^† '6 c _ μ^R Q _ R '6 ÷s _ μ(÷„) '6 Q _ L > - 2 i < ÷„^† '6 Q _ R '6 ÷s _ μ(÷„) '6 c _ μ^L Q _ L > - 2 i < Q _ L '6 ÷s _ μ(÷„)^† '6 ÷„ '6 ÷s _ μ(÷„)^† '6 Q _ R '6 ÷„ > - 2 i < Q _ L '6 ÷„^† '6 Q _ R '6 ÷s _ μ(÷„) '6 ÷„^† '6 ÷s _ μ(÷„) >

Converted by Mathematica  (July 10, 2003)