•The identity ∇ _ μ(H _ (R, L)) = H _ (μ ±)+i/2[u _ μ,H _ (L, R)]

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 _ - >)}.

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

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

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

Converted by Mathematica  (July 10, 2003)