ρ ( x ) = 9 4 + 2Li$_{2}$( x ) + ln x ln(1 − x ) − ln(1 − x ) + ( x − 3 2 ) ln 1 − x x − x 1 − x ln x , (4) f 2 B m 4 B m 2 b e − m 2 $_{B}$/M 2 = 3 4 π 2 s ′ 0 ∫ m 2 b dsse − s/M $^{2}$(1 − x ) 2 [ 1 + α$_{s}$ ( µ$_{IR}$ ) C$_{F}$ π ρ ( x ) ] + e − m 2 $_{b}$/M 2 [ 1 6 〈 α$_{s}$ π GG 〉 − 32 π 27 α$_{s}$ ( µ$_{IR}$ ) 〈 ¯ qq 〉 2 M 2 ( 1 − m 2 b 4 M 2 − m 4 b 12 M 4 ) ] , (5) where s ′ 0 also is the effective threshold. Again, the terms proportional to m$_{q}$ are neglected due to their smallness. It can be found that the sum rules II is simpler in form than that of sum rules I. As a comparison of sum rules I and II, it can be found that by taking proper chiral current in the correlator, one can reduce the theoretical uncertainties to a certain degree. Especially, in sum rules I, the first non-perturbative term and hence the dominant contri- bution of the non-perturbative condensates is the dimension-three quark condensate 〈 ¯ qq 〉 . And, in sum rules II, the first non-perturbative term is the dimension-four gluon condensate 〈 α s π GG 〉 . Therefore, with proper chiral currents in the correlator, one can naturally suppress the non-perturbative sources and hence improve the accuracy of the obtained sum rules. This observation is similar to the case of light-cone sum rules for B → pseudo-scalar tran- sition form factors where with chiral current in the correlator, one can eliminate the most where the dilogarithm function Li$_{2}$( x ) = $^{−}$∫ x 0 dt t ln(1 − t ). Practically, ρ ( x ) is firstly derived under the MS scheme, and then it is transformed to be the present form with the help of the well-known one loop formula for the relation between the MS b -quark mass and the pole quark mass, i.e. ¯ m$_{b}$ ( µ$_{IR}$ ) = m$_{b}$ [ 1 + α s ( µ IR ) C F 4 π ( − 4 + 3 ln m 2 b µ 2 IR )] . In some references [8, 27], it is argued that one should use b -quark running mass other than its pole quark mass to do the numerical analysis. However, we think these two choices are equivalent to each other in principle, since we need to use their relation either to change the bound state part (to be determined by the pole quark mass) with the running mass or to change the hard scattering part (to be calculated by the running mass) with the pole quark mass. Moreover, as a cross check of the present obtained formulae, it is found that the sum rules I agrees with those of Refs.[3, 4, 8–10] when taking the same approximation and the same type of b -quark mass. The sum rules up to NLO for the second correlator can be written as (sum rules II): 4