uncertain twist-3 contributions [24–26] in comparison to the sum rules derived by using the conventional correlator (see Ref.[28] for an explicit example). Schematically, the two sum rules I and II can be rewritten in the following simpler form where the spectral density ρ $^{tot}$( s ) can be read from Eqs.(3,5). It is noted that the continuum threshold s$_{0}$ should be varied with the different choice of correlators. While more practi- cally, the Borel parameter M 2 and the continuum threshold s$_{0}$ are determined in such a combined way that the resulting B -meson decay constant f$_{B}$ does not depend too much on the precise values of these parameters. In addition, the following three criteria are adopted for numerically determining the range of of ( M $^{2}$, s$_{0}$ ): • Criterion (A): The continuum contribution, that is the part of the dispersive integral from s$_{0}$ to ∞ , should not be too large, e.g. less than 30% of the total dispersive integral. As will be found later that this condition is well satisfied, since the resultant ranges for ( M $^{2}$, s$_{0}$ ) shall led this ratio be less than 20% for sum rules I and be less than 15% for sum rules II. • Criterion (B): The contributions from the dimension-six condensate terms shall not exceed 15% for f$_{B}$ . • Criterion (C): The derivative of the logarithm of Eq.(6) with respect to ( − 1 /M $^{2}$) gives the B -meson mass m$_{B}$ , i.e. m 2 B = ∫ s 0 m 2 b ρ $^{tot}$( s ) e − s/M $^{2}$sds / ∫ s 0 m 2 b ρ $^{tot}$( s ) e − s/M $^{2}$ds , and we require its value to be full-filled with high accuracy, i.e. < ∼ 0 . 1%. Here, we shall not give any further constraints on s$_{0}$ , and we only treat it as an effective scale, which is roughly around the squared mass of B -meson first excited state. It is found that the relative error for 〈 α s π GG 〉 is quite large. Further more, we shall use LO anomalous dimensions of the condensates to evaluate them up to the renormalization scale, In numerical evaluation of the sum rules I and II, we take m$_{B}$ = 5 . 279 GeV and use the following values for the condensates [29, 30]: 5 〈 ¯ qq 〉 (1GeV) = − (0 . 246 +0 . 018 − 0 $_{019}$GeV)$^{3}$, 〈 α$_{s}$ π GG 〉 = 0 . 012 +0 . 006 − 0 $_{012}$GeV$^{4}$, 〈 ¯ qgσ · Gq 〉 (1GeV) = (0 . 8 ± 0 . 2) GeV $^{2}$〈 ¯ qq 〉 (1GeV) . (7) f 2 B m 4 B m 2 b e − m 2 $_{B}$/M 2 = ∫ s 0 m 2 b ρ $^{tot}$( s ) e − s/M $^{2}$ds, (6)