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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 

<img src='content_image/362553.jpg'>

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. 

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]: 

<img src='content_image/362552.jpg'>

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, 

