to be their center values. By further varying the non-perturbative condensates within the region of Eq.(7), an extra ± 1 MeV up to ± 4 MeV error should be taken into consideration. For examples, by setting m$_{b}$ = 4 . 68 GeV, an extra ± 2 MeV error shall be caused by those non-perturbative condensates for both sum rules I and II; while by setting m$_{b}$ = 4 . 61 GeV, it changes to be ± 1 MeV. Furthermore, it can be found that the value of f$_{B}$ decreases with the increment of m$_{b}$ , and to compare with the experimental data on f$_{B}$ , both sum rules I and II prefer small b -quark pole mass, i.e. m$_{b}$ = 4 . 68 ± 0 . 07 GeV. It is found that the main uncertainty comes from m$_{b}$ , so a better understanding of m$_{b}$ shall further improve the present sum rules. For example, in Ref.[29], a smaller region of m$_{b}$ ∈ [4 . 55 , 4 . 60] GeV is adopted, which can be obtained by translating the running b -quark mass used there to the present pole mass, and then their result for f$_{B}$ is 214 +7 − 5 MeV. It should be noted that such a small error is also due to a smaller Borel window and a smaller region of s$_{0}$ derived there [29]. However if we treat ( M $^{2}$, s$_{0}$ ) as correlated parameters then their separate region shall be broadened as shown by the left diagrams of Figs.(3,4). 

As a summary, we have presented a detailed discussion on f$_{B}$ from two sum rules, i.e. sum rules I and II, which are derived from the conventional correlator and the correlator with chiral currents respectively. Under proper parameter values, both sum rules I and II can lead to reasonable f$_{B}$ that is consistent with the Belle experiment. At a fixed m$_{b}$ , the estimated uncertainties of f$_{B}$ from both sum rules I and II are less than 10%. If further fixing s$_{0}$ , f$_{B}$ shall be quite steady versus the Borel parameter M $^{2}$. Hence the contributions from the uncertainty sources, e.g. the continuum states and higher dimensional condensates, are well under control. With the help of TABs.(I,II), by varying m$_{b}$ within a small region m$_{b}$ = 4 . 68 ± 0 . 07 GeV, we obtain f$_{B}$ = 172 +23 − 25 MeV for sum rules I and f$_{B}$ = 214 +26 − 34 MeV for sum rules II, where the errors from all the mentioned parameters such as m$_{b}$ , ( M $^{2}$, s$_{0}$ ) and the non-perturbative condensates are added together in quadrature. Furthermore, the comparison of the sum rules I and II shows that we can improve the QCD sum rules in principle by using chiral current in the correlator. At the present, due to the large uncertainty of the dimension-four gluon condensates, such improvement on the sum rules with chiral current is not so clear as that of QCD light-cone sum rules for B → P transition form factors [24–26]. However with a more accurate values for these condensates, sum rules II shall become more accurate. 

