content_image/347827.jpg content_image/347828.jpg content_image/347826.jpg
the correlator, one can not only obtain the right properties of the hadrons but also simplify the theoretical uncertainties effectively. Usually the currents adopted in the correlators are taken to be those with definite quantum numbers, such as those with definite J $^{P}$, where J is the total angular momentum and P is the parity. However such kind of correlator is not the only choice adopted in the literature, e.g. the correlator constructed with chiral current is also adopted, which is firstly proposed by Ref.[19] to study the process B → K ∗ γ and then developed in dealing with various processes [20–26]. For different correlators, we need to check whether the theoretical estimations are consistent with each other or not, or to determine which correlator can lead to more accurate or more appropriate estimation for a particular physical process. And it is one of the purpose of the present paper, to make a comparative study of f$_{B}$ under two typical choices for the two-point correlator. 

The first correlator is the conventional one and is defined as 

<img src='content_image/347827.jpg'>

and the second one is the correlator with chiral current, 

<img src='content_image/347828.jpg'>

Following the standard procedure of SVZ sum rules, we can obtain two sum rules for f$_{B}$ from the above defined correlators. 

The sum rules up to next-to-leading order (NLO) for the first correlator can be written as (sum rules I) 

<img src='content_image/347826.jpg'>

where m$_{b}$ stands for the pole quark mass, µ$_{IR}$ is the renormalization scale, x = m 2 $_{b}$/s and C$_{F}$ = 4 / 3. M and s$_{0}$ stand for the Borel parameter and the effective continuum threshold respectively. Since m$_{q}$ is quite small in comparison to m$_{b}$ or µ$_{IR}$ , so terms proportional to m$_{q}$ have implicitly neglected. The function ρ ( x ) determines the spectral density of the NLO correction to the perturbative part and it takes the following form 

