Analysis of the decay constants of the heavy pseudoscalar mesons with QCD sum rules

In this article, we recalculate the contributions of all vacuum condensates up to dimension-6, in particular the one-loop corrections to the quark condensates ${\alpha }_s<\bar{q}q>$ and partial one-loop corrections to the four-quark condensates ${\alpha }_s^2<\bar{q}q{>}^2$, in the operator product expansion. Then we study the masses and decay constants of the heavy pseudoscalar mesons $D$, $D_s$, $B$ and $B_s$ using the QCD sum rules with two choices: {\bf I} we choose the $\overline{MS}$ masses by setting $m=m(\mu )$ and take perturbative corrections up to the order $\mathcal{O}({\alpha }_s)$; {\bf II} we choose the pole masses $m$, take perturbative corrections up to the order $\mathcal{O}({\alpha }_s^2)$ and set the energy-scale to be the heavy quark pole mass $\mu =m_Q$. In the case of {\bf I}, the predictions $f_D=(208\pm 11)\mathrm{M}\mathrm{e}\mathrm{V}$ and $f_B=(189\pm 15)\mathrm{M}\mathrm{e}\mathrm{V}$ are consistent with the experimental data within uncertainties, while the prediction $f_{D_s}=(241\pm 12)\mathrm{M}\mathrm{e}\mathrm{V}$ is below the lower bound of the experimental data $f_{D_s}=(260.0\pm 5.4)\mathrm{M}\mathrm{e}\mathrm{V}$. In the case of {\bf II}, the predictions $f_D=(211\pm 14)\mathrm{M}\mathrm{e}\mathrm{V}$, $f_B=(190\pm 17)\mathrm{M}\mathrm{e}\mathrm{V}$, $f_{D_s}=(258\pm 13)\mathrm{M}\mathrm{e}\mathrm{V}$ and $f_{D_s}\mathrm{/}{f}_D=1.22\pm 0.08$ are all in excellent agreements with the experimental data within uncertainties.