Accurate decay-constant ratios $f_{B^{*}}/f_B$ and $f_{B_s^{*}}/f_{B_s}$ from Borel QCD sum rules

We present our analysis of the decay constants of the beauty vector mesons $B^{*}$ and $B_s^{*}$ within the framework of dispersive sum rules for the two-point correlator of vector currents in QCD. While the decay constants of the vector mesons $f_{B^{*}}$ and $f_{B_s^{*}}$—similar to the decay constants of the pseudoscalar mesons $f_B$ and $f_{B_s}$—individually have large uncertainties induced by theory parameters not known with a satisfactory precision, these uncertainties almost entirely cancel out in the ratios of vector over pseudoscalar decay constants. These ratios, thus, may be predicted with very high accuracy due to the good control over the systematic uncertainties of the decay constants gained upon application of our hadron-parameter extraction algorithm. Our final results read $f_{B^{*}}/f_B=0.944\pm 0.011_{\mathrm{OPE}}\pm 0.018_{\text{syst}}$ and $f_{B_s^{*}}/f_{B_s}=0.947\pm 0.023_{\mathrm{OPE}}\pm 0.020_{\text{syst}}$. Thus, both $f_{B^{*}}/f_B$ and $f_{B_s^{*}}/f_{B_s}$ are less than unity at $2.5\sigma$ and $2\sigma$ level, respectively.

Figure 1

QCD sum-rule estimates of the $B^{*}$-meson decay constant using the pole-mass OPE (a) and the running-mass OPE at the renormalization scales $\mu =2.5$ (b), $\mu =3$ (c), and $\mu =5\mathrm{GeV}$ (d). The running-mass OPE for ${\overline{m}}_b({\overline{m}}_b)=4.247\mathrm{GeV}$ is shown. The pole-mass OPE employs the corresponding two-loop pole mass $M_b=4.87\mathrm{GeV}$. For each case, separately, a constant effective continuum threshold $s_{\text{eff}}$ is determined by requiring maximal stability of the predicted decay constant in a Borel window of the maximal width $0.05\le \tau ({\mathrm{GeV}}^{-2})\le 0.15$. Bold lines (lilac)—total findings, solid lines (black)—$O(1)$ contributions; dashed lines (red)—$O({\alpha }_{\mathrm{s}})$ contributions; dotted lines (blue)—$O({\alpha }_{\mathrm{s}}^2)$ contributions; dot-dashed lines (green)—power contributions.

Figure 2

Same as Fig. 1 but for the $B$ meson.

Figure 3

Dependence on the Borel parameter $\tau$ of the dual mass (a) and the dual decay constant (b) of the $B^{*}$ meson, obtained by adopting different Ansätze (4.2) for the effective threshold $s_{\text{eff}}(\tau )$ and fixing these thresholds by minimizing (4.3); the results are presented for the central values of all OPE parameters. (c) The $\tau$-dependent effective thresholds as obtained by our algorithm. The integer $n=0,1,2,3$ is the degree of the polynomial in our Ansatz (4.2) for $s_{\text{eff}}(\tau )$: dotted lines (red)—$n=0$; solid lines (green)—$n=1$; dashed lines (blue)—$n=2$; dot-dashed lines (black)—$n=3$.

Figure 4

Renormalization-scale dependence of the predicted decay constants: (a) $f_B^{\text{dual}}(\mu )$ and $f_{B^{*}}^{\text{dual}}(\mu )$, (b) $f_{B_s}^{\text{dual}}(\mu )$ and $f_{B_s^{*}}^{\text{dual}}(\mu )$. For each decay constant, we depict the $\mu$-related uncertainty, i.e., the standard deviation calculated assuming a flat $\mu$ distribution in the range $\mu =3–5\mathrm{GeV}$. Dotted lines (red)—vector beauty mesons; solid lines (blue)—pseudoscalar beauty mesons.

Figure 5

Distributions of the ratios $f_{B^{*}}/f_B$ and $f_{B_s^{*}}/f_{B_s}$ of beauty-meson decay constants, obtained by generating 1000 bootstrap events. For both ratios, their final distributions possess Gaussian-like shapes, with the standard deviations quoted in the plots.