Determination of the Strong Coupling Constant ${\alpha }_s$ from Inclusive Semi-leptonic $B$ Meson Decays

We demonstrate the feasibility of determining the strong coupling constant, ${\alpha }_s$, from the inclusive semileptonic decay width of $B$ mesons. We express the semileptonic $B$ decay width as a function of ${\alpha }_s(5\mathrm{G}\mathrm{e}\mathrm{V})$, the Cabibbo-Kobayashi-Maskawa matrix element $\mathrm{\mid }{V}_{cb}\mathrm{\mid }$, $b$- and $c$-quark masses in the $\overline{\mathrm{M}\mathrm{S}}$ scheme. We fit ${\alpha }_s(5\mathrm{G}\mathrm{e}\mathrm{V})$ to current world averages of the $B^{\pm }$ and $B^0$ semileptonic decay widths. This yields ${\alpha }_s(5\mathrm{G}\mathrm{e}\mathrm{V})=0.245\pm 0.009$, corresponding to a 5-flavor extrapolation of ${\alpha }_s(m_Z)=0.1266\pm 0.0023$. The primary uncertainty contributions arise from the uncertainty on the perturbative expansion and the value of $\mathrm{\mid }{V}_{cb}\mathrm{\mid }$. Future advancements including higher-order perturbative calculations, and precise measurements of $\mathrm{\mid }{V}_{cb}\mathrm{\mid }$ and $B$ decay widths from upcoming $B$ and $Z$ factories, could enable this method to determine ${\alpha }_s(m_Z)$ with a competitive precision of $\mathrm{\Delta }{\alpha }_s(m_Z)\sim 0.0018$. This precision is comparable to the current accuracy of ${\alpha }_s(m_Z)$ measurements from $\tau$-lepton decays, which is regarded as the most precise experimental approach.