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 $\alpha_s(5\mathrm{\,GeV})$, the Cabibbo-Kobayashi-Maskawa matrix element $|V_{cb}|$, $b$- and $c$-quark masses in the $\overline{\mathrm{MS}}$ scheme. We fit $\alpha_s(5\mathrm{\,GeV})$ to current world averages of the $B^{\pm}$ and $B^{0}$ semileptonic decay widths. This yields $\alpha_s(5\mathrm{\,GeV}) = 0.245 \pm 0.009$, 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 $|V_{cb}|$. Future advancements including higher-order perturbative calculations, and precise measurements of $|V_{cb}|$ 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 $\Delta\alpha_s(m_{Z}) \sim 0.0018$. 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.