Determination of the Strong Coupling Constant \(\alpha_s\) from Inclusive Semi-leptonic \(B\) Meson Decays

We present a new methodology for determining the strong coupling constant, \(\alpha_s\), from the inclusive semi-leptonic decay width of \(B\) mesons. We express the semi-leptonic \(B\) decay width as a function of \(\alpha_s\)(5 GeV), the Cabibbo-Kobayashi-Maskawa matrix element \(|V_{cb}|\), \(b\)- and \(c\)-quark masses in the \(\overline{\mathrm{MS}}\) scheme. The method fixes the value of \(|V_{cb}|\) according to the recent measurement from Belle based on exclusive \(B\) decays and uses the PDG averages for the \(b\)- and \(c\)-quark masses. By fitting \(\alpha_s(5\mathrm{\,GeV})\) to current world averages of the \(B^{\pm}\) and \(B^{0}\) semi-leptonic decay widths, the analysis obtains \(\alpha_s(5\mathrm{\,GeV}) = 0.225 \pm 0.012\), corresponding to a 5-flavor extrapolation of \(\alpha_s(m_{Z}) = 0.121 \pm 0.003\). Taking into account future results from higher-order perturbative QCD calculations, heavy quark masses derived from lattice QCD, and measurements of \(|V_{cb}|\) as well as \(B\) decay widths from upcoming \(B\) and \(Z\) factory data, this method could yield a determination of \(\alpha_s(m_{Z})\) 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 \(\tau\) decays, which is regarded as the most precise approach.