New determination of $|V_{\rm cb}|$ using the three-loop QCD corrections for the $B\to D^{\ast}$ semi-leptonic decays

Abstract

We present a new determination of the Cabibbo-Kobayashi-Maskawa matrix element $|V_{\rm cb}|$ by using the three-loop perturbative QCD corrections for the $B\to D^{\ast}$ semi-leptonic decay. The decay width of $B\to D^{\ast}$ semi-leptonic decay can be factorized as perturbatively calculable short-distance part and the non-perturbative but universal long-distance part. We adopt the principle of maximum conformality (PMC) single-scale setting approach to deal with the perturbative series so as to achieve a precise fixed-order prediction for the short-distance parameter $\eta_{A}$. By applying the PMC, an overall effective $\alpha_s$ value is achieved by recursively using the renormalization group equation, which inversely results in a precise scale-invariant pQCD series. Such scale-invariant series also provides a reliable basis for predicting the contributions from uncalculated perturbative terms. We then obtain $\eta_{A}=0.9225^{+0.0117}_{-0.0168}$, where the error is the squared average of those from $\Delta\alpha_{s}(M_Z)=\pm0.0010$ and the uncertainties caused by the uncalculated higher-order perturbative terms. By using the data of $B\to D^{\ast}\ell\bar{\nu}_{\ell}$, we finally obtain $|V_{\rm cb}|_{\rm PMC} =(40.60^{+0.53}_{-0.57})\times10^{-3}$, which is consistent with the PDG value within errors.