New determination of $|V_{\mathrm{cb}}|$ 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 $|V_{\mathrm{cb}}|$ 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 ${\eta }_A=0.9225_{-0.0168}^{+0.0117}$, where the error is the squared average of those from $\Delta {\alpha }_s(M_Z)=\pm 0.0010$ 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 $|V_{\mathrm{cb}}{|}_{\mathrm{PMC}}=(40.60_{-0.57}^{+0.53})\times 10^{-3}$, which is consistent with the PDG value within errors.

Comments: 7 pages,2 figures