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

Jun 2022

We present a new analysis for the Cabibbo-Kobayashi-Maskawa matrix element$|V_{\rm cb}|$ by using the three-loop QCD corrections for the $B\to D^{\ast}$semi-leptonic decays. We adopt the principle of maximum conformality (PMC)scale-setting approach to deal with the perturbative series so as to achieve aprecise scale-invariant prediction for the short-distance parameter $\eta_{A}$.By applying the PMC, an overall effective $\alpha_s$ value is achieved byrecursively using the renormalization group equation, which inversely resultsin a scale-invariant pQCD series. Such scale-invariant series also provides areliable basis for predicting the contributions from uncalculated perturbativeterms. We then obtain $\eta_{A}=0.9225^{+0.0035}_{-0.0036}$, where the error isthe squared average of those from $\Delta\alpha_{s}(M_Z)=\pm0.0010$ and thepredicted N$^4$LO contribution from the Pad$\acute{e}$ approximation approach.By using the data of $B\to D^{\ast}\ell\bar{\nu}_{\ell}$, we then obtain$|V_{\rm cb}|_{\rm PMC} =(40.57\pm0.50)\times10^{-3}$, which is consistent withthe PDG value within errors.

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