Eur. Phys. J. C (2025) 85: 934
https://doi.org/10.1140/epjc/s10052-025-14675-z

Regular Article - Theoretical Physics

Determination of $|V_{\text{cb}}|$ using Bayesian analysis of the $B\to D_{}^{\ast }\ell {\overline{\nu }}_{\ell }$ semileptonic decay width with four-loop QCD corrections

¹ Department of Physics, Chongqing Key Laboratory for Strongly Coupled Physics, Chongqing University, 401331, Chongqing, People’s Republic of China
² School of Mathematics and Physics, Southwest University of Science and Technology, 621010, Mianyang, People’s Republic of China

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Received: 26 June 2025
Accepted: 26 August 2025
Published online: 2 September 2025

Abstract

The Cabibbo–Kobayashi–Maskawa matrix element $|V_{\text{cb}}|$ is an important Standard Model parameter, whose value can be determined by using the semi-leptonic decay $B\to D_{}^{\ast }\ell {\overline{\nu }}_{\ell }$. The perturbative QCD (pQCD) corrections to the $B\to D_{}^{\ast }$ transform form factor $\mathcal{F}(w)$ has been known up to the ${\text{N}}^3$LO level, whose magnitude remains sensitive to the choice of renormalization scale ${\mu }_r$. To improve the precision of $\mathcal{F}(w)$ and hence $|V_{\text{cb}}|$, we first apply the single-scale approach of Principle of Maximum Conformality (PMC) to eliminate the conventional (Conv.) renormalization scale dependence of the short-distance parameter ${\eta }_A$ and then predict the contribution of its unknown ${\text{N}}^4$LO term via Bayesian analysis. In this paper, we adopt two probabilistic models for Bayesian analysis: the Cacciari–Houdeau model (CH model) and the geometric behavior model (GB model). It is shown that by using the PMC series in combination with Bayesian analysis, one can achieve high degree of reliability in estimating unknown higher-order terms. A more convergent behavior is achieved by applying the PMC, confirmed by the predicted ${\text{N}}^4$LO contributions: for the CH model, ${\eta }_A{|}_{\mathrm{Conv}.}^{{\text{N}}^4\text{LO}}=\{-0.0032,+0.0052\}$ and ${\eta }_A{|}_{\text{PMC}}^{{\text{N}}^4\text{LO}}=\{-0.0004,+0.0004\}$; for the GB model, ${\eta }_A{|}_{\mathrm{Conv}.}^{{\text{N}}^4\text{LO}}=\{-0.0049,+0.0075\}$ and ${\eta }_A{|}_{\text{PMC}}^{{\text{N}}^4\text{LO}}=\{-0.0007,+0.0007\}$. Comparing with the latest experimental measurements, we obtain $|V_{\text{cb}}{||}_{\text{PMC}}=(40.{58}_{-0.57}^{+0.53})\times {10}^{-3}$, which is consistent for both CH and GB models and in good agreement with the PDG world average, $|V_{\text{cb}}{|}_{\text{PDG}}=(41.1\pm 1.2)\times {10}^{-3}$ within errors.