$V_{cb}$ puzzle in semi-leptonic $B\to D^*$ decays revisited

Inspired by the newly reported $B\to D^*(\to D\pi)\ell\bar{\nu}_\ell$ differential decay rates by the Belle and Belle II Collaborations, we revisit the $V_{cb}$ puzzle in semi-leptonic $B\to D^*$ decays, considering the latest lattice QCD~(LQCD) simulations and light-cone sum rule~(LCSR) results. We examine the commonly used Caprini-Lellouch-Neubert~(CLN), Boyd-Grinstein-Lebed~(BGL), and heavy quark effective theory~(HQET) parameterizations. We demonstrate that these three parameterizations yield consistent results and reconfirm the $V_{cb}$ puzzle. Then, we use a state-of-the-art Bayesian method to estimate the impact of higher-order terms beyond the present HQET expansion on the uncertainty of $V_{cb}$. We show that higher-order effects cannot eliminate the deviation between the exclusive and inclusive determinations of $V_{cb}$. Finally, utilizing the best-fit results obtained in the HQET parameterization via fitting LQCD and LCSR data only as inputs, we predict the relevant observables, i.e., $R_{D^*}$, $F_L^{D^*}$, and $P_\tau^{D^*}$, sensitive to new physics in the $B\to D^*\ell\bar{\nu}_\ell$ decays. We conclude that lepton-flavour universality violations still exist in the $b\to c\tau\nu$.