The $q^2$ moments in inclusive semileptonic $B$ decays

We compute the first moments of the $q^2$ distribution in inclusive semileptonic $B$ decays as functions of the lower cut on $q^2$, confirming a number of results given in the literature and adding the $O(\alpha_s^2\beta_0)$ BLM contributions. We then include the $q^2$-moments recently measured by Belle and Belle II in a global fit to the moments. The new data are compatible with the other measurements and slightly decrease the uncertainty on the nonperturbative parameters and on $|V_{cb}|$. Our updated value is $|V_{cb}|=(41.97\pm 0.48)\times 10^{-3}$.