A reliable determination of the Cabibbo-Kobayashi-Maskawa matrix element $|V_{cb}|$ is mandatory for precision flavor physics and for the search for CP violating phases from new, heavy particles. In this article, we review the theory of the determina
tion of $|V_{cb}|$ from inclusive semileptonic $B$ decays. We discuss the available measurements of the semileptonic $B$ branching fraction and other inclusive observables in B decays relevant to the determination of $|V_{cb}|$. Finally, we perform a global fit to extract $|V_{cb}|$ and the $b$-quark mass $m_b$.
Using the previous Belle measurement of the inclusive photon energy in $Bto X_sgamma$ decays, we determine the first and second moments of this spectrum for minimum photon energies in the $B$ meson rest frame ranging from 1.8 to 2.3 GeV. Combining th
ese measurements with recent Belle data on the lepton energy and hadronic mass moments in $Bto X_cell u$ decays, we perform fits to theoretical expressions derived in the 1S and kinetic mass schemes and extract the magnitude of the Cabibbo-Kobayashi-Maskawa (CKM) matrix element $V_{cb}$, the $b$-quark mass and other non-perturbative parameters. In the 1S scheme analysis we find $|V_{cb}|=(41.56pm 0.68(mathrm{fit})pm 0.08(tau_B))times 10^{-3}$ and $m_b^mathrm{1S}=(4.723pm 0.055)$ GeV. In the kinetic scheme, we obtain $|V_{cb}|=(41.58pm 0.69(mathrm{fit})pm 0.08(tau_B)pm 0.58(mathrm{th}))times 10^{-3}$ and $m_b^mathrm{kin}=(4.543pm 0.075)$ GeV.
