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The determination of the CKM element $V_{cb}$ from inclusive semileptonic$b\to c \ell \bar\nu$ decays has reached a high precision thanks to acombination of theoretical and experimental efforts. Aiming towards even higherprecision, we discuss two processes that contaminate the inclusive $V_{cb}$determination; the $b\to u$ background and the contribution of the tauonicmode: $b\to c(\tau \to \mu\nu\bar{\nu})\bar{\nu}$. Both of these contributionsare dealt with at the experimental side, using Monte-Carlo methods and momentumcuts. However, these contributions can be calculated with high precision withinthe Heavy-Quark Expansion. In this note, we calculate the theoreticalpredictions for these two processes. The $b\to u$ results are compared withgenerator-level Monte-Carlo results used at Belle and Belle II. We have goodagreement between theory and Monte-Carlo for lepton energy moments, but lessfor hadronic mass moments. Based on our results the uncertainties due to thesebackgrounds processes can basically be eliminated by properly including theminto the analyses.

Impact of background effects on the inclusive $V_{cb}$ determination