A study of semi-inclusive charmless $B\to \pi X$ decays

We study semi-inclusive charmless decays $B\to \pi X$ in detail, such as ${\bar{B}}^0\to {\pi }^{\pm (0)}X$, $B^0\to {\pi }^{\pm (0)}X$, $B^{\pm} \to \pi^{\pm (0)} X$,where$X$ does not contain a charm (anti)quark. We find that the process ${\bar{B}}^0\to {\pi }^{-}X$ ($B^0\to {\pi }^{+}X$) can be particularly useful for determination of the CKM matrix element $\mathrm{\mid }{V}_{ub}\mathrm{\mid }$. We calculate and present the branching ratio (BR) of ${\bar{B}}^0\to {\pi }^{-}X$ as a function of $\mathrm{\mid }{V}_{ub}\mathrm{\mid }$, with an estimate of possible uncertainties. It is expected that the BR is an order of ${10}^{-4}$. Our estimation indicates that one can phenomenologically determine $\mathrm{\mid }{V}_{ub}\mathrm{\mid }$ with reasonable accuracy by measuring the BR of ${\bar{B}}^0\to {\pi }^{-}X$ ($B^0\to {\pi }^{+}X$).