Is There a Real Difference between $|V_{qb}{|}_{\mathrm{excl}}$ vs. $|V_{qb}{|}_{\mathrm{incl}}$ from Semi-leptonic $B$ Decays?

Abstract

For a long time there have been active discussions about the sizable differences between $|V_{ub}{|}_{\mathrm{incl}}$ \& $|V_{ub}{|}_{\mathrm{excl}}$. However, the real $|V_{ub}{|}_{\mathrm{incl}}$ might be smaller than usually stated. The connection of the worlds of hadrons and quarks (\& gluons) is subtle: we cannot ignore final states with a pair of $\bar{K}K$ mesons plus pions: $\left[R(B\to l\nu {\pi }^{\prime }s)+R(B\to l\nu (K\bar{K}+{\pi }^{\prime }s))\right]$ $\simeq$ $R([b\bar{q}]\to l\nu u({\bar{q}}_i{q}_i+{\bar{q}}_j{q}_j{\bar{q}}_k{q}_k...)\bar{q})$ with $\bar{q}=\bar{u},\bar{d}$ \& $q_i=u,d,s$. It might show the limits or violations of duality close to thresholds. While inclusive FS cannot be measured, exclusive ones can be done soon by LHCb and later also by Belle II about $B^{+}\to l^{+}\nu K^{+}{K}^{-}$ \& $B^0\to l^{+}\nu K^{+}{K}^{-}{\pi }^{+}$. Present data have not given us the information about resonances in the region of 1 - 2 GeV that we need to understand the underlying dynamics in $\Delta S=0$. Future data for exclusive $B\to l\nu \pi \pi$ might also narrow the gap between $|V_{ub}{|}_{\mathrm{incl}}$ \& $|V_{ub}{|}_{\mathrm{excl}}$ in the opposite direction. I comment about $B_s^0\to l^{+}\nu X_u^{(s)}(\Delta S=-1)$.

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