How to Measure Kinetic Energy of the Heavy Quark Inside B Mesons?

Abstract

We discuss how one can determine the average kinetic energy of the heavy quark inside heavy mesons from differential distributions in $B$ decays. A new, so-called third, sum rule for the $b\rightarrow c$ transition is derived in the small velocity (SV) limit. Using this sum rule and the existing data on the momentum dependence in the $B\rightarrow D^*$ transition (the slope of the Isgur-Wise function) we obtain a new lower bound on the parameter $\mu_\pi^2 = (2M_B)^{-1}\langle B |\bar b (i\vec{D})^2 b |B\rangle $ proportional to the average kinetic energy of $b$ quark inside $B$ meson. The existing data suggest $\mu_\pi^2 > 0.4$~GeV$^2$ and (from the ``optical'' sum rule) $\overline{\Lambda} > 500$ MeV, albeit with some numerical uncertainties.