Average Kinetic Energy of Heavy Quark in Semileptonic B Decay

Abstract

Within the ACCMM model the average kinetic energy of heavy quark in a heavy-light meson is calculated as $\langle {\bf p}^2 \rangle = {3 \over 2} {p_{_F}}^2$, solely from the fact that the Gaussian momentum probability distribution has been taken in the ACCMM model. Therefore, the Fermi momentum parameter $p_{_F}$ of the ACCMM model is not a truly free parameter, but is closely related to the average kinetic energy of heavy quark, which is theoretically calculable in principle. In this context, we determine $p_{_F}$ by comparing the theoretical prediction of the ACCMM model with the model independent lepton energy spectrum of $B \rightarrow e \nu X$ from the recent CLEO analysis, and find that $p_{_F}=0.54~\pm~^{0.16}_{0.15}$ GeV. We also calculate $p_{_F}$ in the relativistic quark model by applying the quantum mechanical variational method, and obtained $p_{_F}=0.5\sim 0.6$ GeV. We show the correspondences between the relativistic quark model and the heavy quark effective theory. We then clarify the importance of the value of $p_{_F}$ in the determination of $|V_{ub}/V_{cb}|$.