Decay Constants and Semileptonic Decays of Heavy Mesons in Relativistic Quark Model

We investigate the $B$ and $D$ mesons in the relativistic quark model by applying the variational method with the Gaussian wave function. We calculate the Fermi momentum parameter $p_{{}_F}$, and obtain $p_{{}_F}=0.50\sim 0.54$ GeV, which is almost independent of the input parameters, ${\alpha }_s$, $m_b$, $m_c$ and $m_{sp}$. We then calculate the ratio $f_B$/$f_D$, and obtain the result which is larger, by the factor of about 1.3, than $\sqrt{M_D\mathrm{/}{M}_B}$ given by the naive nonrelativistic analogy. This result is in a good agreement with the recent Lattice calculations. We also calculate the ratio $(M_{B^{\ast }}-M_B)$/$(M_{D^{\ast }}-M_D)$. In these calculations the wave function at origin $\psi (0)$ is essential. We also determine $p_{{}_F}$ by comparing the theoretical prediction of the ACCMM model with the lepton energy spectrum of $B \rightarrow e \nu X$fromtherecentARGUSanalysis,andfindthat$p_{F}=0.27~\pm~^{0.22}{0.27}$GeV,whenweuse$m_c=1.5$ GeV. However, this experimentally determined value of $p_{{}_F}$ is strongly dependent on the value of input parameter $m_c$.