Lattice Study of Semi-Leptonic $B$ Decays: I. $\bar B\to D\ell\bar\nu$ Decays

We present a study of semi-leptonic $\bar B\tp D\ell\bar\nu$ decays in quenched lattice QCD through a calculation of the matrix element $\la D|\bar c\gamma_\mu b|\bar B\ra$ on a $24^3\times 48$ lattice at $\beta=6.2$, using an $\ord{a}$-improved fermion action. We perform the calculation for several values of the initial and final heavy-quark masses around the charm mass, and three values of the light-(anti)quark mass around the strange mass. We extract the form factors $h^+$ and $h^-$. After radiative corrections, we find that $h^+$ displays no dependence on the heavy-quark mass, enabling us to identify it with an Isgur-Wise function. Thus, we obtain Isgur-Wise functions relevant for $\bar B_s\to D_s^{(*)}\ell\bar\nu $ and $\bar B\to D^{(*)}\ell\bar\nu $ decays. We then use these functions, in conjunction with heavy-quark effective theory, to extract $V_{cb}$ with no free parameters from the $\bar B\to D^*\ell\bar\nu $ decay rate measured by the ALEPH, ARGUS and CLEO collaborations. We also use our Isgur-Wise functions and heavy-quark effective theory to calculate branching ratios for $\bar B_s\to D_s^{(*)}\ell\bar\nu $ and $\bar B\to D^{(*)}\ell\bar\nu $ decays.