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

Abstract

We present a study of semi-leptonic $\bar{B}\text{tp}D\ell \bar{\nu }$ decays in quenched lattice QCD through a calculation of the matrix element $\text{la}D|\bar{c}{\gamma }_{\mu }b|\bar{B}\text{ra}$ on a $24^3\times 48$ lattice at $\beta =6.2$, using an $\text{ord}a$-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 ${\bar{B}}_s\to D_s^{(\ast )}\ell \bar{\nu }$ and $\bar{B}\to D^{(\ast )}\ell \bar{\nu }$ decays. We then use these functions, in conjunction with heavy-quark effective theory, to extract $V_{cb}$ with no free parameters from the $\bar{B}\to D^{\ast }\ell \bar{\nu }$ 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 ${\bar{B}}_s\to D_s^{(\ast )}\ell \bar{\nu }$ and $\bar{B}\to D^{(\ast )}\ell \bar{\nu }$ decays.

Comments: 53 pages, latex file, requires revtex version 3.0; 11 postscript figures