Lattice study of semileptonic B decays: B ¯→ Dl ν¯ decays

We present a study of semileptonic B\ifmmode\bar\else\textasciimacron\fi{}\ensuremath{\rightarrow}Dl\ensuremath{\nu}\ifmmode\bar\else\textasciimacron\fi{} decays in quenched lattice QCD through a calculation of the matrix element 〈D\ensuremath{\Vert}c\ifmmode\bar\else\textasciimacron\fi{}${\ensuremath{\gamma}}^{\mathrm{\ensuremath{\mu}}}$b\ensuremath{\Vert}B\ifmmode\bar\else\textasciimacron\fi{}〉 on a ${24}^{3}$\ifmmode\times\else\texttimes\fi{}48 lattice at \ensuremath{\beta}=6.2, using an O(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. Because the charm quark has a bare mass which is almost 1/3 the inverse lattice spacing, we study the ensuing mass-dependent discretization errors, and propose a procedure for subtracting at least some of them nonperturbatively. We extract the form factors ${\mathit{h}}^{+}$ and ${\mathit{h}}^{\mathrm{\ensuremath{-}}}$. After radiative corrections, we find that ${\mathit{h}}^{+}$ displays no dependence on the heavy-quark mass, enabling us to identify it with an Isgur-Wise function \ensuremath{\xi}. Interpolating the light-quark mass to that of the strange, we obtain an Isgur-Wise function relevant for B${\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}_{\mathit{s}}$\ensuremath{\rightarrow}${\mathit{D}}_{\mathit{s}}^{(\mathrm{*})}$l\ensuremath{\nu}\ifmmode\bar\else\textasciimacron\fi{} decays which has a slope -${\ensuremath{\xi}}_{\mathit{s}}^{\ensuremath{'}}$=1.${2}_{\mathrm{\ensuremath{-}}2}^{+2}$ (stat${)}_{\mathrm{\ensuremath{-}}1}^{+2}$ (syst) at zero recoil. An extrapolation to a massless light quark enables us to obtain an Isgur-Wise function relevant for B\ifmmode\bar\else\textasciimacron\fi{}\ensuremath{\rightarrow}${\mathit{D}}^{(\mathrm{*})}$l\ensuremath{\nu}\ifmmode\bar\else\textasciimacron\fi{} decays. This function has a slope -${\ensuremath{\xi}}_{\mathit{u},}$${\mathit{d}}^{\ensuremath{'}}$=0.${9}_{\mathrm{\ensuremath{-}}3}^{+2}$ (stat${)}_{\mathrm{\ensuremath{-}}2}^{+4}$ (syst) at zero recoil. We observe a slight decrease in the magnitude of the central value of the slope as the mass of the light quark is reduced; given the errors, however, the significance of this observation is limited.