Form factors for the processes Bc+→D0ℓ+νℓ and Bc+→Ds+ℓ+ℓ−(νν¯) from lattice QCD

We present results of the first lattice QCD calculations of the weak matrix elements for the decays ${B}_{c}^{+}\ensuremath{\rightarrow}{D}^{0}{\ensuremath{\ell}}^{+}{\ensuremath{\nu}}_{\ensuremath{\ell}}$, ${B}_{c}^{+}\ensuremath{\rightarrow}{D}_{s}^{+}{\ensuremath{\ell}}^{+}{\ensuremath{\ell}}^{\ensuremath{-}}$ and ${B}_{c}^{+}\ensuremath{\rightarrow}{D}_{s}^{+}\ensuremath{\nu}\overline{\ensuremath{\nu}}$. Form factors across the entire physical ${q}^{2}$ range are then extracted and extrapolated to the continuum limit with physical quark masses. Results are derived from correlation functions computed on MILC Collaboration gauge configurations with three different lattice spacings and including $2+1+1$ flavors of sea quarks in the highly improved staggered quark (HISQ) formalism. HISQ is also used for all of the valence quarks. The uncertainty on the decay widths from our form factors for ${B}_{c}^{+}\ensuremath{\rightarrow}{D}^{0}{\ensuremath{\ell}}^{+}{\ensuremath{\nu}}_{\ensuremath{\ell}}$ is similar in size to that from the present value for ${V}_{ub}$. We obtain the ratio $\mathrm{\ensuremath{\Gamma}}({B}_{c}^{+}\ensuremath{\rightarrow}{D}^{0}{\ensuremath{\mu}}^{+}{\ensuremath{\nu}}_{\ensuremath{\mu}})/{|{\ensuremath{\eta}}_{\mathrm{EW}}{V}_{ub}|}^{2}=4.43(63)\ifmmode\times\else\texttimes\fi{}{10}^{12}\text{ }\text{ }{\mathrm{s}}^{\ensuremath{-}1}$. Combining our form factors with those found previously by HPQCD for ${B}_{c}^{+}\ensuremath{\rightarrow}J/\ensuremath{\psi}{\ensuremath{\mu}}^{+}{\ensuremath{\nu}}_{\ensuremath{\mu}}$, we find ${|{V}_{cb}/{V}_{ub}|}^{2}\mathrm{\ensuremath{\Gamma}}({B}_{c}^{+}\ensuremath{\rightarrow}{D}^{0}{\ensuremath{\mu}}^{+}{\ensuremath{\nu}}_{\ensuremath{\mu}})/\mathrm{\ensuremath{\Gamma}}({B}_{c}^{+}\ensuremath{\rightarrow}J/\ensuremath{\psi}{\ensuremath{\mu}}^{+}{\ensuremath{\nu}}_{\ensuremath{\mu}})=\phantom{\rule{0ex}{0ex}}0.257(36{)}_{{B}_{c}\ensuremath{\rightarrow}D}(18{)}_{{B}_{c}\ensuremath{\rightarrow}J/\ensuremath{\psi}}$. We calculate the differential decay widths of ${B}_{c}^{+}\ensuremath{\rightarrow}{D}_{s}^{+}{\ensuremath{\ell}}^{+}{\ensuremath{\ell}}^{\ensuremath{-}}$ across the full ${q}^{2}$ range and give integrated results in ${q}^{2}$ bins that avoid possible effects from charmonium and $u\overline{u}$ resonances. For example, we find that the ratio of differential branching fractions integrated over the range ${q}^{2}=1\text{ }\text{ }{\mathrm{GeV}}^{2}--6\text{ }\text{ }{\mathrm{GeV}}^{2}$ for ${B}_{c}^{+}\ensuremath{\rightarrow}{D}_{s}^{+}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}$ and ${B}_{c}^{+}\ensuremath{\rightarrow}J/\ensuremath{\psi}{\ensuremath{\mu}}^{+}{\ensuremath{\nu}}_{\ensuremath{\mu}}$ is $6.31(90{)}_{{B}_{c}\ensuremath{\rightarrow}{D}_{s}}(65{)}_{{B}_{c}\ensuremath{\rightarrow}J/\ensuremath{\psi}}\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$. We also give results for the branching fraction of ${B}_{c}^{+}\ensuremath{\rightarrow}{D}_{s}^{+}\ensuremath{\nu}\overline{\ensuremath{\nu}}$. Prospects for reducing our errors in the future are discussed.