Semileptonic $bc$ to $cc$ baryon decay and heavy quark spin symmetry

We study the semileptonic decays of the lowest-lying $bc$ baryons to the lowest-lying $cc$ baryons (${\Xi }_{bc}^{(\prime *)}\to {\Xi }_{cc}^{(*)}$ and ${\Omega }_{bc}^{(\prime *)}\to {\Omega }_{cc}^{(*)}$), in the limit $m_b,m_c\gg {\Lambda }_{\mathrm{QCD}}$ and close to the zero-recoil point. The separate heavy quark spin symmetries make it possible to describe all these decays using a single form factor. We recover results derived previously by White and Savage in a manner which we think is more straightforward and parallels the method applied later to study $B_c$ semileptonic decays. We further discuss the resemblance between the $bc$ baryon decays and those of $B_c$ mesons to ${\eta }_c$ and $J/\psi$ mesons and comment on the relation between the slopes of the single functions describing each set of decays. Our results can straightforwardly be applied to the decays of $bb$ baryons to $bc$ baryons.

Figure 1

Form factors in the heavy quark limit: $\eta (\omega )$ for cascade $bc$ to cascade $cc$ baryon decays and $\Delta (\omega )$ for $B_c$ to ${\eta }_c$, $J/\psi$ decays, calculated from a nonrelativistic quark model 10, 17 (using the AL1 potential). The solid lines are calculated from the wave function overlaps, illustrated for $\eta (\omega )$ in Eq. (21), while the dashed lines are constructed from appropriate combinations of form factors: for $\eta$ we consider $(F_1+F_2+F_3)/2$, where $F_{1,2,3}$ are defined in Eq. (23) of 10, while for $\Delta$ we use${\Sigma }_1^{(0)}$ defined in Eq. (52) of 17. The solid and dashed curves should agree close to zero recoil ($\omega \to 1$).