Lifetimes of doubly heavy baryons ${\mathcal{B}}_{bb}$ and ${\mathcal{B}}_{bc}$

Lifetimes of the doubly heavy baryons ${\mathcal{B}}_{bb}$ and ${\mathcal{B}}_{bc}$ are analyzed within the framework of the heavy quark expansion (HQE). Lifetime differences arise from the spectator effects such as $W$-exchange and Pauli interference. For doubly bottom baryons, the lifetime pattern is $\tau ({\mathrm{\Omega }}_{bb}^{-})\sim \tau ({\mathrm{\Xi }}_{bb}^{-})>\tau ({\mathrm{\Xi }}_{bb}^0)$. The ${\mathrm{\Xi }}_{bb}^0$ baryon is shortest-lived owing to the $W$-exchange contribution, while ${\mathrm{\Xi }}_{bb}^{-}$ and ${\mathrm{\Omega }}_{bb}^{-}$ have similar lifetimes as they both receive contributions from destructive Pauli interference. We find the lifetime ratio $\tau ({\mathrm{\Xi }}_{bb}^{-})/\tau ({\mathrm{\Xi }}_{bb}^0)=1.26$. The large $W$-exchange contribution to ${\mathrm{\Xi }}_{bc}^0$ through the subprocess $cd\to us\to cd$ and the sizable destructive Pauli interference contribution to ${\mathrm{\Xi }}_{bc}^{+}$ imply a substantial lifetime difference between ${\mathrm{\Xi }}_{bc}^{+}$ and ${\mathrm{\Xi }}_{bc}^0$. In the presence of subleading $1/m_c$ and $1/m_b$ corrections to the spectator effects, we find that $\tau ({\mathrm{\Omega }}_{bc}^0)$ becomes longest-lived. This is because ${\mathrm{\Gamma }}_{+}^{\mathrm{int}}$ and ${\mathrm{\Gamma }}^{\mathrm{semi}}$ for ${\mathrm{\Omega }}_{bc}^0$ are subject to large cancellation between dimension-6 and -7 operators. This implies that the subleading corrections are too large to justify the validity of the HQE. Demanding that ${\mathrm{\Gamma }}_{\mathrm{int}+}^{cs}({\mathrm{\Omega }}_{bc}^0)$, ${\mathrm{\Gamma }}_{\mathrm{int}}^{\mathrm{SL},cs}({\mathrm{\Omega }}_{bc}^0)$ be positive and ${\mathrm{\Gamma }}_{\mathrm{int}-}^{cu}({\mathrm{\Xi }}_{bc}^{+})$ be negative, we conjecture that $1.68\times {10}^{-13}s<\tau ({\mathrm{\Omega }}_{bc}^0)<3.70\times {10}^{-13}\mathrm{s}$, $4.09\times {10}^{-13}s<\tau ({\mathrm{\Xi }}_{bc}^{+})<6.07\times {10}^{-13}\mathrm{s}$, and $0.93\times {10}^{-13}s<\tau ({\mathrm{\Xi }}_{bc}^0)<1.18\times {10}^{-13}\mathrm{s}$. Hence, the lifetime hierarchy of ${\mathcal{B}}_{bc}$ baryons is expected to be $\tau ({\mathrm{\Xi }}_{bc}^{+})>\tau ({\mathrm{\Omega }}_{bc}^0)>\tau ({\mathrm{\Xi }}_{bc}^0)$.

Figure 1

Spectator effects in the nonleptonic decays of the doubly bottom baryons: (a) $W$-exchange in ${\mathrm{\Xi }}_{bb}^0$ decay and destructive Pauli interference in (b) ${\mathrm{\Xi }}_{bb}^{-}$ and (c) ${\mathrm{\Omega }}_{bb}^{-}$ decays.

Figure 2

Spectator effects in the nonleptonic weak decays of the charm-bottom baryons: (a) ${\mathrm{\Xi }}_{bc}^{+}$, (b) ${\mathrm{\Xi }}_{bc}^0$, and (c) ${\mathrm{\Omega }}_{bc}^0$. In (a) and (b) there are two $W$-exchange diagrams, one constructive Pauli interference and one destructive Pauli interference. In (c) there are one $W$-exchange, two constructive Pauli interferences and one destructive Pauli interference.