Lifetimes of Doubly Heavy Baryons ${\cal B}_{bb}$ and ${\cal B}_{bc}$

Lifetimes of the doubly heavy baryons ${\cal B}_{bb}$ and ${\cal B}_{bc}$ 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 $\tau(\Omega_{bb}^-)\sim \tau(\Xi_{bb}^{-})>\tau(\Xi_{bb}^0)$. The $\Xi_{bb}^{0}$ baryon is shortest-lived owing to the $W$-exchange contribution, while $\Xi_{bb}^{-}$ and $\Omega_{bb}^{-}$ have similar lifetimes as they both receive contributions from destructive Pauli interference. We find the lifetime ratio $\tau(\Xi_{bb}^{-})/\tau(\Xi_{bb}^0)=1.26$\,. The large $W$-exchange contribution to $\Xi_{bc}^0$ through the subprocess $cd\to us\to cd$ and the sizable destructive Pauli interference contribution to $\Xi_{bc}^+$ imply a substantial lifetime difference between $\Xi_{bc}^+$ and $\Xi_{bc}^0$. In the presence of subleading $1/m_c$ and $1/m_b$ corrections to the spectator effects, we find that $\tau(\Omega_{bc}^0)$ becomes longest-lived. This is because $\Gamma^{\rm int}_+$ and $\Gamma^{\rm semi}$ for $\Omega_{bc}^0$ 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 $\Gamma^{cs}_{\rm int+}(\Omega_{bc}^0)$, $\Gamma^{{\rm SL},cs}_{\rm int}(\Omega_{bc}^0)$ be positive and $\Gamma^{cu}_{\rm int-}(\Xi^+_{bc})$ be negative, we conjecture that $1.68\times 10^{-13}s \tau(\Omega_{bc}^0)>\tau(\Xi_{bc}^0)$.