Width Difference in the $B_s-\bar{B_s}$ System

We use the heavy quark expansion to investigate the width difference $ΔΓ_{B_s}$ between the $B_s$ mass eigenstates. The corrections of ${\cal O}(Λ_{QCD}/m_b)$ and ${\cal O}(m_s/m_b)$ to the leading order expression in the operator product expansion are derived and estimated to yield a sizable reduction of the leading result for $ΔΓ_{B_s}$ by typically $30\%$. For completeness we also quantify small effects due to penguin operators and CKM suppressed contributions. Based on our results we discuss the prediction for $(ΔΓ/Γ)_{B_s}$ with particular emphasis on theoretical uncertainties. We find $(ΔΓ/Γ)_{B_s}=0.16^{+0.11}_{-0.09}$, where the large error is dominated by the uncertainty in hadronic matrix elements. An accuracy of about $10\%$ in $(ΔΓ/Γ)_{B_s}$ should be within reach, assuming continuing progress in lattice calculations. In addition we address phenomenological issues and implications of a $ΔΓ_{B_s}$ measurement for constraints on $ΔM_{B_s}$ and CKM parameters. We further consider in some detail the lifetime ratio $τ(B_s)/τ(B_d)$ and estimate that, most likely, $|τ(B_s)/τ(B_d)-1|<1\%$.

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