Measurement of the Lifetime Difference of $B_d$ Mesons: Possible and Worthwhile?

We estimate the decay width difference $\mathrm{\Delta }Gamm{a}_d\mathrm{/}{\mathrm{\Gamma }}_d$ in the $B_d$ system including $1\mathrm{/}{m}_b$ contributions and next-to-leading order QCD corrections, and find it to be around 0.3%. We explicitly show that the time measurements of an untagged $B_d$ decaying to a single final state isotropically can only be sensitive to quadratic terms in $\Delta Gamma_d / \Gamma_d$, and hence the use of at least two different final states is desired. We discuss such pairs of candidate decay channels for the final states and explore the feasibility of a $\mathrm{\Delta }Gamm{a}_d\mathrm{/}{\mathrm{\Gamma }}_d$ measurement through them. The measurement of this width difference is essential for an accurate measurement of $\sin (2\beta )$ at the LHC. The nonzero width difference may also be used to identify new physics effects and to resolve a twofold discrete ambiguity in the $B_d$-${\bar{B}}_d$ mixing phase. We also derive an upper bound on the value of $\mathrm{\Delta }Gamm{a}_d\mathrm{/}{\mathrm{\Gamma }}_d$ in the presence of new physics, and point out some differences in the phenomenology of width differences in the $B_s$ and $B_d$ systems.