Width difference in the $B_s$-${\bar{B}}_s$ system with lattice NRQCD

We compute the matrix element of the four-quark operator ${\text{Op}}_S$ = $\bar{b}(1-{\gamma }_5)s\bar{b}(1-{\gamma }_5)s$ in the $B_s-{\bar{B}}_s$ mixing on the lattice using NRQCD, which gives a leading contribution in the calculation of the width difference $\mathrm{\Delta }{\mathrm{\Gamma }}_s$ in the $1\mathrm{/}{m}_b$ expansion. Using the next-to-leading formula of Beneke $\mathit{\text{et}}\mathit{\text{al.}}$, we obtain $(\mathrm{\Delta }\mathrm{\Gamma }\mathrm{/}\mathrm{\Gamma }{)}_s$ = 0.151(37)(36)(17), where the the first error reflects the uncertainty of $f_{B_s}$ = 245(30) MeV, the second error comes from our calculation of the matrix element of ${\text{Op}}_S$, and the third represents unknown $1\mathrm{/}{m}_b$ correction.