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Predicting the $B_s^0-\bar{B}_s^0$ width difference $\Delta\Gamma_s$ relieson the heavy quark expansion and on hadronic matrix elements of $\Delta B=2$operators. We present the first lattice QCD results for matrix elements of thedimension-7 operators $R_{2,3}$ and linear combinations $\tilde{R}_{2,3}$ usingnonrelativistic QCD for the bottom quark and a highly improved staggered quark(HISQ) action for the strange quark. Computations use MILC ensembles of gaugefield configuations with $2+1+1$ flavors of sea quarks with the HISQdiscretization, including lattices with physically light up/down quark masses.We discuss features unique to calculating matrix elements of these operatorsand analyze uncertainties from series truncation, discretization, and quarkmass dependence. Finally we report the first Standard Model determination of$\Delta\Gamma_s$ using lattice QCD results for all hadronic matrix elementsthrough $\mathcal{O}(1/m_b)$. The main result of our calculations yields the$1/m_b$ contribution $\Delta \Gamma_{1/m_b} = -0.022(10)~\mathrm{ps}^{-1}$.Adding this to the leading order contribution, the Standard Model prediction is$\Delta \Gamma_s = 0.092(14)~\mathrm{ps}^{-1}$.

Lattice QCD matrix elements for the ${B_s^0-\bar{B}_s^0}$ width difference beyond leading order