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We present lattice QCD results for the matrix elements of $R_2$ and otherdimension-7, $\Delta B = 2$ operators relevant for calculations of $\Delta\Gamma_s$, the $B_s-\bar{B}_s$ width difference. We have computed correlationfunctions using 5 ensembles of the MILC Collaboration's 2+1+1-flavour gaugefield configurations, spanning 3 lattice spacings and light sea quarks massesdown to the physical point. The HISQ action is used for the valence strangequarks, and the NRQCD action is used for the bottom quarks. Once our analysisis complete, the theoretical uncertainty in the Standard Model prediction for$\Delta \Gamma_s$ will be substantially reduced.

Improving the theoretical prediction for the $B_s-\bar{B}_s$ width difference: matrix elements of next-to-leading order $\Delta B=2$ operators