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