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We complete the calculation of the element $\Gamma_{12}^q$ of the decaymatrix in $B_q-\bar{B}_q$ mixing, $q=d,s$, to order $\alpha_s$ in the leadingpower of the Heavy Quark Expansion. To this end we compute one- and two-loopcontributions involving two four-quark penguin operators. Furthermore, wepresent two-loop QCD corrections involving a chromomagnetic operator and eithera current-current or four-quark penguin operator. Such contributions are oforder $\alpha_s^2$, i.e. next-to-next-to-leading-order. We also presentone-loop and two-loop results involving two chromomagnetic operators which areformally of next-to-next-to-leading and next-to-next-to-next-to-leading-order,respectively. With our new corrections we obtain the Standard-Model prediction$\Delta\Gamma_s/\Delta M_s= (5.20\pm 0.69)\cdot 10^{-3}$ if $\Gamma_{12}^s$ isexpressed in terms of the $\overline{\rm MS}$ b-quark mass, while we find$\Delta \Gamma_s/\Delta M_s= (4.70\pm 0.96)\cdot 10^{-3}$ instead for the useof the pole mass.

The width difference in \bbm\ at order $\alpha_s$ and beyond