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

We complete the calculation of the element $Γ_{12}^q$ of the decay matrix in $B_q-\bar{B}_q$ mixing, $q=d,s$, to order $α_s$ in the leading power of the Heavy Quark Expansion. To this end we compute one- and two-loop contributions involving two four-quark penguin operators. Furthermore, we present two-loop QCD corrections involving a chromomagnetic operator and either a current-current or four-quark penguin operator. Such contributions are of order $α_s^2$, i.e. next-to-next-to-leading-order. We also present one-loop and two-loop results involving two chromomagnetic operators which are formally 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 $ΔΓ_s/ΔM_s= (5.20\pm 0.69)\cdot 10^{-3}$ if $Γ_{12}^s$ is expressed in terms of the $\overline{\rm MS}$ b-quark mass, while we find $ΔΓ_s/ΔM_s= (4.70\pm 0.96)\cdot 10^{-3}$ instead for the use of the pole mass.

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