Two-loop QCD penguin contribution to the width difference in B$_{s}$ − $\overline{B}$$_{s}$ mixing

We consider two-loop QCD corrections to the element ${\Gamma}_{12}^q$ of the decay matrix in B$_{q}$ − ${\overline{B}}_q$ mixing, q = d, s, in the leading power of the Heavy Quark Expansion. The calculated contributions involve one current-current and one penguin operator and constitute the next step towards a theory prediction for the width difference ∆Γs matching the precise experimental data. We present compact analytic results for all matching coefficients in an expansion in m$_{c}$/m$_{b}$ up to second order. Our new corrections are comparable in size to the current experimental error and slightly increase ∆Γs.