Width Difference in the B − $\overline{B}$ System at Next-to-Next-to-Leading Order of QCD

We extend the theoretical prediction for the width difference ΔΓ$_{q}$ in the mixing of neutral B mesons in the Standard Model to next-to-next-to-leading order in α$_{s}$. To this aim, we calculate three-loop diagrams with two |ΔB|=1 current-current operators analytically. In the matching between |ΔB|=1 and |ΔB|=2 effective theories, we regularize the infrared divergences dimensionally and take into account all relevant evanescent operators. Further elements of the calculation are the two-loop renormalization matrix Z$_{ij}$ for the |ΔB|=2 operators and the O(α$^{2}$$_{s}$) corrections to the finite renormalization that ensures the 1/m$_{b}$ suppression of the operator R$_{0}$ at two-loop order. Our theoretical prediction reads ΔΓ$_{s}$/ΔM$_{s}$=(4.33±0.93)×10$^{-3}$ if expressed in terms of the bottom mass in the $\overline{MS}$ scheme and ΔΓ$_{s}$/ΔM$_{s}$=(4.20±0.95)×10$^{-3}$ for the use of the potential-subtracted mass. While the controversy on |V$_{cb}$| affects both ΔΓ$_{s}$ and ΔM$_{s}$, the ratio ΔΓ$_{s}$/ΔM$_{s}$ is not affected by the uncertainty in |V$_{cb}$|.