The width difference in the $B-\bar{B}$ 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 $\mathcal{O}(α_s^2)$ 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 $ΔΓ_s/ΔM_s = {(4.33\pm 0.93)\cdot 10^{-3}}$ if expressed in terms of the bottom mass in the $\overline{\rm MS}$ scheme and $ΔΓ_s/ΔM_s = {(4.20\pm 0.95)\cdot 10^{-3}}$ 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 $|V_{cb}|$ .

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