The width difference in the $B-\bar{B}$ system at next-to-next-to-leading order of QCD

Abstract

We extend the theoretical prediction for the width difference $\Delta {\Gamma }_q$ in the mixing of neutral $B$ mesons in the Standard Model to next-to-next-to-leading order in ${\alpha }_s$. To this aim we calculate three-loop diagrams with two $|\Delta B|=1$ current-current operators analytically. In the matching between $|\Delta B|=1$ and $|\Delta B|=2$ 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 $|\Delta B|=2$ operators and the $\mathcal{O}({\alpha }_s^2)$ 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 $\Delta {\Gamma }_s/\Delta M_s=(4.33\pm 0.93)\cdot 10^{-3}$ if expressed in terms of the bottom mass in the $\overline{\mathrm{MS}}$ scheme and $\Delta {\Gamma }_s/\Delta M_s=(4.20\pm 0.95)\cdot 10^{-3}$ for the use of the potential-subtracted mass. While the controversy on $|V_{cb}|$ affects both $\Delta {\Gamma }_s$ and $\Delta M_s$, the ratio $\Delta {\Gamma }_s/\Delta M_s$ is not affected by the uncertainty in $|V_{cb}|$ .

7 pages, 4 figures

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