Width Difference in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>B</mml:mi><mml:mo>−</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">¯</mml:mo></mml:mrow></mml:mover></mml:mrow></mml:math> System at Next-to-Next-to-Leading Order of QCD
Marvin Gerlach, Ulrich Nierste, Vladyslav Shtabovenko, Matthias Steinhauser
Abstract
We extend the theoretical prediction for the width difference $\mathrm{\ensuremath{\Delta}}{\mathrm{\ensuremath{\Gamma}}}_{q}$ in the mixing of neutral $B$ mesons in the Standard Model to next-to-next-to-leading order in ${\ensuremath{\alpha}}_{s}$. To this aim, we calculate three-loop diagrams with two $|\mathrm{\ensuremath{\Delta}}B|=1$ current-current operators analytically. In the matching between $|\mathrm{\ensuremath{\Delta}}B|=1$ and $|\mathrm{\ensuremath{\Delta}}B|=2$ 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 $|\mathrm{\ensuremath{\Delta}}B|=2$ operators and the $\mathcal{O}({\ensuremath{\alpha}}_{s}^{2})$ 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 $\mathrm{\ensuremath{\Delta}}{\mathrm{\ensuremath{\Gamma}}}_{s}/\mathrm{\ensuremath{\Delta}}{M}_{s}=(4.33\ifmmode\pm\else\textpm\fi{}0.93)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ if expressed in terms of the bottom mass in the $\overline{\mathrm{MS}}$ scheme and $\mathrm{\ensuremath{\Delta}}{\mathrm{\ensuremath{\Gamma}}}_{s}/\mathrm{\ensuremath{\Delta}}{M}_{s}=(4.20\ifmmode\pm\else\textpm\fi{}0.95)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ for the use of the potential-subtracted mass. While the controversy on $|{V}_{cb}|$ affects both $\mathrm{\ensuremath{\Delta}}{\mathrm{\ensuremath{\Gamma}}}_{s}$ and $\mathrm{\ensuremath{\Delta}}{M}_{s}$, the ratio $\mathrm{\ensuremath{\Delta}}{\mathrm{\ensuremath{\Gamma}}}_{s}/\mathrm{\ensuremath{\Delta}}{M}_{s}$ is not affected by the uncertainty in $|{V}_{cb}|$.