Penguin contribution to the width difference and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>C</mml:mi><mml:mi>P</mml:mi></mml:math> asymmetry in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>B</mml:mi><mml:mi>q</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mover accent="true"><mml:mi>B</mml:mi><mml:mo stretchy="false">¯</mml:mo></mml:mover><mml:mi>q</mml:mi></mml:msub></mml:math> mixing at order <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msubsup><mml:mi>α</mml:mi><mml:mi>s</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:msub><mml:mi>N</mml:mi><mml:mi>f</mml:mi></mml:msub></mml:math>

Abstract

We present new contributions to the decay matrix element ${\mathrm{\ensuremath{\Gamma}}}_{12}^{q}$ of the ${B}_{q}\ensuremath{-}{\overline{B}}_{q}$ mixing complex, where $q=d$ or $s$. Our new results constitute the order ${\ensuremath{\alpha}}_{s}^{2}{N}_{f}$ corrections to the penguin contributions to the Wilson coefficients entering ${\mathrm{\ensuremath{\Gamma}}}_{12}^{q}$ with full dependence on the charm quark mass. This is the first step toward the prediction of the $CP$ asymmetry ${a}_{\mathrm{fs}}^{q}$ quantifying $CP$ violation in mixing at next-to-next-to-leading logarithmic order (NNLO) in quantum chromodynamics (QCD) and further improves the prediction of the width difference ${\mathrm{\ensuremath{\Delta}}\mathrm{\ensuremath{\Gamma}}}_{q}$ between the two neutral-meson eigenstates. We find a sizable effect from the nonzero charm mass and our partial NNLO result decreases the NLO penguin corrections to ${a}_{\mathrm{fs}}^{q}$ by 37% and those to ${\mathrm{\ensuremath{\Delta}}\mathrm{\ensuremath{\Gamma}}}_{q}$ by 16%. We further update the Standard-Model NLO predictions for ${a}_{\mathrm{fs}}^{q}$ and the ratio of the width and mass differences of the ${B}_{q}$ eigenstates: If we express the results in terms of the pole mass of the bottom quark, we find ${a}_{\mathrm{fs}}^{s}=(2.07\ifmmode\pm\else\textpm\fi{}0.10)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$, ${a}_{\mathrm{fs}}^{d}=(\ensuremath{-}4.71\ifmmode\pm\else\textpm\fi{}0.24)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$, $\mathrm{\ensuremath{\Delta}}{\mathrm{\ensuremath{\Gamma}}}_{s}/\mathrm{\ensuremath{\Delta}}{M}_{s}=\phantom{\rule{0ex}{0ex}}(4.33\ifmmode\pm\else\textpm\fi{}1.26)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$, and $\mathrm{\ensuremath{\Delta}}{\mathrm{\ensuremath{\Gamma}}}_{d}/\mathrm{\ensuremath{\Delta}}{M}_{d}=(4.48\ifmmode\pm\else\textpm\fi{}1.19)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$. In the $\overline{\mathrm{MS}}$ scheme these numbers read ${a}_{\mathrm{fs}}^{s}=(2.04\ifmmode\pm\else\textpm\fi{}0.11)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$, ${a}_{\mathrm{fs}}^{d}=(\ensuremath{-}4.64\ifmmode\pm\else\textpm\fi{}0.25)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$, $\mathrm{\ensuremath{\Delta}}{\mathrm{\ensuremath{\Gamma}}}_{s}/\mathrm{\ensuremath{\Delta}}{M}_{s}=(4.97\ifmmode\pm\else\textpm\fi{}1.02)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$, and $\mathrm{\ensuremath{\Delta}}{\mathrm{\ensuremath{\Gamma}}}_{d}/\mathrm{\ensuremath{\Delta}}{M}_{d}=(5.07\ifmmode\pm\else\textpm\fi{}0.96)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$.

Topics & Concepts