Penguin contribution to width difference and CP asymmetry in $B_q$-${\bar{B}}_q$ mixing at order ${\alpha }_s^2{N}_f$

We present new contributions to the decay matrix element ${\mathrm{\Gamma }}_{12}^q$ of the $B_q$-${\bar{B}}_q$ mixing complex, where $q=d$ or $s$. Our new results constitute the order ${\alpha }_s^2{N}_f$ corrections to the penguin contributions to the Wilson coefficients entering ${\mathrm{\Gamma }}_{12}^q$ with full dependence on the charm quark mass. This is the first step towards the prediction of the CP asymmetry $a_{\mathrm{f}\mathrm{s}}^q$ 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 $\mathrm{\Delta }{\mathrm{\Gamma }}_q$ between the two neutral-meson eigenstates. We find a sizable effect from the non-zero charm mass and our partial NNLO result decreases the NLO penguin corrections to $a_{\mathrm{f}\mathrm{s}}^q$ by 37% and those to $\mathrm{\Delta }{\mathrm{\Gamma }}_q$ by 16%. We further update the Standard-Model NLO predictions for $a_{\rm fs}^q$andtheratioofthewidthandmassdifferencesofthe$B_q$ eigenstates: If we express the results in terms of the pole mass of the bottom quark we find $a_{\mathrm{f}\mathrm{s}}^s=(2.07\pm 0.10)\cdot {10}^{-5}$, $a_{\rm fs}^d=(-4.71 \pm 0.24)\cdot 10^{-4}$,$\Delta{\Gamma}_s/\Delta{M}_s = (4.33 \pm 1.26)\cdot 10^{-3}$,and$\Delta{\Gamma}d/\Delta{M}d = (4.48 \pm 1.19)\cdot 10^{-3}$. In the $\overline{\mathrm{M}\mathrm{S}}$ scheme these numbers read $a^s{\rm fs} =(2.04 \pm 0.11)\cdot 10^{-5}$,$a^d{\rm fs} = (-4.64 \pm 0.25)\cdot 10^{-4}$,$\Delta{\Gamma}_s/\Delta{M}_s = (4.97 \pm 1.02)\cdot 10^{-3}$,and$\Delta{\Gamma}_d/\Delta{M}_d = (5.07 \pm 0.96)\cdot 10^{-3}$.