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Penguin contribution to width difference and CP asymmetry in $B_q$-$\bar B_q$ mixing at order $α_s^2 N_f$

We present new contributions to the decay matrix element $Γ_{12}^q$ of the $B_q$-$\bar B_q$ mixing complex, where $q=d$ or $s$. Our new results constitute the order $α_s^2 N_f$ corrections to the penguin contributions to the Wilson coefficients entering $Γ_{12}^q$ with full dependence on the charm quark mass. This is the first step towards the prediction of the CP asymmetry $a_{\rm fs}^q$ 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 $ΔΓ_q$ 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 $a_{\rm fs}^q$ by 37\% and those to $ΔΓ_q$ by 16\%. We further update the Standard-Model NLO predictions for $a_{\rm fs}^q$ 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 $a_{\rm fs}^s=(2.07 \pm 0.10)\cdot 10^{-5}$, $a_{\rm fs}^d=(-4.71 \pm 0.24)\cdot 10^{-4}$, $ΔΓ_s/Δ{M}_s = (4.33 \pm 1.26)\cdot 10^{-3}$, and $ΔΓ_d/Δ{M}_d = (4.48 \pm 1.19)\cdot 10^{-3}$. In the $\overline{\rm MS}$ scheme these numbers read $a^s_{\rm fs} =(2.04 \pm 0.11)\cdot 10^{-5}$, $a^d_{\rm fs} = (-4.64 \pm 0.25)\cdot 10^{-4}$, $ΔΓ_s/Δ{M}_s = (4.97 \pm 1.02)\cdot 10^{-3}$, and $ΔΓ_d/Δ{M}_d = (5.07 \pm 0.96)\cdot 10^{-3}$.