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

Abstract

We present new contributions to the decay matrix element ${\Gamma }_{12}^q$ 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 ${\Gamma }_{12}^q$ with full dependence on the charm quark mass. This is the first step towards the prediction of the CP asymmetry $a_{\mathrm{fs}}^q$ 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 $\Delta {\Gamma }_q$ 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 $a_{\mathrm{fs}}^q$ by 37\% and those to $\Delta {\Gamma }_q$ 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 $a_{\mathrm{fs}}^s=(2.07\pm 0.10)\cdot 10^{-5}$, $a_{\mathrm{fs}}^d=(-4.71\pm 0.24)\cdot 10^{-4}$, $\Delta {\Gamma }_s/\Delta M_s=(4.33\pm 1.26)\cdot 10^{-3}$, and $\Delta {\Gamma }_d/\Delta M_d=(4.48\pm 1.19)\cdot 10^{-3}$. In the $\overline{\mathrm{MS}}$ scheme these numbers read $a_{\mathrm{fs}}^s=(2.04\pm 0.11)\cdot 10^{-5}$, $a_{\mathrm{fs}}^d=(-4.64\pm 0.25)\cdot 10^{-4}$, $\Delta {\Gamma }_s/\Delta M_s=(4.97\pm 1.02)\cdot 10^{-3}$, and $\Delta {\Gamma }_d/\Delta M_d=(5.07\pm 0.96)\cdot 10^{-3}$.

11 pages, 1 figure

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Cited by in corpus (1)

• Mixing and $CP$ violation in the charm system