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

Understand

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 $a_{\rm fs}^q$ 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 $\Delta\Gamma_q$ 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 $a_{\rm fs}^q$ by 37\% and those to $\Delta\Gamma_q$ by 16\%.