Current-current operator contribution to the decay matrix in $B$-meson mixing at next-to-next-to-leading order of QCD

Abstract

We compute next-to-next-to-leading order perturbative corrections to the decay width difference of mass eigenstates and the charge-parity asymmetry $a_{\mathrm{fs}}$ in flavour-specific decays of neutral $B$ mesons. In our calculation we take into account the full dependence on the charm and bottom quark masses for the current-current operator contributions up to three-loop order. Special emphasis is put on the proper construction of the so-called $|\Delta B|=2$ theory such that Fierz symmetry is preserved. We provide updated phenomenological predictions, for $\Delta \Gamma$, $\Delta \Gamma /\Delta M$ and $a_{\mathrm{fs}}$ for the $B_d$ and $B_s$ system, including a detailed analysis of the uncertainties of our predictions. The calculated NNLO correction reduce the perturbative uncertainty of the leading term of the $1/m_b$ expansion of the width difference ${\text{dg}}_s$ in the $B_s$ system to the level of the current experimental error. The uncertainty of our prediction ${\text{dg}}_s=(0.077\pm 0.016)\text{mbox}{ps}^{-1}$ is dominated by the sub-leading term of this expansion. We further illustrate how better future measurements of ${\text{dg}}_d$ and $a_{\mathrm{fs}}^d$ will help to gain a better understanding of $B_d$-${\bar{B}}_d$ mixing.

Comments: 59 pages