The width difference in the $B-\bar{B}$ system at next-to-next-to-leading order of QCD

TL;DR

This paper provides a precise theoretical prediction for the width difference in neutral B meson mixing at NNLO in QCD, reducing uncertainties and clarifying the ratio's independence from certain CKM matrix element uncertainties.

Contribution

The authors extend the calculation of the B meson width difference to NNLO in QCD, including three-loop diagrams and detailed operator renormalization, improving the precision of theoretical predictions.

Findings

Predicted ratio ΔΓ_s/ΔM_s ≈ 4.2-4.3 x 10^{-3} with uncertainties.

The ratio is unaffected by the |V_{cb}| uncertainty.

Enhanced theoretical framework for B meson mixing at NNLO.

Abstract

We extend the theoretical prediction for the width difference $\Delta \Gamma_q$ in the mixing of neutral $B$ mesons in the Standard Model to next-to-next-to-leading order in $\alpha_s$. To this aim we calculate three-loop diagrams with two $|\Delta B|=1$ current-current operators analytically. In the matching between $|\Delta B|=1$ and $|\Delta B|=2$ effective theories we regularize the infrared divergences dimensionally and take into account all relevant evanescent operators. Further elements of the calculation are the two-loop renormalization matrix $Z_{ij}$ for the $|\Delta B|=2$ operators and the $\mathcal{O}(\alpha_s^2)$ corrections to the finite renormalization that ensures the $1/m_b$ suppression of the operator $R_0$ at two-loop order. Our theoretical prediction reads $\Delta\Gamma_s/\Delta M_s = {(4.33\pm 0.93)\cdot 10^{-3}}$ if expressed in terms of the bottom mass in the $\overline{\rm MS}$ scheme and $\Delta\Gamma_s/\Delta M_s = {(4.20\pm 0.95)\cdot 10^{-3}}$ for the use of the potential-subtracted mass. While the controversy on $|V_{cb}|$ affects both $\Delta\Gamma_s$ and $\Delta M_s$, the ratio $\Delta\Gamma_s/\Delta M_s$ is not affected by the uncertainty in $|V_{cb}|$ .