Complete next-to-next-to-leading order QCD corrections to the decay matrix in $\boldsymbol{B}$-meson mixing at leading power

TL;DR

This work delivers the complete next-to-next-to-leading order QCD corrections to the leading-power term of the B_q decay-width difference Γ_{12}^q and the CP asymmetry a_{fs}^q, incorporating both current-current and penguin operators at three loops with a thorough expansion in the charm-to-bottom mass ratio z = [m_c/m_b]^2. Using an automated amplitude-generation pipeline and a deep HQE treatment, the authors provide precise predictions for ΔΓ_s, ΔΓ_d, a_{fs}^s, and a_{fs}^d in the B_s and B_d systems, including robust uncertainty analyses and valuable double ratios that cancel hadronic inputs. They demonstrate that the double ratio (ΔΓ_d/ΔM_d)/(ΔΓ_s/ΔM_s) is predicted with high precision, enabling ΔΓ_d to be inferred from ΔM_d, ΔM_s, and ΔΓ_s, and they explore the resulting constraints on the CKM unitarity triangle from mixing observables alone. Additionally, the paper supplies ready-to-use formulae and coefficient tables to accommodate updated lattice inputs, highlighting the ongoing impact of these precise SM predictions on searches for new physics in flavor.

Abstract

We compute next-to-next-to-leading order corrections to the decay width difference of mass eigenstates and the charge-parity (CP) asymmetry $a_{\rm fs}$ in flavour-specific decays of neutral $B$ mesons. We include both current-current and penguin operators at three-loop order. All input integrals in the transition amplitude are reduced to a small set of master integrals which depend on the ratio of the charm and bottom quark masses. The latter are computed using semi-analytic methods which provide deep expansions around properly selected values of $m_c/m_b$. We provide numerical results for $ΔΓ$ and $ΔΓ/ΔM$, both for the $B_d$ and $B_s$ system, including a detailed uncertainty analysis. Using the experimental value for the mass difference $ΔM_s$ we predict $ΔΓ_s=(0.078 \pm 0.015) ~\mbox{ps}^{-1}$. For the CP asymmetries we find $ a_{\rm fs}^s = (2.27 \pm 0.13 ) \times 10^{-5}$ and $a_{\rm fs}^d = -(5.19 \pm 0.30)~\times~10^{-4}$. Furthermore, we show that the ratios $(ΔΓ_s/ΔM_s) / (ΔΓ_d/ΔM_d)$ and $ΔΓ_d / ΔΓ_s$ can be predicted with high precision. The former quantity permits the prediction $ΔΓ_d=(0.00215\pm 0.00013)~\mbox{ps}^{-1}$ from the measurements of $ΔM_{d,s}$ and $ΔΓ_s$. We further discuss the impact of $ΔΓ_d/ΔΓ_s$ on the CKM unitarity triangle and present ready-to-use formulae which permit improved predictions once updated results for the operator matrix elements are available.

Figures (11)

• Figure 1: Leading-order diagrams contributing to $\Gamma_{12}^q$ with two current-current operators (left) and one current-current and one penguin operator (right). $\Gamma_{12}^q$ originates from decays into final states into which both $B_q$ and $\bar{B}_q$ can decay, indicated by the dashed line in the left figure.
• Figure 2: CKM unitarity triangle with constraints from $B\!-\!\bar{B}\,$ mixing observables.
• Figure 3: Two-loop Feynman diagrams of the $\Delta B = 2$ transition operator. The blue dot stands for the insertion of one of the operators $Q$, $\widetilde{Q}_S$ or $R_0$.
• Figure 4: Three-loop Feynman diagrams with two $\Delta B = 1$ insertions shown in orange.
• Figure 5: Renormalisation scale dependence at LO (short dashes), NLO (long dashes) and NNLO (solid) for $\Delta\Gamma_s/\Delta M_s$. The plot shows the scale variation of the leading-power terms where the scales $\mu_1=\mu_b=\mu_c$ are varied together. The grey band shows the experimental value of Eq. (\ref{['eq:dgsexp']}).