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

Abstract

We extend the theoretical prediction for the width difference ∆Γq in the mixing of neutral B mesons in the Standard Model to next-to-next-to-leading order in αs. To this aim we calculate threeloop diagrams with two |∆B| = 1 current-current operators analytically. In the matching between |∆B| = 1 and |∆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 Zij for the |∆B| = 2 operators and the O(α 2 s ) corrections to the finite renormalization that ensures the 1/m b suppression of the operator R0 at two-loop order. Our theoretical prediction reads ∆Γs/∆Ms = (4.33 ± 0.93) • 10 −3 if expressed in terms of the bottom mass in the MS scheme and ∆Γs/∆Ms = (4.20 ± 0.95) • 10 −3 for the use of the potential-subtracted mass. While the controversy on |V cb | affects both ∆Γs and ∆Ms, the ratio ∆Γs/∆Ms is not affected by the uncertainty in |V cb | .

Summary of Working Group 4: Mixing and mixing-related $CP$ violation in the $B$ system: $Δm$, $ΔΓ$, $ϕ_s$, $ϕ_{1}/β$, $ϕ_{2}/α$, $ϕ_{3}/γ$

“…Furthermore, the two-loop double chromomagnetic penguin insertion 𝑄 8 × 𝑄 8 is formally of N 3 LO. The analytic two-loop results can be found in [18,19], while the three-loop calculation has just appeared [20].…”

$$\tau _{B_{s}}/\tau _{B_{d}}$$ and $$\Delta \Gamma _{s}$$ confront new physics in $$b\rightarrow s\tau \tau $$

“…Using the current SM prediction for s [12], and the HFLAV average [1] we obtain for the ratio of the decay width difference…”