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Theoretical update of Bs-Bs-bar mixing

Alexander Lenz, Ulrich Nierste

TL;DR

This work refines the theory of $B_s$–$\overline{B}_s$ mixing by deriving a more accurate $Γ_{12}^s$ through a redesigned operator basis and by summing charm-mass logarithms to all orders. The resulting predictions for $ΔM_s$, $ΔΓ_s$, and $a_{fs}^s$ exhibit substantially reduced hadronic and perturbative uncertainties, enabling tighter constraints on new physics via observables such as $ΔM_s$, $ΔΓ_s$, $a_{fs}^s$, and CP-violating asymmetries like $A_{CP}^{\text{mix}}(B_s\to J/ψφ)$. The paper reports a SM prediction of $ΔΓ_s/ΔM_s=(49.7±9.4)×10^{-4}$ and a refined $ΔΓ_s$ expression with bag parameters, while finding a 2σ deviation in the $B_s$ mixing phase when applying DØ data. It also updates the $B_d$ system and outlines a roadmap—combining experimental measurements with lattice inputs—to robustly constrain or uncover new short-distance physics in $B_s$ mixing.

Abstract

We update the theory predictions for the mass difference $\dm_s$, the width difference $\dg_s$ and the CP asymmetry in flavour-specific decays, $a_{\rm fs}^{s}$, for the \bbs system. In particular we present a new expression for the element $Γ_{12}^s$ of the decay matrix, which enters the predictions of $\dg_s$ and $a_{\rm fs}^{s}$. To this end we introduce a new operator basis, which reduces the troublesome sizes of the $1/m_b$ and $α_s$ corrections and diminishes the hadronic uncertainty in $\dg_s/\dm_s$ considerably. Logarithms of the charm quark mass are summed to all orders. We find $\dg_s/\dm_s= (49.7 \pm 9.4) \cdot 10^{-4}$ and $\dg_s =(f_{B_s}/240 {\rm MeV})^2 [(0.105 \pm 0.016) B + (0.024 \pm 0.004) \tilde{B}_S' - 0.027 \pm 0.015] {ps}^{-1}$ in terms of the bag parameters $B$, $\tilde{B}_S'$ in the NDR scheme and the decay constant $f_{B_s}$. The improved result for $Γ_{12}^s$ also permits the extraction of the CP-violating \bbms phase from $a_{\rm fs}^{s}$ with better accuracy. We show how the measurements of $ΔM_s$, $ΔΓ_s$, $a_{\rm fs}^{s}$, $A_{\rm CP}^{\rm mix}(B_s\to J/ψφ)$ and other observables can be efficiently combined to constrain new physics. Applying our new formulae to data from the DØexperiment, we find a 2$σ$ deviation of the \bbms phase from its Standard Model value. We also briefly update the theory predictions for the \bbd system and find $\dg_d/\dm_d = \lt(52.6 \epm{11.5}{12.8} \rt) \cdot 10^{-4}$ and $a_{\rm fs}^d = \lt(-4.8\epm{1.0}{1.2} \rt) \cdot 10^{-4}$ in the Standard Model.

Theoretical update of Bs-Bs-bar mixing

TL;DR

This work refines the theory of – mixing by deriving a more accurate through a redesigned operator basis and by summing charm-mass logarithms to all orders. The resulting predictions for , , and exhibit substantially reduced hadronic and perturbative uncertainties, enabling tighter constraints on new physics via observables such as , , , and CP-violating asymmetries like . The paper reports a SM prediction of and a refined expression with bag parameters, while finding a 2σ deviation in the mixing phase when applying DØ data. It also updates the system and outlines a roadmap—combining experimental measurements with lattice inputs—to robustly constrain or uncover new short-distance physics in mixing.

Abstract

We update the theory predictions for the mass difference , the width difference and the CP asymmetry in flavour-specific decays, , for the \bbs system. In particular we present a new expression for the element of the decay matrix, which enters the predictions of and . To this end we introduce a new operator basis, which reduces the troublesome sizes of the and corrections and diminishes the hadronic uncertainty in considerably. Logarithms of the charm quark mass are summed to all orders. We find and in terms of the bag parameters , in the NDR scheme and the decay constant . The improved result for also permits the extraction of the CP-violating \bbms phase from with better accuracy. We show how the measurements of , , , and other observables can be efficiently combined to constrain new physics. Applying our new formulae to data from the DØexperiment, we find a 2 deviation of the \bbms phase from its Standard Model value. We also briefly update the theory predictions for the \bbd system and find and in the Standard Model.

Paper Structure

This paper contains 16 sections, 84 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Improved prediction of $\Gamma_{12}^s$
  3. New operator basis
  4. A closer look at $\mathbf{1/m_b}$ corrections
  5. Summing terms of order $\alpha_s^n z \ln^n z$
  6. Numerical predictions
  7. Input
  8. $\mathbf{\Delta M_s}$ within the SM
  9. $\Delta \Gamma_s$, $\Delta \Gamma_s/\Delta M_s$ and $a_{\rm fs}^s$ within the SM
  10. $\Delta M_d$, $\Delta \Gamma_d$ and $a_{\rm fs}^d$ within the SM
  11. Constraining new physics with $B_s\!-\!\,\overline{\!B}{}_s\,$ mixing
  12. $\Delta \Gamma_s$, $\Delta \Gamma_s/\Delta M_s$ and $a_{\rm fs}^s$ beyond the SM
  13. Basic observables
  14. Current experimental constraints on $\Delta_s$
  15. A road map for $B_s\!-\!\,\overline{\!B}{}_s\,$ mixing
  16. ...and 1 more sections

Figures (7)

  • Figure 1: In the lowest order $M_{12}^s$ is calculated from the dispersive parts of the box diagrams on the left. It is dominated by the top contribution. The result involves only one local $|\Delta B|=2$ operator, shown in the right picture. The leading contribution to $\Gamma_{12}^s$ is obtained from the absorptive parts of the box diagrams on the left, to which only diagrams without top quark line contribute. To lowest order in the heavy quark expansion two $|\Delta B|=2$ operators occur, the $\overline{\Lambda}/m_b$ corrections involve five more.
  • Figure 2: Leading-order CKM-favoured contribution to $\Gamma_{12}^s$, which arises from $\raisebox{7.7pt}{$(\space)$} \space\!\,\overline{\!B}_{s}$ decays to final states (indicated by the dashed lines) with a $(c,\overline{c})$ pair and zero strangeness. The crosses denote any of the operators $Q_{1-6}$ of the $|\Delta B| =1$ hamiltonian. The Cabibbo-suppressed contributions correspond to diagrams with one or both $c$ quarks replaced by $u$ quarks.
  • Figure 3: Uncertainty budget for the theory prediction of $\Delta \Gamma_s$. The largest uncertainties stem from $f_{B_s}$, the renormalisation scale $\mu_1$ of the $\Delta B=1$ operators and the bag parameter of the $1/m_b$--suppressed operator $\widetilde{R}_2$. The transparent segment of the right pie chart shows the improvement with respect to the old result on the left.
  • Figure 4: Uncertainty budget for $\Delta \Gamma_s/\Delta M_s$. See Fig. \ref{['kuchendg']} for explanations. The ratio $\Delta \Gamma_s/\Delta M_s$ does not depend on $f_{B_s}$ and the progress due to the new operator basis is more substantial than in $\Delta \Gamma_s$.
  • Figure 5: $a_{\rm fs}^s$ as a function of the new phase $\phi^\Delta_s$ from Eq. (\ref{['boundafs']}) for the range $-\pi \leq \phi^\Delta_s \leq \pi$. The thick blue lines show the prediction in the new basis, while thin red lines correspond to the old operator basis. The solid lines display the central values of our predictions and the dashed lines show the uncertainties, which are much larger for the old result. The standard model value $a_{\rm fs}^s (\phi^\Delta_s=0) = 2.1 \cdot 10^{-5}$ is too close to zero to be visible in the plot.
  • ...and 2 more figures