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Bag Parameters for Heavy Meson Lifetimes

Matthew Black, Robert V. Harlander, Jonas T. Kohnen, Fabian Lange, Antonio Rago, Andrea Shindler, Oliver Witzel

Abstract

We calculate the dimension-six $ΔQ=0$ four-quark matrix elements describing heavy-meson lifetime ratios using the gradient flow with its short flow-time expansion as a renormalization procedure. On six RBC/UKQCD 2+1-flavor domain-wall fermion ensembles, we determine flowed bag parameters for physical charm and strange quarks and match to the $\overline{\text{MS}}$ scheme with perturbative short flow-time expansion coefficients through next-to-next-to-leading order (NNLO). A multi-scale matching procedure using renormalization-group running improves the extrapolation to zero flow time. For the operators relevant to $τ(D_s)/τ(D^0)$ at the SU(3)$_{\rm F}$ symmetric point, we obtain $B_1^{\overline{\text{MS}}}(3\,{\rm GeV})=1.0524(97)$,$B_2^{\overline{\text{MS}}}(3\,{\rm GeV})=0.9621(70)$, $ε_1^{\overline{\text{MS}}}(3\,{\rm GeV})=-0.2275(76)$, and $ε_2^{\overline{\text{MS}}}(3\,{\rm GeV})=-0.0005(8)$ using a specific choice of evanescent operators. This is the first lattice-QCD determination of $ΔQ=0$ four-quark operators with a full error budget. It opens the path towards higher-precision predictions of heavy-meson lifetimes and similar quantities exhibiting operator mixing under renormalization.

Bag Parameters for Heavy Meson Lifetimes

Abstract

We calculate the dimension-six four-quark matrix elements describing heavy-meson lifetime ratios using the gradient flow with its short flow-time expansion as a renormalization procedure. On six RBC/UKQCD 2+1-flavor domain-wall fermion ensembles, we determine flowed bag parameters for physical charm and strange quarks and match to the scheme with perturbative short flow-time expansion coefficients through next-to-next-to-leading order (NNLO). A multi-scale matching procedure using renormalization-group running improves the extrapolation to zero flow time. For the operators relevant to at the SU(3) symmetric point, we obtain ,, , and using a specific choice of evanescent operators. This is the first lattice-QCD determination of four-quark operators with a full error budget. It opens the path towards higher-precision predictions of heavy-meson lifetimes and similar quantities exhibiting operator mixing under renormalization.

Paper Structure

This paper contains 10 sections, 13 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction ---
  2. Operators and bag parameters ---
  3. Lattice calculation ---
  4. Gradient flow ---
  5. Numerical Analysis ---
  6. Zero-flow-time limit ---
  7. Results ---
  8. Conclusion ---
  9. Acknowledgments ---
  10. Data availability ---

Figures (3)

  • Figure 1: (a) diagram for the four-quark operator contribution to heavy-meson decay in ; (b) eye diagram (not considered in this paper). The diagrams were produced with the help of FeynGameHarlander:2020cyhBundgen:2025utt.
  • Figure 2: Quark-line diagram for the lattice calculation of $\Delta Q=0$ operators.
  • Figure 3: Zero-flow-time extrapolation of the $\Delta Q=0$ bag parameters -- $B_1$ (top left), $B_2$ (top right), $\epsilon_1$ (bottom left), $\epsilon_2$ (bottom right). The matching to $\overline{{\scalefont{.9}\text{MS}}}$ is performed at both NLO and NNLO using $\mu\in\{2.5,3.0,3.5,4.0,4.5\}\,\text{Ge\spaceV}$ before running back to $\mu_0=3\,\text{Ge\spaceV}$ and uses the flow-time equation to set flow-time logarithms to zero and improve the extrapolations. The pink and orange shaded areas indicate the spread of all fits with $p$-value $> 0.05$ at NLO and NNLO, resulting in the pink and brown data points in the left panel respectively. The gray data points estimate systematic effects due to discarding our coarsest ensembles; the black data points show our final result with all errors added in quadrature. The dashed lines indicate = $\tau_{\rm min}=0.08\,\text{Ge\spaceV}^{-2}$, whereas dash-dotted lines mark the maximum flow time included in our fit results.