Heavy-Meson Bag Parameters using Gradient Flow

Abstract

We demonstrate the use of the gradient flow combined with the short flow-time expansion (GF+SFTX) as a renormalization procedure for four-quark operator matrix elements and associated bag parameters relevant to neutral heavy-meson mixing ($ΔQ=2$) and heavy-meson lifetimes ($ΔQ=0$). Using six RBC/UKQCD 2+1-flavor domain-wall fermion ensembles, we calculate for a charm-strange system with physical quark masses flowed bag parameters and match them to the $\overline{\text{MS}}$ scheme using perturbative SFTX coefficients up to next-to-next-to-leading order in QCD. We employ a multi-scale matching strategy and a renormalization-group improved flow-time evolution which allows for a reliable estimate of systematic uncertainties. For a fictitious neutral $D_s$ meson, we obtain the $ΔQ=2$ $\overline{\text{MS}}$ bag parameter ${\cal B}^{\overline{\text{MS}}}_1(3\,{\rm GeV})=0.7673(123)$, consistent with existing short-distance $D^0$ mixing determinations. For the $ΔQ=0$ lifetime-ratio operator basis, we find the $\overline{\text{MS}}$ results $B^{\overline{\text{MS}}}_1(3\,{\rm GeV})=1.0524(97)$, $B^{\overline{\text{MS}}}_2(3\,{\rm GeV})=0.9621(71)$, $ε^{\overline{\text{MS}}}_1(3\,{\rm GeV})=-0.2275(76)$, and $ε^{\overline{\text{MS}}}_2(3\,{\rm GeV})=-0.0005(8)$. We provide conversion formulae to re-express these results for an arbitrary choice of evanescent operators. These results demonstrate that GF+SFTX can deliver precise determinations of dimension-six four-quark operators and establish a framework for future lattice computations including more complex operator bases, where the challenge of power-divergent mixing is shifted to the continuum and handled in the SFTX.

Figures (18)

• Figure 1: .9SM box diagrams contributing to neutral meson mixing (top). After integrating out the heavy electroweak degrees of freedom, this is represented by a point-like $\Delta Q=2$ four-quark operator (bottom). The diagrams were produced with the help of FeynGameHarlander:2020cyhBundgen:2025utt.
• Figure 2: Imaginary part of a $\Delta Q=0$ box diagram in the .9SM (top left). After integrating out the $W$ bosons, this is described by a double insertion of the resulting $\Delta Q=1$ effective weak Hamiltonian (top right) which can be matched to local $\Delta Q=0$ four-quark operators of the HQE (bottom).
• Figure 3: Eye diagram (left) and contributions from lower-dimensional operators (right). The quark $q'$ forming the "eye" can be any light quark flavor.
• Figure 4: Quark-line diagram to calculate $\Delta Q=2$ operators on the lattice for neutral meson mixing.
• Figure 5: Quark-line diagrams to calculate $\Delta Q=0$ operators on the lattice for valence-quark contributions to lifetimes.