Heavy-meson bag parameters using gradient flow

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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 ( Δ Q = 2 ) and heavy-meson lifetimes ( Δ Q = 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 MS ¯ scheme using perturbative SFTX coefficients up to next-to-next-to-leading order in QCD. We employ a multiscale 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 Δ Q = 2 MS ¯ bag parameter B 1 MS ¯ ( 3 GeV ) = 0.7673 ( 123 ) , consistent with existing short-distance D 0 mixing determinations. For the Δ Q = 0 lifetime-ratio operator basis, we find the MS ¯ results B 1 MS ¯ ( 3 GeV ) = 1.0524 ( 97 ) , B 2 MS ¯ ( 3 GeV ) = 0.9621 ( 71 ) , ε 1 MS ¯ ( 3 GeV ) = − 0.2275 ( 76 ) , and ε 2 MS ¯ ( 3 GeV ) = − 0.0005 ( 8 ) . We provide conversion formulae to reexpress 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.
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Funding Information
  • Deutsche Forschungsgemeinschaft (396021762)
  • Deutsche Forschungsgemeinschaft (513989149)
  • Science and Technology Facilities Council (ST/X000494/1)
  • Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung (TMSGI2_211209)
  • National Science Foundation (PHY-2209185)
  • Nuclear Physics (DE-SC0023663)
  • Germany’s Excellence Strategy (533766364)

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