- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Letters

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Bag Parameters for Heavy Meson Lifetimes from a Lattice QCD Calculation at the SU(3)F-symmetric Point

Matthew Black1,2,*, Robert V. Harlander3, Jonas T. Kohnen3, Fabian Lange4,5, Antonio Rago6, Andrea Shindler3,7,8, and Oliver Witzel9

  • *Contact author: matthew.black@ed.ac.uk
PDF

Phys. Rev. Lett. 137, 191905 – Published 5 November, 2026

DOI: https://doi.org/10.1103/2vg9-34y9

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 (SFTX) 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 MS¯ scheme with perturbative SFTX coefficients through next-to-next-to-leading order (NNLO). A multiscale matching procedure using renormalization-group running improves the extrapolation to zero flow time. For the operators relevant to τ(Ds)/τ(D0) at the SU(3)F symmetric point, we obtain B1MS¯(3  GeV)=1.0524(97), B2MS¯(3  GeV)=0.9621(70), ε1MS¯(3  GeV)=−0.2275(76), and ε2MS¯(3  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 toward higher-precision predictions of heavy-meson lifetimes and similar quantities exhibiting operator mixing under renormalization.

View figure in article

Physics Subject Headings (PhySH)

See Also

Heavy-meson bag parameters using gradient flow

Matthew Black, Robert V. Harlander, Jonas T. Kohnen, Fabian Lange, Antonio Rago, Andrea Shindler, and Oliver Witzel
Phys. Rev. D 114, 094503 (2026)

Article Text

References (73)

  1. S. Banerjee et al. (Heavy Flavor Averaging Group (HFLAV), Averages of b-hadron, c-hadron, and τ-lepton properties as of 2023, Phys. Rev. D 113, 012008 (2026).
  2. S. Navas et al. (Particle Data Group), Review of particle physics, Phys. Rev. D 110, 030001 (2024).
  3. J. Albrecht, F. Bernlochner, A. Lenz, and A. Rusov, Lifetimes of b-hadrons and mixing of neutral B-mesons: Theoretical and experimental status, Eur. Phys. J. Special Topics 233, 359 (2024).
  4. M. Kirk, A. Lenz, and T. Rauh, Dimension-six matrix elements for meson mixing and lifetimes from sum rules, J. High Energy Phys. 12 (2017) 068; 06 (2020) 162(E).
  5. M. J. Kirk, Charming New Physics in Beautiful Processes?, Ph.D. thesis, Durham University, IPPP, 2018, 10.1007/978-3-030-19197-9.
  6. D. King, A. Lenz, M. L. Piscopo, T. Rauh, A. V. Rusov, and C. Vlahos, Revisiting inclusive decay widths of charmed mesons, J. High Energy Phys. 08 (2022) 241.
  7. D. King, A. Lenz, and T. Rauh, SU(3) breaking effects in B and D meson lifetimes, J. High Energy Phys. 06 (2022) 134.
  8. D. J. King, Three loop corrections to HQET Sum Rules for the Bag Parameter, Ph.D. thesis, Durham University, 2022, https://etheses.durham.ac.uk/id/eprint/14487/.
  9. M. Black, M. Lang, A. Lenz, and Z. Wüthrich, HQET sum rules for matrix elements of dimension-six four-quark operators for meson lifetimes within and beyond the Standard Model, J. High Energy Phys. 04 (2025) 081.
  10. N. Carrasco et al., D0−D¯0 mixing in the standard model and beyond from Nf=2 twisted mass QCD, Phys. Rev. D 90, 014502 (2014).
  11. N. Carrasco, P. Dimopoulos, R. Frezzotti, V. Lubicz, G. C. Rossi, S. Simula, and C. Tarantino (ETM Collaboration), ΔS=2 and ΔC=2 bag parameters in the standard model and beyond from Nf=2+1+1 twisted-mass lattice QCD, Phys. Rev. D 92, 034516 (2015).
  12. A. Bazavov et al., Short-distance matrix elements for D0-meson mixing for Nf=2+1 lattice QCD, Phys. Rev. D 97, 034513 (2018).
  13. N. Carrasco et al. (ETM Collaboration), B-physics from Nf=2 tmQCD: The Standard Model and beyond, J. High Energy Phys. 03 (2014) 016.
  14. Y. Aoki, T. Ishikawa, T. Izubuchi, C. Lehner, and A. Soni, Neutral B meson mixings and B meson decay constants with static heavy and domain-wall light quarks, Phys. Rev. D 91, 114505 (2015).
  15. E. Gamiz, C. T. H. Davies, G. P. Lepage, J. Shigemitsu, and M. Wingate (HPQCD Collaboration), Neutral B meson mixing in unquenched lattice QCD, Phys. Rev. D 80, 014503 (2009).
  16. A. Bazavov et al. (Fermilab Lattice Collaboration and MILC Collaboration), B(s)0-mixing matrix elements from lattice QCD for the Standard Model and beyond, Phys. Rev. D 93, 113016 (2016).
  17. R. J. Dowdall, C. T. H. Davies, R. R. Horgan, G. P. Lepage, C. J. Monahan, J. Shigemitsu, and M. Wingate, Neutral B-meson mixing from full lattice QCD at the physical point, Phys. Rev. D 100, 094508 (2019).
  18. P. Boyle, F. Erben, A. Jüttner, T. Kaneko, M. Marshall, A. Portelli, O. Witzel, L. Del Debbio, and J. T. Tsang, BSM B−B¯ mixing on JLQCD and RBC/UKQCD Nf=2+1 DWF ensembles, Proc. Sci. LATTICE2021 (2022) 224 [arXiv:2111.11287].
  19. M. Di Pierro and C. T. Sachrajda (UKQCD Collaboration), A Lattice study of spectator effects in inclusive decays of B mesons, Nucl. Phys. B534, 373 (1998).
  20. M. Di Pierro, C. T. Sachrajda, and C. Michael (UKQCD Collaboration), An Exploratory lattice study of spectator effects in inclusive decays of the Λb baryon, Phys. Lett. B 468, 143 (1999); 525, 360(E) (2002).
  21. D. Becirevic, Theoretical progress in describing the B meson lifetimes, Proc. Sci. HEP2001 (2001) 098 [arXiv:hep-ph/0110124].
  22. J. Lin, W. Detmold, and S. Meinel, Lattice study of spectator effects in b-hadron decays, Proc. Sci. LATTICE2022 (2023) 417 [arXiv:2212.09275].
  23. M. Black, R. V. Harlander, J. T. Kohnen, F. Lange, A. Rago, A. Shindler, and O. Witzel, companion paper, Heavy-meson bag parameters using gradient flow, Phys. Rev. D 114, 094503 (2026).
  24. M. Black, R. Harlander, F. Lange, A. Rago, A. Shindler, and O. Witzel, Using gradient flow to renormalise matrix elements for meson mixing and lifetimes, Proc. Sci. LATTICE2023 (2024) 263 [arXiv:2310.18059].
  25. M. Black, Improving Precision for Hadronic Observables in the Standard Model and Beyond, Ph.D. thesis, Universität Siegen, 2024, 10.25819/ubsi/10603
  26. M. Black, R. Harlander, F. Lange, A. Rago, A. Shindler, and O. Witzel, Gradient flow renormalisation for meson mixing and lifetimes, Proc. Sci. LATTICE2024 (2025) 243 [arXiv:2409.18891].
  27. N. N. Nikolaev, Gauge model for weak interaction and supercharged hadrons, Pis’ma Zh. Eksp. Teor. Fiz. 18, 447 (1973), https://inspirehep.net/literature/87709.
  28. M. Neubert, Heavy quark symmetry, Phys. Rep. 245, 259 (1994).
  29. M. L. Piscopo, Higher order corrections to the lifetime of heavy hadrons, Ph.D. thesis, Siegen University, 2021, 10.25819/ubsi/10024.
  30. R. Harlander, S. Klein, and M. Lipp, feyngame, Comput. Phys. Commun. 256, 107465 (2020).
  31. L. Bündgen, R. V. Harlander, S. Y. Klein, and M. C. Schaaf, feyngame 3.0, Comput. Phys. Commun. 314, 109662 (2025).
  32. P. A. Boyle, L. Del Debbio, N. Garron, A. Jüttner, A. Soni, J. T. Tsang, and O. Witzel (RBC/UKQCD Collaboration), SU(3)-breaking ratios for D(s) and B(s) mesons, arXiv:1812.08791.
  33. H. Suzuki, Energy–momentum tensor from the Yang–Mills gradient flow, Prog. Theor. Exp. Phys. 2013, 083B03 (2013); 2015, 079201(E) (2015).
  34. H. Makino and H. Suzuki, Lattice energy–momentum tensor from the Yang–Mills gradient flow—inclusion of fermion fields, Prog. Theor. Exp. Phys. 2014, 063B02 (2014); 2015, 079202(E) (2015).
  35. Y. Taniguchi, S. Ejiri, K. Kanaya, M. Kitazawa, H. Suzuki, and T. Umeda (WHOT-QCD Collaboration), Nf=2+1 QCD thermodynamics with gradient flow using two-loop matching coefficients, Phys. Rev. D 102, 014510 (2020); 102, 059903(E) (2020).
  36. H. Suzuki and H. Takaura, t→0 extrapolation function in the small flow time expansion method for the energy–momentum tensor, Prog. Theor. Exp. Phys. 2021, 073B02 (2021).
  37. M. Black, R. V. Harlander, A. Hasenfratz, A. Rago, and O. Witzel, Renormalized quark masses using gradient flow, Phys. Rev. D 113, 094504 (2026).
  38. A. Francis, P. Fritzsch, R. Karur, J. Kim, G. Pederiva, D. A. Pefkou, A. Rago, A. Shindler, A. Walker-Loud, and S. Zafeiropoulos, Moments of parton distributions functions of the pion from lattice QCD using gradient flow, Phys. Rev. D 113, 074520 (2026).
  39. A. Francis et al., Gradient flow for parton distribution functions: First application to the pion, Phys. Rev. Lett. 136, 171903 (2026).
  40. R. Narayanan and H. Neuberger, Infinite N phase transitions in continuum Wilson loop operators, J. High Energy Phys. 03 (2006) 064.
  41. M. Lüscher, Trivializing maps, the Wilson flow and the HMC algorithm, Commun. Math. Phys. 293, 899 (2010).
  42. M. Lüscher, Properties and uses of the Wilson flow in lattice QCD, J. High Energy Phys. 08 (2010) 071; 03 (2014) 92.
  43. M. Lüscher, Chiral symmetry and the Yang–Mills gradient flow, J. High Energy Phys. 04 (2013) 123.
  44. M. Lüscher and P. Weisz, Perturbative analysis of the gradient flow in non-abelian gauge theories, J. High Energy Phys. 02 (2011) 051.
  45. M. Lüscher, Future applications of the Yang-Mills gradient flow in lattice QCD, Proc. Sci. LATTICE2013 (2014) 016 [arXiv:1308.5598].
  46. R. V. Harlander and F. Lange, Effective electroweak Hamiltonian in the gradient-flow formalism, Phys. Rev. D 105, L071504 (2022).
  47. J. Borgulat, R. V. Harlander, J. T. Kohnen, and F. Lange, Short-flow-time expansion of quark bilinears through next-to-next-to-leading order QCD, J. High Energy Phys. 05 (2024) 179.
  48. C. Allton et al. (RBC-UKQCD Collaboration), Physical results from 2+1 flavor domain wall QCD and SU(2) chiral perturbation theory, Phys. Rev. D 78, 114509 (2008).
  49. Y. Aoki et al. (RBC Collaboration and UKQCD Collaboration), Continuum limit physics from 2+1 flavor domain wall QCD, Phys. Rev. D 83, 074508 (2011).
  50. T. Blum et al. (RBC Collaboration and UKQCD Collaboration), Domain wall QCD with physical quark masses, Phys. Rev. D 93, 074505 (2016).
  51. C. McNeile and C. Michael (UKQCD Collaboration), Decay width of light quark hybrid meson from the lattice, Phys. Rev. D 73, 074506 (2006).
  52. P. A. Boyle, A. Jüttner, C. Kelly, and R. D. Kenway, Use of stochastic sources for the lattice determination of light quark physics, J. High Energy Phys. 08 (2008) 086.
  53. S. Güsken, U. Löw, K. H. Mütter, R. Sommer, A. Patel, and K. Schilling, Nonsinglet axial vector couplings of the baryon octet in lattice QCD, Phys. Lett. B 227, 266 (1989).
  54. D. B. Kaplan, A method for simulating chiral fermions on the lattice, Phys. Lett. B 288, 342 (1992).
  55. Y. Shamir, Chiral fermions from lattice boundaries, Nucl. Phys. B406, 90 (1993).
  56. V. Furman and Y. Shamir, Axial symmetries in lattice QCD with Kaplan fermions, Nucl. Phys. B439, 54 (1995).
  57. C. Morningstar and M. J. Peardon, Analytic smearing of SU(3) link variables in lattice QCD, Phys. Rev. D 69, 054501 (2004).
  58. R. C. Brower, H. Neff, and K. Orginos, The Möbius domain wall fermion algorithm, Comput. Phys. Commun. 220, 1 (2017).
  59. Y.-G. Cho, S. Hashimoto, A. Jüttner, T. Kaneko, M. Marinkovic, J.-I. Noaki, and J. T. Tsang, Improved lattice fermion action for heavy quarks, J. High Energy Phys. 05 (2015) 072.
  60. P. A. Boyle et al., Grid: A next generation data parallel C++ QCD library, Proc. Sci. LATTICE2015 (2016) 023.
  61. A. Portelli et al., Hadrons, 10.5281/zenodo.4293902.
  62. A. Portelli et al., 10.5281/zenodo.4063666 (2022).
  63. R. V. Harlander, F. Lange, and T. Neumann, Hadronic vacuum polarization using gradient flow, J. High Energy Phys. 08 (2020) 109.
  64. A. J. Buras and P. H. Weisz, QCD nonleading corrections to weak decays in dimensional regularization and ’t Hooft-Veltman schemes, Nucl. Phys. B333, 66 (1990).
  65. M. Ciuchini, E. Franco, V. Lubicz, G. Martinelli, I. Scimemi, and L. Silvestrini, Next-to-leading order QCD corrections to ΔF=2 effective Hamiltonians, Nucl. Phys. B523, 501 (1998).
  66. A. J. Buras, M. Misiak, and J. Urban, Two loop QCD anomalous dimensions of flavor changing four quark operators within and beyond the standard model, Nucl. Phys. B586, 397 (2000); B1002, 116529(E) (2024).
  67. J. Aebischer, P. Morell, M. Pesut, and J. Virto, Two-loop anomalous dimensions in the LEFT: Dimension-six four-fermion operators in NDR, J. High Energy Phys. 05 (2026) 043.
  68. K. G. Chetyrkin, J. H. Kühn, and M. Steinhauser, RunDec: A Mathematica package for running and decoupling of the strong coupling and quark masses, Comput. Phys. Commun. 133, 43 (2000).
  69. F. Herren and M. Steinhauser, Version 3 of RunDec and CRunDec, Comput. Phys. Commun. 224, 333 (2018).
  70. R. V. Harlander, Y. Kluth, and F. Lange, The two-loop energy–momentum tensor within the gradient-flow formalism, Eur. Phys. J. C 78, 944 (2018); 79, 858(E) (2019).
  71. M. Fael, J. Jenkins, E. Lunghi, and Z. Polonsky, The Standard Model prediction for the rare decay B→Xsνν¯, J. High Energy Phys. 03 (2026) 217.
  72. F. Moretti, U. Nierste, P. Reeck, and M. Steinhauser, Next-to-next-to-leading QCD corrections to the B+-Bd0, D+-D0, and Ds+-D0 lifetime ratios, J. High Energy Phys. 09 (2026) 250.
  73. M. Black, R. Harlander, J. Kohnen, F. Lange, A. Rago, A. Shindler, and O. Witzel, Dataset: Heavy-meson bag parameters using gradient flow, 10.25819/67d6vsvpskcu5g9w (2026).

Outline

Information

Phys. Rev. Lett. 137, 191905– Published 5 November, 2026
  • Received 16 April 2026
  • Revised 15 July 2026
  • Accepted 23 September 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation