Open Access

Heavy-meson bag parameters using gradient flow

Matthew Black ^1,2,*, Robert V. Harlander ³, Jonas T. Kohnen ³, Fabian Lange ^4,5, Antonio Rago ⁶, Andrea Shindler ^3,7,8, and Oliver Witzel ⁹

¹Higgs Centre for Theoretical Physics, School of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom
²Theoretische Physik 1, Center for Particle Physics Siegen, Naturwissenschaftlich-Technische Fakultät, Universität Siegen, 57068 Siegen, Germany
³Institute for Theoretical Particle Physics and Cosmology, RWTH Aachen University, 52056 Aachen, Germany
⁴Physik-Institut, Universität Zürich, Winterthurerstrasse 190, 8057 Zürich, Switzerland
⁵PSI Center for Neutron and Muon Sciences, 5232 Villigen PSI, Switzerland
⁶IMADA and Quantum Theory Center, University of Southern Denmark, Odense, Denmark
⁷Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
⁸Department of Physics, University of California, Berkeley, California 94720, USA
⁹Theoretische Physik 1, Center for Particle Physics Siegen, Naturwissenschaftlich-Technische Fakultät, and Cluster of Excellence “Color meets Flavor,” Universität Siegen, 57068 Siegen, Germany

^*Contact author: matthew.black@ed.ac.uk

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Phys. Rev. D 114, 094503 – Published 5 November, 2026

DOI: https://doi.org/10.1103/47j9-h8s5

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 $\bar{MS}$ 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 $Δ Q = 2$ $\bar{MS}$ bag parameter $B_{1}^{\bar{MS}} (3 GeV) = 0.7673 (123)$ , consistent with existing short-distance $D^{0}$ mixing determinations. For the $Δ Q = 0$ lifetime-ratio operator basis, we find the $\bar{MS}$ results $B_{1}^{\bar{MS}} (3 GeV) = 1.0524 (97)$ , $B_{2}^{\bar{MS}} (3 GeV) = 0.9621 (71)$ , $ε_{1}^{\bar{MS}} (3 GeV) = - 0.2275 (76)$ , and $ε_{2}^{\bar{MS}} (3 GeV) = - 0.0005 (8)$ . 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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References (272)

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).
S. Dürr et al. (BMW Collaboration), Precision computation of the kaon bag parameter, Phys. Lett. B 705, 477 (2011).
B. J. Choi et al. (SWME Collaboration), Kaon BSM B-parameters using improved staggered fermions from $N_{f} = 2 + 1$ unquenched QCD, Phys. Rev. D 93, 014511 (2016).
T. Blum et al. (RBC Collaboration and UKQCD Collaboration), Domain wall QCD with physical quark masses, Phys. Rev. D 93, 074505 (2016).
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).
P. A. Boyle, F. Erben, J. M. Flynn, N. Garron, J. Kettle, R. Mukherjee, and J. T. Tsang (RBC Collaboration and UKQCD Collaboration), Kaon mixing beyond the standard model with physical masses, Phys. Rev. D 110, 034501 (2024).
V. Bertone et al. (ETM Collaboration), Kaon Mixing Beyond the SM from $N_{f} = 2$ tmQCD and model independent constraints from the UTA, J. High Energy Phys. 03 (2013) 089; 07 (2013) 143.
N. Carrasco et al. (ETM Collaboration), B-physics from $N_{f} = 2$ tmQCD: The Standard Model and beyond, J. High Energy Phys. 03 (2014) 016.
N. Carrasco et al., $D^{0} - {\bar{D}}^{0}$ mixing in the standard model and beyond from $N_{f} = 2$ twisted mass QCD, Phys. Rev. D 90, 014502 (2014).
N. Carrasco, P. Dimopoulos, R. Frezzotti, V. Lubicz, G. C. Rossi, S. Simula, and C. Tarantino (ETM Collaboration), $Δ S = 2$ and $Δ C = 2$ bag parameters in the standard model and beyond from $N_{f} = 2 + 1 + 1$ twisted-mass lattice QCD, Phys. Rev. D 92, 034516 (2015).
J. Laiho and R. S. Van de Water, Pseudoscalar decay constants, light-quark masses, and $B_{K}$ from mixed-action lattice QCD, Proc. Sci. LATTICE2011 (2011) 293.
A. Bazavov et al., Short-distance matrix elements for $D^{0}$ -meson mixing for $N_{f} = 2 + 1$ lattice QCD, Phys. Rev. D 97, 034513 (2018).
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).
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).
R. S. Van de Water and S. R. Sharpe, $B_{K}$ in staggered chiral perturbation theory, Phys. Rev. D 73, 014003 (2006).
J. A. Bailey, H.-J. Kim, W. Lee, and S. R. Sharpe, Kaon mixing matrix elements from beyond-the-Standard-Model operators in staggered chiral perturbation theory, Phys. Rev. D 85, 074507 (2012).
G. Martinelli, C. Pittori, C. T. Sachrajda, M. Testa, and A. Vladikas, A general method for non-perturbative renormalization of lattice operators, Nucl. Phys. B445, 81 (1995).
C. Sturm, Y. Aoki, N. H. Christ, T. Izubuchi, C. T. C. Sachrajda, and A. Soni, Renormalization of quark bilinear operators in a momentum-subtraction scheme with a nonexceptional subtraction point, Phys. Rev. D 80, 014501 (2009).
P. A. Boyle, N. Garron, R. J. Hudspith, C. Lehner, and A. T. Lytle (RBC Collaboration and UKQCD Collaboration), Neutral kaon mixing beyond the Standard Model with $n_{f} = 2 + 1$ chiral fermions. Part 2: Non perturbative renormalisation of the $Δ F = 2$ four-quark operators, J. High Energy Phys. 10 (2017) 054.
Y. Aoki et al. (Flavour Lattice Averaging Group (FLAG), FLAG review 2024, Phys. Rev. D 113, 014508 (2026).
N. Garron, R. J. Hudspith, and A. T. Lytle (RBC/UKQCD Collaboration), Neutral Kaon mixing beyond the Standard Model with $n_{f} = 2 + 1$ chiral fermions Part 1: Bare matrix elements and physical results, J. High Energy Phys. 11 (2016) 001.
P. Boyle, F. Erben, A. Jüttner, T. Kaneko, M. Marshall, A. Portelli, O. Witzel, L. Del Debbio, and J. T. Tsang, BSM $B - \bar{B}$ mixing on JLQCD and RBC/UKQCD $N_{f} = 2 + 1$ DWF ensembles Proc. Sci., LATTICE2021 (2022) 224.
L. Giusti, T. Harris, A. Nada, and S. Schaefer, Frequency-splitting estimators of single-propagator traces, Eur. Phys. J. C 79, 586 (2019).
P. A. Boyle, F. Erben, J. M. Flynn, V. Gülpers, R. C. Hill, R. Hodgson, A. Jüttner, F. Ó. hÓgáin, A. Portelli, and C. T. Sachrajda (RBC Collaboration and UKQCD Collaboration), Simulating rare kaon decays $K^{+} \to π^{+} ℓ^{+} ℓ^{-}$ using domain wall lattice QCD with physical light quark masses, Phys. Rev. D 107, L011503 (2023).
G. Spanoudes, M. Costa, K. Mitsidi, and H. Panagopoulos, Classification of four-quark operators with $Δ F \leq 2$ under flavor symmetry and their renormalization in a gauge-invariant scheme, Phys. Rev. D 113, 034504 (2026).
J. Lin, W. Detmold, and S. Meinel, Position-space renormalization schemes for four-quark operators in HQET, J. High Energy Phys. 07 (2024) 188.
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).
M. Di Pierro, C. T. Sachrajda, and C. Michael (UKQCD Collaboration), An Exploratory lattice study of spectator effects in inclusive decays of the $Λ_{b}$ baryon, Phys. Lett. B 468, 143 (1999); 525, 360(E) (2002).
D. Becirevic, Theoretical progress in describing the B meson lifetimes, Proc. Sci. HEP2001 (2001) 098 [arXiv:hep-ph/0110124].
J. Lin, W. Detmold, and S. Meinel, Lattice study of spectator effects in $b$ -hadron decays, Proc. Sci. LATTICE2022 (2023) 417 [arXiv:2212.09275].
R. Narayanan and H. Neuberger, Infinite $N$ phase transitions in continuum Wilson loop operators, J. High Energy Phys. 03 (2006) 064.
M. Lüscher, Trivializing maps, the Wilson flow and the HMC algorithm, Commun. Math. Phys. 293, 899 (2010).
M. Lüscher, Properties and uses of the Wilson flow in lattice QCD, J. High Energy Phys. 08 (2010) 071; 03 (2014) 092.
M. Lüscher, Chiral symmetry and the Yang–Mills gradient flow, J. High Energy Phys. 04 (2013) 123.
M. Lüscher and P. Weisz, Perturbative analysis of the gradient flow in non-Abelian gauge theories, J. High Energy Phys. 02 (2011) 051.
H. Suzuki, Energy–momentum tensor from the Yang–Mills gradient flow, Prog. Theor. Exp. Phys. 2013, 083B03 (2013); 2015, 079201 (2015).
M. Lüscher, Future applications of the Yang-Mills gradient flow in lattice QCD, Proc. Sci. LATTICE2013 (2014) 016 [arXiv:1308.5598].
Y. Taniguchi, Non-perturbative renormalization of Kaon B parameter using gradient flow, in Contribution to the 37th International Symposium on Lattice Field Theory, https://indico.cern.ch/event/764552/contributions/3420565/ (2019).
M. Black, R. V. Harlander, A. Hasenfratz, A. Rago, and O. Witzel, Renormalized quark masses using gradient flow, Phys. Rev. D 113, 094504 (2026).
A. Shindler, Moments of parton distribution functions of any order from lattice QCD, Phys. Rev. D 110, L051503 (2024).
R. V. Harlander, J. T. Kohnen, and A. Shindler, Short-flow-time expansion of non-singlet twist-two operators at next-to-next-to-leading order QCD, Eur. Phys. J. C 86, 439 (2026).
A. Francis et al., Gradient flow for parton distribution functions: first application to the pion, Phys. Rev. Lett. 136, 171903 (2026).
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).
R. Edwards, J. Karpie, L. Maio, C. J. Monahan, K. Orginos, D. Richards, A. M. Sturzu, and S. Zafeiropoulos (HadStruc Collaboration), Accessing the gluon momentum fraction of nucleons through the gradient flow, Phys. Rev. D 114, 014516 (2026).
T. Iritani, M. Kitazawa, H. Suzuki, and H. Takaura, Thermodynamics in quenched QCD: Energy–momentum tensor with two-loop order coefficients in the gradient-flow formalism, Prog. Theor. Exp. Phys. 2019, 023B02 (2019).
Y. Taniguchi, S. Ejiri, K. Kanaya, M. Kitazawa, H. Suzuki, and T. Umeda (WHOT-QCD Collaboration), $N_{f} = 2 + 1$ QCD thermodynamics with gradient flow using two-loop matching coefficients, Phys. Rev. D 102, 014510 (2020); 102, 059903(E) (2020).
P. Boyle, L. Del Debbio, and A. Khamseh, Massive momentum-subtraction scheme, Phys. Rev. D 95, 054505 (2017).
L. Del Debbio, F. Erben, J. M. Flynn, R. Mukherjee, and J. T. Tsang (RBC Collaboration and UKQCD Collaboration), Absorbing discretization effects with a massive renormalization scheme: The charm-quark mass, Phys. Rev. D 110, 054512 (2024).
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.
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].
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].
M. Black, Improving Precision for Hadronic Observables in the Standard Model and Beyond, Ph.D. thesis, Universität Siegen, 2024, 10.25819/ubsi/10603.
M. Black, R. V. Harlander, J. T. Kohnen, F. Lange, A. Rago, A. Shindler, and O. Witzel, companion Letter, Bag parameters for heavy meson lifetimes from a lattice QCD calculation at the $SU (3)_{F}$ -symmetric point, Phys. Rev. Lett. 137, 191905 (2026).
V. A. Khoze and M. A. Shifman, Heavy quarks, Sov. Phys. Usp. 26, 387 (1983).
M. A. Shifman and M. B. Voloshin, Preasymptotic effects in inclusive weak decays of charmed particles, Sov. J. Nucl. Phys. 41, 120 (1985).
K. G. Wilson, Nonlagrangian models of current algebra, Phys. Rev. 179, 1499 (1969).
K. G. Wilson and J. B. Kogut, The Renormalization group and the epsilon expansion, Phys. Rep. 12, 75 (1974).
D. Buskulic et al. (ALEPH Collaboration), Improved measurement of the $B_{d}^{0} - {\bar{B}}_{d}^{0}$ oscillation frequency, Z. Phys. C 75, 397 (1997).
P. Abreu et al. (DELPHI Collaboration), Measurement of $B_{d}^{0} - {\bar{B}}_{d}^{0}$ oscillations, Z. Phys. C 76, 579 (1997).
J. Abdallah et al. (DELPHI Collaboration), Search for $B_{s}^{0} - {\bar{B}}_{s}^{0}$ oscillations and a measurement of $B_{d}^{0} - {\bar{B}}_{d}^{0}$ oscillations using events with an inclusively reconstructed vertex, Eur. Phys. J. C 28, 155 (2003).
K. Ackerstaff et al. (OPAL Collaboration), An updated study of B meson oscillations using dilepton events, Z. Phys. C 76, 417 (1997).
K. Ackerstaff et al. (OPAL Collaboration), A study of $B$ meson oscillations using hadronic $Z^{0}$ decays containing leptons, Z. Phys. C 76, 401 (1997).
G. Abbiendi et al. (OPAL Collaboration), Measurement of the $B^{0}$ lifetime and oscillation frequency using ${\bar{B}}^{0} \to D^{* +} ℓ^{-}$ anti-neutrino decays, Phys. Lett. B 493, 266 (2000).
F. Abe et al. (CDF Collaboration), Measurement of the $B^{0} - {\bar{B}}^{0}$ oscillation frequency using $π - B$ meson charge-flavor correlations in $p \bar{p}$ collisions at $\sqrt{s} = 1.8 TeV$ , Phys. Rev. Lett. 80, 2057 (1998).
F. Abe et al. (CDF Collaboration), Measurement of the $B_{d}^{0} - {\bar{B}}_{d}^{0}$ flavor oscillation frequency and study of same side flavor tagging of $B$ mesons in $p \bar{p}$ collisions, Phys. Rev. D 59, 032001 (1999).
F. Abe et al. (CDF Collaboration), Measurement of the $B_{d}^{0} {\bar{B}}_{d}^{0}$ oscillation frequency using dimuon data in $p \bar{p}$ collisions at $\sqrt{s} = 1.8 TeV$ , Phys. Rev. D 60, 051101 (1999).
F. Abe et al. (CDF Collaboration), Measurement of $B^{0} - {\bar{B}}^{0}$ flavor oscillations using jet-charge and lepton flavor tagging in $p \bar{p}$ collisions at $\sqrt{s} = 1.8 TeV$ , Phys. Rev. D 60, 072003 (1999).
T. Affolder et al. (CDF Collaboration), Measurement of the $B^{0} {\bar{B}}^{0}$ oscillation frequency using $ℓ^{-} D^{* +}$ pairs and lepton flavor tags, Phys. Rev. D 60, 112004 (1999).
M. Acciarri et al. (L3 Collaboration), Measurement of the $B_{d}^{0} - {\bar{B}}_{d}^{0}$ oscillation frequency, Eur. Phys. J. C 5, 195 (1998).
B. Aubert et al. (BABAR Collaboration), Measurement of $B^{0} - {\bar{B}}^{0}$ flavor oscillations in hadronic $B^{0}$ decays, Phys. Rev. Lett. 88, 221802 (2002).
B. Aubert et al. (BABAR Collaboration), Measurement of the $B^{0} - {\bar{B}}^{0}$ oscillation frequency with inclusive dilepton events, Phys. Rev. Lett. 88, 221803 (2002).
B. Aubert et al. (BABAR Collaboration), A study of time dependent $C P$ -violating asymmetries and flavor oscillations in neutral $B$ decays at the $ϒ (4 S)$ , Phys. Rev. D 66, 032003 (2002).
B. Aubert et al. (BABAR Collaboration), Simultaneous measurement of the $B^{0}$ meson lifetime and mixing frequency with $B^{0} \to D^{* -} ℓ^{+} ν_{ℓ}$ decays, Phys. Rev. D 67, 072002 (2003).
B. Aubert et al. (BABAR Collaboration), Measurement of the ${\bar{B}}^{0}$ lifetime and the $B^{0} {\bar{B}}^{0}$ oscillation frequency using partially reconstructed ${\bar{B}}^{0} \to D^{* +} ℓ^{-} {\bar{ν}}_{ℓ}$ decays, Phys. Rev. D 73, 012004 (2006).
N. C. Hastings et al. (Belle Collaboration), Studies of $B^{0} - {\bar{B}}^{0}$ mixing properties with inclusive dilepton events, Phys. Rev. D 67, 052004 (2003).
Y. Zheng et al. (Belle Collaboration), Measurement of the $B^{0} - {\bar{B}}^{0}$ mixing rate with $B^{0} ({\bar{B}}^{0}) \to D^{* \mp} π^{\pm}$ partial reconstruction, Phys. Rev. D 67, 092004 (2003).
K. Abe et al. (Belle Collaboration), Improved measurement of $C P$ -violation parameters $\sin 2 ϕ_{1}$ and $| λ |$ , $B$ meson lifetimes, and $B^{0} - {\bar{B}}^{0}$ mixing parameter $Δ m_{d}$ , Phys. Rev. D 71, 072003 (2005); 71, 079903(E) (2005).
F. Abudinén et al. (Belle-II Collaboration), Measurement of the $B^{0}$ lifetime and flavor-oscillation frequency using hadronic decays reconstructed in 2019–2021 Belle II data, Phys. Rev. D 107, L091102 (2023).
R. Aaij et al. (LHCb Collaboration), Measurement of the $B^{0} - {\bar{B}}^{0}$ oscillation frequency $Δ m_{d}$ with the decays $B^{0} \to D^{-} π^{+}$ and $B^{0} \to J / ψ K^{* 0}$ , Phys. Lett. B 719, 318 (2013).
R. Aaij et al. (LHCb Collaboration), Observation of $B_{s}^{0} - {\bar{B}}_{s}^{0}$ mixing and measurement of mixing frequencies using semileptonic B decays, Eur. Phys. J. C 73, 2655 (2013).
R. Aaij et al. (LHCb Collaboration), A precise measurement of the $B^{0}$ meson oscillation frequency, Eur. Phys. J. C 76, 412 (2016).
A. Abulencia et al. (CDF Collaboration), Observation of $B_{s}^{0} - {\bar{B}}_{s}^{0}$ Oscillations, Phys. Rev. Lett. 97, 242003 (2006).
R. Aaij et al. (LHCb Collaboration), Measurement of the $B_{s}^{0} - {\bar{B}}_{s}^{0}$ oscillation frequency $Δ m_{s}$ in $B_{s}^{0} \to D_{s}^{-} (3) π$ decays, Phys. Lett. B 709, 177 (2012).
R. Aaij et al. (LHCb Collaboration), Precision measurement of the $B_{s}^{0} - {\bar{B}}_{s}^{0}$ oscillation frequency with the decay $B_{s}^{0} \to D_{s}^{-} π^{+}$ , New J. Phys. 15, 053021 (2013).
R. Aaij et al. (LHCb Collaboration), Precision measurement of $C P$ violation in $B_{s}^{0} \to J / ψ K^{+} K^{-}$ decays, Phys. Rev. Lett. 114, 041801 (2015).
R. Aaij et al. (LHCb Collaboration), Measurement of the CKM angle $γ$ and $B_{s}^{0} - {\bar{B}}_{s}^{0}$ mixing frequency with $B_{s}^{0} \to D_{s}^{\mp} h^{\pm} π^{\pm} π^{\mp}$ decays, J. High Energy Phys. 03 (2021) 137.
R. Aaij et al. (LHCb Collaboration), Precise determination of the $B_{s}^{0} - {\bar{B}}_{s}^{0}$ oscillation frequency, Nat. Phys. 18, 1 (2022).
I. Bezshyiko et al. (LHCb Collaboration), Improved measurement of $C P$ violation parameters in $B_{s}^{0} \to J / ψ K^{+} K^{-}$ decays in the vicinity of the $ϕ (1020)$ resonance, Phys. Rev. Lett. 132, 051802 (2024).
A. M. Sirunyan et al. (CMS Collaboration), Measurement of the $C P$ -violating phase $ϕ_{s}$ in the $B_{s}^{0} \to J / ψ ϕ (1020) \to μ^{+} μ^{-} K^{+} K^{-}$ channel in proton-proton collisions at $s = 13 TeV$ , Phys. Lett. B 816, 136188 (2021).
M. Staric et al. (Belle Collaboration), Evidence for $D^{0} - {\bar{D}}^{0}$ mixing, Phys. Rev. Lett. 98, 211803 (2007).
B. Aubert et al. (BABAR Collaboration), Evidence for $D^{0} - {\bar{D}}^{0}$ mixing, Phys. Rev. Lett. 98, 211802 (2007).
T. Aaltonen et al. (CDF Collaboration), Evidence for $D^{0} - {\bar{D}}^{0}$ mixing using the CDF II detector, Phys. Rev. Lett. 100, 121802 (2008).
R. Aaij et al. (LHCb Collaboration), Observation of $D^{0} - {\bar{D}}^{0}$ oscillations, Phys. Rev. Lett. 110, 101802 (2013).
T. A. Aaltonen et al. (CDF Collaboration), Observation of $D^{0} - {\bar{D}}^{0}$ mixing using the CDF II detector, Phys. Rev. Lett. 111, 231802 (2013).
B. R. Ko et al. (Belle Collaboration), Observation of $D^{0} - {\bar{D}}^{0}$ mixing in $e^{+} e^{-}$ collisions, Phys. Rev. Lett. 112, 111801 (2014); 112, 139903(A) (2014).
R. Aaij et al. (LHCb Collaboration), First observation of $D^{0} - {\bar{D}}^{0}$ oscillations in $D^{0} \to K^{+} π^{-} π^{+} π^{-}$ decays and measurement of the associated coherence parameters, Phys. Rev. Lett. 116, 241801 (2016).
R. Aaij et al. (LHCb Collaboration), Measurement of the charm-mixing parameter $y_{C P}$ , Phys. Rev. Lett. 122, 011802 (2019).
R. Aaij et al. (LHCb Collaboration), Measurement of the mass difference between neutral charm-meson eigenstates, Phys. Rev. Lett. 122, 231802 (2019).
R. Aaij et al. (LHCb Collaboration), Observation of the mass difference between neutral charm-meson eigenstates, Phys. Rev. Lett. 127, 111801 (2021).
R. Aaij et al. (LHCb Collaboration), Simultaneous determination of CKM angle $γ$ and charm mixing parameters, J. High Energy Phys. 12 (2021) 141.
R. Aaij et al. (LHCb Collaboration), Measurement of the charm mixing parameter $y_{C P} - y_{C P}^{K π}$ using two-body $D^{0}$ meson decays, Phys. Rev. D 105, 092013 (2022).
R. Aaij et al. (LHCb Collaboration), Measurement of $D^{0} - {\bar{D}}^{0}$ mixing and search for $C P$ violation with $D^{0} \to K^{+} π$ decays, Phys. Rev. D 111, 012001 (2025).
LHCb Collaboration, Simultaneous determination of the CKM angle $γ$ and parameters related to mixing and $C P$ violation in the charm sector 10.17181/CERN.8CUC.W3FT (2024).
R. Aaij et al. (LHCb Collaboration), Search for charge-parity violation in semileptonically tagged $D^{0} \to K^{+} π^{-}$ decays, J. High Energy Phys. 03 (2025) 149.
I. Adachi et al. (Belle Collaboration and Belle-II Collaboration), Model-independent measurement of $D^{0} - {\bar{D}}^{0}$ mixing parameters in $D^{0} \to K_{S}^{0} π^{+} π^{-}$ decays at Belle and Belle II, Phys. Rev. D 111, 112011 (2025).
S. Banerjee et al. (Heavy Flavor Averaging Group (HFLAV) Collaboration), Averages of $b$ -hadron, $c$ -hadron, and $τ$ -lepton properties as of 2023, Phys. Rev. D 113, 012008 (2026).
A. Lenz and G. Wilkinson, Mixing and $C P$ violation in the charm system, Annu. Rev. Nucl. Part. Sci. 71, 59 (2021).
M. Di Carlo, F. Erben, and M. T. Hansen, Long distance contributions to neutral D-meson mixing from lattice QCD, J. High Energy Phys. 07 (2025) 229.
M. Artuso, G. Borissov, and A. Lenz, $C P$ violation in the $B_{s}^{0}$ system, Rev. Mod. Phys. 88, 045002 (2016); 91, 049901(A) (2019).
R. Harlander, S. Klein, and M. Lipp, FeynGame, Comput. Phys. Commun. 256, 107465 (2020).
L. Bündgen, R. V. Harlander, S. Y. Klein, and M. C. Schaaf, FeynGame 3.0, Comput. Phys. Commun. 314, 109662 (2025).
T. Inami and C. S. Lim, Effects of superheavy quarks and leptons in low-energy weak processes $K_{L} \to μ \bar{μ}$ , $K^{+} \to π^{+} ν \bar{ν}$ and $K^{0} \leftrightarrow {\bar{K}}^{0}$ , Prog. Theor. Phys. 65, 297 (1981); 65, 1772(E) (1981).
A. J. Buras, M. Jamin, and P. H. Weisz, Leading and next-to-leading QCD corrections to $ε$ parameter and $B^{0} - {\bar{B}}^{0}$ mixing in the presence of a heavy top quark, Nucl. Phys. B347, 491 (1990).
S. Navas et al. (Particle Data Group), Review of particle physics, Phys. Rev. D 110, 030001 (2024).
A. G. Grozin, R. Klein, T. Mannel, and A. A. Pivovarov, $B^{0} - {\bar{B}}^{0}$ mixing at next-to-leading order, Phys. Rev. D 94, 034024 (2016).
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).
M. J. Kirk, Charming new physics in beautiful processes?, Ph.D. thesis, Durham University, IPPP, 2018, 10.1007/978-3-030-19197-9.
D. King, A. Lenz, and T. Rauh, $B_{s}$ mixing observables and $| V_{t d} / V_{t s} |$ from sum rules, J. High Energy Phys. 05 (2019) 034.
C. T. H. Davies, J. Harrison, G. P. Lepage, C. J. Monahan, J. Shigemitsu, and M. Wingate (HPQCD Collaboration), Lattice QCD matrix elements for the $B_{s}^{0} - {\bar{B}}_{s}^{0}$ width difference beyond leading order, Phys. Rev. Lett. 124, 082001 (2020).
P. Abreu et al. (DELPHI Collaboration), A measurement of $B^{+}$ and $B^{0}$ lifetimes using $\bar{D} ℓ^{+}$ events, Z. Phys. C 68, 13 (1995).
W. Adam et al. (DELPHI Collaboration), Lifetimes of charged and neutral B hadrons using event topology, Z. Phys. C 68, 363 (1995).
P. Abreu et al. (DELPHI Collaboration), A precise measurement of the $B_{d}^{0}$ meson lifetime using a new technique, Z. Phys. C 74, 19 (1997); 75, 579(E) (1997).
P. Abreu et al. (DELPHI Collaboration), Study of $B_{s}^{0} - {\bar{B}}_{s}^{0}$ oscillations and $B_{s}^{0}$ lifetimes using hadronic decays of $B_{s}^{0}$ mesons, Eur. Phys. J. C 18, 229 (2000).
J. Abdallah et al. (DELPHI Collaboration), A precise measurement of the $B^{+}$ , $B^{0}$ and mean $b$ hadron lifetime with the DELPHI detector at LEP1, Eur. Phys. J. C 33, 307 (2004).
P. Abreu et al. (DELPHI Collaboration), Measurement of the $B_{s}^{0}$ lifetime and study of $B_{s}^{0} - {\bar{B}}_{s}^{0}$ oscillations using $D_{s} ℓ$ events, Eur. Phys. J. C 16, 555 (2000).
D. Buskulic et al. (ALEPH Collaboration), Improved measurement of the ${\bar{B}}^{0}$ and $B^{-}$ meson lifetimes, Z. Phys. C 71, 31 (1996).
R. Barate et al. (ALEPH Collaboration), Study of $B_{s}^{0}$ oscillations and lifetime using fully reconstructed $D_{s}$ —decays, Eur. Phys. J. C 4, 367 (1998).
R. Barate et al. (ALEPH Collaboration), Measurement of the ${\bar{B}}^{0}$ and $B^{-}$ meson lifetimes, Phys. Lett. B 492, 275 (2000).
R. Barate et al. (ALEPH Collaboration), A study of the decay width difference in the $B_{s}^{0} - {\bar{B}}_{s}^{0}$ system using $ϕ ϕ$ correlations, Phys. Lett. B 486, 286 (2000).
M. Acciarri et al. (L3 Collaboration), Upper limit on the lifetime difference of shortlived and longlived $B_{s}^{0}$ mesons, Phys. Lett. B 438, 417 (1998).
R. Akers et al. (OPAL Collaboration), Improved measurements of the $B^{0}$ and $B^{+}$ meson lifetimes, Z. Phys. C 67, 379 (1995).
K. Ackerstaff et al. (OPAL Collaboration), A measurement of the $B_{s}^{0}$ lifetime using reconstructed $D_{s}^{-}$ —mesons, Eur. Phys. J. C 2, 407 (1998).
K. Ackerstaff et al. (OPAL Collaboration), Measurements of the $B_{s}^{0}$ and $Λ_{b}^{0}$ lifetimes, Phys. Lett. B 426, 161 (1998).
G. Abbiendi et al. (OPAL Collaboration), Measurement of the $B^{+}$ and $B^{0}$ lifetimes and search for $CP (T)$ violation using reconstructed secondary vertices, Eur. Phys. J. C 12, 609 (2000).
K. Abe et al. (SLD Collaboration), Measurement of the $B^{+}$ and $B^{0}$ lifetimes using topological reconstruction of inclusive and semileptonic decays, Phys. Rev. Lett. 79, 590 (1997).
F. Abe et al. (CDF Collaboration), Measurement of $B$ hadron lifetimes using $J / ψ$ final states at CDF, Phys. Rev. D 57, 5382 (1998).
F. Abe et al. (CDF Collaboration), Improved measurement of the $B^{-}$ and ${\bar{B}}^{0}$ meson lifetimes using semileptonic decays, Phys. Rev. D 58, 092002 (1998).
F. Abe et al. (CDF Collaboration), Measurement of the $B_{s}^{0}$ meson lifetime using semileptonic decays, Phys. Rev. D 59, 032004 (1999).
D. Acosta et al. (CDF Collaboration), Measurement of $B$ meson lifetimes using fully reconstructed $B$ decays produced in $p \bar{p}$ collisions at $\sqrt{s} = 1.8 TeV$ , Phys. Rev. D 65, 092009 (2002).
T. Aaltonen et al. (CDF Collaboration), Measurement of $b$ hadron lifetimes in exclusive decays containing a $J / ψ$ in $p^{-}$ pbar collisions at $\sqrt{s} = 1.96 TeV$ , Phys. Rev. Lett. 106, 121804 (2011).
T. Aaltonen et al. (CDF Collaboration), Measurement of the $B^{-}$ lifetime using a simulation free approach for trigger bias correction, Phys. Rev. D 83, 032008 (2011).
T. Aaltonen et al. (CDF Collaboration), Measurement of the $B_{s}$ lifetime in fully and partially reconstructed $B_{s} \to D_{s}^{-} (ϕ π^{-}) X$ decays in $\bar{p} - p$ collisions at $\sqrt{s} = 1.96 TeV$ , Phys. Rev. Lett. 107, 272001 (2011).
T. Aaltonen et al. (CDF Collaboration), Measurement of branching ratio and $B_{s}^{0}$ lifetime in the decay $B_{s}^{0} \to J / ψ f_{0} (980)$ at CDF, Phys. Rev. D 84, 052012 (2011).
T. Aaltonen et al. (CDF Collaboration), Measurement of the bottom-strange meson mixing phase in the full CDF data set, Phys. Rev. Lett. 109, 171802 (2012).
V. M. Abazov et al. (D0 Collaboration), Measurement of the ratio of $B^{+}$ and $B^{0}$ meson lifetimes, Phys. Rev. Lett. 94, 182001 (2005).
V. M. Abazov et al. (D0 Collaboration), Measurement of the $B_{s}^{0}$ lifetime in the exclusive decay channel $B_{s}^{0} \to J / ψ ϕ$ , Phys. Rev. Lett. 94, 042001 (2005).
V. M. Abazov et al. (D0 Collaboration), Measurement of the angular and lifetime parameters of the decays $B_{d}^{0} \to J / ψ K^{* 0}$ and $B_{s}^{0} \to J / ψ ϕ$ , Phys. Rev. Lett. 102, 032001 (2009).
V. M. Abazov et al. (D0 Collaboration), Measurement of the $C P$ -violating phase $ϕ_{s}^{J / ψ ϕ}$ using the flavor-tagged decay $B_{s}^{0} \to J / ψ ϕ$ in $8 {fb}^{- 1}$ of $p \bar{p}$ collisions, Phys. Rev. D 85, 032006 (2012).
V. M. Abazov et al. (D0 Collaboration), Measurement of the $B_{s}^{0}$ lifetime in the flavor-specific decay channel $B_{s}^{0} \to D_{s}^{-} μ^{+} ν X$ , Phys. Rev. Lett. 114, 062001 (2015).
V. M. Abazov et al. (D0 Collaboration), $B_{s}^{0}$ lifetime measurement in the $C P$ -odd decay channel $B_{s}^{0} \to J / ψ f_{0} (980)$ , Phys. Rev. D 94, 012001 (2016).
B. Aubert et al. (BABAR Collaboration), Measurement of the $B^{0}$ and $B^{+}$ meson lifetimes with fully reconstructed hadronic final states, Phys. Rev. Lett. 87, 201803 (2001).
B. Aubert et al. (BABAR Collaboration), Measurement of the $B^{0}$ lifetime with partially reconstructed $B^{0} \to D^{* -} ℓ + ν_{ℓ}$ decays, Phys. Rev. Lett. 89, 011802 (2002); 89, 169903(E) (2002).
B. Aubert et al. (BABAR Collaboration), Measurement of the $B^{0}$ meson lifetime with partial reconstruction of $B^{0} \to D^{* -} π^{+}$ and $B^{0} \to D^{* -} ρ^{+}$ decays, Phys. Rev. D 67, 091101 (2003).
G. Aad et al. (ATLAS Collaboration), Flavor tagged time-dependent angular analysis of the $B_{s} \to J / ψ ϕ$ decay and extraction of $Δ Γ_{s}$ and the weak phase $ϕ_{s}$ in ATLAS, Phys. Rev. D 90, 052007 (2014).
G. Aad et al. (ATLAS Collaboration), Measurement of the $C P$ -violating phase $ϕ_{s}$ and the $B_{s}^{0}$ meson decay width difference with $B_{s}^{0} \to J / ψ ϕ$ decays in ATLAS, J. High Energy Phys. 08 (2016) 147.
G. Aad et al. (ATLAS Collaboration), Measurement of the $C P$ -violating phase $ϕ_{s}$ in $B_{s}^{0} \to J / ψ ϕ$ decays in ATLAS at 13 TeV, Eur. Phys. J. C 81, 342 (2021).
V. Khachatryan et al. (CMS Collaboration), Measurement of the $C P$ -violating weak phase $ϕ_{s}$ and the decay width difference $Δ Γ_{s}$ using the $B_{s}^{0} \to J / ψ ϕ (1020)$ decay channel in pp collisions at $\sqrt{s} = 8 TeV$ , Phys. Lett. B 757, 97 (2016).
A. M. Sirunyan et al. (CMS Collaboration), Measurement of b hadron lifetimes in pp collisions at $\sqrt{s} = 8 TeV$ , Eur. Phys. J. C 78, 457 (2018); 78, 561(E) (2018).
A. M. Sirunyan et al. (CMS Collaboration), Measurement of properties of $B_{s}^{0} \to μ^{+} μ^{-}$ decays and search for $B^{0} \to μ^{+} μ^{-}$ with the CMS experiment, J. High Energy Phys. 04 (2019) 188.
R. Aaij et al. (LHCb Collaboration), Measurement of the effective $B_{s}^{0} \to K^{+} K^{-}$ lifetime, Phys. Lett. B 707, 349 (2012).
R. Aaij et al. (LHCb Collaboration), Measurement of the effective $B_{s}^{0} \to J / ψ K_{S}^{0}$ lifetime, Nucl. Phys. B873, 275 (2013).
R. Aaij et al. (LHCb Collaboration), Measurement of the ${\bar{B}}_{s}^{0} \to D_{s}^{-} D_{s}^{+}$ and ${\bar{B}}_{s}^{0} \to D^{-} D_{s}^{+}$ effective lifetimes, Phys. Rev. Lett. 112, 111802 (2014).
R. Aaij et al. (LHCb Collaboration), Measurement of $C P$ violation and the $B_{s}^{0}$ meson decay width difference with $B_{s}^{0} \to J / ψ K^{+} K^{-}$ and $B_{s}^{0} \to J / ψ π^{+} π^{-}$ decays, Phys. Rev. D 87, 112010 (2013).
R. Aaij et al. (LHCb Collaboration), Measurements of the $B^{+}, B^{0}, B_{s}^{0}$ meson and $Λ_{b}^{0}$ baryon lifetimes, J. High Energy Phys. 04 (2014) 114.
R. Aaij et al. (LHCb Collaboration), Effective lifetime measurements in the $B_{s}^{0} \to K^{+} K^{-}$ , $B^{0} \to K^{+} π^{-}$ and $B_{s}^{0} \to π^{+} K^{-}$ decays, Phys. Lett. B 736, 446 (2014).
R. Aaij et al. (LHCb Collaboration), Measurement of the ${\bar{B}}_{s}^{0}$ meson lifetime in $D_{s}^{+} π^{-}$ decays, Phys. Rev. Lett. 113, 172001 (2014).
R. Aaij et al. (LHCb Collaboration), Measurement of the $B_{s}^{0} \to J / ψ η$ lifetime, Phys. Lett. B 762, 484 (2016).
R. Aaij et al. (LHCb Collaboration), First study of the $C P$ -violating phase and decay-width difference in $B_{s}^{0} \to ψ (2 S) ϕ$ decays, Phys. Lett. B 762, 253 (2016).
R. Aaij et al. (LHCb Collaboration), Measurement of $B_{s}^{0}$ and $D_{s}^{-}$ meson lifetimes, Phys. Rev. Lett. 119, 101801 (2017).
R. Aaij et al. (LHCb Collaboration), Resonances and $C P$ violation in $B_{s}^{0}$ and ${\bar{B}}_{s}^{0} \to J / ψ K^{+} K^{-}$ decays in the mass region above the $ϕ (1020)$ , J. High Energy Phys. 08 (2017) 037.
R. Aaij et al. (LHCb Collaboration), Measurement of the $C P$ -violating phase $ϕ_{s}$ from $B_{s}^{0} \to J / ψ π^{+} π^{-}$ decays in 13 TeV $p p$ collisions, Phys. Lett. B 797, 134789 (2019).
R. Aaij et al. (LHCb Collaboration), Updated measurement of time-dependent $C P$ -violating observables in $B_{s}^{0} \to J / ψ K^{+} K^{-}$ decays, Eur. Phys. J. C 79, 706 (2019); 80, 601(E) (2020).
R. Aaij et al. (LHCb Collaboration), First measurement of the $C P$ -violating phase in $B_{s}^{0} \to J / ψ (\to e^{+} e^{-}) ϕ$ decays, Eur. Phys. J. C 81, 1026 (2021).
R. Aaij et al. (LHCb Collaboration), Measurement of the $B_{s}^{0} \to μ^{+} μ^{-}$ decay properties and search for the $B^{0} \to μ^{+} μ^{-}$ and $B_{s}^{0} \to μ^{+} μ^{-} γ$ decays, Phys. Rev. D 105, 012010 (2022).
J. R. Raab et al. (Tagged Photon Spectrometer Collaboration), Measurement of the $D^{0}, D^{+}$ and $D_{s}^{+}$ lifetimes, Phys. Rev. D 37, 2391 (1988).
P. L. Frabetti et al. (E687 Collaboration), A precise measurement of the $D_{S}^{\pm}$ meson lifetime, Phys. Rev. Lett. 71, 827 (1993).
P. L. Frabetti et al. (E687 Collaboration), Precise measurements of the $D^{0}$ and $D^{+}$ meson lifetimes, Phys. Lett. B 323, 459 (1994).
E. M. Aitala et al. (E791 Collaboration), Measurement of the $D_{s}$ lifetime, Phys. Lett. B 445, 449 (1999).
E. M. Aitala et al. (E791 Collaboration), Measurements of lifetimes and a limit on the lifetime difference in the neutral D meson system, Phys. Rev. Lett. 83, 32 (1999).
G. Bonvicini et al. (CLEO Collaboration), Measurement of charm meson lifetimes, Phys. Rev. Lett. 82, 4586 (1999).
A. Kushnirenko et al. (SELEX Collaboration), Precision measurements of the $Λ_{c}^{+}$ and $D^{0}$ lifetimes, Phys. Rev. Lett. 86, 5243 (2001).
M. Iori et al. (SELEX Collaboration), Measurement of the $D_{s}^{\pm}$ lifetime, Phys. Lett. B 523, 22 (2001).
J. M. Link et al. (FOCUS Collaboration), New measurements of the $D^{0}$ and $D^{+}$ lifetimes, Phys. Lett. B 537, 192 (2002).
J. M. Link et al. (FOCUS Collaboration), A measurement of the $D_{s}^{+}$ lifetime, Phys. Rev. Lett. 95, 052003 (2005).
F. Abudinén et al. (Belle-II Collaboration), Precise measurement of the $D^{0}$ and $D^{+}$ lifetimes at Belle II, Phys. Rev. Lett. 127, 211801 (2021).
I. Adachi et al. (Belle-II Collaboration), Precise measurement of the $D_{s}^{+}$ lifetime at Belle II, Phys. Rev. Lett. 131, 171803 (2023).
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 (2021) 241.
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).
M. L. Piscopo, Higher order corrections to the lifetime of heavy hadrons, Ph.D. thesis, Siegen University, 2021, 10.25819/ubsi/10024.
B. Guberina, S. Nussinov, R. D. Peccei, and R. Rückl, D meson lifetimes and decays, Phys. Lett. B 89, 111 (1979).
M. A. Shifman and M. B. Voloshin, Hierarchy of lifetimes of charmed and beautiful hadrons, Sov. Phys. JETP 64, 698 (1986).
N. G. Uraltsev, On the problem of boosting nonleptonic b baryon decays, Phys. Lett. B 376, 303 (1996).
M. Neubert and C. T. Sachrajda, Spectator effects in inclusive decays of beauty hadrons, Nucl. Phys. B483, 339 (1997).
M. Beneke, G. Buchalla, C. Greub, A. Lenz, and U. Nierste, The $B^{+} - B_{d}^{0}$ lifetime difference beyond leading logarithms, Nucl. Phys. B639, 389 (2002).
E. Franco, V. Lubicz, F. Mescia, and C. Tarantino, Lifetime ratios of beauty hadrons at the next-to-leading order in QCD, Nucl. Phys. B633, 212 (2002).
A. Lenz and T. Rauh, D-meson lifetimes within the heavy quark expansion, Phys. Rev. D 88, 034004 (2013).
F. Gabbiani, A. I. Onishchenko, and A. A. Petrov, $Λ_{b}$ lifetime puzzle in heavy quark expansion, Phys. Rev. D 68, 114006 (2003).
F. Gabbiani, A. I. Onishchenko, and A. A. Petrov, Spectator effects and lifetimes of heavy hadrons, Phys. Rev. D 70, 094031 (2004).
D. King, A. Lenz, and T. Rauh, SU(3) breaking effects in B and D meson lifetimes, J. High Energy Phys. 06 (2022) 134.
D. J. King, Three loop corrections to HQET sum rules for the bag parameter, Ph.D. thesis, Durham University, 2022.
J. Artz, R. V. Harlander, F. Lange, T. Neumann, and M. Prausa, Results and techniques for higher order calculations within the gradient-flow formalism, J. High Energy Phys. 06 (2019) 121; 10 (2019) 032(E).
K. Hieda, H. Makino, and H. Suzuki, Proof of the renormalizability of the gradient flow, Nucl. Phys. B918, 23 (2017).
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).
J. Kim, T. Luu, M. D. Rizik, and A. Shindler (SymLat Collaboration), Nonperturbative renormalization of the quark chromoelectric dipole moment with the gradient flow: Power divergences, Phys. Rev. D 104, 074516 (2021).
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).
M. J. Dugan and B. Grinstein, On the vanishing of evanescent operators, Phys. Lett. B 256, 239 (1991).
S. Herrlich and U. Nierste, Evanescent operators, scheme dependences and double insertions, Nucl. Phys. B455, 39 (1995).
M. Fierz, Zur Fermischen Theorie des $β$ -Zerfalls, Z. Phys. 104, 553 (1937).
See Supplemental Material at http://link.aps.org/supplemental/10.1103/47j9-h8s5 for the perturbative results of this work.
R. V. Harlander and F. Lange, Effective electroweak Hamiltonian in the gradient-flow formalism, Phys. Rev. D 105, L071504 (2022).
S. G. Gorishnii, S. A. Larin, and F. V. Tkachov, The algorithm for OPE coefficient functions in the MS scheme, Phys. Lett. B 124, 217 (1983).
S. G. Gorishnii and S. A. Larin, Coefficient functions of asymptotic operator expansions in minimal subtraction scheme, Nucl. Phys. B283, 452 (1987).
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).
P. Nogueira, Automatic Feynman graph generation, J. Comput. Phys. 105, 279 (1993).
P. Nogueira, Abusing QGRAF, Nucl. Instrum. Methods Phys. Res., Sect. A 559, 220 (2006).
R. Harlander, T. Seidensticker, and M. Steinhauser, Corrections of $O (α α_{s})$ to the decay of the Z boson into bottom quarks, Phys. Lett. B 426, 125 (1998).
T. Seidensticker, Automatic application of successive asymptotic expansions of Feynman diagrams, in Proceedings of AIHENP 99, Crete, Greece (1999).
M. Gerlach, F. Herren, and M. Lang, tapir: A tool for topologies, amplitudes, partial fraction decomposition and input for reductions, Comput. Phys. Commun. 282, 108544 (2023).
J. A. M. Vermaseren, New features of FORM, arXiv:math-ph/0010025.
B. Ruijl, T. Ueda, and J. Vermaseren, FORM version 4.2, arXiv:1707.06453.
T. van Ritbergen, A. N. Schellekens, and J. A. M. Vermaseren, Group theory factors for Feynman diagrams, Int. J. Mod. Phys. A 14, 41 (1999).
F. V. Tkachov, A theorem on analytical calculability of 4-loop renormalization group functions, Phys. Lett. B 100, 65 (1981).
K. G. Chetyrkin and F. V. Tkachov, Integration by parts: The algorithm to calculate $β$ -functions in 4 loops, Nucl. Phys. B192, 159 (1981).
S. Laporta, High-precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A 15, 5087 (2000).
P. Maierhöfer, J. Usovitsch, and P. Uwer, Kira—A Feynman integral reduction program, Comput. Phys. Commun. 230, 99 (2018).
J. Klappert, F. Lange, P. Maierhöfer, and J. Usovitsch, Integral reduction with Kira 2.0 and finite field methods, Comput. Phys. Commun. 266, 108024 (2021).
J. Klappert and F. Lange, Reconstructing rational functions with FireFly, Comput. Phys. Commun. 247, 106951 (2020).
J. Klappert, S. Y. Klein, and F. Lange, Interpolation of dense and sparse rational functions and other improvements in FireFly, Comput. Phys. Commun. 264, 107968 (2021).
Ò. L. Crosas and P. Stoffer, One-loop matching of the LEFT to the QCD gradient flow, J. High Energy Phys. 07 (2026) 047.
A. J. Buras, M. Gorbahn, U. Haisch, and U. Nierste, Charm quark contribution to $K^{+} \to π^{+} ν \bar{ν}$ at next-to-next-to-leading order, J. High Energy Phys. 11 (2006) 002; 11 (2012) 167(E).
A. Suzuki, Y. Taniguchi, H. Suzuki, and K. Kanaya, Four quark operators for kaon bag parameter with gradient flow, Phys. Rev. D 102, 034508 (2020).
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).
Y. Aoki et al. (RBC/UKQCD Collaboration), Continuum limit physics from $2 + 1$ flavor domain wall QCD, Phys. Rev. D 83, 074508 (2011).
T. Blum et al. (RBC/UKQCD Collaboration), Domain wall QCD with physical quark masses, Phys. Rev. D 93, 074505 (2016).
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.
Y. Iwasaki, Renormalization group analysis of lattice theories and improved lattice action. II. Four-dimensional non-Abelian SU(N) gauge model, arXiv:1111.7054.
D. B. Kaplan, A method for simulating chiral fermions on the lattice, Phys. Lett. B 288, 342 (1992).
Y. Shamir, Chiral fermions from lattice boundaries, Nucl. Phys. B406, 90 (1993).
V. Furman and Y. Shamir, Axial symmetries in lattice QCD with Kaplan fermions, Nucl. Phys. B439, 54 (1995).
C. Morningstar and M. J. Peardon, Analytic smearing of SU(3) link variables in lattice QCD, Phys. Rev. D 69, 054501 (2004).
R. C. Brower, H. Neff, and K. Orginos, The Möbius domain wall fermion algorithm, Comput. Phys. Commun. 220, 1 (2017).
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.
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).
P. A. Boyle et al., Grid: A next generation data parallel c++ QCD library, Proc. Sci. LATTICE2015 (2016) 023.
A. Portelli et al., github.com/aportelli/Hadrons: Hadrons v1.2 10.5281/zenodo.4293902.
A. Portelli et al., github.com/aportelli/Hadrons: Hadrons v1.3 (2022) 10.5281/zenodo.6382460.
B. Efron, Bootstrap methods: Another look at the jackknife, Ann. Stat. 7, 1 (1979).
U. Wolff (ALPHA Collaboration), Monte Carlo errors with less errors, Comput. Phys. Commun. 156, 143 (2004); 176, 383(E) (2007).
S. Schaefer, R. Sommer, and F. Virotta (ALPHA Collaboration), Critical slowing down and error analysis in lattice QCD simulations, Nucl. Phys. B845, 93 (2011).
A. Ramos, Automatic differentiation for error analysis of Monte Carlo data, Comput. Phys. Commun. 238, 19 (2019).
F. Joswig, S. Kuberski, J. T. Kuhlmann, and J. Neuendorf, pyerrors: A python framework for error analysis of Monte Carlo data, Comput. Phys. Commun. 288, 108750 (2023).
M. Bruno and R. Sommer, On fits to correlated and auto-correlated data, Comput. Phys. Commun. 285, 108643 (2023).
T. Endo, K. Hieda, D. Miura, and H. Suzuki, Universal formula for the flavor non-singlet axial-vector current from the gradient flow, Prog. Theor. Exp. Phys. 2015, 053B03 (2015); 2024, 039202(E) (2024).
K. Hieda and H. Suzuki, Small flow-time representation of fermion bilinear operators, Mod. Phys. Lett. A 31, 1650214 (2016).
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.
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).
F. Herren and M. Steinhauser, Version 3 of RunDec and CRunDec, Comput. Phys. Commun. 224, 333 (2018).
M. Ciuchini, E. Franco, V. Lubicz, G. Martinelli, I. Scimemi, and L. Silvestrini, Next-to-leading order QCD corrections to $Δ F = 2$ effective Hamiltonians, Nucl. Phys. B523, 501 (1998).
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).
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.
R. V. Harlander, F. Lange, and T. Neumann, Hadronic vacuum polarization using gradient flow, J. High Energy Phys. 08 (2020) 109.
Y. Aoki, B. Colquhoun, H. Fukaya, S. Hashimoto, T. Kaneko, R. Kellermann, J. Koponen, and E. Kou (JLQCD Collaboration), $B \to D * ℓ ν ℓ$ semileptonic form factors from lattice QCD with Möbius domain-wall quarks, Phys. Rev. D 109, 074503 (2024).
J.-W. Lee, Matrix elements of $Δ B = 0$ operators in heavy hadron chiral perturbation theory, Phys. Rev. D 91, 094511 (2015).
M. Beneke, G. Buchalla, C. Greub, A. Lenz, and U. Nierste, Next-to-leading order QCD corrections to the lifetime difference of $B_{s}$ mesons, Phys. Lett. B 459, 631 (1999).
T. Huber, T. Hurth, J. Jenkins, E. Lunghi, Q. Qin, and K. K. Vos, Phenomenology of inclusive $\bar{B} \to X_{s} ℓ^{+} ℓ^{-}$ for the Belle II era, J. High Energy Phys. 10 (2020) 088.
M. Fael, J. Jenkins, E. Lunghi, and Z. Polonsky, The standard model prediction for the rare decay $B \to X_{s} ν \bar{ν}$ , J. High Energy Phys. 03 (2026) 217.
A. Hasenfratz, C. J. Monahan, M. D. Rizik, A. Shindler, and O. Witzel, A novel nonperturbative renormalization scheme for local operators, Proc. Sci. LATTICE2021 (2022) 155.
A. Carosso, A. Hasenfratz, and E. T. Neil, Nonperturbative renormalization of operators in near-conformal systems using gradient flows, Phys. Rev. Lett. 121, 201601 (2018).
A. Hasenfratz and O. Witzel, Continuous renormalization group $β$ function from lattice simulations, Phys. Rev. D 101, 034514 (2020).
F. Moretti, U. Nierste, P. Reeck, and M. Steinhauser, Next-to-next-to-leading QCD corrections to the $B^{+} - B_{d}^{0}$ , $D^{+} - D^{0}$ , and $D_{s}^{+} - D^{0}$ lifetime ratios, J. High Energy Phys. 09 (2026) 250.
M. Black, R. Harlander, J. Kohnen, F. Lange, A. Rago, A. Shindler, and O. Witzel, Dataset: Heavy-meson bag parameters using gradient flow (2026) 10.25819/67d6vsvpskcu5g9w.
M. Gorbahn and U. Haisch, Effective Hamiltonian for non-leptonic $| Δ F | = 1$ decays at NNLO in QCD, Nucl. Phys. B713, 291 (2005).

Phys. Rev. D 114, 094503– Published 5 November, 2026

Vol. 114, Iss. 9 — 1 November 2026

Received 16 April 2026
Accepted 24 September 2026

DOI: https://doi.org/10.1103/47j9-h8s5

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.

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Heavy-meson bag parameters using gradient flow

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

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Heavy-meson bag parameters using gradient flow

Abstract

Physics Subject Headings (PhySH)

See Also

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

Article Text

Supplemental Material

References (272)

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