Next-to-next-to-leading QCD corrections to the B^+-B_d⁰, D^+-D⁰, and D_s^+-D⁰ lifetime ratios

Francesco Moretti , Ulrich Nierste , Pascal Reeck , Matthias Steinhauser

Authors on Pith no claims yet

Pith reviewed 2026-05-08 02:52 UTC · model grok-4.3

classification ✦ hep-ph

keywords mathbftermsalphacontributionselementsfindhadronicheavy

0 comments

The pith

Three-loop perturbative corrections to B and D meson lifetime ratios are calculated, producing values that agree with experiment when using HQET sum rules or lattice inputs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Heavy mesons containing a bottom or charm quark decay at rates that differ slightly between charged and neutral versions. These differences arise mainly from effects suppressed by three powers of the heavy quark mass. The authors calculate the next-to-next-to-leading perturbative QCD pieces of these effects, which involve three-loop diagrams. They do this under exact isospin or V-spin symmetry. The resulting coefficients are then multiplied by hadronic matrix elements taken from separate calculations using heavy quark effective theory sum rules or lattice QCD. The final numbers for the lifetime ratios are 1.072 for B plus over B zero, 2.344 for D plus over D zero, and 1.289 for D_s plus over D zero. These match measured values within uncertainties. The agreement lets the authors estimate the size of still-missing higher-order terms.

Core claim

Combining our perturbative coefficients with hadronic matrix elements calculated from Heavy Quark Effective Theory sum rules, we find τ(B⁺)/τ(B_d⁰)=1.072±0.024. Using hadronic matrix elements from a recent lattice QCD calculation we find τ(D⁺)/τ(D⁰)=2.344±0.170 and τ(D_s⁺)/τ(D⁰)=1.289±0.042. We find good agreement of our predictions with experimental data.

Load-bearing premise

The hadronic matrix elements from HQET sum rules and lattice QCD are accurate enough that the 1/m_Q^4 and V-spin breaking contributions remain smaller than the quoted uncertainties; the calculation is performed in the exact isospin and V-spin symmetric limit.

read the original abstract

The total decay widths of heavy mesons can be systematically calculated in terms of an expansion in the two parameters $1/m_Q$ and $\alpha_s(m_Q)$, where $Q=c,b$ denotes the heavy quark. The dominant contributions to meson lifetime splittings stem from terms which are suppressed by $1/m_Q^3$ with respect to the leading universal contribution to the total decay width. We calculate three-loop contributions of order $\alpha_s^2/m_q^3$ to the lifetime ratios $\tau(B^+)/\tau(B_d^0)$, $\tau(D^+)/\tau(D^0)$, and $\tau(D_s^+)/\tau(D^0)$ in the limit of exact isospin and V-spin symmetry, respectively. Furthermore, we present new $\alpha_s/m_q^3$ corrections to the Cabibbo-suppressed terms in $\tau(B^+)/\tau(B_d^0)$. Combining our perturbative coefficients with hadronic matrix elements calculated from Heavy Quark Effective Theory sum rules, we find $\tau(B^+)/\tau(B_d^0)= {1.072 \pm 0.024 }$. Using hadronic matrix elements from a recent lattice QCD calculation we find $\tau(D^+)/\tau(D^0)={2.344 \pm 0.170} $ and $\tau(D_s^+)/\tau(D^0)={1.289 \pm 0.042}$. We find good agreement of our predictions with experimental data, which constitutes a successful probe of the calculations of hadronic matrix elements and permits estimates of the unknown $1/m_Q^4$ contributions as well as the V-spin breaking terms in $\tau(D_s^+)/\tau(D^0)$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper delivers new three-loop perturbative coefficients for B and D meson lifetime ratios that look technically solid, but the headline predictions still rest on external hadronic matrix elements whose uncertainties may not fully cover omitted higher-order terms.

read the letter

The core advance here is the calculation of the three-loop α_s²/m_q³ pieces for the lifetime ratios in the isospin and V-spin symmetric limits, plus the new α_s/m_q³ corrections to the Cabibbo-suppressed contributions in the B sector. Those coefficients were not available before, and the authors combine them with existing HQET sum-rule and lattice matrix elements to produce numerical predictions that sit close to the measured values: 1.072 ± 0.024 for B⁺/B_d⁰, 2.344 ± 0.170 for D⁺/D⁰, and 1.289 ± 0.042 for D_s⁺/D⁰. That is useful concrete output for the heavy-flavor community. The perturbative work itself appears careful and fills a clear gap in the literature. The agreement with data then serves as a consistency check on the matrix elements and gives a rough handle on the size of missing 1/m_Q⁴ pieces. The main limitation is that the quoted errors are driven almost entirely by the external hadronic inputs. For the charm sector the 1/m_c expansion parameter is not small, V-spin breaking is known to matter in other observables, and those effects are left out of the present calculation. The B-sector result is on firmer ground because the expansion converges better and isospin breaking is smaller. Readers who need the new coefficients for their own work will find them directly usable; the paper is aimed at people doing precision calculations in HQET or lattice QCD for lifetimes, mixing, or rare decays. It is technically grounded enough to deserve a serious referee who can check the diagram evaluation and the matching to the matrix elements.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The calculation rests on the heavy-quark expansion in 1/m_Q and α_s, exact isospin and V-spin symmetry, and the accuracy of external hadronic matrix elements; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Heavy quark expansion in powers of 1/m_Q and α_s(m_Q) is valid for bottom and charm mesons
Invoked in the first sentence of the abstract as the systematic framework for total decay widths.
domain assumption Exact isospin symmetry for B mesons and exact V-spin symmetry for D mesons
Explicitly stated as the limit in which the three-loop contributions are computed.

pith-pipeline@v0.9.0 · 5662 in / 1484 out tokens · 105049 ms · 2026-05-08T02:52:16.910883+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

58 extracted references · 48 canonical work pages · 2 internal anchors

[1]

V. A. Khoze and M. A. Shifman,Heavy quarks in:Heavy Quarksed. V. A. Khoze and M. A. Shifman,Soviet Physics Uspekhi26(1983) 387

1983
[2]

M. A. Shifman and M. B. Voloshin,Preasymptotic Effects in Inclusive Weak Decays of Charmed Particles,Sov. J. Nucl. Phys.41(1985) 120

1985
[3]

V. A. Khoze, M. A. Shifman, N. G. Uraltsev and M. B. Voloshin,On Inclusive Hadronic Widths of Beautiful Particles,Sov. J. Nucl. Phys.46(1987) 112

1987
[4]

M. A. Shifman and M. B. Voloshin,Hierarchy of Lifetimes of Charmed and Beautiful Hadrons,Sov. Phys. JETP64(1986) 698. – 31 –

1986
[5]

I. I. Y. Bigi, N. G. Uraltsev and A. I. Vainshtein,Nonperturbative corrections to inclusive beauty and charm decays: QCD versus phenomenological models,Phys. Lett. B293 (1992) 430 [hep-ph/9207214]

work page arXiv 1992
[6]

Effective Hamiltonian for Non-Leptonic |Delta F| = 1 Decays at NNLO in QCD

M. Gorbahn and U. Haisch,Effective Hamiltonian for non-leptonic|∆F|= 1decays at NNLO in QCD,Nucl. Phys. B713(2005) 291 [hep-ph/0411071]

work page Pith review arXiv 2005
[7]

Beneke, G

M. Beneke, G. Buchalla, C. Greub, A. Lenz and U. Nierste,TheB +−B0 d Lifetime Difference Beyond Leading Logarithms,Nucl. Phys. B639(2002) 389 [hep-ph/0202106]

work page arXiv 2002
[8]

Franco, V

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(2002) 212 [hep-ph/0203089]. [9]Heavy Flavor A veraging Group (HFLA V)collaboration,Averages ofb-hadron, c-hadron, andτ-lepton properties as of 2023,2411.18639

work page arXiv 2002
[9]

M. Kirk, A. Lenz and T. Rauh,Dimension-six matrix elements for meson mixing and lifetimes from sum rules,JHEP12(2017) 068 [1711.02100]

work page arXiv 2017
[10]

Black, M

M. Black, M. Lang, A. Lenz and Z. W¨ uthrich,HQET sum rules for matrix elements of dimension-six four-quark operators for meson lifetimes within and beyond the Standard Model,JHEP04(2025) 081 [2412.13270]

work page arXiv 2025
[11]

Black, R

M. Black, R. V. Harlander, J. T. Kohnen, F. Lange, A. Rago, A. Shindler et al., Heavy-Meson Bag Parameters using Gradient Flow,2603.28517

work page arXiv
[12]

Black, R

M. Black, R. V. Harlander, J. T. Kohnen, F. Lange, A. Rago, A. Shindler et al.,Bag Parameters for Heavy Meson Lifetimes,2603.28516. [14]UKQCDcollaboration,A Lattice study of spectator effects in inclusive decays of B mesons,Nucl. Phys. B534(1998) 373 [hep-lat/9805028]. [15]UKQCDcollaboration,An Exploratory lattice study of spectator effects in inclusive d...

work page arXiv 1998
[13]

Becirevic,Theoretical progress in describing the B meson lifetimes,PoSHEP2001 (2001) 098 [hep-ph/0110124]

D. Becirevic,Theoretical progress in describing the B meson lifetimes,PoSHEP2001 (2001) 098 [hep-ph/0110124]

work page internal anchor Pith review arXiv 2001
[14]

J. Lin, W. Detmold and S. Meinel,Lattice Study of Spectator Effects inb-hadron Decays, PoSLA TTICE2022(2023) 417 [2212.09275]

work page arXiv 2023
[15]

Beneke, G

M. Beneke, G. Buchalla and I. Dunietz,Width Difference in theB s−¯Bs System,Phys. Rev. D54(1996) 4419 [hep-ph/9605259]

work page arXiv 1996
[16]

Beneke, G

M. Beneke, G. Buchalla, A. Lenz and U. Nierste,CP Asymmetry in Flavor Specific B Decays beyond Leading Logarithms,Phys. Lett. B576(2003) 173 [hep-ph/0307344]

work page arXiv 2003
[17]

Lenz and U

A. Lenz and U. Nierste,Theoretical update ofB s−¯Bs mixing,JHEP06(2007) 072 [hep-ph/0612167]

work page arXiv 2007
[18]

Gerlach, U

M. Gerlach, U. Nierste, V. Shtabovenko and M. Steinhauser,The width difference in B−¯Bmixing at orderα s and beyond,JHEP04(2022) 006 [2202.12305]. – 32 –

work page arXiv 2022
[19]

Gerlach, U

M. Gerlach, U. Nierste, P. Reeck, V. Shtabovenko and M. Steinhauser,Current-current operator contribution to the decay matrix inB-meson mixing at next-to-next-to-leading order of QCD,2505.22740

work page arXiv
[20]

Nierste, P

U. Nierste, P. Reeck, V. Shtabovenko and M. Steinhauser,Complete next-to-next-to-leading order QCD corrections to the decay matrix in B-meson mixing at leading power,JHEP03(2026) 094 [2512.07949]

work page arXiv 2026
[21]

Egner, M

M. Egner, M. Fael, K. Sch¨ onwald and M. Steinhauser,Nonleptonic B-meson decays to next-to-next-to-leading order,JHEP10(2024) 144 [2406.19456]

work page arXiv 2024
[22]

Egner, M

M. Egner, M. Fael, A. Lenz, M. L. Piscopo, A. V. Rusov, K. Sch¨ onwald et al.,Total decay rates of B mesons at NNLO-QCD,JHEP04(2025) 106 [2412.14035]

work page arXiv 2025
[23]

Neubert and C

M. Neubert and C. T. Sachrajda,Spectator effects in inclusive decays of beauty hadrons, Nucl. Phys. B483(1997) 339 [hep-ph/9603202]

work page arXiv 1997
[24]

Beneke, G

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. B459(1999) 631 [hep-ph/9808385]

work page arXiv 1999
[25]

Ciuchini, E

M. Ciuchini, E. Franco, V. Lubicz and F. Mescia,Next-to-leading order QCD corrections to spectator effects in lifetimes of beauty hadrons,Nucl. Phys. B625(2002) 211 [hep-ph/0110375]

work page arXiv 2002
[26]

F. J. Gilman and M. B. Wise,Effective Hamiltonian for Delta s = 1 Weak Nonleptonic Decays in the Six Quark Model,Phys. Rev. D20(1979) 2392

1979
[27]

A. J. Buras and P. H. Weisz,QCD Nonleading Corrections to Weak Decays in Dimensional Regularization and ’t Hooft-Veltman Schemes,Nucl. Phys. B333(1990) 66

1990
[28]

Keum and U

Y.-Y. Keum and U. Nierste,Probing penguin coefficients with the lifetime ratio τ(Bs)/τ(Bd),Phys. Rev. D57(1998) 4282 [hep-ph/9710512]

work page arXiv 1998
[29]

Herrlich and U

S. Herrlich and U. Nierste,Evanescent operators, scheme dependences and double insertions,Nucl. Phys. B455(1995) 39 [hep-ph/9412375]

work page arXiv 1995
[30]

H. M. Asatrian, A. Hovhannisyan, U. Nierste and A. Yeghiazaryan,Towards next-to-next-to-leading-log accuracy for the width difference in theB s−¯Bs system: fermionic contributions to order(m c/mb)0 and(m c/mb)1,JHEP10(2017) 191 [1709.02160]

work page arXiv 2017
[31]

A. J. Buras, M. Gorbahn, U. Haisch and U. Nierste,Charm quark contribution to K+→π+ν¯νat next-to-next-to-leading order,JHEP11(2006) 002 [hep-ph/0603079]

work page arXiv 2006
[32]

K. G. Chetyrkin, M. Misiak and M. Munz,|∆F|= 1nonleptonic effective Hamiltonian in a simpler scheme,Nucl. Phys. B520(1998) 279 [hep-ph/9711280]

work page arXiv 1998
[33]

A. J. Buras, M. Jamin, M. E. Lautenbacher and P. H. Weisz,Effective Hamiltonians for ∆S= 1and∆B= 1nonleptonic decays beyond the leading logarithmic approximation, Nucl. Phys. B370(1992) 69. – 33 –

1992
[34]

A. J. Buras, M. Jamin, M. E. Lautenbacher and P. H. Weisz,Two loop anomalous dimension matrix for∆S= 1weak nonleptonic decays I:O(α 2 s),Nucl. Phys. B400 (1993) 37 [hep-ph/9211304]

work page arXiv 1993
[35]

Ciuchini, E

M. Ciuchini, E. Franco, G. Martinelli and L. Reina,The Delta S = 1 effective Hamiltonian including next-to-leading order QCD and QED corrections,Nucl. Phys. B 415(1994) 403 [hep-ph/9304257]

work page arXiv 1994
[36]

Black, R

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,PoS LA TTICE2023(2024) 263 [2310.18059]

work page arXiv 2024
[37]

Black, R

M. Black, R. Harlander, F. Lange, A. Rago, A. Shindler and O. Witzel,Gradient Flow Renormalisation for Meson Mixing and Lifetimes,PoSLA TTICE2024(2025) 243 [2409.18891]

work page arXiv 2025
[38]

D. King, A. Lenz and T. Rauh,SU(3) breaking effects in B and D meson lifetimes,JHEP 06(2022) 134 [2112.03691]

work page arXiv 2022
[39]

Two-Loop Anomalous Dimensions in the LEFT: Dimension-Six Four-Fermion Operators in NDR

J. Aebischer, P. Morell, M. Pesut and J. Virto,Two-Loop Anomalous Dimensions in the LEFT: Dimension-Six Four-Fermion Operators in NDR,2501.08384

work page internal anchor Pith review Pith/arXiv arXiv
[40]

Ancillary files at:https://www.ttp.kit.edu/preprints/2026/ttp26-014/

2026
[41]

Kuipers, T

J. Kuipers, T. Ueda, J. A. M. Vermaseren and J. Vollinga,FORM version 4.0,Comput. Phys. Commun.184(2013) 1453–1467 [1203.6543]

work page arXiv 2013
[42]

Davies, T

J. Davies, T. Kaneko, C. Marinissen, T. Ueda and J. A. M. Vermaseren,FORM Version 5.0,2601.19982

work page arXiv
[43]

Nogueira,Abusing QGRAF,Nucl

P. Nogueira,Abusing QGRAF,Nucl. Instrum. Meth. A559(2006) 220

2006
[44]

Gerlach, F

M. Gerlach, F. Herren and M. Lang,tapir: A tool for topologies, amplitudes, partial fraction decomposition and input for reductions,Comput. Phys. Commun.282(2023) 108544 [2201.05618]

work page arXiv 2023
[45]

Reeck, V

P. Reeck, V. Shtabovenko and M. Steinhauser,B meson mixing at NNLO: technical aspects,JHEP08(2024) 002 [2405.14698]

work page arXiv 2024
[46]

Kira - A Feynman Integral Reduction Program

P. Maierh¨ ofer, J. Usovitsch and P. Uwer,Kira—A Feynman integral reduction program, Comput. Phys. Commun.230(2018) 99 [1705.05610]

work page Pith review arXiv 2018
[47]

Klappert, F

J. Klappert, F. Lange, P. Maierh¨ ofer and J. Usovitsch,Integral reduction with Kira 2.0 and finite field methods,Comput. Phys. Commun.266(2021) 108024 [2008.06494]

work page arXiv 2021
[48]

Lange, J

F. Lange, J. Usovitsch and Z. Wu,Kira 3: Integral reduction with efficient seeding and optimized equation selection,Comput. Phys. Commun.322(2026) 109999. [52]Particle Data Groupcollaboration,Review of particle physics,Phys. Rev. D110 (2024) 030001. – 34 –

2026
[49]

Chetyrkin, J

K. Chetyrkin, J. H. Kuhn, A. Maier, P. Maierhofer, P. Marquard, M. Steinhauser et al., Precise Charm- and Bottom-Quark Masses: Theoretical and Experimental Uncertainties, Theor. Math. Phys.170(2012) 217–228 [1010.6157]

work page arXiv 2012
[50]

Charm and bottom quark masses: An update

K. G. Chetyrkin, J. H. Kuhn, A. Maier, P. Maierhofer, P. Marquard, M. Steinhauser et al.,Addendum to “Charm and bottom quark masses: An update”,1710.04249. [55]Flavour Lattice A veraging Group (FLAG)collaboration,FLAG Review 2024, 2411.04268

work page arXiv 2024
[51]

Charleset al., C P violation and the CKM matrix: Assessing the impact of the asymmetric B factories, Eur

CKMfitter Group (J. Charles et al.), updated results and plots available at: http://ckmfitter.in2p3.fr,Eur. Phys. JC41(2005) 1–131 [hep-ph/0406184]

work page arXiv 2005
[52]

Beneke,A Quark mass definition adequate for threshold problems,Phys

M. Beneke,A Quark mass definition adequate for threshold problems,Phys. Lett. B434 (1998) 115 [hep-ph/9804241]

work page arXiv 1998
[53]

D. King, A. Lenz, M. L. Piscopo, T. Rauh, A. V. Rusov and C. Vlahos,Revisiting inclusive decay widths of charmed mesons,JHEP08(2022) 241 [2109.13219]

work page arXiv 2022
[54]

A. Lenz, M. L. Piscopo and A. V. Rusov,Disintegration of beauty: a precision study, JHEP01(2023) 004 [2208.02643]

work page arXiv 2023
[55]

Heavy Quark Effective Theory beyond Perturbation Theory: Renormalons, the Pole Mass and the Residual Mass Term

M. Beneke and V. M. Braun,Heavy quark effective theory beyond perturbation theory: Renormalons, the pole mass and the residual mass term,Nucl. Phys. B426(1994) 301 [hep-ph/9402364]

work page Pith review arXiv 1994
[56]

I. I. Y. Bigi, M. A. Shifman, N. G. Uraltsev and A. I. Vainshtein,The Pole mass of the heavy quark. Perturbation theory and beyond,Phys. Rev. D50(1994) 2234 [hep-ph/9402360]

work page arXiv 1994
[57]

Lenz and T

A. Lenz and T. Rauh,D-meson lifetimes within the heavy quark expansion,Phys. Rev. D 88(2013) 034004 [1305.3588]. [63]Belle-IIcollaboration,Precise measurement of theD 0 andD + lifetimes at Belle II, Phys. Rev. Lett.127(2021) 211801 [2108.03216]

work page arXiv 2013
[58]

A. S. Kronfeld and S. M. Ryan,Remark on the Theoretical Uncertainty inB 0 - ¯B0 Mixing,Phys. Lett. B543(2002) 59 [hep-ph/0206058]. – 35 –

work page arXiv 2002