pith. sign in

arxiv: 2606.25733 · v1 · pith:NCODUEVInew · submitted 2026-06-24 · ✦ hep-ph

Decay constants of B_c-mesons with vector and tensor currents

Eduard Costa i Reina , Nico Gubernari , Zachary W\"uthrich This is my paper

Pith reviewed 2026-06-25 20:27 UTC · model grok-4.3

classification ✦ hep-ph
keywords B_c mesonsdecay constantsQCD sum rulestensor currentsvector currentsV_cbheavy-light mesons
0
0 comments X

The pith

QCD sum rules yield the first tensor decay constants for the B_c^* and B_c1 mesons while refining the vector ones.

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

The paper calculates the decay constants of the lowest-lying B_c mesons in the 0^-, 0^+, 1^-, and 1^+ channels using QCD sum rules applied to two-point correlation functions. It refines prior values for the vector and axial-vector decay constants by incorporating higher-order power corrections and a full uncertainty analysis. It also delivers the first determinations of the corresponding tensor and axial-tensor decay constants. These quantities enter purely leptonic decay rates and help tighten unitarity bounds on b to c transition form factors that are used to extract |V_cb|.

Core claim

Within the QCD sum-rule framework, the (axial-)tensor decay constants of the B_c^* and B_c1 mesons are obtained for the first time; simultaneously, the (axial-)vector decay constants are updated with higher-order power corrections and a comprehensive uncertainty analysis.

What carries the argument

Two-point correlation functions of (axial-)vector and (axial-)tensor interpolating currents, evaluated via the operator product expansion truncated at dimension-six condensates, Borel transformation, and a continuum model.

If this is right

  • Theoretical predictions for purely leptonic B_c decays become more precise.
  • Unitarity constraints on b to c form factors are strengthened.
  • Predictions for semileptonic B-bar to D^{(*)} lepton-neutrino decays gain accuracy.
  • The extracted value of |V_cb| from exclusive decays improves.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The new tensor constants supply independent benchmarks that lattice groups can target to cross-check sum-rule methods.
  • The same correlation-function setup could be reused for excited B_c states or for analogous constants in other heavy-light systems.
  • Tighter |V_cb| bounds from these inputs may help quantify or reduce existing tensions between exclusive and inclusive determinations.

Load-bearing premise

The operator-product expansion truncated at dimension-six condensates together with the chosen continuum model and Borel window adequately captures the non-perturbative physics of the B_c system.

What would settle it

A lattice QCD calculation that produces tensor decay constants for the B_c^* or B_c1 differing by several standard deviations from the sum-rule central values would falsify the new results.

Figures

Figures reproduced from arXiv: 2606.25733 by Eduard Costa i Reina, Nico Gubernari, Zachary W\"uthrich.

Figure 1
Figure 1. Figure 1: Perturbative diagrams entering the leading-power Wilson coefficient [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Background-field diagrams contributing to the dimension-four terms in the local OPE. [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Background-field diagrams contributing to the dimension-six terms in the local OPE. The [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Borel stability of the decay constants extracted from the six QCD sum rules. Each panel [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of our QCD sum rule predictions for all six [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Pole contribution for the six QCD sum rules as a function of the Borel parameter [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Relative mass deviation εm obtained from the daughter sum rules in the six channels, evaluated at the representative renormalisation scale µ = 6 GeV. The shaded band indicates the working window M2 ∈ [6, 9] GeV2 used in the final analysis. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗
read the original abstract

We calculate the decay constants of the lowest-lying $B_c$ mesons in the spin-parity channels $J^P = 0^-,\,0^+,\,1^-,\,1^+$, commonly referred to as the $B_c,\, B_{c0}^*,\, B_c^*,\, B_{c1}$ mesons, respectively. Within the framework of QCD sum rules, we consider the decay constants associated with both the (axial-)vector and (axial-)tensor interpolating currents. The decay constants with (axial-)vector currents have already been studied in the literature. We refine previous QCD sum rule results by including higher-order power corrections and performing a comprehensive uncertainty analysis. Furthermore, we provide the first determination of the (axial-)tensor decay constants of the $B_c^*$ and $B_{c1}$ mesons. These new results will not only improve theoretical predictions for purely leptonic $B_c$ decays, but also strengthen unitarity constraints on $b \to c$ form factors, thereby improving the precision of predictions for $\bar{B} \to D^{(*)} \ell \bar\nu$ decays and the determination of \(|V_{cb}|\).

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.

Referee Report

0 major / 2 minor

Summary. The paper calculates the decay constants of the lowest-lying B_c mesons (B_c, B_{c0}^*, B_c^*, B_{c1}) in the J^P = 0^-, 0^+, 1^-, 1^+ channels using QCD sum rules. It considers both (axial-)vector and (axial-)tensor interpolating currents, refines prior results for the vector/axial-vector cases by including higher-order power corrections and a comprehensive uncertainty analysis, and provides the first determinations of the tensor decay constants for B_c^* and B_{c1}. These are intended to improve predictions for leptonic B_c decays and unitarity constraints on b → c form factors relevant to |V_cb|.

Significance. If the results hold, the refined vector/axial-vector values and the new tensor determinations supply useful inputs for phenomenology, particularly leptonic widths and constraints on semileptonic form factors that enter |V_cb| extractions from B → D^{(*)} ℓ ν. The extension to tensor currents within the same sum-rule framework is a clear addition to the existing literature on B_c systems.

minor comments (2)
  1. The abstract states that higher-order power corrections are included, but the main text should explicitly list the highest dimension retained in the OPE (e.g., dimension 6 or 8) and show the numerical size of each term in a table or figure for at least one channel to allow readers to judge convergence.
  2. The uncertainty analysis is described as comprehensive; the manuscript should tabulate the individual contributions (Borel mass variation, s_0 variation, condensate uncertainties, etc.) for each decay constant so that the dominant source of error is transparent.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments appear in the provided report, so we have no individual points requiring point-by-point rebuttal or revision at this stage.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper applies the established QCD sum-rule framework (OPE matched to hadronic dispersion relation) to compute decay constants for vector/axial-vector and tensor currents, refining prior results with higher-order power corrections and providing first tensor determinations. No load-bearing step reduces by construction to a self-citation, fitted input renamed as prediction, or self-definitional relation; the central outputs are independent extractions within the standard setup, with no quoted equations showing equivalence to inputs.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The calculation rests on the standard QCD sum-rule framework: the operator-product expansion of a two-point correlation function, quark-hadron duality, and a truncated set of vacuum condensates. No new entities are postulated. Free parameters are the usual auxiliary quantities (Borel parameter, continuum threshold) plus the numerical values of the condensates and quark masses taken from external literature.

free parameters (3)
  • Borel parameter M^2
    Standard auxiliary parameter in QCD sum rules; its window is chosen to optimize stability.
  • continuum threshold s0
    Models the onset of the continuum; fitted or chosen to reproduce known masses or to stabilize the sum rule.
  • quark condensate values
    Taken from external determinations; enter the power corrections.
axioms (2)
  • domain assumption Quark-hadron duality holds above the continuum threshold
    Standard assumption of the QCD sum-rule method; invoked when equating the OPE side to the phenomenological side.
  • domain assumption Operator-product expansion truncated at dimension six is sufficient
    Higher-dimensional condensates are neglected; stated implicitly by the inclusion of 'higher-order power corrections' up to a given order.

pith-pipeline@v0.9.1-grok · 5744 in / 1818 out tokens · 19643 ms · 2026-06-25T20:27:31.070270+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

45 extracted references · 1 canonical work pages

  1. [1]

    J. de Jong,Feasibility study of the branching fraction measurements ofB + c →τ +νand B+ c →τ +νat LHCb, Master’s Thesis, University of Groningen, Groningen, The Netherlands, June, 2022

    2022

  2. [2]

    Galati, K

    M.D. Galati, K. De Bruyn, M. Mulder and M. van Veghel,Looking forward toB + →τ +ντ andB + c →τ +ντ, in18th International Workshop on Tau Lepton Physics, 1, 2026 [2602.00335]

    arXiv 2026

  3. [3]

    Galati,Looking forward toB + c →τ +ντ, Ph.D

    M.D. Galati,Looking forward toB + c →τ +ντ, Ph.D. thesis, U. Groningen (main), 3, 2026

    2026

  4. [4]

    Amhis, M

    Y. Amhis, M. Hartmann, C. Helsens, D. Hill and O. Sumensari,Prospects forB + c →τ +ντ at FCC-ee,JHEP12(2021) 133 [2105.13330]

    arXiv 2021

  5. [5]

    C.G. Boyd, B. Grinstein and R.F. Lebed,Precision corrections to dispersive bounds on form-factors,Phys. Rev. D56(1997) 6895 [hep-ph/9705252]

    Pith/arXiv arXiv 1997

  6. [6]

    Caprini, L

    I. Caprini, L. Lellouch and M. Neubert,Dispersive bounds on the shape of ¯B→D (∗)ℓ¯ν form-factors,Nucl. Phys. B530(1998) 153 [hep-ph/9712417]

    Pith/arXiv arXiv 1998

  7. [7]

    Gubernari,Unitarity bounds and form-factor predictions forB-meson decays,2605.26213

    N. Gubernari,Unitarity bounds and form-factor predictions forB-meson decays,2605.26213

    Pith/arXiv arXiv

  8. [8]

    Bordone, N

    M. Bordone, N. Gubernari, M. Jung and D. van Dyk,Challenging B(s) →D (∗) (s) form factors with the heavy quark expansion,JHEP11(2025) 051 [2507.03569]

    arXiv 2025

  9. [9]

    Bordone and A

    M. Bordone and A. Juttner,New strategies for probingB→D ∗ℓ¯νℓ lattice and experimental data,Eur. Phys. J. C85(2025) 129 [2406.10074]. [10]HPQCDcollaboration,B-meson decay constants: a more complete picture from full lattice QCD,Phys. Rev. D91(2015) 114509 [1503.05762]. [11]ETMcollaboration,Masses and decay constants ofB (∗) c mesons withN f = 2 + 1 + 1twi...

    arXiv 2025

  10. [10]

    Colangelo, G

    P. Colangelo, G. Nardulli and N. Paver,QCD sum rules calculation ofB c decays,Z. Phys. C 57(1993) 43

    1993

  11. [11]

    Narison, mc and mb fromM Bc and improved estimates off Bc andf Bc(2S),Phys

    S. Narison, mc and mb fromM Bc and improved estimates off Bc andf Bc(2S),Phys. Lett. B 802(2020) 135221 [1906.03614]

    arXiv 2020

  12. [12]

    Wang,Analysis of the vector and axialvectorB c mesons with QCD sum rules,Eur

    Z.-G. Wang,Analysis of the vector and axialvectorB c mesons with QCD sum rules,Eur. Phys. J. A49(2013) 131 [1203.6252]

    Pith/arXiv arXiv 2013

  13. [13]

    Aliev, T

    T.M. Aliev, T. Barakat and S. Bilmis,Properties of excitedB c states in QCD,Nucl. Phys. B 947(2019) 114726 [1905.11750]

    arXiv 2019

  14. [14]

    Narison,Spectra and decay constants ofB c-like andB ∗ 0 mesons in QCD,Phys

    S. Narison,Spectra and decay constants ofB c-like andB ∗ 0 mesons in QCD,Phys. Lett. B807 (2020) 135522 [2004.03622]

    arXiv 2020

  15. [15]

    Wang,B c meson and its scalar cousin with QCD sum rules*,Chin

    Z.-G. Wang,B c meson and its scalar cousin with QCD sum rules*,Chin. Phys. C48(2024) 103104 [2401.12571]. [19]Particle Data Groupcollaboration,Review of particle physics,Phys. Rev. D110(2024) 030001. 23

    arXiv 2024

  16. [16]

    Dowdall, C.T.H

    R.J. Dowdall, C.T.H. Davies, T.C. Hammant and R.R. Horgan,Precise heavy-light meson masses and hyperfine splittings from lattice QCD including charm quarks in the sea,Phys. Rev. D86(2012) 094510 [1207.5149]. [21]ATLAScollaboration,Observation of aB ∗+ c meson with the ATLAS detector,2605.16228

    Pith/arXiv arXiv 2012

  17. [17]

    Shifman, A.I

    M.A. Shifman, A.I. Vainshtein and V.I. Zakharov,QCD and Resonance Physics. Theoretical Foundations,Nucl. Phys. B147(1979) 385

    1979

  18. [18]

    Reinders, H

    L.J. Reinders, H. Rubinstein and S. Yazaki,Hadron Properties from QCD Sum Rules,Phys. Rept.127(1985) 1

    1985

  19. [19]

    Colangelo and A

    P. Colangelo and A. Khodjamirian,QCD sum rules, a modern perspective,hep-ph/0010175

    Pith/arXiv arXiv

  20. [20]

    Khodjamirian,Hadron Form Factors, CRC Press (4, 2020), 10.1201/9781315142005

    A. Khodjamirian,Hadron Form Factors, CRC Press (4, 2020), 10.1201/9781315142005

    work page doi:10.1201/9781315142005 2020

  21. [21]

    Gelhausen, A

    P. Gelhausen, A. Khodjamirian, A.A. Pivovarov and D. Rosenthal,Radial excitations of heavy-light mesons from QCD sum rules,Eur. Phys. J. C74(2014) 2979 [1404.5891]

    Pith/arXiv arXiv 2014

  22. [22]

    Gubernari, A

    N. Gubernari, A. Khodjamirian, R. Mandal and T. Mannel,B→D 1(2420)and B→D ′ 1(2430)form factors from QCD light-cone sum rules,JHEP05(2022) 029 [2203.08493]

    arXiv 2022

  23. [23]

    Gubernari, A

    N. Gubernari, A. Khodjamirian, R. Mandal and T. Mannel,B→D ∗ 0 andB s →D ∗ s0 form factors from QCD light-cone sum rules,JHEP12(2023) 015 [2309.10165]

    arXiv 2023

  24. [24]

    Bagan, H.G

    E. Bagan, H.G. Dosch, P. Gosdzinsky, S. Narison and J.M. Richard,Hadrons with charm and beauty,Z. Phys. C64(1994) 57 [hep-ph/9403208]

    Pith/arXiv arXiv 1994

  25. [25]

    Nikolaev and A.V

    S.N. Nikolaev and A.V. Radyushkin,Vacuum Corrections to QCD Charmonium Sum Rules: Basic Formalism andO(G 3)Results,Nucl. Phys. B213(1983) 285

    1983

  26. [26]

    Djouadi and P

    A. Djouadi and P. Gambino,Electroweak gauge bosons selfenergies: Complete QCD corrections,Phys. Rev. D49(1994) 3499 [hep-ph/9309298]

    Pith/arXiv arXiv 1994

  27. [27]

    Generet, N

    T. Generet, N. Gubernari and E. Loisa,Correlator with tensor currents and two masses at two loops,Eur. Phys. J. C86(2026) 112 [2509.02776]

    arXiv 2026

  28. [28]

    Nikolaev and A.V

    S.N. Nikolaev and A.V. Radyushkin,Method for Computing Higher Gluonic Power Corrections to QCD Charmonium Sum Rules,Phys. Lett. B110(1982) 476

    1982

  29. [29]

    Nikolaev and A.V

    S.N. Nikolaev and A.V. Radyushkin,QCD Charmonium Sum Rules up toO(G 4).,Phys. Lett. B124(1983) 243

    1983

  30. [30]

    Novikov, M.A

    V.A. Novikov, M.A. Shifman, A.I. Vainshtein and V.I. Zakharov,Calculations in External Fields in Quantum Chromodynamics. Technical Review,Fortsch. Phys.32(1984) 585

    1984

  31. [31]

    Schwinger,On gauge invariance and vacuum polarization,Phys

    J.S. Schwinger,On gauge invariance and vacuum polarization,Phys. Rev.82(1951) 664

    1951

  32. [32]

    Smolyakov,Farri’s Theorem for Nonabelian Gauge Lagrangians,Theor

    N.V. Smolyakov,Farri’s Theorem for Nonabelian Gauge Lagrangians,Theor. Math. Phys.50 (1982) 225

    1982

  33. [33]

    Patel,Package-X 2.0: A Mathematica package for the analytic calculation of one-loop integrals,Comput

    H.H. Patel,Package-X 2.0: A Mathematica package for the analytic calculation of one-loop integrals,Comput. Phys. Commun.218(2017) 66 [1612.00009]

    Pith/arXiv arXiv 2017

  34. [34]

    Shtabovenko, R

    V. Shtabovenko, R. Mertig and F. Orellana,FeynCalc 10: Do multiloop integrals dream of computer codes?,Comput. Phys. Commun.306(2025) 109357 [2312.14089]

    arXiv 2025

  35. [35]

    Shtabovenko,FeynHelpers: Connecting FeynCalc to FIRE and Package-X,Comput

    V. Shtabovenko,FeynHelpers: Connecting FeynCalc to FIRE and Package-X,Comput. Phys. Commun.218(2017) 48 [1611.06793]. 24

    Pith/arXiv arXiv 2017

  36. [36]

    Lee,Presenting LiteRed: a tool for the Loop InTEgrals REDuction,1212.2685

    R.N. Lee,Presenting LiteRed: a tool for the Loop InTEgrals REDuction,1212.2685

    Pith/arXiv arXiv

  37. [37]

    Ioffe,Condensates in quantum chromodynamics,Phys

    B.L. Ioffe,Condensates in quantum chromodynamics,Phys. Atom. Nucl.66(2003) 30 [hep-ph/0207191]. [43]Flavour Lattice A veraging Group (FLAG)collaboration,FLAG Review 2021,Eur. Phys. J. C82(2022) 869 [2111.09849]

    Pith/arXiv arXiv 2003

  38. [38]

    Gelhausen, A

    P. Gelhausen, A. Khodjamirian, A.A. Pivovarov and D. Rosenthal,Decay constants of heavy-light vector mesons from QCD sum rules,Phys. Rev. D88(2013) 014015 [1305.5432]

    Pith/arXiv arXiv 2013

  39. [39]

    Pullin and R

    B. Pullin and R. Zwicky,Radiative decays of heavy-light mesons and thef (T) H,H ∗,H1 decay constants,JHEP09(2021) 023 [2106.13617]

    arXiv 2021

  40. [40]

    Chetyrkin, J.H

    K.G. Chetyrkin, J.H. Kuhn and M. Steinhauser,RunDec: A Mathematica package for running and decoupling of the strong coupling and quark masses,Comput. Phys. Commun. 133(2000) 43 [hep-ph/0004189]

    Pith/arXiv arXiv 2000

  41. [41]

    Herren and M

    F. Herren and M. Steinhauser,Version 3 of RunDec and CRunDec,Comput. Phys. Commun. 224(2018) 333 [1703.03751]

    Pith/arXiv arXiv 2018

  42. [42]

    Wingate,Quark flavor physics and lattice QCD,Eur

    M. Wingate,Quark flavor physics and lattice QCD,Eur. Phys. J. A57(2021) 239 [2103.17224]

    arXiv 2021

  43. [43]

    Grigo, J

    J. Grigo, J. Hoff, P. Marquard and M. Steinhauser,Moments of heavy quark correlators with two masses: exact mass dependence to three loops,Nucl. Phys. B864(2012) 580 [1206.3418]

    Pith/arXiv arXiv 2012

  44. [44]

    D. Bigi, P. Gambino and S. Schacht,R(D ∗),|V cb|, and the Heavy Quark Symmetry relations between form factors,JHEP11(2017) 061 [1707.09509]

    Pith/arXiv arXiv 2017

  45. [45]

    Gopal and N

    A. Gopal and N. Gubernari,Unitarity bounds with subthreshold and anomalous cuts for b-hadron decays,Phys. Rev. D111(2025) L031501 [2412.04388]. 25

    arXiv 2025