pith. sign in

arxiv: 2606.02182 · v1 · pith:S3QX7K7Lnew · submitted 2026-06-01 · ✦ hep-ph · hep-ex

Semi-analytical two-loop QCD corrections to e^+e^-to J/psi+chi_(cJ) at B factories

Hao-Yang Liu , Cong Li , Wen-Long Sang This is my paper

Pith reviewed 2026-06-28 13:44 UTC · model grok-4.3

classification ✦ hep-ph hep-ex
keywords NNLO QCD correctionscharmonium productionNRQCD factorizationB factoriesJ/psi chi_cJhelicity amplitudesasymptotic expansionsangular distributions
0
0 comments X

The pith

NNLO QCD corrections to J/ψ + χ_c0 production match Belle data while cutting scale uncertainties.

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

This paper calculates the next-to-next-to-leading order QCD corrections to electron-positron production of J/ψ plus χ_cJ charmonium pairs at B-factory energies inside the NRQCD factorization framework. The authors obtain the helicity amplitudes through asymptotic expansions around the mass-ratio limits r=0 and r=1, which reproduce exact numerical results to better than 10^{-5} relative accuracy and supply the large logarithms analytically. The O(α_s²) term amounts to 33 percent of the leading-order cross section for the χ_c0 channel and substantially reduces renormalization-scale dependence; the corresponding corrections for χ_c1 and χ_c2 are minus 15 percent and minus 38 percent. The resulting NNLO rate for χ_c0 agrees with Belle measurements, yet the predicted angular parameter α^0_θ differs sharply from the same data.

Core claim

Within NRQCD factorization the NNLO short-distance coefficients for e⁺e⁻ → J/ψ + χ_cJ are evaluated via asymptotic expansions of the helicity amplitudes around r=0 and r=1. These expansions reproduce the exact results across 0 ≤ r ≤ 1 with relative error below 10^{-5}. The unpolarized cross sections receive O(α_s) corrections of order 25–35 percent; the O(α_s²) pieces are +33 percent for χ_c0, −15 percent for χ_c1, and −38 percent for χ_c2. The NNLO χ_c0 rate is consistent with Belle data and lies within 2σ of BaBar data, while the angular parameters α^J_θ, which are free of long-distance matrix elements, exhibit a sharp discrepancy for J=0.

What carries the argument

Asymptotic expansions of the two-loop helicity amplitudes around r=0 and r=1 that deliver semi-analytical NNLO short-distance coefficients with controlled logarithmic terms.

If this is right

  • The NNLO cross section for χ_c0 production lies within experimental errors of the Belle measurement.
  • Inclusion of the two-loop term reduces the renormalization-scale uncertainty for the χ_c0 rate.
  • The angular parameters α^J_θ are independent of nonperturbative matrix elements and can be compared directly to data.
  • Large cancellations between NLO and NNLO corrections for χ_c2 bring the NNLO cross section close to the leading-order value.
  • Future angular-distribution measurements for χ_c1 and χ_c2 will furnish independent tests of the factorization framework.

Where Pith is reading between the lines

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

  • If the α^0_θ discrepancy survives improved data, it may indicate that polarization observables receive larger higher-order or relativistic corrections than total rates.
  • The asymptotic-expansion technique can be reused for other double-charmonium channels where direct two-loop integration remains difficult.
  • Agreement on rates combined with disagreement on angles suggests total cross sections are more stable under the factorization assumption than angular observables.
  • Tighter experimental bounds on the χ_c1 and χ_c2 rates at NNLO would help extract the relevant long-distance matrix elements with smaller theoretical error.

Load-bearing premise

NRQCD factorization continues to hold at NNLO, so the short-distance coefficients can be multiplied by the long-distance matrix elements without large additional power-suppressed corrections.

What would settle it

A new measurement of the cross section for e⁺e⁻ → J/ψ + χ_c0 at 10.58 GeV that lies outside the NNLO band after scale variation, or a confirmation of the α^0_θ discrepancy at >3σ with higher statistics, would test the claim.

read the original abstract

In this work, we compute the next-to-next-to-leading-order (NNLO) QCD corrections to the process $e^+e^-\to J/\psi+\chi_{cJ}$ at B factories within the NRQCD factorization framework. The helicity amplitudes are obtained via asymptotic expansions around $r=0$ and $r=1$, with $r=16m_c^2/s$. Our asymptotic expressions reproduce the exact numerical results with high accuracy across the entire range $0\le r \le 1$, achieving a relative error below $10^{-5}$, which is sufficient for phenomenological applications. Notably, the large logarithmic terms are obtained analytically. We compute the unpolarized cross sections. The $\mathcal{O}(\alpha_s)$ correction is found to be large, while the $\mathcal{O}(\alpha_s^2)$ correction for $\chi_{c0}$ production amounts to $33\%$ of the leading-order (LO) cross section, significantly reducing the scale uncertainties. For $\chi_{c1}$, the $\mathcal{O}(\alpha_s)$ and $\mathcal{O}(\alpha_s^2)$ corrections correspond to $35\%$ and $-15\%$, respectively. For $\chi_{c2}$, the corresponding corrections are $25\%$ and $-38\%$. The large cancellation between the corrections for $\chi_{c2}$ brings the NNLO cross section close to the LO prediction. Our prediction for $\chi_{c0}$ is consistent with the {\tt Belle} measurement and agrees with the {\tt BaBar} data within $2\sigma$. We also predict the angular distribution parameters $\alpha^J_\theta$, which are independent of nonperturbative inputs. A sharp discrepancy between the theory and the {\tt Belle} measurement is observed for $\alpha^0_\theta$, calling for further experimental and theoretical investigations. Moreover, future measurements of the angular distribution parameters for $\chi_{c1}$ and $\chi_{c2}$ will provide important tests of the theoretical framework.

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 / 3 minor

Summary. The paper computes NNLO QCD corrections to e+e− → J/ψ + χcJ at B factories in NRQCD factorization. Helicity amplitudes are obtained via asymptotic expansions around r=0 and r=1 (with r=16mc²/s), which the authors state reproduce exact numerical results to relative accuracy better than 10^{-5} across 0 ≤ r ≤ 1, including analytic capture of large logarithms. Unpolarized cross sections are reported with O(αs) corrections of 25–35% and O(αs²) corrections of +33%, −15%, and −38% for J=0,1,2 respectively; the NNLO results reduce scale dependence and the χc0 prediction is stated to be consistent with Belle while α0θ shows a sharp discrepancy with data. Angular parameters αJθ are predicted independently of LDMEs.

Significance. If the stated validation of the asymptotic expansions holds, this supplies the first NNLO short-distance coefficients for this process, materially improving perturbative precision and scale stability for charmonium-associated production at B factories. The LDME-independent angular-distribution predictions constitute a clean test of the framework and the explicit size of the NNLO terms (e.g., the 33% correction for χc0) quantifies the importance of higher-order QCD effects.

minor comments (3)
  1. The manuscript should include an explicit table or figure comparing the asymptotic results to the independent numerical integration for a representative set of r values (including near the endpoints) so that the quoted 10^{-5} relative accuracy can be directly inspected.
  2. Notation for the renormalization and factorization scales (μR, μF) and the precise choice of central scale should be stated once in §2 or §3 rather than only in the numerical-results section.
  3. A brief statement on the treatment of the charm-quark mass (on-shell vs. MS-bar) and its numerical value would remove ambiguity when readers reproduce the quoted percentage corrections.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments are listed in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation proceeds from standard Feynman-diagram evaluation of short-distance coefficients in NRQCD, followed by asymptotic expansions around r=0 and r=1 that are explicitly validated against independent numerical integration to relative accuracy <10^{-5}. The resulting O(α_s²) corrections (e.g., +33% for χ_c0) and angular parameters α^J_θ are direct outputs of this perturbative computation; LDMEs are external inputs, and no parameters are fitted to the Belle/BaBar data that are later compared. No self-citation, ansatz smuggling, or renaming of known results reduces any load-bearing step to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of NRQCD factorization at NNLO (standard domain assumption) and on the accuracy of the asymptotic expansions matching the full two-loop integrals (ad-hoc to the paper). No new free parameters are introduced beyond the usual renormalization scale; no new particles or forces are postulated.

axioms (2)
  • domain assumption NRQCD factorization holds at NNLO for the process e+e- -> J/psi + chi_cJ, allowing separation of short-distance coefficients from long-distance matrix elements.
    Invoked when the computed perturbative corrections are folded with LDMEs to obtain cross sections and when angular parameters are stated to be independent of non-perturbative inputs.
  • ad hoc to paper The asymptotic expansions around r=0 and r=1 capture all large logarithmic terms and reproduce the exact two-loop result to the stated precision.
    Central technical assumption stated in the abstract as achieving relative error below 10^-5.

pith-pipeline@v0.9.1-grok · 5919 in / 1873 out tokens · 24101 ms · 2026-06-28T13:44:10.420071+00:00 · methodology

discussion (0)

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

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Semi-analytical results for $e^+e^-\to J/\psi + X_{{\rm non\,}c\bar{c}}$ up to $\mathcal{O}(\alpha_s v^2)$ at B factories

    hep-ph 2026-06 unverdicted novelty 6.0

    NRQCD calculation of O(α_s), O(v²) and first-time O(α_s v²) corrections to e⁺e⁻ → J/ψ + X_non c c̄, with r = m_c/√s expansions to r⁴⁰, yielding cross section 0.530 pb consistent with Belle but angular parameter deviating >2σ.

Reference graph

Works this paper leans on

50 extracted references · 32 linked inside Pith · cited by 1 Pith paper

  1. [1]

    Braaten and J

    E. Braaten and J. Lee,Exclusive Double Charmonium Production frome +e− Annihilation into a Virtual Photon,Phys. Rev. D67(2003) 054007 [hep-ph/0211085]

    Pith/arXiv arXiv 2003

  2. [2]

    Hagiwara, E

    K. Hagiwara, E. Kou and C.-F. Qiao,ExclusiveJ/ψproductions ate +e− colliders,Phys. Lett. B570(2003) 39 [hep-ph/0305102]

    Pith/arXiv arXiv 2003

  3. [3]

    Liu, Z.-G

    K.-Y. Liu, Z.-G. He and K.-T. Chao,Problems of double charm production in e+ e- annihilation at s**(1/2) = 10.6-GeV,Phys. Lett. B557(2003) 45 [hep-ph/0211181]

    Pith/arXiv arXiv 2003

  4. [4]

    Zhang, Y.-j

    Y.-J. Zhang, Y.-j. Gao and K.-T. Chao,Next-to-leading order QCD correction to e+ e- —> J / psi + eta(c) at s**(1/2) = 10.6-GeV,Phys. Rev. Lett.96(2006) 092001 [hep-ph/0506076]

    Pith/arXiv arXiv 2006

  5. [5]

    Gong and J.-X

    B. Gong and J.-X. Wang,QCD corrections toJ/ψplusη c production ine +e− annihilation atS (1/2) = 10.6-GeV,Phys. Rev. D77(2008) 054028 [0712.4220]

    Pith/arXiv arXiv 2008

  6. [6]

    Z.-G. He, Y. Fan and K.-T. Chao,Relativistic corrections to J/psi exclusive and inclusive double charm production at B factories,Phys. Rev. D75(2007) 074011 [hep-ph/0702239]

    Pith/arXiv arXiv 2007

  7. [7]

    H.-R. Dong, F. Feng and Y. Jia,O(α sv2)correction toe +e− →J/ψ+η c atBfactories, Phys. Rev. D85(2012) 114018 [1204.4128]. – 27 –

    Pith/arXiv arXiv 2012

  8. [8]

    Li and J.-X

    X.-H. Li and J.-X. Wang,O(α sυ2)correction toJ/ψplusη c production ine +e− annihilation at √s=10.6 GeV,Chin. Phys. C38(2014) 043101 [1301.0376]

    Pith/arXiv arXiv 2014

  9. [9]

    Bodwin, J

    G.T. Bodwin, J. Lee and C. Yu,Resummation of Relativistic Corrections to e+ e- —>J/psi + eta(c),Phys. Rev. D77(2008) 094018 [0710.0995]. [12]Bellecollaboration,Study of double charmonium production in e+ e- annihilation at s**(1/2) ˜ 10.6-GeV,Phys. Rev. D70(2004) 071102 [hep-ex/0407009]

    Pith/arXiv arXiv 2008

  10. [10]

    Zhang, Y.-Q

    Y.-J. Zhang, Y.-Q. Ma and K.-T. Chao,Factorization and NLO QCD correction in e+e− →J/ψ(ψ(2S)) +χ c0 at B Factories,Phys. Rev. D78(2008) 054006 [0802.3655]

    Pith/arXiv arXiv 2008

  11. [11]

    Wang, Y.-Q

    K. Wang, Y.-Q. Ma and K.-T. Chao,QCD corrections to e+e− →J/ψ(ψ(2S)) +χ cj(J= 0,1,2)at B Factories,Phys. Rev. D84(2011) 034022 [1107.2646]

    Pith/arXiv arXiv 2011

  12. [12]

    H.-R. Dong, F. Feng and Y. Jia,O(α s)corrections toJ/ψ+χ cJ production atBfactories, JHEP10(2011) 141 [1107.4351]

    Pith/arXiv arXiv 2011

  13. [13]

    Wang, X.-G

    S.-Q. Wang, X.-G. Wu, X.-C. Zheng, J.-M. Shen and Q.-L. Zhang,J/ψ+χ cJ Production at theBFactories under the Principle of Maximum Conformality,Nucl. Phys. B876(2013) 731 [1301.2992]

    Pith/arXiv arXiv 2013

  14. [14]

    Jiang and Z

    Y. Jiang and Z. Sun,Further studies on the exclusive productions ofJ/ψ+χ cJ (J= 0,1,2) viae +e− annihilation at theBfactories,Eur. Phys. J. C78(2018) 892 [1809.09071]

    Pith/arXiv arXiv 2018

  15. [15]

    Sun,Next-to-leading-order study ofJ/ψangular distributions ine +e− →J/ψ+η c, χcJ at√s≈10.6GeV,JHEP09(2021) 073 [2107.02047]

    Z. Sun,Next-to-leading-order study ofJ/ψangular distributions ine +e− →J/ψ+η c, χcJ at√s≈10.6GeV,JHEP09(2021) 073 [2107.02047]

    arXiv 2021

  16. [16]

    L.-B. Chen, Y. Liang and C.-F. Qiao,NNLO QCD corrections toγ+η c(ηb)exclusive production in electron-positron collision,JHEP01(2018) 091 [1710.07865]

    Pith/arXiv arXiv 2018

  17. [17]

    W.-L. Sang, F. Feng and Y. Jia,Next-to-next-to-leading-order radiative corrections toe+e− →χ cJ +γat B factory,JHEP10(2020) 098 [2008.04898]

    arXiv 2020

  18. [18]

    Li, W.-L

    C. Li, W.-L. Sang and H.-F. Zhang,The next-to-next-to-leading-order QCD corrections to e+e− →η c/χcJ +γat B factories,JHEP05(2026) 163 [2512.04758]

    arXiv 2026

  19. [19]

    F. Feng, Y. Jia, Z. Mo, W.-L. Sang and J.-Y. Zhang,Next-to-next-to-leading-order QCD corrections toe +e− →J/ψη c at B factories,Phys. Lett. B850(2024) 138506 [1901.08447]

    arXiv 2024

  20. [20]

    W.-L. Sang, F. Feng, Y. Jia, Z. Mo and J.-Y. Zhang,O(α 2 s)corrections toJ/ψ+χ c0,1,2 production atBfactories,Phys. Lett. B843(2023) 138057 [2202.11615]

    arXiv 2023

  21. [21]

    Huang, B

    X.-D. Huang, B. Gong and J.-X. Wang,Next-to-next-to-leading-order QCD corrections to J/ψplusη c production at the B factories,JHEP02(2023) 049 [2212.03631]

    arXiv 2023

  22. [22]

    Li, X.-D

    C. Li, X.-D. Huang and W.-L. Sang,Two loop QCD corrections to e+e−→J/ψ+ηc in asymptotic expansion,Phys. Lett. B873(2026) 140150 [2506.16317]

    arXiv 2026

  23. [23]

    X. Chen, X. Guan, C.-Q. He, Y.-Q. Ma, J. Wang and D.-J. Zhang,Analytical two-loop amplitudes of e+e-→J/ψ+ηc at B factories,Phys. Rev. D113(2026) 074023 [2508.20777]

    arXiv 2026

  24. [24]

    W.-L. Sang, F. Feng, Y. Jia, Z. Mo, J. Pan and J.-Y. Zhang,Optimized O(αs2) Correction to Exclusive Double-J/ψProduction at B Factories,Phys. Rev. Lett.131(2023) 161904 [2306.11538]

    arXiv 2023

  25. [25]

    Huang, B

    X.-D. Huang, B. Gong, R.-C. Niu, H.-M. Yu and J.-X. Wang,Next-to-next-to-leading-order QCD corrections to double J/ψproduction at the B factories,JHEP02(2024) 055 [2311.04751]. – 28 –

    arXiv 2024

  26. [26]

    Bodwin, E

    G.T. Bodwin, E. Braaten and G.P. Lepage,Rigorous QCD analysis of inclusive annihilation and production of heavy quarkonium,Phys. Rev. D51(1995) 1125 [hep-ph/9407339]

    Pith/arXiv arXiv 1995

  27. [27]

    Petrelli, M

    A. Petrelli, M. Cacciari, M. Greco, F. Maltoni and M.L. Mangano,NLO production and decay of quarkonium,Nucl. Phys. B514(1998) 245 [hep-ph/9707223]

    Pith/arXiv arXiv 1998

  28. [28]

    Bodwin and A

    G.T. Bodwin and A. Petrelli,Order-v 4 corrections toS-wave quarkonium decay,Phys. Rev. D66(2002) 094011 [hep-ph/0205210]

    Pith/arXiv arXiv 2002

  29. [29]

    Zhang, W.-L

    Y.-D. Zhang, W.-L. Sang and H.-F. Zhang,Higher-Order QCD Corrections toΥDecay into Double Charmonia,Phys. Rev. Lett.129(2022) 112002 [2205.06124]

    arXiv 2022

  30. [30]

    Zhang, F

    Y.-D. Zhang, F. Feng, W.-L. Sang and H.-F. Zhang,Next-to-leading-order QCD corrections to a vector bottomonium radiative decay into a charmonium,JHEP12(2021) 189 [2109.15223]

    arXiv 2021

  31. [31]

    Czarnecki and K

    A. Czarnecki and K. Melnikov,Two loop QCD corrections to the heavy quark pair production cross-section in e+ e- annihilation near the threshold,Phys. Rev. Lett.80(1998) 2531 [hep-ph/9712222]

    Pith/arXiv arXiv 1998

  32. [32]

    Beneke, A

    M. Beneke, A. Signer and V.A. Smirnov,Two loop correction to the leptonic decay of quarkonium,Phys. Rev. Lett.80(1998) 2535 [hep-ph/9712302]

    Pith/arXiv arXiv 1998

  33. [33]

    Hoang and P

    A.H. Hoang and P. Ruiz-Femenia,Heavy pair production currents with general quantum numbers in dimensionally regularized NRQCD,Phys. Rev. D74(2006) 114016 [hep-ph/0609151]

    Pith/arXiv arXiv 2006

  34. [34]

    W.-L. Sang, F. Feng, Y. Jia and S.-R. Liang,Next-to-next-to-leading-order QCD corrections toχ c0,2 →γγ,Phys. Rev. D94(2016) 111501 [1511.06288]

    Pith/arXiv arXiv 2016

  35. [35]

    Hahn,Generating Feynman diagrams and amplitudes with FeynArts 3,Comput

    T. Hahn,Generating Feynman diagrams and amplitudes with FeynArts 3,Comput. Phys. Commun.140(2001) 418 [hep-ph/0012260]

    Pith/arXiv arXiv 2001

  36. [36]

    Mertig, M

    R. Mertig, M. Bohm and A. Denner,FEYN CALC: Computer algebraic calculation of Feynman amplitudes,Comput. Phys. Commun.64(1991) 345

    1991

  37. [37]

    Feng and R

    F. Feng and R. Mertig,FormLink/FeynCalcFormLink : Embedding FORM in Mathematica and FeynCalc,1212.3522

    Pith/arXiv arXiv

  38. [38]

    The CalcLoop package:https://gitlab.com/multiloop-pku/calcloop

  39. [39]

    Liu, Y.-Q

    X. Liu, Y.-Q. Ma and C.-Y. Wang,A Systematic and Efficient Method to Compute Multi-loop Master Integrals,Phys. Lett. B779(2018) 353 [1711.09572]

    Pith/arXiv arXiv 2018

  40. [40]

    Liu and Y.-Q

    X. Liu and Y.-Q. Ma,AMFlow: A Mathematica package for Feynman integrals computation via auxiliary mass flow,Comput. Phys. Commun.283(2023) 108565 [2201.11669]

    arXiv 2023

  41. [41]

    Liu and Y.-Q

    Z.-F. Liu and Y.-Q. Ma,Determining Feynman Integrals with Only Input from Linear Algebra,Phys. Rev. Lett.129(2022) 222001 [2201.11637]

    arXiv 2022

  42. [42]

    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]

    Pith/arXiv arXiv 2021

  43. [43]

    X. Guan, X. Liu, Y.-Q. Ma and W.-H. Wu,Blade: A package for block-triangular form improved Feynman integrals decomposition,Comput. Phys. Commun.310(2025) 109538 [2405.14621]

    arXiv 2025

  44. [44]

    Smirnov,FIRE5: A C++ implementation of Feynman Integral REduction,Comput

    A.V. Smirnov,FIRE5: A C++ implementation of Feynman Integral REduction,Comput. Phys. Commun.189(2015) 182 [1408.2372]. – 29 –

    Pith/arXiv arXiv 2015

  45. [45]

    Binosi, J

    D. Binosi, J. Collins, C. Kaufhold and L. Theussl,JaxoDraw: A Graphical user interface for drawing Feynman diagrams. Version 2.0 release notes,Comput. Phys. Commun.180(2009) 1709 [0811.4113]

    Pith/arXiv arXiv 2009

  46. [46]

    X. Chen, X. Guan, C.-Q. He and Y.-Q. Ma,Two-loop QCD corrections to e+e− →Z ∗ →J/ψ+J/ψ,Phys. Rev. D113(2026) 094017. [50]Particle Data Groupcollaboration,Review of particle physics,Phys. Rev. D110(2024) 030001

    2026

  47. [47]

    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

  48. [48]

    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

  49. [49]

    Bodwin, H.S

    G.T. Bodwin, H.S. Chung, D. Kang, J. Lee and C. Yu,Improved determination of color-singlet nonrelativistic QCD matrix elements for S-wave charmonium,Phys. Rev. D77 (2008) 094017 [0710.0994]

    Pith/arXiv arXiv 2008

  50. [50]

    Chung, J

    H.S. Chung, J. Lee and C. Yu,Exclusive heavy quarkonium + gamma production from e+ e- annihilation into a virtual photon,Phys. Rev. D78(2008) 074022 [0808.1625]. – 30 –

    Pith/arXiv arXiv 2008