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REVIEW 3 major objections 5 minor 37 references

$B_c$, $B_s$ and $D_s$ production at lepton-hadron colliders

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper claims that electroproduced $B_c$ mesons should be observable at the EIC, predicting 5–28 reconstructed events per year, while HERA cannot accumulate a reconstructable sample.

desk verdict First NRQCD electroproduction estimates for Bc, Bs, Ds at ep colliders; the Bc EIC rate is new but rests on an unverified LO amplitude, so treat the event numbers as indicative. read the letter →

arxiv 2501.00550 v1 pith:T4FP24VS submitted 2024-12-31 hep-ph

Qiwei Hu , Cong-Feng Qiao , Li-Ping Sun This is my paper · ORCID
classification hep-ph PACS 13.60.-r12.38.Bx
keywords B_cmesonelectroproductionNRQCDElectron-IonColliderHERAB_sD_sleading-ordercrosssection
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper predicts that the $B_c$ meson — a bound state of a bottom and a charm quark, and the only meson whose decay is purely weak — should be producible in observable numbers at the future Electron-Ion Collider (EIC), but not at the earlier HERA collider. Working at leading order in nonrelativistic QCD, the authors compute electroproduction cross sections of $B_c$ and $B_c^*$ along with the single-heavy-flavour mesons $B_s$ and $D_s$ at $ep$ colliders. At EIC energies the $B_c$ cross section is $\sigma(B_c)=5.44$ fb, which through the $B_c\to J/\psi\pi$, $J/\psi\to l^+l^-$ chain yields an estimated 5 to 28 reconstructed events per year — enough to make the first electroproduced $B_c$ observable. At HERA the $B_c$ rate is too small for reconstruction, while $B_s$ and $D_s$ are produced abundantly but with leading-order predictions that vary by more than a factor of two with quark mass and renormalisation scale.

What carries the argument

The calculation rests on the NRQCD factorisation formula $d\sigma = \int dx\, f_{g/p}(x,\mu^2)\, d\hat\sigma\, \langle O_M(n)\rangle$, which splits the process into a gluon PDF, a perturbative hard part, and a long-distance matrix element $\langle O_M\rangle = 2(2J+1)N_c |R(0)|^2/4\pi$. The hard part is built from 24 tree-level $\gamma^* g \to q\bar q' + q + \bar q'$ diagrams evaluated with covariant projection operators ($\gamma_5$ for pseudoscalar, $\not\epsilon$ for vector), with the leptonic tensor folded in and the four-body phase space decomposed into sequential two-body phase spaces for Monte Carlo efficiency. The long-distance matrix elements are fixed at leading order by $|R(0)|^2$ from a potential-model wave function for $B_c$ and from the decay constants $f_{B_s}$, $f_{D_s}$ for the single-heavy mesons; heavy-quark spin symmetry gives vector and pseudoscalar mesons the same $R(0)$. The comparison numbers come from the CT10 gluon PDF, HERA cuts ($p_T>1$ GeV, $0.3<z<0.9$, $2<Q^2<100$ GeV$^2$) and EIC Yellow Report cuts ($Q^2>1$ GeV$^2$, $20<W<80$ GeV, $0.05<z<0.9$).

What would settle it

Run the EIC for one nominal year (about 315 fb$^{-1}$) and search for $e+p\to e+B_c+\bar c+b$ with $B_c^\pm\to J/\psi\pi^\pm$ and $J/\psi\to l^+l^-$: fewer than about five reconstructed events, or a central cross-section value more than a factor of three below 5.44 fb, would falsify the central claim. Before any collider data exist, a lattice-QCD determination of the $B_c$ decay constant that contradicts the potential-model input $|R_{B_c}(0)|^2 = 1.642$ GeV$^3$ would undercut the same prediction.

Watch

Extended reading notes

Core claim

The central claim is that leading-order NRQCD electroproduction of the $B_c$ meson at the EIC is observable: with all $B_c^*$ decays feeding $B_c$, the predicted cross sections $\sigma(B_c)=5.44$ fb and $\sigma(B_c^*)=22.12$ fb translate into 3922–19209 $B_c$ mesons per year and, after the $B_c\to J/\psi\pi^\pm$ (0.5%) and $J/\psi\to l^+l^-$ (12%) branching cuts and counting both charge states, about 5–28 reconstructed events per year. The same calculation at HERA gives a larger cross section (42.13 fb) but insufficient accumulated events for reconstruction, while at the proposed EicC the cross section is below 0.03 fb. For $B_s$ and $D_s$, the paper reports that HERA produces them abundantly ($\sigma(B_s)=0.29$ pb, $\sigma(D_s)=9.35$ pb at leading order) but that these predictions are highly sensitive to the strange-quark mass and renormalisation scale, a sensitivity the authors read as a possible experimental window rather than as a failure.

Load-bearing premise

The whole rate estimate stands or falls on the assumption that a leading-order, colour-singlet NRQCD calculation, seeded by a potential-model wave function at the origin with $|R_{B_c}(0)|^2 = 1.642$ GeV$^3$ and assuming every $B_c^*$ decays to $B_c$, is accurate to within a factor of about three.

Editorial extensions

If this is right

  • At the EIC, the predicted 3922–19209 $B_c$ mesons per year translate into roughly 5–28 reconstructed $B_c^\pm$ events per year through $B_c\to J/\psi\pi^\pm$ with $J/\psi\to l^+l^-$, making the $B_c$ a reachable target for the first time in electroproduction.
  • At HERA the $B_c$ cross section is larger in absolute terms (42.13 fb) but the integrated luminosity is insufficient for reconstruction, so HERA cannot test the prediction.
  • The proposed EicC collider is ruled out for this purpose: its $B_c$ cross section is below 0.03 fb.
  • At HERA, $B_s$ and $D_s$ production is plentiful (roughly 48–189 $B_s^0$, 119–476 $B_s^*$, 1575–5100 $D_s^+$, and 3465–11440 $D_s^{*+}$ events per 315 pb$^{-1}$), enough for the main decay channels.
  • An order-of-magnitude luminosity upgrade (HL-EIC) would raise the expected $B_c$ yield correspondingly, strengthening the case for observation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The event-number claim is effectively a leading-order, colour-singlet estimate; if next-to-leading-order or colour-octet corrections shift the cross section down by roughly a factor of three, the 5–28 event prediction would essentially vanish, so the estimate is best read as an order-of-magnitude target awaiting NLO confirmation.
  • Because the $B_c^*$ cross section is about four times the $B_c$ one and feed-down is assumed complete, the same calculation predicts the EIC sample to be dominated by $B_c$ from vector-meson decay; a polarisation or momentum-distribution measurement of reconstructed $B_c$ would test that feed-down assumption directly.
  • The dramatic quark-mass sensitivity of the $B_s$ and $D_s$ cross sections (greater than 200% for a 0.1 GeV mass shift) could be inverted: precision $D_s$ electroproduction data at a future $ep$ collider would constrain the strange-quark mass or delimit where NRQCD's velocity expansion breaks down for single-heavy mesons.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper presents leading-order (LO) NRQCD predictions for the electroproduction of Bc, Bc*, Bs, Bs*, Ds, and Ds* in electron-proton collisions at HERA and at the EIC. The calculation uses standard color-singlet factorization with a gluon PDF, covariant projection operators, and long-distance matrix elements taken from potential-model wave functions for Bc and from decay constants for Bs and Ds. The central phenomenological claim is that Bc production is observable at the EIC, with a predicted cross section of 5.44 fb plus a Bc* cross section of 22.12 fb (Section III, Eq. 10), yielding roughly 5 to 28 reconstructed Bc -> J/psi pi -> l+l- events per year. The paper also reports much larger but strongly parameter-sensitive cross sections for Bs and Ds at HERA.

Significance. If the numerical predictions are correct, the paper provides a concrete, falsifiable observable for the EIC and fills a gap in LO electroproduction calculations for mesons containing one or two heavy quarks. The work has several strengths: the formalism is standard NRQCD; the input parameters are taken from external sources rather than fitted to the target observables; the paper explicitly separates quark-mass and scale uncertainties; and it gives differential distributions in cos(theta) and pT^2. The central event-rate estimate, however, rests on an amplitude calculation that the authors report required manual patching of FeynCalc's Dirac contractions, and no independent verification or code is provided. Since the EIC observability claim scales linearly with the LO cross section, the numerical security of that claim is presently unverified.

major comments (3)
  1. [Section II and Eq. (10)] The central Bc cross-section prediction, and hence the 5 to 28 reconstructed events per year at the EIC, depends entirely on the LO squared amplitude for gamma* + g -> Bc + cbar + b. In Section II the authors state that 'Feyncalc seems to give wrong results when contracting long chains of Dirac gamma matrix, and can be solved by manually add contracting rules.' No code, ancillary file, or independent amplitude check is provided, so the reader cannot verify that the manual contraction rules do not introduce a sign or factor error. An O(1) error would change the quoted cross sections in Eq. (10) by a factor that could remove or greatly inflate the observability claim. I therefore request an independent verification, for example a comparison of the squared amplitude in a known kinematic limit, a helicity-amplitude check, or release of the calculation code, before the central claim can be assessed.
  2. [Section III, Eq. (10)] The quoted Bc and Bc* cross sections are LO color-singlet predictions only; NLO QCD corrections and color-octet contributions are omitted. Because the event-rate estimate is directly proportional to the LO cross section, the absence of any estimate of these corrections leaves the central claim with an unquantified systematic uncertainty. The large scale variation in Eq. (10) (the +6.36/-1.91 fb band on the Bc cross section) already spans a factor of several, and NLO corrections in comparable quarkonium leptoproduction calculations are known to be substantial. The authors should either provide an estimate of the expected size of NLO/octet contributions or explicitly restrict the conclusion to 'an LO estimate suggests observability.'
  3. [Section III, Eq. (11) and Bs/Ds discussion] For Bs and Ds, the NRQCD velocity expansion is applied to mesons containing a strange quark with ms = 0.5 GeV, for which v is not parametrically small. The paper itself demonstrates the fragility of this assumption: Eq. (11) shows that a 0.1 GeV change in quark mass can alter the Bs cross section by more than 200%, and the total uncertainty bands are very wide. The abstract and conclusion describe these Bs and Ds cross sections as 'notably significant,' but without a discussion of the convergence of the NRQCD expansion for light strange quarks, the quoted event numbers are difficult to interpret as more than order-of-magnitude estimates. I recommend either adding a quantitative discussion of the v-expansion validity for cs and bs systems or softening the claims accordingly.
minor comments (5)
  1. [Abstract and affiliations] There are several typos: 'chromodyn amics' should be 'chromodynamics', and the affiliation 'University of Chinese Academy of Science s' has an extra 's'.
  2. [Eq. (11)] The lower uncertainty for sigma_Ds is printed as '-3,03 pb' and should presumably be '-3.03 pb'.
  3. [Section III, figure captions] The figure captions for Figs. 2-4 describe the upper/lower bounds in terms of specific quark mass combinations, but it would be clearer to state that the upper/lower curves correspond to the full mass-and-scale envelope rather than to a single combination.
  4. [Section III, EicC discussion] The statement that the EicC cross section is 'less than or similar to 0.03 fb' is given without a formula or reference; a brief derivation or citation would improve transparency.
  5. [Eq. (3)] The phase-space decomposition notation in Eq. (3) is hard to follow; in particular the meaning of the integration limits 'E2/(4m^2)' and 'M^2/m^2' is not explained. A sentence defining M and the limits would help.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the cross-section predictions use external LDMEs and decay constants and are not fitted to the target observables.

full rationale

The derivation chain is: NRQCD factorization (Eq. 1), the leptonic tensor formula (Eq. 2), covariant projection for the q-qbar pair (Eq. 4), LO Feynman diagrams generated with FeynArts/FeynCalc, and numerical integration. The inputs are the gluon PDF, quark masses, the one-loop running coupling, and the LDMEs. For Bc the LDME is taken from a potential-model calculation, |R_Bc(0)|^2 = 1.642 GeV^3 (Eq. 8); for Bs and Ds the LDMEs come from PDG decay constants through Eq. (7). None of these inputs is derived from the ep -> e + Bc + cbar + b (or Bs/Ds) cross sections being computed, and the predicted HERA and EIC rates (Eqs. 9-11) are not used to adjust any parameter. The quark-mass and renormalization-scale variations are uncertainty estimates, not fits to data. The event-number estimate additionally uses external branching ratios, BR(Bc -> J/psi pi) = 0.5% and BR(J/psi -> l+l-) = 12%, which are independent of the computed cross sections. The passage noting that FeynCalc seemed to give wrong results for long Dirac chains and was handled by manually adding contraction rules is a numerical verification concern, not a circularity, because it does not make the output equivalent to an input. The self-citations, Ref. [12] and Ref. [13], appear only in the historical review of Bc hadroproduction and are not load-bearing for the electroproduction predictions. No circular step can be exhibited, so the circularity score is 0.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central predictions depend on standard NRQCD factorization, a leading-order color-singlet calculation, potential-model or decay-constant LDMEs, and four adjustable inputs (three quark masses and the scale). The Bc rate is most sensitive to the wave-function input and neglected higher-order terms; the Bs/Ds rates are highly sensitive to the strange-quark mass and the scale, as the paper documents.

free parameters (4)
  • charm quark mass mc = 1.5 GeV (varied +/-0.1 GeV)
    Set in Eq. (5) and varied in Section III to estimate uncertainty; a leading-order input.
  • bottom quark mass mb = 4.9 GeV (varied +/-0.1 GeV)
    Set in Eq. (5) and varied; strongly affects phase space and the scale choice.
  • strange quark mass ms = 0.5 GeV (varied +/-0.03 GeV)
    Set in Eq. (5) and varied by +/-30 MeV; cross sections for Bs/Ds change by more than 200% for a 0.1 GeV variation, so this parameter dominates the prediction.
  • renormalization/factorization scale mu = mT (varied from mT/2 to 2mT)
    Default mu = mT in Section III; the scale is varied to produce the second uncertainty in Eqs. (9)-(11).
assumptions (5)
  • domain assumption NRQCD factorization for electroproduction of heavy mesons (Eq. (1))
    The cross section is written as the gluon-PDF fold with short-distance coefficient and LDME; this assumes the relative velocity of the q-qbar pair is small enough for a velocity expansion, which is standard for Bc but questionable for Bs/Ds with a 0.5 GeV strange quark.
  • domain assumption Heavy-quark spin symmetry: vector and pseudoscalar mesons share |R(0)| at leading order in v (Section III after Eq. (8))
    Used to set |R(Bc*)|=|R(Bc)|, |R(Bs*)|=|R(Bs)|, |R(Ds*)|=|R(Ds)|; this is an NRQCD symmetry valid to leading order in the heavy-quark velocity.
  • domain assumption Buchmuller-Tye potential model determines |R_Bc(0)|^2 = 1.642 GeV^3 (Eq. (8))
    The Bc LDME is taken from a potential-model wave function rather than from data; the uncertainty in this model input is not propagated into the quoted errors.
  • domain assumption Decay-constant extraction of |R_Bs(0)|^2 = 0.299 GeV^3 and |R_Ds(0)|^2 = 0.124 GeV^3 from PDG f_Bs and f_Ds (Eqs. (6)-(8))
    The LDMEs for Bs and Ds are derived from measured and lattice decay constants, treated as external inputs.
  • standard math One-loop running coupling with Lambda_QCD = 297 MeV (nf=4) and 214 MeV (nf=5) and CT10 PDFs
    Standard LO choices from Refs. [32,35]; they are external inputs, not derived in the paper.

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Cite this review

Pith. "Pith review of $B_c$, $B_s$ and $D_s$ production at lepton-hadron colliders." pith.science (2026). https://pith.science/paper/T4FP24VS

@misc{pith2026250100550,
  author       = {Pith},
  title        = {Pith review of: $B_c$, $B_s$ and $D_s$ production at lepton-hadron colliders},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T4FP24VS}},
  note         = {Machine review of arXiv:2501.00550}
}
abstract

Within the framework of nonrelativistic quantum chromodynamics, this study examines the electroproduction processes $e+p\to e+B_c+\overline{c}+b$, $e+p\to e+B_s+\overline{s}+b$, and $e+p\to e+D_s+\overline{c}+s$ at lepton-hadron colliders. The differential cross sections in $\cos\theta$ and $p_T^2$ at HERA are presented. The results indicate that the production of $B_c$ is feasible at the EIC, whereas it is not at HERA. The cross sections for $B_s$ and $D_s$ are notably significant at HERA, yet they exhibit sensitivity to variations in quark mass and renormalization scale.

Figures

Figures reproduced from arXiv: 2501.00550 by the authors.

Figure 1
Figure 1. FIG. 1. Typical LO Feynman diagrams of [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The differential cross section in (a) cos [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The differential cross section in (a) cos [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The differential cross section in (a) cos [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

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Pith tools

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