Expected statistical uncertainties at future $e^+e^-$ colliders

Hieu Minh Tran; Sang Quang Dinh; Trang Quynh Trieu

REVIEW 2 major objections 4 minor 13 references

Expected statistical uncertainties at future $e^+e^-$ colliders

T0 review · 2 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read A single LEP reference point predicts muon-pair statistical uncertainties at FCC-ee and CEPC as low as 0.0016 per mille.

desk verdict A clean but textbook Poisson-scaling exercise whose headline numbers are off by about a factor of two due to an internally inconsistent LEP reference luminosity; the qualitative conclusion survives, but the tables need correction before this is citable. read the letter →

arxiv 2504.13734 v1 pith:EQT5T4JK submitted 2025-04-18 hep-ph hep-exnucl-ex

Hieu Minh Tran , Sang Quang Dinh , Trang Quynh Trieu This is my paper · ORCID

classification hep-phhep-exnucl-ex

keywords expectedstatisticaluncertaintiesfuturee+e-collidersFCC-eeCEPCmuonpairproductionPoissonstatisticscrosssectionmeasurementLEPreferencedata

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 proposes that the expected statistical uncertainty of a cross-section measurement at a future $e^+e^-$ collider can be obtained from one known reference measurement and the ratio of the theoretical cross section to the planned luminosity. The formula is Eq. (12), $\Delta\sigma_{\rm est}=\Delta\sigma_0\sqrt{(\sigma/L)/(\sigma_0/L_0)}$, which follows from Poisson counting statistics once the reference and future experiments are assumed to have similar overall efficiencies. The method is checked against DELPHI data at LEP, where estimated and recorded uncertainties agree to within 5%. Applied to $e^+e^-\to\mu^+\mu^-$ at FCC-ee and CEPC, it gives relative statistical uncertainties as small as 0.0016 per mille at the Z pole at FCC-ee and 0.0014 per mille at CEPC, values about a thousand times smaller than the corresponding LEP uncertainties. The authors use these numbers as targets for systematic-uncertainty improvement and as input for precision standard-model tests and new-physics constraints.

What carries the argument

The central object is the scaling relation Eq. (12), $\Delta\sigma_{\rm est}=\Delta\sigma_0\sqrt{(\sigma/L)/(\sigma_0/L_0)}$. It is derived from Poisson statistics: with $N$ signal events one has $\Delta N\simeq\sqrt{N}$, and since $\sigma=N/(\epsilon L)$, the statistical uncertainty scales with the square root of the cross-section-to-luminosity ratio. The unknown efficiency $\epsilon$ cancels only under the assumption $\epsilon_0\simeq\epsilon_1$, which is why the reference and future experiments must be considered for the same process. The relation is anchored to the chosen LEP 2 point and uses ZFITTER version 6.42 to compute $\sigma$ at each FCC-ee and CEPC energy with radiative corrections included.

What would settle it

Recompute Eqs. (9) and (12) using the combined LEP 2 luminosity reported by the four experiments (about 40 pb$^{-1}$) instead of $L_0=10$ pb$^{-1}$ while keeping the same $\sigma_0$ and $\Delta\sigma_0$; if the resulting uncertainties in Tables 2 and 3 are close to twice the published values, the numerical projections do not hold. A second check is to repeat the exercise with a different reference point, such as the LEP 1 Z-pole measurement, and see whether the projected Z-pole uncertainties move by more than the quoted precision.

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Extended reading notes

Core claim

The central claim is that Eq. (12), anchored to the LEP 2 combined result at $\sqrt{s_0}=172$ GeV with $L_0=10$ pb$^{-1}$, $\sigma_0=3.562$ pb, and $\Delta\sigma_0=0.331$ pb, gives the expected statistical uncertainties for $e^+e^-\to\mu^+\mu^-$ at FCC-ee and CEPC. The standard-model cross section $\sigma$ at each planned energy is computed with ZFITTER, the luminosity $L$ is taken from the colliders' design reports, and the same acceptance cut $\sqrt{s'}/s>0.85$ as in the LEP 2 analysis is applied. The predicted relative statistical uncertainties reach 0.0016 per mille at the Z pole at FCC-ee and 0.0014 per mille at CEPC, and rise to a few per mille in the $t\bar{t}$ mode; the paper concludes that these estimates set the level to which systematic uncertainties should be reduced and provide a basis for testing the standard model and constraining new physics.

Load-bearing premise

The load-bearing premise is that the chosen LEP 2 reference numbers are physically consistent with the counting formula: the quoted cross section and uncertainty imply an efficiency-luminosity product of about 32.5 inverse picobarns, which would require an efficiency larger than one if the luminosity were only 10 inverse picobarns; if the true luminosity is the full combined value of about 40 inverse picobarns, all projected uncertainties should be doubled.

Editorial extensions

If this is right

At the Z pole, FCC-ee and CEPC would measure the $e^+e^-\to\mu^+\mu^-$ cross section with relative statistical uncertainties of about 0.0016 and 0.0014 per mille, so systematic uncertainties at or below this level become the bottleneck for a precision measurement.
The estimated statistical uncertainties are roughly a thousand times smaller than the corresponding LEP uncertainties, meaning the planned luminosity upgrades directly translate into much tighter standard-model tests.
In the $t\bar{t}$ mode at FCC-ee, the relative statistical uncertainty is about 3.7 per mille, making it the least statistically precise of the modes considered, so that measurement will need the most care in systematics.
Equation (12) can be reused with any known reference measurement and any future collider luminosity to set the statistical floor for other processes, provided the efficiency assumption holds.

Reading between the lines

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

A consistency check not reported in the paper: Eq. (9) applied to the LEP 2 reference implies $\epsilon_0 L_0 \approx 32.5$ pb$^{-1}$, which cannot hold with $L_0=10$ pb$^{-1}$ unless $\epsilon_0>1$; if the true combined luminosity is about 40 pb$^{-1}$, all projected uncertainties in Tables 2 and 3 increase by a factor of about two.
The same formula could be applied to other final states, such as $\tau^+\tau^-$ or hadronic final states, as long as the efficiency assumption holds; the paper does not make those predictions.
A direct test of the method's robustness would be to anchor Eq. (12) at the LEP 1 Z-pole measurement instead of the LEP 2 $W^+W^-$-threshold point and compare the projected Z-pole uncertainty; if the result changes substantially, the reference-point choice is a dominant source of error.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

This is a simple application of textbook Poisson statistics: Eq. (12) is just the scaling of sqrt(sigma/(eps L)) from a reference point, and the paper applies it to muon pair production at FCC-ee and CEPC. Nothing here is new methodologically, but that's fine if the numbers are right and the assumptions are transparent.

The DELPHI validation is a reasonable sanity check: using one DELPHI energy as reference and predicting the others gives ratios close to 1. But it is a within-experiment consistency check, not an independent verification. The real soft spot is the LEP 2 reference point used for the future projections. With L0 = 10 pb^-1, sigma0 = 3.562 pb, and Delta_sigma0 = 0.331 pb, Eq. (9) requires epsilon0*L0 = 32.5 pb^-1, which is impossible for epsilon0 <= 1. The combined LEP luminosity at 172 GeV is actually about 40 pb^-1, so the stated L0 is off by roughly a factor of four. Since Eq. (12) scales as sqrt(L0), every projected Delta_sigma_est in Tables 2 and 3 is too small by about a factor of two. The Z-pole relative uncertainty should read about 0.003 per mille, not 0.0016. The qualitative claim—that statistical errors at FCC-ee and CEPC will be orders of magnitude below LEP—is untouched; only the specific benchmark numbers are wrong.

Other caveats: the equal-efficiency assumption (eps0 = eps1) is a rough guess, and at the Z pole the muon pair cross section is large enough that the Poisson approximation still holds but the extrapolation from 172 GeV deserves a comment about acceptance differences. These are minor compared to the luminosity issue.

Bottom line: this is a useful back-of-the-envelope note for planning systematic uncertainty targets, once the reference luminosity is corrected and the tables regenerated. With that fix, it's a legitimate short letter. I would not cite the current numbers without the correction.

Recommendation: send it to peer review, but the referee should insist on the corrected reference luminosity before acceptance.

Referee Report

2 major / 4 minor

Summary. The paper derives a simple scaling formula, Eq. (12), that estimates the expected statistical uncertainty of a future cross-section measurement from a reference measurement at an earlier collider, assuming equal overall efficiencies. The authors validate the formula against DELPHI data at LEP and then apply it to e+e- -> mu+mu- at FCC-ee and CEPC, using ZFITTER Standard Model cross sections and luminosities from the conceptual design reports. They report relative statistical uncertainties as small as about 0.0016 per mille at the Z pole and about 0.1 per mille in the WW mode, and argue that these numbers set useful targets for systematic uncertainties and for new-physics sensitivity studies.

Significance. If the numerical application is corrected, the paper provides a transparent, essentially parameter-free estimator that complements full detector simulations and is useful for quick sensitivity studies and for setting systematic-uncertainty goals. The derivation from binomial/Poisson statistics is clear, and the DELPHI comparison gives encouraging support for the scaling formula. The future projections are not circular, because they use external LEP data as a reference and ZFITTER cross sections rather than fitting the target quantities. However, the quantitative claims in Tables 2 and 3 are currently compromised by an internally inconsistent LEP 2 reference luminosity, so the paper requires revision before those numbers can be used.

major comments (2)

[§4 (Statistical uncertainties at FCC-ee and CEPC), Eq. (12)] The LEP 2 reference point quoted in §4 is internally inconsistent with Eq. (9). Inserting sigma0 = 3.562 pb, Delta_sigma0 = 0.331 pb, and L0 = 10 pb^-1 into Eq. (9) gives epsilon0*L0 = sigma0/Delta_sigma0^2 = 3.562/0.331^2 pb^-1 ≈ 32.5 pb^-1. Since epsilon0 <= 1, the combined luminosity must be at least about 32.5 pb^-1, not 10 pb^-1; the actual combined LEP 2 luminosity at 172 GeV is approximately 40 pb^-1. This is load-bearing because Eq. (12) scales as sqrt(L0), so every Delta_sigma_est in Tables 2 and 3 is underestimated by about a factor sqrt(40/10) ≈ 2. For example, the Z-pole relative uncertainty in Table 2 would be about 0.0032 per mille rather than 0.0016 per mille. The authors should either replace L0 with the self-consistent combined luminosity or choose a reference point for which (L0, sigma0, Delta_sigma0) satisfy Eq. (9).
[§3 (Statistical uncertainty of cross section), Table 1] The DELPHI validation in Table 1 does not fully validate the procedure used for the future predictions unless the origin of the cross sections sigma entering Eq. (12) is stated. If those sigma values are the measured DELPHI cross sections, the comparison checks only the statistical scaling formula, not the combination of a ZFITTER theoretical cross section with a CDR luminosity that is the actual future application. Please specify the source of sigma in Table 1 and, ideally, repeat the comparison using the same ZFITTER predictions that are used for Tables 2 and 3.

minor comments (4)

[Introduction] The introduction refers to the 'Circular Electron Position Collider'; the correct name is the Circular Electron-Positron Collider.
[§3, Eq. (11)] The assumption epsilon0 ≈ epsilon1 is stated but not quantified; a short sensitivity discussion showing that Delta_sigma_est scales as sqrt(epsilon0/epsilon1) would help readers assess how much the numbers depend on this assumption.
[Tables 2 and 3] The luminosities are quoted from the conceptual design reports without specifying whether they correspond to one interaction point or to the total integrated luminosity used in the report; this should be clarified for reproducibility.
[Figure 1] The figure shows the relative difference between the Poisson and binomial standard deviations over the full range of p, while Eq. (4) is a small-p approximation; marking the range where Eq. (4) is accurate would make the presentation clearer.

Circularity Check

1 steps flagged · score 2.0 of 10

FCC-ee/CEPC predictions are independent; DELPHI 'verification' is a Poisson self-consistency check.

self definitional [Section 'Statistical uncertainty of cross section', Eq. (12), Table 1]
"The data at the center-of-mass energy √s0 = 192 GeV is taken to be the reference point with the measured cross section σ0(e+e−→ µ+µ−) = 7.37 pb, the statistical uncertainty ∆σ0 = 0.61 pb, and the luminosity L0 = 25.79 pb−1. ... The estimated statistical uncertainties (∆σest) calculated using Eq. (12) are presented in the third column."

For the non-reference rows of Table 1, the σ and L needed in Eq. (12) are the DELPHI measured cross sections and luminosities from [6]; the paper gives no other source for these inputs. Under the paper's own Eq. (9), the statistical uncertainty of a measured cross section is ∆σ = sqrt(σ/(εL)). The reference point fixes ε via ∆σ0^2 = σ0/(εL0). Substituting this ε into Eq. (12) cancels the reference quantities and gives ∆σ_est = sqrt(σ/(εL)), exactly the Poisson uncertainty associated with the measured σ and L. Hence the 'predicted' uncertainties in Table 1 are the same Poisson uncertainties that the recorded ∆σ values already represent; the agreement is a consistency check of the Poisson formula, not an independent validation of the extrapolation to future colliders.

full rationale

The central chain leading to the FCC-ee and CEPC predictions in Tables 2 and 3 is not circular. Eq. (12) combines an external LEP 2 reference point (σ0, ∆σ0, L0) with ZFITTER-computed Standard Model cross sections and conceptual-design-report luminosities. No parameter is fitted to the target results, and the predicted uncertainties are not defined in terms of the quantities they claim to predict. The L0=10 pb−1 versus ε0L0≈32.5 pb−1 inconsistency is a correctness concern about the internal consistency of the chosen reference luminosity, but it is not a circularity because the reference values are not derived from the future predictions. The only step with a definitional character is the DELPHI validation in Table 1, where the estimated uncertainties are computed from the same measured cross sections and luminosities that determine the Poisson uncertainties, making the comparison a self-consistency check rather than an independent test. Because that self-referential validation is auxiliary and does not affect the central future-collider numbers, the overall circularity score is low at 2.

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

The paper introduces no free parameters fitted to data and no new entities. The load-bearing inputs are the LEP reference measurement, the assumed efficiency equality, the Poisson approximation, and the ZFITTER Standard Model cross sections.

assumptions (4)

standard math The binomial distribution is approximated by the Poisson distribution (Eq. 5), requiring the per-collision probability p = N/n to be small.
Invoked in 'Statistics of Bernoulli trials' before Eq. (7). At Z-pole running, p for muon pairs is O(10^-2), so the approximation error is O(p/2), which the paper does not quantify.
ad hoc to paper The overall efficiencies of the reference and future experiments are equal, epsilon0 = epsilon1 (Eq. 11).
Introduced to remove unknown future detector efficiencies. If future detectors have different efficiency, the projected statistical uncertainties change by the square root of the efficiency ratio.
domain assumption The quoted LEP 2 combined reference values are self-consistent: L0=10 pb^-1, sigma0=3.562 pb, Delta_sigma0=0.331 pb.
Used as the anchor of Eq. (12). Under Eq. (9), these numbers imply epsilon0 L0 = 32.5 pb^-1, so the consistency of this input is load-bearing for all projected numbers.
domain assumption ZFITTER v6.42 with the same s'/s > 0.85 cut provides the Standard Model cross sections at future collider energies.
The paper relies on ZFITTER for Tables 2 and 3, but does not list the input parameters, electroweak constants, or run settings used.

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Cite this review

Pith. "Pith review of Expected statistical uncertainties at future $e^+e^-$ colliders." pith.science (2026). https://pith.science/paper/EQT5T4JK

@misc{pith2026250413734,
  author       = {Pith},
  title        = {Pith review of: Expected statistical uncertainties at future $e^+e^-$ colliders},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EQT5T4JK}},
  note         = {Machine review of arXiv:2504.13734}
}

read the original abstract

In future colliders, the frontiers of luminosity and energy are extended to explore the physics of elementary particles at extremely high precision, and to discover new phenomena suggested from current experimental anomalies. In this letter, we present a simple method to estimate the expected statistical uncertainties of scattering cross sections at future colliders using their conceptual design reports. In particular, the expected statistical uncertainties of muon pair production cross section at the Future Circular Collider (FCC-ee) and the Circular Electron-Positron Collider (CEPC) are calculated. The results can be used to set a goal for systematic uncertainty improvement, to determine the standard model parameters accurately, and to identify the viable parameter space of new physics models.

Figures

Figures reproduced from arXiv: 2504.13734 by the authors.

**Figure 2.** The ratio between the estimated statistical uncertainty of [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

13 extracted references · 1 canonical work pages

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

FCC collaboration, FCC Physics Opportunities: Future Circular Collider Conceptual Design Report Volume 1, Eur. Phys. J. C 79 (2019) 474

2019

[2] [2]

FCC collaboration, FCC-ee: The Lepton Collider: Future Circular Collider Conceptual Design Report Volume 2, Eur. Phys. J. ST 228 (2019) 261

2019

[3] [3]

CEPC Study Group collaboration, CEPC Conceptual Design Report: Volume 1 - Accelerator , 1809.00285

arXiv

[4] [4]

CEPC Study Group collaboration, CEPC Technical Design Report: Accelerator , Radiat. Detect. Technol. Methods 8 (2024) 1 [ 2312.14363]

arXiv 2024

[5] [5]

Campbell et al., Event generators for high-energy physics experiments , SciPost Phys

J.M. Campbell et al., Event generators for high-energy physics experiments , SciPost Phys. 16 (2024) 130 [2203.11110]

arXiv 2024

[6] [6]

DELPHI collaboration, Measurement and interpretation of fermion-pair production at LEP energies above the Z resonance , Eur. Phys. J. C 45 (2006) 589 [ hep-ex/0512012]

arXiv 2006

[7] [7]

Shiltsev and F

V. Shiltsev and F. Zimmermann, Modern and Future Colliders , Rev. Mod. Phys. 93 (2021) 015006 [2003.09084]

arXiv 2021

[8] [8]

Prepared for the US Snowmass Community Planning Exercise (Snowmass 2021) , in Snowmass 2021, 5, 2022 [ 2205.08553]

CEPC Physics Study Group collaboration, The Physics potential of the CEPC. Prepared for the US Snowmass Community Planning Exercise (Snowmass 2021) , in Snowmass 2021, 5, 2022 [ 2205.08553]

arXiv 2021