{"id":"83ab9150-aedb-41fe-957e-b0da05275d0a","arxiv_id":"2504.13734","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":2.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Using a LEP reference measurement and design luminosities, the paper estimates statistical uncertainties for e+e- to mu+mu- at FCC-ee and CEPC, finding relative uncertainties as low as about 0.0016 per mille at the Z pole.","lead":"This paper proposes a simple formula from Poisson statistics to estimate the expected statistical uncertainty of cross-section measurements at future electron-positron colliders, and applies it to muon pair production at FCC-ee and CEPC. It matters because the resulting benchmark numbers help define how precisely the Standard Model can be tested and how much systematic precision future detectors will need.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"LEP 2 reference luminosity L0=10 pb^-1 is internally inconsistent with the quoted sigma0 and Delta_sigma0; all projected uncertainties scale by sqrt(L0_true/L0_assumed).","rationale":"The reader's weakest_assumption identifies exactly the LEP 2 reference luminosity inconsistency, and my independent calculation confirms it: Eq. (9) applied to the stated numbers forces epsilon0*L0 = 32.5 pb^-1, incompatible with L0=10 pb^-1 and epsilon0 <= 1. This is not a disagreement with external consensus but an internal consistency failure of the numerical input to the central formula. The projected relative statistical uncertainties in Tables 2 and 3, which the abstract and conclusions advertise as the main quantitative results, are therefore uncertain by a factor of about 2. The paper has genuine merits: the scaling method is transparent, the Poisson-to-binomial check is a useful pedagogical reminder, and the DELPHI test (Table 1) shows the formula reproduces that experiment's uncertainties to within 5% when a consistent reference point is used. However, that test does not validate the LEP 2 reference choice, and the claimed Z-pole precision of 0.0016 per mille is the specific number most affected. The fix is straightforward: correct L0 to the actual combined luminosity or cite the correct subset, then regenerate the tables and replace the relative uncertainties. Since the method and qualitative conclusions are robust, this does not warrant rejection, but the current quantitative claims should not be accepted verbatim. Thus CONDITIONAL is appropriate, matching the reader's verdict.","tokens_in":5475,"tokens_out":2161,"duration_ms":16549,"concrete_test":"Recover the original LEP 2 combined data in Ref. [9] at sqrt_s=172 GeV, specifically the combined cross section, statistical uncertainty, and total integrated luminosity after applying the cut sqrt(s')/s > 0.85. Recompute each entry of Tables 2 and 3 using Eq. (12) with the directly quoted luminosity from the combination (which should be near 40 pb^-1) instead of the value 10 pb^-1. If the resulting Delta_sigma_est values are about a factor of 2 larger, the tables in the paper are wrong by that factor and the text should state that the reference L0=10 pb^-1 was a misprint. Alternatively, if a legitimate reason exists for using only a 10 pb^-1 subset (e.g., a single experiment or a restricted data sample at 172 GeV), then the paper should cite the source for that subset; otherwise the inconsistency stands.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central numerical predictions in Tables 2 and 3 follow from Eq. (12) with the LEP 2 reference point (sqrt_s0=172 GeV, sigma0=3.562 pb, Delta_sigma0=0.331 pb, L0=10 pb^-1). Eq. (9) requires epsilon0*L0 = sigma0/Delta_sigma0^2. Inserting the quoted numbers gives epsilon0*L0 = 3.562/0.331^2 pb^-1 = 32.5 pb^-1. Since epsilon0 <= 1, the reference luminosity must be at least 32.5 pb^-1, not 10 pb^-1. For a realistic efficiency 0.5-1, the implied L0 is 32.5-65 pb^-1. This is consistent with the actual combined LEP 2 luminosity at 172 GeV, which is about 40 pb^-1, but it contradicts the stated L0=10 pb^-1. Because Eq. (12) scales as sqrt(sigma/L)/(sigma0/L0)-type, the projected uncertainties in Tables 2 and 3 are inversely proportional to sqrt(L0): using the internally consistent L0 ~ 40 pb^-1 increases every Delta_sigma_est by about a factor sqrt(40/10)=2. The relative uncertainties delta_est would then be about 0.003 per mille at the Z pole rather than 0.0016, and the W-pair entries would roughly double. The central qualitative conclusion (statistical uncertainties orders of magnitude below LEP) survives, but the headline quantitative claim, that 0.0016 per mille can be reached, is not supported until the reference luminosity is corrected or justified. The DELPHI validation in Table 1 does not resolve this because it uses the DELPHI reference point (L0=25.79 pb^-1, sigma0=7.37 pb, Delta_sigma0=0.61 pb; then epsilon0*L0 = 7.37/0.61^2 = 19.8 pb^-1, which is consistent for an efficiency ~0.77; note the stated L0=25.79 pb^-1 is self-consistent only if epsilon0~0.77), and it validates only the scaling formula, not the choice of the LEP 2 reference point used for the future projections.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":5892,"tokens_out":8215,"duration_ms":79668,"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":[{"comment":"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).","section":"§4 (Statistical uncertainties at FCC-ee and CEPC), Eq. (12)"},{"comment":"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.","section":"§3 (Statistical uncertainty of cross section), Table 1"}],"minor_comments":[{"comment":"The introduction refers to the 'Circular Electron Position Collider'; the correct name is the Circular Electron-Positron Collider.","section":"Introduction"},{"comment":"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.","section":"§3, Eq. (11)"},{"comment":"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.","section":"Tables 2 and 3"},{"comment":"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.","section":"Figure 1"}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern lands: the LEP 2 reference luminosity L0 = 10 pb^-1 is not self-consistent with the quoted sigma0 and Delta_sigma0, and the projected uncertainties in Tables 2 and 3 scale as sqrt(L0). This appears to be a localized numerical error, likely from using a per-experiment luminosity together with the combined LEP 2 cross-section and uncertainty. If the authors correct L0 to the combined value (about 40 pb^-1) and recompute the tables, the paper would be a useful and concise contribution; in its current form, the headline precision is overestimated by about a factor of two."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis 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.\n\nThe 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.\n\nOther 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.\n\nBottom 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.\n\nRecommendation: send it to peer review, but the referee should insist on the corrected reference luminosity before acceptance.","headline":"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.","tokens_in":6505,"tokens_out":2049,"would_cite":false,"duration_ms":20207,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A single LEP reference point predicts muon-pair statistical uncertainties at FCC-ee and CEPC as low as 0.0016 per mille.","keywords":["expected statistical uncertainties","future e+e- colliders","FCC-ee","CEPC","muon pair production","Poisson statistics","cross section measurement","LEP reference data"],"falsifier":"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.","tokens_in":5241,"feed_emoji":"🎯","tokens_out":10516,"duration_ms":83384,"temperature":0.7,"pith_summary":"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.","feed_headline":"Muon pairs at FCC-ee: statistical errors as low as 0.0016 per mille","feed_subtitle":"The same formula that matched DELPHI data sets the precision goal for FCC-ee and CEPC.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the LEP 2 combined reference point $\\sqrt{s_0}=172$ GeV, $L_0=10$ pb$^{-1}$, $\\sigma_0=3.562$ pb, and $\\Delta\\sigma_0=0.331$ pb that anchors Eq. (12).","marker":"[9]"},{"why":"Provides the DELPHI muon-pair data used to validate the method by comparing estimated and recorded statistical uncertainties.","marker":"[6]"},{"why":"The ZFITTER package used to compute the standard-model $e^+e^-\\to\\mu^+\\mu^-$ cross section at each FCC-ee and CEPC energy.","marker":"[10–13]"},{"why":"Supplies the energies and integrated luminosities for the FCC-ee running modes in the projection.","marker":"[2]"},{"why":"Supplies the energies and integrated luminosities for the CEPC running modes in the projection.","marker":"[4]"}],"fun_headline_variants":["How precise will FCC-ee and CEPC measure muon pairs?","A simple method to forecast collider statistical errors","From LEP 2 data to FCC-ee precision goals","Statistical error limits for future e+e− colliders","Predicting muon pair uncertainty at FCC-ee and CEPC"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["How precise will FCC-ee and CEPC measure muon pairs?","A simple method to forecast collider statistical errors","From LEP 2 data to FCC-ee precision goals","Statistical error limits for future e+e− colliders","Predicting muon pair uncertainty at FCC-ee and CEPC"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000439,"raw_usage":{"total_tokens":2205,"prompt_tokens":899,"completion_tokens":1306,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":515,"completion_tokens_details":{"reasoning_tokens":1220}},"tokens_in":515,"tokens_out":1306,"duration_ms":11423,"temperature":1.0,"reasoning_tokens":1220,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T12:01:30.020789+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}