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Born rule survives its first collider test, bounded to ε < 0.042

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2026-07-09 22:33 UTC pith:O2HJUIUN

load-bearing objection First collider bound on Born-rule violations — real but thin, limited by 1986 binning the 3 major comments →

arxiv 2607.06938 v1 pith:O2HJUIUN submitted 2026-07-08 hep-ph hep-thquant-ph

First constraint on Born-rule violations at high-energy colliders

Antony Valentini , Mira Varma This is my paper
classification hep-ph hep-thquant-ph PACS 13.66.De03.65.Ta
keywords Born ruleBhabha scatteringquantum equilibriumpilot-wave theorydifferential cross sectionangular smearingPEP colliderquantum foundations
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The Born rule—the foundational postulate of quantum mechanics that measurement probabilities equal the squared modulus of the wave function—has never been experimentally tested at high-energy colliders. This paper performs that test by modeling a hypothetical Born-rule violation as an angular smearing of the predicted differential cross section for Bhabha scattering (electron-positron to electron-positron), implemented by convolving the standard QED prediction with a Gaussian of width ε in the scattering angle variable cos θ. Using 1986 data from the PEP collider at SLAC—22 angular bins of large-angle Bhabha scattering at a center-of-mass energy of 29 GeV—the authors perform a likelihood scan over ε, profiling over a luminosity nuisance parameter, and obtain an upper bound of ε < 0.042 at 95% confidence. The bound is physically meaningful because the smearing full-width at half-maximum at this limit is approximately twice the experimental bin width (0.05 in cos θ), meaning the data exclude any anomalous angular smearing at or above the scale at which the experiment could detect it. The best-fit value ε = 0.024 yields a smearing width comparable to half the bin width, consistent with the expected slight improvement from matching the bin-averaging scale rather than from new physics. The paper also surveys alternative datasets (ALEPH, DELPHI, TASSO, BESIII) and detector resolution capabilities (ATLAS, CMS, FCC-ee), concluding that improvements of less than two orders of magnitude are realistically possible with current or forthcoming technology, and that substantially better tests would require new experimental techniques specifically designed for Born-rule testing.

Core claim

The paper establishes the first experimental upper limit on Born-rule violations at high-energy colliders: ε < 0.042 at 95% confidence, where ε parameterizes a Gaussian angular smearing of the Bhabha scattering differential cross section. This bound, derived from 22 bins of PEP data at √s = 29 GeV, excludes smearing widths whose full-width at half-maximum exceeds twice the experimental bin width, placing a constraint at the threshold of detectability rather than deep in an unobservable regime.

What carries the argument

The central mechanism is the smeared cross section (Eq. 4): the standard QED differential cross section dσ/dx (with x = cos θ) is convolved with a normalized Gaussian δ_ε(x' − x) of width ε, integrated over a range parameter a = 0.4 to avoid edge effects. This smeared prediction is then compared bin-by-bin to the HRS/PEP data via a profile likelihood scan over ε, with a Gaussian-constrained nuisance parameter A (combining luminosity and radiative-correction uncertainties, σ_A = 0.0117). Confidence intervals use Wilks' theorem on the Δχ² profile. The robustness of the integration range was checked by varying a from 0.25 to 0.40 with no change in results to four decimals.

Load-bearing premise

The entire analysis rests on modeling Born-rule violations as a Gaussian angular smearing of the cross section. This is a phenomenological choice, not derived from an underlying theory, so the bound ε < 0.042 only constrains violations that happen to take this specific functional form.

What would settle it

If a future experiment with finer angular bins and higher statistics observed a differential cross section whose angular shape departed from the standard QED prediction in a manner consistent with a Gaussian smearing of width ε > 0.042, this bound would be falsified.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Modern high-luminosity experiments like BESIII, with event counts exceeding 10⁸, could in principle reduce the bin width—and thus the probeable smearing scale—by orders of magnitude, but only if detector angular resolution improves correspondingly.
  • Current detector angular resolutions (ATLAS ~0.5–1 mrad, CMS ~0.5 mrad, FCC-ee target ~0.1 mrad) are less than two orders of magnitude better than the 6.5 mrad HRS resolution used here, capping realistic near-term improvement.
  • The phenomenological Gaussian-smearing ansatz could be applied to other scattering processes and energies, potentially yielding complementary constraints if datasets with finer and more uniform angular binning become available.
  • The analysis framework is model-independent within the smearing ansatz, so it can be directly re-applied to future data without requiring a specific underlying theory of Born-rule violation.

Where Pith is reading between the lines

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

  • If Born-rule violations exist but manifest as energy-dependent or process-dependent smearing rather than a universal Gaussian, this bound would not constrain them—a null result here does not rule out all forms of Born-rule violation.
  • The fact that the best-fit ε = 0.024 is non-zero and corresponds to roughly half the bin width suggests the fit is capturing binning artifacts rather than a physical signal, but a dedicated unbinned analysis could provide a cleaner test.
  • The gap between detector resolution (δx ≈ 0.0065) and bin width (Δx = 0.05) implies that the current bound is statistics-limited by the analysis choice of binning, not by detector capability—a reanalysis of the same raw data with finer bins could potentially tighten the bound.
  • If a future theory of Born-rule violation predicts a specific non-Gaussian smearing kernel, the framework would need to be generalized beyond the Gaussian ansatz to constrain that theory.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The manuscript presents what the authors claim is the first experimental constraint on Born-rule violations at high-energy colliders. The approach is phenomenological: Born-rule violations are modeled as a Gaussian angular smearing of the differential scattering cross section, parameterized by a width ε (Eq. 2, 4). This smeared cross section is fit to 22 angular bins of large-angle Bhabha scattering data from the HRS experiment at PEP (Ref. [5], Table XII, √s = 29 GeV). A profile likelihood scan over ε, with a normalization nuisance parameter A constrained by luminosity and systematic uncertainties, yields a best-fit ε = 0.024 and a 95% confidence upper bound of ε < 0.042. The authors discuss future prospects for improvement, noting that the bound is currently limited by the experimental bin width rather than detector angular resolution.

Significance. The paper addresses a genuinely novel question: whether the Born probability rule can be tested in the high-energy collider regime. The statistical methodology is standard and transparent (profile likelihood, Wilks' theorem, Gaussian nuisance parameter), the fit quality is good (χ²/ndof ≈ 1.06), and the authors are commendably honest about the limitations of their result, including the fact that the best-fit ε is comparable to the bin width and likely a binning artifact. The use of a single-energy dataset with publicly available bin-by-bin corrections is a reasonable starting point. The bound ε < 0.042 is a concrete, falsifiable result, though its physical reach is modest. The discussion of future detector improvements (ATLAS, CMS, FCC-ee) provides useful context.

major comments (3)
  1. The central tension in the paper is that the bound ε < 0.042 is almost entirely determined by the experimental bin width Δx = 0.05, rather than by the statistical power of the data or the underlying physics. The authors themselves acknowledge (page 4) that the best-fit ε = 0.024 has a FWHM ≈ 0.058 that 'closely matches the bin width Δx = 0.05' and that this is expected from bin averaging. The 95% upper limit (FWHM ≈ 0.10 ≈ 2Δx) is then the scale at which smearing becomes visibly inconsistent with the data. This means the constraint is essentially 'ε must be smaller than ~2Δx,' which is a statement about experimental resolution rather than about Born-rule violations per se. The authors should more clearly distinguish what is genuinely constrained by the data (i.e., that no anomalous smearing beyond the binning scale is present) from what is merely a restatement of the binning choice. The
  2. The physical motivation for the Gaussian smearing ansatz (Eq. 2, 4) is thin. The authors cite the de Broglie-Bohm pilot-wave framework and node regularization as a possible origin (page 1–2), but the connection from that theoretical motivation to a Gaussian smearing of the angular cross section in x = cos θ is not derived. The bound is therefore only meaningful for violations that take this specific functional form. A Born-rule violation with a different signature (non-Gaussian, energy-dependent, normalization-only, or affecting specific angular structures) would not be constrained. The authors acknowledge this is phenomenological, but the abstract and conclusion could be read as overclaiming a 'constraint on Born-rule violations' when the constraint is on a specific model of violations. The authors should qualify the central claim accordingly.
  3. The choice of a Gaussian smearing in x = cos θ (rather than in θ, or in a different angular variable) is presented without justification. Different choices of smearing variable would generally yield different bounds on the corresponding width parameter. The authors should briefly justify why x = cos θ is the natural variable for the pilot-wave motivation cited, or at minimum acknowledge that the bound is variable-dependent.
minor comments (5)
  1. Eq. (3): the electroweak amplitude definitions use notation that could be clearer — for instance, the coupling notation g_V, g_A is standard but their numerical values are not stated, making it difficult for a reader to reproduce the prediction without consulting Ref. [5].
  2. Fig. 2: the y-axis label 'Δχ²' is clear, but the horizontal line indicating the 95% confidence level is not explicitly drawn or labeled on the figure, making it slightly harder to read off the bound visually.
  3. Page 4: the statement 'This is nearly half the experimental angular bin width, Δx/2 = 0.025' is correct but could be misread; clarifying that ε = 0.024 is the best-fit smearing width (not the FWHM) would help.
  4. The robustness check with a = 0.25, 0.3, 0.35, 0.4 (page 3) is mentioned but the results are only summarized qualitatively. A small table or explicit statement of the best-fit ε and upper bound for each value of a would strengthen this claim.
  5. Reference [13] is listed as a 2026 Oxford Research Encyclopedia entry; the authors should verify this is publicly available at the time of submission or mark it as forthcoming.

Circularity Check

0 steps flagged

No significant circularity: the bound is obtained by fitting a phenomenological parameter to external experimental data.

full rationale

The paper's central result — the upper bound ε < 0.042 at 95% CL — is obtained by fitting a phenomenological smearing model (Eq. 4, with free parameter ε) to external experimental data (HRS/PEP, Ref. [5], Table XII). The Standard Model cross section (Eq. 3) uses standard QED/electroweak parameters from the literature. The smearing ansatz (Eq. 2, 4) is the authors' construct but introduces ε as a genuinely free parameter that is independently constrained by the data via a profiled likelihood scan. No step in the derivation chain reduces to its own inputs by construction: ε is not defined in terms of the bound, and the bound is not a renamed fit input. The self-citations to Valentini's prior work (refs [2], [4], [11–13]) provide physical motivation for why Born-rule violations might exist, but the central result does not depend on them — the fit stands alone as a phenomenological parameter estimation against external data. The skeptic's concern that the bound is numerically close to 2Δx (twice the bin width) is a question about physical significance and experimental resolution, not about circularity of the derivation. The bound is a genuine output of the statistical analysis, not a tautological restatement of the binning choice.

Axiom & Free-Parameter Ledger

3 free parameters · 3 axioms · 1 invented entities

The paper introduces one primary free parameter (ε) constrained by data, one nuisance parameter (A) with a Gaussian prior, and one analysis choice (a) with robustness checks. The key ad hoc axiom is that Born-rule violations take the form of Gaussian angular smearing — this is not derived from theory but is a phenomenological ansatz. No new physical entities are postulated beyond the smearing function itself.

free parameters (3)
  • ε (smearing width) = best-fit 0.024, upper bound 0.042
    The central free parameter of the model, representing the Gaussian smearing width in cos θ space. Constrained by fit to 22 bins of HRS/PEP data.
  • A (normalization nuisance parameter) = best-fit 0.99
    Dimensionless nuisance parameter for luminosity and O(α³) QED systematic uncertainty, constrained by Gaussian prior with σ_A = 0.0117. Profiled over in the likelihood scan.
  • a (integration range parameter) = 0.4
    Range parameter in Eq. (4) chosen to avoid edge effects. Robness checked with a = 0.25, 0.3, 0.35, 0.4 with unchanged results to four decimals.
axioms (3)
  • ad hoc to paper Born-rule violations at high energies can be modeled as a Gaussian angular smearing of the differential cross section (Eq. 2, 4).
    This is the central modeling assumption. The paper states 'we take a phenomenological approach' and does not derive this form from an underlying theory. The physical significance of the bound depends on the generality of this ansatz.
  • domain assumption Standard QED/electroweak prediction for Bhabha scattering (Eq. 3) is correct at O(α²) at √s = 29 GeV.
    The analysis subtracts O(α³) radiative effects from the data and compares to the O(α²) prediction. This is standard and well-established in the literature.
  • standard math Wilks' theorem applies for constructing confidence intervals from the profile likelihood.
    Used to obtain the 95% confidence upper bound from the Δχ² scan. Standard statistical practice for this type of analysis.
invented entities (1)
  • Gaussian smearing function δ_ε(x' − x) no independent evidence
    purpose: Models Born-rule violations as angular smearing of the cross section
    This is a phenomenological construct introduced by the authors. It is not derived from a specific underlying theory of Born-rule violations. The paper acknowledges this is a model assumption. The bound is only meaningful for violations taking this specific functional form.

pith-pipeline@v1.1.0-glm · 12886 in / 2975 out tokens · 349866 ms · 2026-07-09T22:33:27.098494+00:00 · methodology

0 comments
read the original abstract

We obtain an experimental constraint on possible Born-rule violations at high-energy colliders. We model Born-rule violations with differential scattering cross sections $d\sigma/d\Omega$ subject to an angular smearing by a narrow Gaussian of width $\varepsilon$ (with respect to $x=\cos\theta$ for scattering angle $\theta$). For large-angle Bhabha ($e^{+}e^{-} \rightarrow e^{+}e^{-}$) scattering, at a centre-of-mass energy $\sqrt{s}=29\, \mathrm{GeV}$, data from the PEP collider at SLAC allow us to set an upper bound of $\varepsilon<0.042$ at $95\%$ confidence. This corresponds to a Gaussian smearing over an angular range of twice the experimental bin width, and hence provides a physically meaningful limit on deviations from the Born rule. Future prospects for improving this limit are discussed.

Figures

Figures reproduced from arXiv: 2607.06938 by Antony Valentini, Mira Varma.

Figure 1
Figure 1. Figure 1: FIG. 1. HRS Bhabha scattering data with [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. HRS Bhabha scattering data with [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

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Reference graph

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