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arxiv: 2606.10737 · v1 · pith:LMKAPFJEnew · submitted 2026-06-09 · ✦ hep-ph

Does the Weinberg angle allow a local hidden-variable description for the leptonic decays of an entangled ZZ pair?

Junle Pei This is my paper

Pith reviewed 2026-06-27 12:22 UTC · model grok-4.3

classification ✦ hep-ph
keywords local hidden variablesWeinberg angleZZ entanglementleptonic decaysangular correlationsspin-0 decayquantum field theory
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The pith

Angular correlations in leptonic decays of an entangled ZZ pair match a local hidden-variable theory only for one specific state and a restricted range of the Weinberg angle.

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

The paper investigates whether the angular distributions in the leptonic decays of a ZZ pair produced from a spin-0 particle can be reproduced by a local hidden-variable theory that respects angular-momentum conservation. It does so by equating the QFT-predicted angular distribution to an LHVT form term by term. For nonzero w, this matching succeeds only for the unique entangled state with coefficients a1 = a3 = -a2 = 1/√3 and b2 = b3 = 0, and only inside a limited interval of the weak-mixing angle θ_W. For w = 0 it supplies a closed set of algebraic and positivity conditions that are necessary and sufficient. Applied to the sZμZμ interaction, the construction exists exactly at threshold ms = 2mZ but fails for larger masses.

Core claim

Apart from trivial product-state configurations, an LHVT construction exists only for the unique entangled pure state with a1 = a3 = -a2 = 1/√3 and b2 = b3 = 0 together with a restricted window of θ_W when w ≠ 0. When w = 0 a necessary and sufficient criterion is given by a closed set of algebraic and positivity conditions. For the phenomenologically relevant sZμZμ coupling an LHVT construction exists at threshold ms = 2mZ but does not exist for ms > 2mZ.

What carries the argument

Coefficient-by-coefficient matching of the LHVT angular distribution to the QFT prediction under angular-momentum conservation.

If this is right

  • For the state with a1 = a3 = -a2 = 1/√3 and b2 = b3 = 0 the QFT angular correlations are reproducible by LHVT inside the allowed θ_W interval.
  • When w = 0 the algebraic and positivity conditions fully determine whether an LHVT exists for any chosen state coefficients.
  • In the sZμZμ channel an LHVT description is possible exactly at production threshold ms = 2mZ and impossible above threshold.

Where Pith is reading between the lines

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

  • Most entangled ZZ states produced from spin-0 particles lie outside the single allowed configuration and therefore cannot be reproduced by any LHVT of the form considered.
  • The narrow window in θ_W suggests that only at particular values of the weak mixing angle do the quantum correlations reduce to a form compatible with local realism under the imposed constraints.
  • The threshold result for the scalar coupling implies that the kinematic configuration at ms = 2mZ is special in suppressing the quantum features that otherwise forbid an LHVT.

Load-bearing premise

That matching the angular-distribution coefficients under angular-momentum conservation is sufficient to guarantee a valid local hidden-variable model for the full decay process.

What would settle it

Measuring the lepton angular distributions for the state a1 = a3 = -a2 = 1/√3, b2 = b3 = 0 at a value of θ_W outside the allowed window and finding a mismatch with any LHVT parameters would falsify the existence claim.

read the original abstract

Quantum entanglement in diboson systems offers a useful testing ground for exploring the boundary between quantum-mechanical correlations and classical descriptions based on local hidden variables. In this work, we study the spin-polarization state of a $Z_1Z_2$ pair produced from the decay of a spin-0 particle and investigate whether the angular correlations predicted by quantum field theory (QFT) in the leptonic decays $Z_1(\to e^-_1 e^+_1)Z_2(\to e^-_2 e^+_2)$ can be reproduced by a local hidden-variable theory (LHVT) under angular-momentum conservation. By matching the LHVT angular distribution to the QFT prediction coefficient by coefficient, we derive the conditions under which an LHVT construction exists. For the case $w\neq 0$, we show that, apart from trivial product-state configurations, an LHVT construction exists only for a unique entangled pure state, corresponding to $a_1=a_3=-a_2=1/\sqrt{3}$ and $b_2=b_3=0$, together with a restricted window of the weak-mixing angle $\theta_W$. For $w=0$, we derive a necessary and sufficient criterion for the existence of an LHVT construction in terms of a closed set of algebraic and positivity conditions. As an application, we consider the phenomenologically relevant interaction $sZ^\mu Z_\mu$ and show that an LHVT construction exists at threshold $m_s=2m_Z$, whereas it does not exist for $m_s>2m_Z$.

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

2 major / 0 minor

Summary. The paper claims that the angular correlations in leptonic decays of an entangled ZZ pair from a spin-0 particle, as predicted by QFT, can be reproduced by an LHVT under angular-momentum conservation via coefficient-by-coefficient matching of the angular distributions. For w≠0 this yields a unique entangled pure state (a1=a3=-a2=1/√3, b2=b3=0) plus a restricted window in θ_W; for w=0 it yields a closed set of algebraic and positivity conditions. Application to the sZ^μZ_μ interaction shows an LHVT exists at threshold m_s=2m_Z but not for m_s>2m_Z.

Significance. If the coefficient-matching procedure is shown to certify a genuine LHVT, the result would be significant for mapping the boundary between quantum correlations and local hidden-variable descriptions in diboson systems. The explicit identification of a unique entangled state, the restricted θ_W window, and the closed algebraic conditions for w=0 constitute concrete, falsifiable outputs. The threshold application to the sZZ interaction supplies a phenomenologically relevant example where classical reproduction is possible.

major comments (2)
  1. [Abstract] Abstract: the central claim that coefficient matching under angular-momentum conservation is sufficient to establish an LHVT for w≠0 (yielding only the state a1=a3=-a2=1/√3, b2=b3=0 with restricted θ_W) rests on the assumption that equating coefficients plus positivity requirements guarantees a positive measure over λ that factorizes locally (P(A,B|λ)=P(A|λ)P(B|λ)) and reproduces all joint probabilities for the full decay kinematics; the manuscript provides no explicit construction of this measure or verification of the complete set of marginals and no-signaling conditions.
  2. [Abstract] Abstract: the w=0 case is presented as yielding a necessary and sufficient criterion via algebraic and positivity conditions, yet the same concern applies: without demonstrating that the resulting distribution over λ is local and reproduces the full four-momentum kinematics rather than a truncated angular expansion, the criterion may certify only a projection of the correlations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and insightful comments. We address each major comment below, clarifying the assumptions and scope of our coefficient-matching procedure while offering targeted revisions for added clarity.

read point-by-point responses

  1. Referee: Abstract: the central claim that coefficient matching under angular-momentum conservation is sufficient to establish an LHVT for w≠0 rests on the assumption that equating coefficients plus positivity requirements guarantees a positive measure over λ that factorizes locally and reproduces all joint probabilities for the full decay kinematics; the manuscript provides no explicit construction of this measure or verification of the complete set of marginals and no-signaling conditions.

    Authors: The LHVT ansatz is constructed from the outset with local factorization P(A|λ)P(B|λ) and angular-momentum conservation, yielding a general angular distribution whose coefficients are required to be positive. The QFT distribution is expanded in the same complete angular basis; equating coefficients therefore enforces exact reproduction of all joint probabilities for the lepton angles. Because the basis spans the full angular dependence of the leptonic decays and the marginal single-Z distributions are independent of the distant measurement by construction, no-signaling holds automatically. An explicit parametrization of λ is not supplied because the algebraic conditions already certify existence of a suitable measure; we will add a clarifying paragraph in Section 2 explaining this point and confirming the marginals. revision: partial

  2. Referee: Abstract: the w=0 case is presented as yielding a necessary and sufficient criterion via algebraic and positivity conditions, yet the same concern applies: without demonstrating that the resulting distribution over λ is local and reproduces the full four-momentum kinematics rather than a truncated angular expansion, the criterion may certify only a projection of the correlations.

    Authors: For w=0 the same complete angular basis is employed, so the matching is not a truncation. In the Z rest frames the four-momenta of the leptons are completely determined by the two angles; hence reproducing the full angular distribution reproduces the observable kinematics. Locality is again built into the product form of the LHVT. We will revise the text to state explicitly that the conditions apply to the complete angular distribution and therefore to the full decay kinematics. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation is direct coefficient equating between independent LHVT ansatz and QFT distribution

full rationale

The paper derives existence conditions for an LHVT by explicitly constructing an angular distribution from local hidden variables (under angular-momentum conservation) and setting its coefficients equal to those of the QFT prediction. This is a standard algebraic matching procedure that solves for parameter values (state coefficients a_i, b_i and heta_W window) at which the two expressions coincide. No parameter is fitted from data and then re-used as a 'prediction'; no self-citation supplies a load-bearing uniqueness theorem; the target QFT distribution is taken from standard electroweak theory and is independent of the LHVT construction. The resulting conditions (unique entangled state for w eq0, algebraic/positivity criteria for w=0) are therefore genuine outputs of the equating step rather than re-statements of the inputs. The paper remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated beyond the standard assumption of angular-momentum conservation and the QFT angular distribution taken as given.

pith-pipeline@v0.9.1-grok · 5825 in / 1236 out tokens · 20130 ms · 2026-06-27T12:22:06.929251+00:00 · methodology

discussion (0)

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Works this paper leans on

31 extracted references · 21 canonical work pages · 1 internal anchor

  1. [1]

    Einstein, B

    A. Einstein, B. Podolsky and N. Rosen,Can quantum-mechanical description of physical reality be considered complete?,Phys. Rev.47(1935) 777

    1935

  2. [2]

    Bell,On the einstein podolsky rosen paradox,Physics Physique Fizika1(1964) 195

    J.S. Bell,On the einstein podolsky rosen paradox,Physics Physique Fizika1(1964) 195

    1964

  3. [3]

    Freedman and J.F

    S.J. Freedman and J.F. Clauser,Experimental test of local hidden-variable theories,Phys. Rev. Lett.28(1972) 938

    1972

  4. [4]

    Aspect, P

    A. Aspect, P. Grangier and G. Roger,Experimental tests of realistic local theories via bell’s theorem,Phys. Rev. Lett.47(1981) 460

    1981

  5. [5]

    Aspect, J

    A. Aspect, J. Dalibard and G. Roger,Experimental test of bell’s inequalities using time-varying analyzers,Phys. Rev. Lett.49(1982) 1804

    1982

  6. [6]

    Bouwmeester, J.-W

    D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter and A. Zeilinger, Experimental quantum teleportation,Nature390(1997) 575–579

    1997

  7. [7]

    Tittel, J

    W. Tittel, J. Brendel, H. Zbinden and N. Gisin,Violation of bell inequalities by photons more than 10 km apart,Phys. Rev. Lett.81(1998) 3563

    1998

  8. [8]

    Boschi, S

    D. Boschi, S. Branca, F. De Martini, L. Hardy and S. Popescu,Experimental realization of teleporting an unknown pure quantum state via dual classical and einstein-podolsky-rosen channels,Phys. Rev. Lett.80(1998) 1121. – 13 –

    1998

  9. [9]

    Riebe et al.,Deterministic quantum teleportation with atoms,Nature429(2004) 734

    M. Riebe et al.,Deterministic quantum teleportation with atoms,Nature429(2004) 734

    2004

  10. [10]

    Polarization and Entanglement in Baryon-Antibaryon Pair Production in Electron-Positron Annihilation

    M.D. Barrett et al.,Deterministic quantum teleportation of atomic qubits,Nature429(2004) 737. [11]BESIIICollaboration,Polarization and Entanglement in Baryon-Antibaryon Pair Production in Electron-Positron Annihilation,Nature Phys.15(2019) 631 [1808.08917]

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  11. [11]

    Fabbrichesi, R

    M. Fabbrichesi, R. Floreanini and G. Panizzo,Testing Bell Inequalities at the LHC with Top-Quark Pairs,Phys. Rev. Lett.127(2021) 161801 [2102.11883]

    work page arXiv 2021

  12. [12]

    Afik and J

    Y. Afik and J.R.M. de Nova,Entanglement and quantum tomography with top quarks at the LHC,Eur. Phys. J. Plus136(2021) 907 [2003.02280]

    work page arXiv 2021

  13. [13]

    T. Han, M. Low and T.A. Wu,Quantum entanglement and Bell inequality violation in semi-leptonic top decays,JHEP07(2024) 192 [2310.17696]

    work page arXiv 2024

  14. [14]

    Cheng, T

    K. Cheng, T. Han and M. Low,Optimizing fictitious states for Bell inequality violation in bipartite qubit systems with applications to the tt¯system,Phys. Rev. D109(2024) 116005 [2311.09166]

    work page arXiv 2024

  15. [15]

    Subba and R

    A. Subba and R. Rahaman,On bipartite and tripartite entanglement at present and future particle colliders,2404.03292

    work page arXiv

  16. [16]

    Subba, R.K

    A. Subba, R.K. Singh and R.M. Godbole,Looking into the quantum entanglement in H→ZZ ⋆ at LHC within SMEFT framework,2411.19171

    work page arXiv

  17. [17]

    Aadet al.(ATLAS), Nature633, 542 (2024), arXiv:2311.07288 [hep-ex]

    ATLAS Collaboration,Observation of quantum entanglement with top quarks at the ATLAS detector,Nature633(2024) 542 [2311.07288]. [19]CMSCollaboration,Observation of quantum entanglement in top quark pair production in proton–proton collisions at √s= 13TeV,Rept. Prog. Phys.87(2024) 117801 [2406.03976]

    work page arXiv 2024

  18. [18]

    T. Han, M. Low and Y. Su,Entanglement and Bell Nonlocality inτ +τ − at the BEPC, 2501.04801

    work page arXiv

  19. [19]

    von Kuk, K

    R. von Kuk, K. Lee, J.K.L. Michel and Z. Sun,Towards a Quantum Information Theory of Hadronization: Dihadron Fragmentation and Neutral Polarization in Heavy Baryons, 2503.22607

    work page arXiv

  20. [20]

    J. Pei, Y. Fang, L. Wu, D. Xu, M. Biyabi and T. Li,Quantum Entanglement Theory and Its Generic Searches in High Energy Physics,2505.09280

    work page arXiv

  21. [21]

    J. Pei, X. Hao, X. Wang and T. Li,Observation of quantum entanglement inΛ ¯Λpair production via electron-positron annihilation,2505.09931

    work page arXiv

  22. [22]

    Lin, M.-J

    S.-J. Lin, M.-J. Liu, D.Y. Shao and S.-Y. Wei,Spin correlations and Bell nonlocality inΛ ¯Λ pair production frome +e− collisions with a thrust cut,2507.15387

    work page arXiv

  23. [23]

    S. Wu, C. Qian, Q. Wang and Y.-G. Yang,Quantum steering and discord in hyperon-antihyperon system in electron-positron annihilation,2509.14990. [26]BESIIICollaboration,Test of local realism via entangledΛ ¯Λsystem,2505.14988

    work page arXiv

  24. [24]

    Cheng and B

    K. Cheng and B. Yan,Bell Inequality Violation of Light Quarks in Dihadron Pair Production at Lepton Colliders,Phys. Rev. Lett.135(2025) 011902 [2501.03321]

    work page arXiv 2025

  25. [25]

    S. Abel, M. Dittmar and H. Dreiner,Testing locality at colliders via bell’s inequality?, Physics Letters B280(1992) 304. – 14 –

    1992

  26. [26]

    S. Li, W. Shen and J.M. Yang,Can Bell inequalities be tested via scattering cross-section at colliders ?,Eur. Phys. J. C84(2024) 1195 [2401.01162]

    work page arXiv 2024

  27. [27]

    Bechtle, C

    P. Bechtle, C. Breuning, H.K. Dreiner and C. Duhr,A critical appraisal of tests of locality and of entanglement versus non-entanglement at colliders,2507.15947

    work page arXiv

  28. [28]

    Abel, H.K

    S.A. Abel, H.K. Dreiner, R. Sengupta and L. Ubaldi,Colliders are Testing neither Locality via Bell’s Inequality nor Entanglement versus Non-Entanglement,2507.15949

    work page arXiv

  29. [29]

    Low, Phys

    M. Low,Addressing local realism through Bell tests at colliders,Phys. Rev. D112(2025) 096008 [2508.10979]

    work page arXiv 2025

  30. [30]

    J. Pei, L. Wu, T. Li and X. Hao,Excluding Local Hidden Variables inΛ ¯ΛProduction: The Incompatibility with Angular-Momentum Conservation and CPT Invariance,2601.15747

    work page arXiv

  31. [31]

    Pei and L

    J. Pei and L. Wu,Can Mirror Symmetry Challenge Local Realism? Probing Photon Entanglement from Positronium via Compton Scattering,2602.08541. [35]Particle Data GroupCollaboration,Review of particle physics,Phys. Rev. D110(2024) 030001. – 15 –

    work page arXiv 2024