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arXiv:2606.10737

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

Junle Pei

Published Jun 9, 2026• 0 citations
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Abstract

Quantum entanglement in di-boson 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, I 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, I derive the conditions under which an LHVT construction exists. For the case $w\neq 0$, I show that, apart from trivial product-state configurations, an LHVT construction exists only for a unique entangled state, corresponding to $a_1=a_3=-a_2=1/\sqrt{3}$ and $b_2=b_3=0$, together with restricted windows of the weak-mixing angle $θ_W$. For $w=0$, I 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, I consider the phenomenologically relevant interaction $sZ^μZ_μ$ and show that an LHVT construction exists at threshold $m_s=2m_Z$, whereas it does not exist for $m_s>2m_Z$.

DOI
10.48550/arxiv.2606.10737

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Does the Weinberg angle allow a local hidden-variable description for the leptonic decays of an entangled ZZZZZZ pair?
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Excluding local hidden variables in ΛΛ‾ \Lambda \overline{\Lambda} ΛΛ production: the incompatibility with angular-momentum conservation and CPT invariance
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This study investigates photon entanglement generated from parapositronium decay by analyzing azimuthal correlations after the double Compton scattering with stationary electrons. We introduce a normalized correlation observable O 1 = cos ( 2 ϕ 1 − 2 ϕ 2 ) / C 1 to witness entanglement. In the absence of decoherence, ⟨ O 1 ⟩ = − 1 , corresponding to a maximally entangled Bell state. With decoherence parametrized by ρ , the expectation becomes − ( 1 − ρ ) , allowing direct experimental quantification of coherence loss. A prior symmetry analysis of the Compton scattering process within the quantum field theory (QFT) is provided, which establishes the mirror-symmetric nature of the single-photon angular distribution. We further examine a local hidden-variable theory (LHVT) under the angular-momentum conservation and the rotation invariance. Imposing the mirror symmetry with respect to the plane defined by the photon spin and momentum leads to a non-negative LHVT prediction for ⟨ sin 2 θ 1 sin 2 θ 2 cos ( 2 ϕ 1 − 2 ϕ 2 ) ⟩ , contradicting the negative QFT prediction value for any ρ < 1 . Thus, mirror symmetry serves as a novel criterion to exclude LHVT descriptions of the entangled state, whereas without preserving this symmetry, LHVTs can reproduce the correlations.
Entanglement is studied in the framework of Dyson's S-matrix theory in relativistic quantum field theory, which leads to a natural definition of entangled states of a particle-antiparticle pair and the spin operator from a Noether current. As an explicit example, the decay of a massive pseudo-scalar particle into a pair of electron and positron is analyzed. Two spin operators are extracted from the Noether current. The Wigner spin operator characterizes spin states at the rest frame of each fermion and, although not measurable in the laboratory, gives rise to a straightforward generalization of low-energy analysis of entanglement to the ultrarelativistic domain. In contrast, if one adopts a (modified) Dirac spin operator, the entanglement measured by spin correlation becomes maximal near the threshold of the decay, while the entanglement is replaced by the classical correlation for the ultrarelativistic electron-positron pair by analogy to the case of neutrinos, for which a hidden-variables type of description is possible. Chiral symmetry differentiates the spin angular momentum and the magnetic moment. The use of weak interaction that can measure helicity is suggested in the analysis of entanglement at high energies instead of a Stern-Gerlach apparatus, which is useless for the electron. A difference between the electron spin at high energies and the photon linear polarization is also noted. The Standard Model can describe all of the observable properties of leptons.
A bstract Recently there has been an increased interest in possible tests of locality via Bell’s inequalities, as well as separately tests of entanglement at colliders, in particular at the LHC. These have involved various physical processes, such as tt‾ t\overline{t} tt t t ¯ , or τ + τ − production, or the decay of a Higgs boson to two vector bosons H → VV ∗ . We argue that none of these proposals constitute a test of locality via Bell’s inequality or promise unconditional observational evidence of entanglement. In all cases what is measured are the momenta of the final state particles. Using the construction proposed by Kasday (1971) in a different context, and adapted to collider scenarios by Abel, Dittmar, and Dreiner (1992), it is straightforward to construct a local hidden variable theory (LHVT) which exactly reproduces the data. This construction is only possible as the final state momenta all commute. We show that this LHVT satisfies Bell’s inequality or the related CHSH inequality as appropriate for all the proposed LHC collider tests in the literature. Thus a test of locality via Bell’s inequality is not possible at colliders. The LHVT is also by construction local, i.e. all correlations are separable. Thus an unconditional proof of entanglement is also inherently not possible at colliders. It can only be shown that the entanglement within the Standard Model consistently describes the data.
Abstract Quantum entanglement of weak interaction gauge bosons produced at colliders can be explored by computing the corresponding polarization density matrix. To this end, we consider the Higgs boson decays H→WW∗H\rightarrow W W^*H→WW∗ H → W W ∗ and H→ZZ∗H\rightarrow Z Z^*H→ZZ∗ H → Z Z ∗ , in which W∗W^*W∗ W ∗ and Z∗Z^*Z∗ Z ∗ are off-shell states, and the WW , WZ and ZZ di-boson production in proton collisions. The polarization density matrix of the di-boson state is determined by the amplitude of the production process and can be experimentally reconstructed from the angular distribution of the momenta of the final states into which the gauge bosons decay. We show that a suitable instance of the Bell inequality is violated in H→ZZ∗H\rightarrow Z Z^*H→ZZ∗ H → Z Z ∗ to a degree that can be tested at the LHC with future data. The same Bell inequality is violated in the production of WW and ZZ boson pairs for invariant masses above 900 GeV and scattering angles close to π/2\pi /2π/2 π / 2 in the center of mass frame. LHC data in this case are not sufficient to establish the violation of the Bell inequality. We also analyze the prospects for detecting Bell inequality violations in di-boson final states at future e+e−e^+e^-e+e− e + e - and muon colliders. A further observable that provides a lower bound on the amount of polarization entanglement in the di-boson system is computed for each of the examined processes. The analytic expressions for the polarization density matrices are presented in full in an Appendix. We also provide the unitary matrices required in the optimization procedure necessary in testing the Bell inequalities.
Spin correlations of e + e − pair productions of two colliding photons are investigated and explicit expressions for their corresponding probabilities are derived and found to be energy (speed) dependent, for initially linearly and circularly polarized photons, different from those obtained by simply combining the spins of the relevant particles, for initially polarized photons. These expressions also depend on the angles of spin of e + (and/or of e −), for initially linearly polarized photons, but not for circularly polarized photons, as a function of the energy. It is remarkable that these explicit results obtained from quantum field theory show a clear violation of Bell's inequality of Local Hidden Variables theories at all energies beyond that of the threshold one for particle production, in support of quantum field theory in the relativistic regime. We hope that our explicit expression will lead to experiments, of the type described in the bulk of this paper, which can monitor energy (and speed) in polarization correlation experiments.