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

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 $\theta_W$. 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 $sZ^\mu Z_\mu$ and show that an LHVT construction exists at threshold $m_s=2m_Z$, whereas it does not exist for $m_s>2m_Z$.