Can Bell inequalities be tested via scattering cross-section at colliders ?

In current studies for testing Bell inequalities at colliders, the reconstruction of spin correlations from scattering cross-sections relies on the bilinear form of the spin correlations, but not all local hidden variable models (LHVMs) have such a property. To demonstrate that a general LHVM cannot be rule out via scattering cross-section data, we propose a specific LHVM, which can exactly duplicate the same scattering cross-section for particle production and decay as the standard quantum theory, making it indistinguishable at colliders in principle. Despite of this, we find that reconstructing spin correlations through scattering cross-sections can still exclude a broad class of LHVMs, e.g., those models employing classical spin correlations as a surrogate for quantum spin correlations.