Colliders are not Testing Locality via Bell's Inequality nor Providing an Unconditional Proof of Entanglement
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 $t\bar t$, or $τ^+τ^-$ production, or the decay of a Higgs boson to two vector bosons $H\to VV^*$. We argue that $\textit{none}$ 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, $\textit{i.e.}$ all correlations are separable. Thus an unconditional proof of entanglement is also inherently $\textit{not}$ possible at colliders. It can only be shown that the entanglement within the Standard Model consistently describes the data.