Entangled baryons: violation of Inequalities based on local realism assuming dependence of decays on hidden variables

Bell inequalities are consequences of local realism while violated by quantum mechanics. In particle physics, entangled high energy particles can be produced from a common source, and the decay of each particle plays the role of measurement. However, in a hidden variable theory, the decay could be determined by hidden variables. This loophole killed such approaches to Bell test in particle physics. It is a special form of measurement-setting or free-will loophole, which also exists in other systems. Using entangled baryons, we present new inequalities of local realism with the explicit assumption of the dependence of the decays on hidden variables, as well as the consideration of the statistical mixture of polarizations and the separation of local hidden variables for objects with spacelike distances. These violations closes the measurement-setting loophole once and for all. We propose to use the processes $\eta _c\to \Lambda \bar{\Lambda}$ and $\chi _{c0} \to \Lambda \bar{\Lambda}$ to test our inequalities, and show that their violations are likely to be observed with the data already collected in BESIII.