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Bell inequalities are consequences of local realism while violated by quantummechanics. In particle physics, entangled high energy particles can be producedfrom a common source, and the decay of each particle plays the role ofmeasurement. However, in a hidden variable theory, the decay could bedetermined by hidden variables. This loophole killed such approaches to Belltest in particle physics. It is a special form of measurement-setting orfree-will loophole, which also exists in other systems. Using entangledbaryons, we present new inequalities of local realism with the explicitassumption of the dependence of the decays on hidden variables, as well as theconsideration of the statistical mixture of polarizations and the separation oflocal hidden variables for objects with spacelike distances. These violationscloses the measurement-setting loophole once and for all. We propose to use theprocesses $\eta _c\to \Lambda \bar{\Lambda}$ and $\chi _{c0} \to \Lambda\bar{\Lambda}$ to test our inequalities, and show that their violations arelikely to be observed with the data already collected in BESIII.

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