Excluding Local Hidden Variables in $\Lambda\bar{\Lambda}$ Production: The Incompatibility with Angular-Momentum Conservation and CPT Invariance

We analyze spin entanglement in $\Lambda\bar{\Lambda}$ pairs produced in the decays of scalar ($h$) and pseudoscalar ($a$) particles, contrasting predictions from quantum field theory (QFT) with those of local hidden-variable theories (LHVTs). Using the self-analyzing weak decays $\Lambda \to p\pi^-$ and $\bar{\Lambda} \to \bar{p}\pi^+$, we derive the joint angular distributions within QFT. Our key findings are: For scalar production $h \to \Lambda\bar{\Lambda}$, no LHVT respecting locality and angular-momentum conservation can reproduce the QFT distribution. For pseudoscalar production $a \to \Lambda\bar{\Lambda}$, a CPT-symmetric LHVT is excluded by positivity constraints given the measured analyzing powers; however, if CPT symmetry is relaxed, an explicit LHVT construction, with uniform hidden-variable measure and response functions satisfying $b_1 c_1 = 3\alpha_{\Lambda}\alpha_{\bar{\Lambda}}$, can match the QFT result. These distinct signatures provide clear, experimentally testable criteria to discriminate between QFT and LHVT in $\Lambda\bar{\Lambda}$ systems across different production mechanisms.