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

We analyze spin entanglement in $\mathrm{\Lambda }\bar{\mathrm{\Lambda }}$ 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 $\mathrm{\Lambda }\to p{\pi }^{-}$ and $\bar{\mathrm{\Lambda }}\to \bar{p}{\pi }^{+}$, we derive the joint angular distributions within QFT. Our key findings are: For scalar production $h\to \mathrm{\Lambda }\bar{\mathrm{\Lambda }}$, no LHVT respecting locality and angular-momentum conservation can reproduce the QFT distribution. For pseudoscalar production $a\to \mathrm{\Lambda }\bar{\mathrm{\Lambda }}$, 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 $b_1{c}_1=3{\alpha }_{\mathrm{\Lambda }}{\alpha }_{\bar{\mathrm{\Lambda }}}$, can match the QFT result. These distinct signatures provide clear, experimentally testable criteria to discriminate between QFT and LHVT in $\mathrm{\Lambda }\bar{\mathrm{\Lambda }}$ systems across different production mechanisms.