Excluding local hidden variables in $$ \Lambda \overline{\Lambda} $$ production: the incompatibility with angular-momentum conservation and CPT invariance

A bstract We analyze spin entanglement in $$ \Lambda \overline{\Lambda} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Λ</mml:mi> <mml:mover> <mml:mi>Λ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> pairs produced in the decays of spin-zero particles, contrasting predictions from quantum field theory (QFT) with those of local hidden-variable theories (LHVTs). Using the self-analyzing weak decays Λ → pπ − and $$ \overline{\Lambda}\to \overline{p}{\pi}^{+} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>Λ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mo>→</mml:mo> <mml:mover> <mml:mi>p</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:msup> <mml:mi>π</mml:mi> <mml:mo>+</mml:mo> </mml:msup> </mml:math> , we derive the joint angular distributions within QFT. Our key findings are: for scalar production $$ h\to \Lambda \overline{\Lambda} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>h</mml:mi> <mml:mo>→</mml:mo> <mml:mi>Λ</mml:mi> <mml:mover> <mml:mi>Λ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> , no LHVT respecting locality and angular-momentum conservation can reproduce the QFT distribution. For pseudoscalar production $$ a\to \Lambda \overline{\Lambda} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>a</mml:mi> <mml:mo>→</mml:mo> <mml:mi>Λ</mml:mi> <mml:mover> <mml:mi>Λ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> , 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 $$ {b}_1{c}_1=3{\alpha}_{\Lambda}{\alpha}_{\overline{\Lambda}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>b</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:msub> <mml:mi>c</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>Λ</mml:mi> </mml:msub> <mml:msub> <mml:mi>α</mml:mi> <mml:mover> <mml:mi>Λ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:msub> </mml:math> — can match the QFT result. For the most general spin-zero decay $$ s\to \Lambda \overline{\Lambda} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>s</mml:mi> <mml:mo>→</mml:mo> <mml:mi>Λ</mml:mi> <mml:mover> <mml:mi>Λ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> with arbitrary scalar-pseudoscalar mixing, we, under CPT invariance, identify the regions of parameter space where the QFT joint angular distribution does or does not admit an LHVT realization. These distinct signatures provide clear, experimentally testable criteria to discriminate between QFT and LHVT in $$ \Lambda \overline{\Lambda} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Λ</mml:mi> <mml:mover> <mml:mi>Λ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> systems across different production mechanisms.