Bell inequality is violated in charmonium decays

The experimental data on the helicity amplitudes of charmonium decays allow us to measure entanglement in final state spin correlations and test possible violations of the Bell inequality. We find that the Bell inequality is violated with a significance of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mn>5</a:mn><a:mi>σ</a:mi></a:mrow></a:math> or more in the decays <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mrow><c:msub><c:mrow><c:mi>η</c:mi></c:mrow><c:mrow><c:mi>c</c:mi></c:mrow></c:msub><c:mo>,</c:mo><c:msubsup><c:mrow><c:mi>χ</c:mi></c:mrow><c:mrow><c:mi>c</c:mi></c:mrow><c:mrow><c:mn>0</c:mn></c:mrow></c:msubsup><c:mo>,</c:mo><c:mi>J</c:mi><c:mo>/</c:mo><c:mi>ψ</c:mi><c:mo stretchy="false">→</c:mo><c:mi mathvariant="normal">Λ</c:mi><c:mo>+</c:mo><c:mover accent="true"><c:mrow><c:mi mathvariant="normal">Λ</c:mi></c:mrow><c:mrow><c:mo stretchy="false">¯</c:mo></c:mrow></c:mover></c:mrow></c:math>, <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:mrow><j:mi>J</j:mi><j:mo>/</j:mo><j:mi>ψ</j:mi><j:mo stretchy="false">→</j:mo><j:msup><j:mrow><j:mi mathvariant="normal">Ξ</j:mi></j:mrow><j:mrow><j:mo>−</j:mo></j:mrow></j:msup><j:mo>+</j:mo><j:msup><j:mrow><j:mover accent="true"><j:mrow><j:mi mathvariant="normal">Ξ</j:mi></j:mrow><j:mrow><j:mo stretchy="false">¯</j:mo></j:mrow></j:mover></j:mrow><j:mrow><j:mo>+</j:mo></j:mrow></j:msup><j:mo>,</j:mo><j:msup><j:mrow><j:mi mathvariant="normal">Ξ</j:mi></j:mrow><j:mrow><j:mn>0</j:mn></j:mrow></j:msup><j:mo>+</j:mo><j:msup><j:mrow><j:mover accent="true"><j:mrow><j:mi mathvariant="normal">Ξ</j:mi></j:mrow><j:mrow><j:mo stretchy="false">¯</j:mo></j:mrow></j:mover></j:mrow><j:mrow><j:mn>0</j:mn></j:mrow></j:msup><j:mo>,</j:mo><j:msup><j:mrow><j:mi mathvariant="normal">Σ</j:mi></j:mrow><j:mrow><j:mo>−</j:mo></j:mrow></j:msup><j:mo>+</j:mo><j:msup><j:mrow><j:mover accent="true"><j:mrow><j:mi mathvariant="normal">Σ</j:mi></j:mrow><j:mrow><j:mo stretchy="false">¯</j:mo></j:mrow></j:mover></j:mrow><j:mrow><j:mo>+</j:mo></j:mrow></j:msup><j:mo>,</j:mo><j:msup><j:mrow><j:mi mathvariant="normal">Σ</j:mi></j:mrow><j:mrow><j:mn>0</j:mn></j:mrow></j:msup><j:mo>+</j:mo><j:msup><j:mrow><j:mover accent="true"><j:mrow><j:mi mathvariant="normal">Σ</j:mi></j:mrow><j:mrow><j:mo stretchy="false">¯</j:mo></j:mrow></j:mover></j:mrow><j:mrow><j:mn>0</j:mn></j:mrow></j:msup></j:mrow></j:math>, <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:mrow><cb:mi>ψ</cb:mi><cb:mo stretchy="false">(</cb:mo><cb:mn>3686</cb:mn><cb:mo stretchy="false">)</cb:mo><cb:mo stretchy="false">→</cb:mo><cb:msup><cb:mrow><cb:mi mathvariant="normal">Ξ</cb:mi></cb:mrow><cb:mrow><cb:mo>−</cb:mo></cb:mrow></cb:msup><cb:mo>+</cb:mo><cb:msup><cb:mrow><cb:mover accent="true"><cb:mrow><cb:mi mathvariant="normal">Ξ</cb:mi></cb:mrow><cb:mrow><cb:mo stretchy="false">¯</cb:mo></cb:mrow></cb:mover></cb:mrow><cb:mrow><cb:mo>+</cb:mo></cb:mrow></cb:msup><cb:mo>,</cb:mo><cb:msup><cb:mrow><cb:mi mathvariant="normal">Σ</cb:mi></cb:mrow><cb:mrow><cb:mo>−</cb:mo></cb:mrow></cb:msup><cb:mo>+</cb:mo><cb:msup><cb:mrow><cb:mover accent="true"><cb:mrow><cb:mi mathvariant="normal">Σ</cb:mi></cb:mrow><cb:mrow><cb:mo stretchy="false">¯</cb:mo></cb:mrow></cb:mover></cb:mrow><cb:mrow><cb:mo>+</cb:mo></cb:mrow></cb:msup><cb:mo>,</cb:mo><cb:msup><cb:mrow><cb:mi mathvariant="normal">Σ</cb:mi></cb:mrow><cb:mrow><cb:mn>0</cb:mn></cb:mrow></cb:msup><cb:mo>+</cb:mo><cb:msup><cb:mrow><cb:mover accent="true"><cb:mrow><cb:mi mathvariant="normal">Σ</cb:mi></cb:mrow><cb:mrow><cb:mo stretchy="false">¯</cb:mo></cb:mrow></cb:mover></cb:mrow><cb:mrow><cb:mn>0</cb:mn></cb:mrow></cb:msup></cb:mrow></cb:math>, <tb:math xmlns:tb="http://www.w3.org/1998/Math/MathML" display="inline"><tb:mrow><tb:msubsup><tb:mrow><tb:mi>χ</tb:mi></tb:mrow><tb:mrow><tb:mi>c</tb:mi></tb:mrow><tb:mrow><tb:mn>0</tb:mn></tb:mrow></tb:msubsup><tb:mo>,</tb:mo><tb:msubsup><tb:mrow><tb:mi>χ</tb:mi></tb:mrow><tb:mrow><tb:mi>c</tb:mi></tb:mrow><tb:mrow><tb:mn>1</tb:mn></tb:mrow></tb:msubsup><tb:mo stretchy="false">→</tb:mo><tb:mi>ϕ</tb:mi><tb:mo>+</tb:mo><tb:mi>ϕ</tb:mi></tb:mrow></tb:math>. The decays <wb:math xmlns:wb="http://www.w3.org/1998/Math/MathML" display="inline"><wb:mi>ψ</wb:mi><wb:mo stretchy="false">(</wb:mo><wb:mn>3686</wb:mn><wb:mo stretchy="false">)</wb:mo><wb:mo stretchy="false">→</wb:mo><wb:mi mathvariant="normal">Λ</wb:mi><wb:mo>+</wb:mo><wb:mover accent="true"><wb:mi mathvariant="normal">Λ</wb:mi><wb:mo stretchy="false">¯</wb:mo></wb:mover></wb:math> and <fc:math xmlns:fc="http://www.w3.org/1998/Math/MathML" display="inline"><fc:msup><fc:mi mathvariant="normal">Ξ</fc:mi><fc:mn>0</fc:mn></fc:msup><fc:mo>+</fc:mo><fc:msup><fc:mover accent="true"><fc:mi mathvariant="normal">Ξ</fc:mi><fc:mo stretchy="false">¯</fc:mo></fc:mover><fc:mn>0</fc:mn></fc:msup></fc:math> show the same violation but with less significance. The decay <lc:math xmlns:lc="http://www.w3.org/1998/Math/MathML" display="inline"><lc:mrow><lc:mi>ψ</lc:mi><lc:mo stretchy="false">(</lc:mo><lc:mn>3686</lc:mn><lc:mo stretchy="false">)</lc:mo><lc:mo stretchy="false">→</lc:mo><lc:msup><lc:mrow><lc:mi mathvariant="normal">Ω</lc:mi></lc:mrow><lc:mrow><lc:mo>−</lc:mo></lc:mrow></lc:msup><lc:mo>+</lc:mo><lc:msup><lc:mrow><lc:mover accent="true"><lc:mrow><lc:mi mathvariant="normal">Ω</lc:mi></lc:mrow><lc:mrow><lc:mo stretchy="false">¯</lc:mo></lc:mrow></lc:mover></lc:mrow><lc:mrow><lc:mo>+</lc:mo></lc:mrow></lc:msup></lc:mrow></lc:math> displays entanglement. These results firmly establish the presence of entanglement and quantum nonseparability at high energies, in a setting with particles of different spins and interacting through electroweak and strong interactions. In addition, the relatively long lifetime of some of the strange baryons produced in the decays provides a natural probe to test whether quantum spin correlations remain after the particles have interacted with the beam pipe and the first few layers of the detector. Published by the American Physical Society 2024