image: Coordinate system used to analyze the decay ψ(3686)→ΛΛ̄→pπ-pπ+ . Because the weak decays of Λ and Λ̄ violate parity, the angular distributions of the proton and antiproton carry information about the spins of their parent hyperons. A five-dimensional joint angular analysis allows the complete spin state of the ΛΛ̄ pair to be reconstructed.
Credit: ©Science Bulletin/BESIII Collaboration
How completely can scientists describe the quantum state of a particle-antiparticle pair?
For a pair of spin-1/2 particles, measuring the polarization of each particle tells only part of the story. A complete description must also include the correlations between their spins. These quantities can all be encoded in a spin density matrix — effectively a complete quantum spin map of the two-particle system.
The BESIII Collaboration has now achieved the first complete determination of the spin density matrix in the decay ψ(3686)→ΛΛ̄ . Using (448.1 ± 2.9) million ψ(3686) events collected with the BESIII detector, the researchers used the same experimental framework to investigate both how the hyperon-antihyperon pair is produced and how its spins are quantum mechanically correlated.
The results are published in Science Bulletin.
Hyperons as natural spin analyzers
The Λ hyperon has a special property that makes this measurement possible.
The Λ and its antiparticle decay through the weak interaction. Because the weak interaction violates parity, their decay products are not emitted isotropically: the directions of the proton from Λ→pπ- and the antiproton from Λ̄ →p̄π+ carry information about the spins of their parent particles. In this sense, Λ and Λ̄ act as natural “spin analyzers.”
The BESIII researchers reconstructed the full decay chain ψ(3686)→ΛΛ̄ →pπ-pπ+ and performed a five-dimensional joint angular analysis, simultaneously using the Λ production angle and the decay angles of the proton and antiproton. This approach gives direct access to the full spin structure of the ΛΛ̄ system, rather than only partially integrated angular information.
A significant transverse polarization
The analysis determined the angular-distribution parameter to be αψ = 0.82 ± 0.02 (stat.) ± 0.01 (syst.) and measured, for the first time in this decay, a relative phase between the production amplitudes of ΔΦ = (20.6 ± 3.6 (stat.) ± 0.6 (syst.)) degrees. This nonzero phase produces a transverse polarization of the Λ hyperon. The maximum polarization was measured to be Py = 0.074 ± 0.013 (stat.) ± 0.003 (syst.), with a significance of 6.3 standard deviations relative to the non-polarization hypothesis.
The spin density matrix also contains correlations between different spin components of the Λ and Λ̄ . Measuring these correlations opens another window: the study of quantum entanglement in a system of massive, strongly interacting particles.
Testing quantum nonlocality with hyperons
Quantum entanglement has been extensively studied using photons, atoms and solid-state systems. Experimental studies involving strongly interacting massive particles, however, remain comparatively scarce.
The parity-violating weak decays of Λ and Λ̄ provide a way to access their spin information and make the hyperon-antihyperon system particularly suitable for such studies.
Using the measured polarization and spin-correlation coefficients, the BESIII researchers evaluated Wootters' concurrence, a measure of two-particle entanglement, and the Bell nonlocality measure m12, which can be used to test the Clauser-Horne-Shimony-Holt (CHSH) inequality.
Local hidden-variable theories require m12 ≤ 1. The experiment obtained an average value of m12 = 1.266 ± 0.078 (stat.) ± 0.049 (syst.).
A likelihood-ratio test using simulated experiments yielded a 3.7 sigma violation of the CHSH inequality, providing evidence for quantum nonlocality in the ΛΛ̄ system produced in vector-charmonium decay.
One framework, two frontiers
A key feature of the study is that the investigation of particle-production dynamics and quantum correlations does not require two separate experimental descriptions. Both are contained in the same spin density matrix.
On one side, the angular-distribution parameter and relative phase probe the dynamics by which ψ(3686) produces the ΛΛ̄ pair. On the other, the polarization and spin-correlation coefficients provide access to entanglement and Bell nonlocality.
The researchers also used the measured production parameters to study the relative S- and D-wave contributions. Comparisons with ΛΛ̄ and Σ0Σ̄0 production at other charmonium resonances reveal different phase behaviors, highlighting the sensitivity of spin observables to the internal spin-flavor structure of hyperons.
The results establish ψ(3686)→ΛΛ̄ as a laboratory in which hadron production dynamics, hyperon structure and quantum correlations can be studied within a unified experimental framework.