Determining the Spin-Analyzing Powers via Invariants of the Spin Correlation Matrices and Probing the Bell Non-Locality at the Lepton Colliders

TL;DR

This paper introduces a method to determine spin-analyzing powers and probe Bell non-locality at lepton colliders using invariants of spin correlation matrices, applicable to new physics searches.

Contribution

It proves that the trace of the spin correlation matrix is an invariant quantity, enabling the reconstruction of spin correlations and Bell non-locality testing at colliders.

Findings

The trace of the spin correlation matrix is invariant under basis rotations.

The method allows determination of spin-analyzing powers via this invariant.

Application to $ ext{Lambda}ar{ ext{Lambda}}$ production at BESIII demonstrates the approach.

Abstract

We consider the two-fermion $F_a F_b$ productions and decays via one mediator exchange at the $e^+e^-$ collider. With the assumption that the spin is defined via the Lorentz symmetry, or considering the implicit symmetry in the spin density matrix, we prove that the trace ${\rm Tr} [C]$ of the spin correlation matrix $C$ is an invariant quantity, and is invariant under basis rotations. Thus, for the exchanges of one mediator such as scalar and gauge boson, we can determine the product of the spin-analyzing powers for $F_a F_b$ via ${\rm Tr} [C]$, and reconstruct the spin correlation matrix. With the CHSH-Horodecki criterion, we can probe the Bell non-locality, and evade the no-go theorem. To be concrete, we study the Bell non-locality for the $\Lambda \bar \Lambda$ productions and decays at the BESIII experiment. In addition, the invariant ${\rm Tr} [C]$ is a new physics observable to probe the new physics beyond the Standard Model (SM) and study the SM precision measurements. Moreover, for the scalar exchanges, we discuss the general invariants of the spin correlation matrices and the related phenomenological consequences.