Unveiling a Universal Formalism for Quantum Entanglement in Arbitrary Spin Decays

Abstract

We present a comprehensive theoretical framework for probing quantum entanglement in the decay angular distributions of a spin-$S$ particle-antiparticle pair $A\bar{A}$, where each particle decays sequentially into a two-body final state, $A\to B+C$ and $\bar{A}\to\bar{B}+\bar{C}$, with $B(\bar{B})$ carrying spin $b$ and $C(\bar{C})$ being spinless. Starting from the most general polarized initial state, we derive the fully differential angular distribution $\mathcal{W}(\theta_1,\theta_2,\phi_1,\phi_2)$ and identify observables $\langle\cos(2S(\phi_1\mp\phi_2))\rangle$ whose expectation values directly depend on the entanglement-sensitive coefficients $\text{Re}\left(\alpha_{-S,\mp S}\alpha_{S,\pm S}^*\right)$ of the initial state. The proportionality factor $\mathcal{C}(S,b)$ in these relations is computed explicitly. For bosonic decays ($b=0,1,2,\ldots$), $\mathcal{C}(S,b)$ is universal and independent of decay dynamics; in particular, $\mathcal{C}(S,0)=1/2$ for any $S$, and $\mathcal{C}(1,1)=1/8$, matching known results for $W^+W^-$ decays. For fermionic decays ($b=\frac{1}{2},\frac{3}{2},\frac{5}{2}\ldots$), $\mathcal{C}(S,b)$ depends explicitly on the spin analysis powers $\alpha_{A/\bar{A}}$, making entanglement extraction more decay-dependent. We further demonstrate, within the context of $e^+e^-\to\gamma^*\to A\bar{A}$ production, how $\alpha_{A/\bar{A}}$ can be determined experimentally using specific angular observables restricted to the beam-axis region. Our results highlight the special role of bosonic decays in providing clean, model-independent tests of quantum entanglement at colliders, while outlining a pathway for entanglement measurement in fermionic cases through supplementary polarization information.