Entanglement suppression for $ΩΩ$ scattering

We study entanglement suppression in $s$-wave $ΩΩ$ scattering, where each baryon has spin $3/2$. By treating the $S$-matrix as a quantum operator acting on the spin states, we quantify its ability to generate entanglement and identify the conditions on the phase shifts of the spin channels that minimize entanglement generation in the system. In $ΩΩ$ scattering, only antisymmetric spin channels are allowed due to Fermi-Dirac statistics. Applying the entanglement-suppression framework to $ΩΩ$ scattering, we find two solutions for the phase shifts: one leading to a spin SU(4) symmetry and the other to a nonrelativistic conformal symmetry. We show that the solution associated with the nonrelativistic conformal symmetry originates from the specific structure of the Clebsch-Gordan coefficients in the $3/2 \otimes 3/2$ system.

💡 Research Summary

The paper investigates how the principle of entanglement suppression can be used to uncover emergent symmetries in hadronic scattering, focusing on the s‑wave scattering of two Ω baryons, each carrying spin 3/2. The authors begin by reviewing the entanglement‑suppression framework that was previously applied to nucleon‑nucleon (NN) scattering. In that context, the S‑matrix is written as a sum of projection operators onto the symmetric (J = 1) and antisymmetric (J = 0) spin channels, weighted by phase shifts δ₁ and δ₀. By treating the S‑matrix as a quantum gate acting on a product of two spin‑½ states, they define the entanglement power (EP) as the average linear entropy generated from all possible product inputs. The EP for NN scattering is found to be EP = (1/6) sin²(2Δδ) with Δδ = δ₀ − δ₁. The EP vanishes when Δδ = 0 (identity gate) or Δδ = π/2 (SWAP gate). The former corresponds to identical interactions in the two spin channels and realizes an emergent SU(4) spin‑flavor symmetry; the latter corresponds to one channel at the unitarity limit and the other non‑interacting, giving rise to a non‑relativistic conformal symmetry.

The authors then extend the analysis to ΩΩ scattering. Because each Ω has spin 3/2, the total spin decomposition is 3/2 ⊗ 3/2 = 0 ⊕ 1 ⊕ 2 ⊕ 3. Fermi‑Dirac statistics require the overall wavefunction to be antisymmetric; for s‑wave scattering this forces the spin part to be antisymmetric, allowing only the J = 0 and J = 2 channels. The low‑energy S‑matrix is therefore ˆS_ΩΩ = e^{2iδ₀} J₀ + e^{2iδ₂} J₂, where J₀ and J₂ are projection operators that can be expressed as polynomials of the SU(2) Casimir t·t in the four‑dimensional spin‑3/2 representation.

A direct application of the NN EP formula fails because the final state is not normalized when the symmetric channels (J = 1, 3) are absent. To resolve this, the authors introduce the antisymmetric projector A = J₀ + J₂ and define a normalized outgoing state |ψ_out⟩ = ˆS_ΩΩ|ψ_in⟩ /⟨ψ_in|A|ψ_in⟩. They then define a weighted EP, E₂, which averages the linear entropy with a weight proportional to the square of the normalization factor. After performing the angular integration over the product of single‑particle spin coherent states (using the Fubini–Study measure), they obtain a compact expression:

E₂(ˆS_ΩΩ) =