How does the maximal vacuum Bell-CHSH violation decay with spacelike separation?

Statement

Let $\beta (L)$ denote the supremum, over all admissible dichotomic observable pairs $(A_1,A_2;B_1,B_2)$ localized respectively in two spacelike-separated regions ${\mathcal{O}}_A,{\mathcal{O}}_B$ at separation $L$, of the Bell-CHSH functional in the vacuum of a free relativistic field. What is the functional form and decay rate of $\beta (L)$ as $L\to \infty$?

This is the quantitative half of IQOQI Open Quantum Problem #12 (R. Verch, Bell inequalities for long range vacuum correlations): vacuum fluctuations maximally violate CHSH for suitable spacelike-separated observables, and the violation tends to the classical bound $2$ as the two localization regions are moved apart. The rate of that decay — and explicit observables realizing it — is open.

Why It Matters

The $L\to 0$ (region-tangency) endpoint is PROVED: Summers–Werner showed that the type III₁ structure together with the Reeh–Schlieder property forces maximal violation $\beta =2\sqrt{2}$ (Tsirelson) for the algebras of tangent wedges/double cones. The group’s Weyl-operator + Tomita–Takesaki machinery supplies the body of the curve at finite $L$. Pinning the decay class — exponential $e^{-mL}$ (massive) versus power-law $L^{-p}$ (massless) — would settle a named open problem and tie the falloff directly to the mass gap and the split property.

What We Know

• Lower-bound machinery (group results). The Weyl + modular-conjugation construction (Phys. Rev. D 108, 085026) gives $\beta \ge 2\sqrt{1+|⟨W(f)W(g)⟩{|}^2}$, with the correlator fixed by the smeared Pauli–Jordan / two-point function. Causal-diamond results (Eur. Phys. J. C 85) and the optimal bounded-Hermitian operator class (Phys. Rev. D 112) make $\beta (L)$ a genuine supremum rather than a single construction. The numerical framework (Phys. Rev. D 110) maps it across mass and dimension.
• Upper-bound route (open). Bounding the supremum over all observables requires controlling the maximal correlation between $\mathcal{A}({\mathcal{O}}_A)$ and $\mathcal{A}({\mathcal{O}}_B)$ via Buchholz–Wichmann–Yngvason nuclearity / the split property — not yet carried out for this quantity.

Possible Approaches

1. Explicit lower bound ${\beta }_{\text{lower}}(L)$ for two diamonds, extracting the leading $L$-law (massive exponential / massless power-law).
2. Nuclearity/split upper bound: for massive free fields the split property yields exponential clustering, giving $\beta (L)-2\le C{e}^{-mL}$.
3. Massless dichotomy: tie the slower (power-law) decay to the failure of the split property.
4. Higher spin: reuse the massless-spinor Haar-wavelet construction (Phys. Rev. D 108, L081701).

A full execution plan lives at long-range-bell-decay-project.

Related Questions

• higher-spin-bell-inequalities — the spin-dependence of the decay constant
• impossible-measurements-qft — whether the optimal observables are operationally realizable
• bell-inequalities-with-gribov-horizon — the gauge-theory analogue of the same vacuum-correlation question
• bell-chsh-across-traversable-wormhole — long-range vacuum correlations in the TFD / wormhole setting