Draft — Phase 0/1: the β(L) decay law of the vacuum Bell–CHSH violation

Decay of the vacuum Bell–CHSH violation with spacelike separation

Working draft (Phase 0 problem statement + Phase 1 analysis) for long-range-bell-decay-project. Grounded in the group’s own formalism: PRD 108 (2023) 085026 [arXiv:2309.02941] and EPJC 85 (2025) 334 [arXiv:2411.03485].

⚠ Correction (2026-06-19, v2). The v1 of this note claimed the violation decays at rate $m$ (linear in the cross-correlator). That was wrong. The Bell violation is quadratic in the residual cross-correlation, so the rate is $2m$, not $m$ — and even that is a conjectured sharp rate, not a derived one. The error was treating the leading $L$-dependence as the linear term of the four-Gaussian sum; in fact that linear term does not produce a genuine CHSH violation (it is killed by the Horodecki threshold, §4). The exact reorganization in §3 survives; §§4–5 are rewritten. See the companion rebuilt theory sections (…-theory-v2-sections-3-5.md).

Evidence labels: [Exact], [Derived], [Conjectured], [Open], [Stated — ref].

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1. Problem statement (Phase 0)

Free real scalar of mass $m$ in $(1+d)$D Minkowski space. For two spacelike-separated regions ${\mathcal{O}}_A,{\mathcal{O}}_B$ at separation $L$, define

$$\beta (L)=\sup |⟨0|\mathcal{C}|0⟩|,\mathcal{C}=A_1{B}_1+A_1{B}_2+A_2{B}_1-A_2{B}_2,$$

over bounded Hermitian $A_i\in \mathcal{A}({\mathcal{O}}_A)$, $B_j\in \mathcal{A}({\mathcal{O}}_B)$ with $\Vert A_i\Vert ,\Vert B_j\Vert \le 1$.

Known endpoint [Proved]. Summers–Werner: for tangent regions ($L\to 0^{+}$), type III₁ + Reeh–Schlieder force $\beta =2\sqrt{2}$.

Open (Verch, OQP #12). $\beta (L)$ for $L>0$: that $\beta (L)\to 2$ as $L\to \infty$, the rate, and — a subtlety surfaced in v2 — whether $\beta (L)>2$ strictly for all finite $L$ or terminates at a finite threshold $L_0$.

The concrete gap. In arXiv:2411.03485 the separation is not a control variable (Table 1 fixes geometry; isolated violations $\approx 2.07$–$2.10$). This note makes $L$ the variable.

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2. Setup and notation

Following arXiv:2309.02941:

• Lorentz-invariant inner product (their Eq. 13): $⟨h|h^{\prime }⟩=\int \frac{d^{d}k}{(2\pi {)}^d}\frac{1}{2{\omega }_k}{\hat{h}}^{\ast }{\hat{h}}^{\prime }$, ${\omega }_k=\sqrt{{\vec{k}}^2+m^2}$. Symmetric part $H(h,h^{\prime })=\mathrm{Re}⟨h|h^{\prime }⟩$ (smeared Wightman/Hadamard); antisymmetric part the Pauli–Jordan $\Delta (h,h^{\prime })=2\mathrm{Im}⟨h|h^{\prime }⟩$.
• Microcausality. Spacelike supports $\Rightarrow \Delta (h,h^{\prime })=0\Rightarrow ⟨h|h^{\prime }⟩=H(h,h^{\prime })$ real — the Weyl phase drops, correlators are real Gaussians.
• Weyl operators (weyl-operators, ${\mathcal{A}}_h=e^{i\varphi (h)}$) with $⟨0|{\mathcal{A}}_h|0⟩=e^{-\frac{1}{2}\Vert h{\Vert }^2}$, $\Vert h{\Vert }^2\equiv H(h,h)$.
• Bob via modular conjugation $J$: $J{\mathcal{A}}_{h}J=e^{-i\varphi (jh)}$, $jh$ in the complementary region. This modular reflection is what creates the genuine cross-region entanglement — see §4.
• Geometry (arXiv:2411.03485): Alice’s diamond carries $f,f^{\prime }$; Bob’s $g,g^{\prime }$; compact bumps.

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3. The CHSH correlator as a function of $L$ (exact reorganization — unchanged)

The master formula (arXiv:2411.03485, Eq. 17), with $\Vert f_i+g_j{\Vert }^2=\Vert f_i{\Vert }^2+\Vert g_j{\Vert }^2+2H(f_i,g_j)$:

$$⟨0|\mathcal{C}|0⟩=e^{-\frac{1}{2}\Vert f+g{\Vert }^2}+e^{-\frac{1}{2}\Vert f^{\prime }+g{\Vert }^2}+e^{-\frac{1}{2}\Vert f+g^{\prime }{\Vert }^2}-e^{-\frac{1}{2}\Vert f^{\prime }+g^{\prime }{\Vert }^2}.\text{\hspace{1em}(3.1)}$$

Translate Bob’s diamond by $L$: self-norms are $L$-independent; only the cross terms move,

$${\chi }_{ij}(L)\equiv H(f_i,g_{j,L})=\iint f_i(x)\mathsf{W}(x-y)g_j(y-L\hat{e})d^{D}x{d}^{D}y.\text{\hspace{1em}(3.2)}$$

With equal self-norms $\nu$,

$$⟨0|\mathcal{C}(L)|0⟩=e^{-\nu }[e^{-{\chi }_{11}}+e^{-{\chi }_{12}}+e^{-{\chi }_{21}}-e^{-{\chi }_{22}}].\text{\hspace{1em}(3.3)}$$

[Exact within this Weyl family.] This reorganization is correct and useful — but, as §4 shows, this particular family (bare Weyl, equal norms) does not by itself produce a violation; it is not the lower bound.

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4. The decay law (corrected)

(a) Cluster limit [Exact]. ${\chi }_{ij}(L)\to 0$, so $⟨\mathcal{C}(L)⟩\to 2e^{-\nu }\le 2$. The construction (3.3) extinguishes — consistent with Verch’s claim, but note it tends to $2e^{-\nu }<2$, i.e. below the classical bound: this family does not violate at large $L$.

(b) The linear term is not the violation [Corrected]. Expanding (3.3), $⟨\mathcal{C}(L)⟩=2e^{-\nu }-e^{-\nu }X(L)+O(X^2)$ with $X={\chi }_{11}+{\chi }_{12}+{\chi }_{21}-{\chi }_{22}$. The v1 note read off "$\beta -2\sim X\sim e^{-mL}$" from this. This is wrong: the $L$-independent piece is $2e^{-\nu }<2$, so the linear term must first cover the deficit $2(1-e^{-\nu })$ before any violation appears — and a numerical scan confirms (3.3) never exceeds 2 (exactly as the cos/sin family in the v1 theory draft fails the Horodecki condition $t_1^2+t_2^2>1$). Genuine violation lives in the correlation-matrix singular-value structure, $\beta =2\sqrt{t_1^2+t_2^2}$, and is therefore quadratic in the residual cross-correlation $c(L)$.

(c) Quadratic rate [Conjectured]. Writing the surviving violation as $\beta (L)-2\sim c(L{)}^2$ with $c(L)$ the cross-region two-point function (3.2), the spacelike asymptotics of $\mathsf{W}$ give

$$\boxed{\beta (L)-2\sim C{e}^{-2mL}(\text{massive}),\sim C{L}^{-2(d-1)}(\text{massless},d\ge 2).}\text{\hspace{1em}(4.1)}$$

The rate is $2m$ — twice the mass — because $\beta -2$ is quadratic in a correlator that decays at rate $m$. [Conjectured — see §5 for the rigorous bound that forces it, and the open lower bound.]

Field	$\mathsf{W}(s)$, large spacelike $s$	$c(L)$	$\beta -2$
massive, $d=1$	$\frac{1}{2\pi }{K}_0(ms)\sim \frac{e^{-ms}}{2\sqrt{2\pi ms}}$	$\sim e^{-mL}$	$\sim e^{-2mL}$
massive, $d\ge 2$	$\sim e^{-ms}/s^{(d-1)/2}$	$\sim e^{-mL}/L^{(d-1)/2}$	$\sim e^{-2mL}$
massless, $d\ge 2$	$\sim s^{-(d-1)}$	$\sim L^{-(d-1)}$	$\sim L^{-2(d-1)}$

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5. The two bounds (corrected: upper bound is the spine)

Upper bound — rigorous backbone [Derived modulo standard clustering]. Tsirelson/Landau bounds the CHSH supremum by the vacuum maximal correlation $\mathcal{M}({\omega }_0;L)=\sup \{|{\omega }_0(AB)-{\omega }_0(A){\omega }_0(B)|:\Vert A\Vert ,\Vert B\Vert \le 1\}$; Buchholz–Wichmann–Yngvason nuclearity gives exponential clustering $\mathcal{M}({\omega }_0;L)\le C{e}^{-mL}$ (massive). Hence $\beta (L)\to 2$, gap-controlled. Under the quadratic Tsirelson form $\beta \le 2\sqrt{1+{\mathcal{M}}^2}$ this sharpens to $\beta (L)-2\le C^{\prime }{e}^{-2mL}$ [Conjectured-leaning]; the safe rigorous statement is $\beta (L)-2=O(e^{-mL})$.

Lower bound — [Open]. No closed-form construction yet achieves $\beta (L)>2$ at finite $L$ with a known $L$-scaling. What is known: (i) tangency $\beta \to 2\sqrt{2}$ [Proved]; (ii) numerics $\approx 2.07$–$2.10$ at small $L$ [Numerical, EPJC 85]; (iii) the equal-norm bare-Weyl family (§4b) and the cos/sin family do not work. Open question, sharper than Verch’s: does the violation persist for all finite $L$ (tail $\sim e^{-2mL}$), or terminate at a threshold $L_0$ where $t_1^2+t_2^2$ drops below 1? This is the genuine content of the lower bound and the target of the modular-construction computation.

Massless dichotomy [Stated]. Polynomial clustering (no gap) $\Rightarrow \beta -2\lesssim L^{-2(d-1)}$. Caveat: the contrast is exponential vs. polynomial clustering, i.e. the rate, which tracks the mass gap — not the existence of the split property (which holds for the free massless field in $d\ge 3$). The clean punchline survives: the Bell-violation falloff measures the mass gap.

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6. Concrete next computations (Phase 1 execution)

1. Build the modular-conjugation correlation matrix $T_{ij}(L)$ for two diamonds and compute its singular values $t_1(L),t_2(L)$ — this is the object that actually violates (not (3.3)).
2. Determine whether $t_1(L{)}^2+t_2(L{)}^2>1$ for all finite $L$ (tail) or only $L<L_0$ (threshold). This settles the open lower bound.
3. Fit the resulting $\beta (L)-2$ to $e^{-2mL}$ with the existing pipeline (arXiv:2406.20033); sweep $m,d$ (arXiv:2511.20244).
4. Phase 2: make the upper bound (Tsirelson maximal-correlation $\to$ CHSH-pair) rigorous with Verch/Fewster.

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References

1. R. Verch, Bell inequalities for long range vacuum correlations, IQOQI Open Quantum Problem #12.
2. S. J. Summers, R. Werner, Commun. Math. Phys. 110 (1987) 247; J. Math. Phys. 28 (1987) 2440, 2448.
3. P. De Fabritiis et al., Phys. Rev. D 108 (2023) 085026, arXiv:2309.02941.
4. M. S. Guimarães, I. Roditi, S. P. Sorella, Eur. Phys. J. C 85 (2025) 334, arXiv:2411.03485.
5. M. S. Guimarães, I. Roditi, S. P. Sorella, Phys. Rev. D 112 (2025) 085009, arXiv:2506.00504.
6. P. De Fabritiis et al., Phys. Rev. D 110 (2024) 065006, arXiv:2406.20033.
7. B. S. Cirel’son (Tsirelson), Lett. Math. Phys. 4 (1980) 93; L. J. Landau, Phys. Lett. A 120 (1987) 54.
8. D. Buchholz, E. H. Wichmann, Commun. Math. Phys. 106 (1986) 321.

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Draft v2 (corrected), 2026-06-19. The §3 reorganization is exact; the decay rate is $2m$ (conjectured sharp), with the lower bound open. Companion to long-range-bell-decay-project.