The Vacuum–Matter Coupling from Finite-Density QCD: Sigma Terms, Chiral Condensate, and the Origin of α

 Paper #3a in the research program "What If the Vacuum Gravitates Locally?"

Programme context. This 17-paper programme investigates the hypothesis that quantum vacuum energy and the cosmological constant are physically distinct, with the vacuum responding to local matter density through ρ_vac(ρ_m) = Λ₀ − α ρ_m. Paper #3 derived this functional form from one-loop QFT in curved spacetime but found a 10³² gap between the perturbative coefficient (α ~ 10⁻³⁴, gravitationally suppressed) and the phenomenological requirement (α ~ 10⁻³). Paper #8 established the observational constraints from structure growth data. Paper #9 discovered an unexpected nonlinear self-screening in N-body simulations.

What this paper does. Paper #3a closes the 10³² gap. The derivation uses finite-density QCD — the standard framework for nuclear matter — rather than quantum gravity. Three standard, experimentally verified ingredients are combined:

1. The Gell-Mann–Oakes–Renner relation (tested to ~5%).
2. The in-medium chiral condensate shift at finite baryon density (Drukarev & Levin 1991, Cohen et al. 1992).
3. The nucleon sigma terms σ_π ≈ 50 MeV and σ_s ≈ 40 MeV (measured in πN scattering and on the lattice).

The result: α = (σ_π + σ_s)/m_N ≈ 0.096. No new physics is invoked.

The equation-of-state argument. The chiral condensate ⟨q̄q⟩ is a Lorentz scalar. Its energy therefore has stress-energy T^μν ∝ g^μν, giving equation of state w = −1 exactly. This is vacuum energy in the precise general-relativistic sense — distinguished from ordinary matter (w = 0) by Lorentz structure, not by convention.

The strangeness sigma term. σ_s ≈ 40 MeV measures the nucleon's coupling to virtual s̄s pairs in the QCD vacuum. Strange quarks are not valence constituents of the nucleon — they exist only as vacuum fluctuations. This is an unambiguous vacuum effect with no separation ambiguity, providing a model-independent lower bound α_s = σ_s/m_N ≈ 0.04.

The factor-of-1.7 agreement. Two physical reduction factors bring the bare QCD prediction into agreement with observations:

• Dark matter does not carry color charge → only baryons couple → ×(Ω_b/Ω_m) = ×0.156
• Nonlinear screening from Paper #9 → ÷3

Result: α_predicted = 0.005. Paper #8 best fit: α_observed = 0.003. Ratio: 1.7. No free parameters.

Position within the programme. If this derivation holds, it transforms the gravitating vacuum model from a phenomenological ansatz into a consequence of QCD. The coupling α is no longer a free parameter — it is determined by measured sigma terms, the Planck baryon fraction, and the nonlinear screening from N-body simulations. Paper #3a is the theoretical foundation; Papers #8–#9 provide the observational and computational verification.

Programme links:

• Full programme page: https://interdisciplinary-research.institute/2026/03/11/research-program-invitation/
• Paper #8 (Structure Growth): https://www.researchgate.net/publication/401979975
• Paper #9 (N-body Simulations): https://www.researchgate.net/publication/401992447

Recommended for submission to Foundations of Physics.

Author: Boris Kriger ORCID: 0009-0001-0034-2903

Affiliations:

1. Information Physics Institute, Department of Theoretical Astrophysics and Cosmology
2. Institute of Integrative and Interdisciplinary Research, Toronto

Upload type: Preprint License: CC BY 4.0

Keywords: vacuum energy, sigma term, chiral condensate, finite-density QCD, vacuum–matter coupling, equation of state, cosmological constant, dark energy, trace anomaly, S8 tension, running vacuum model