Emergent Gravity and QCD from PEPS Tensor Networks: Glueball Spectrum, Beta Function, and Entanglement Thermodynamics (v10 – Critical Revision)

This work presents a unified theoretical framework deriving key properties of Quantum Chromodynamics (QCD) and emergent gravity from Projected Entangled Pair States (PEPS) tensor networks on a two-dimensional lattice. The QCD vacuum is modeled as a PEPS state with SU(3) gauge invariance, where lattice connectivity is encoded in tensor indices and dynamics emerges from entropy flows in the Multiscale Entanglement Renormalization Ansatz (MERA) modular Hamiltonian. Key results (version 10 – critical revision): • Glueball spectrum: An anharmonic potential U(Λ)U(Λ) derived from the PEPS modular Hamiltonian, with a single effective parameter θeff=π/4θeff=π/4, describes four pure glueball channels (0++0++, 2++2++, 0++∗0++∗, 0−+0−+) using standard lattice values at r0−1=410r0−1=410 MeV. We explicitly acknowledge that anharmonicity is only ~0.3%, meaning the spectrum is essentially harmonic; lattice mass ratios arise from quantum-number-to-level assignments rather than genuine anharmonic effects. • 1−+1−+ channel correction: The v8 value of 1040 MeV is re-identified as a hybrid (qqˉgqqˉg) state, not a pure glueball. The pure 1−+1−+ glueball is predicted at ~2980 MeV, consistent with recent lattice estimates (2800–3200 MeV). • Beta function: The renormalization function Zren(χ)=1/(1+1.536 a2+0.787 a4)Zren(χ)=1/(1+1.536a2+0.787a4), with a=χ−1/2a=χ−1/2, reproduces the effective coefficient b0effb0eff of the QCD beta function, consistent with PDG values at bond dimension χ=64χ=64–128. However, the underlying MERA parameters αα and ββ are not universal: they change by 16–34% between SU(2) and SU(3). • String tension: σ=0.194 GeV2σ=0.194 GeV2 is obtained from the area law for PEPS entanglement entropy, agreeing with lattice calculations within 1% at χ=128χ=128. • Emergent gravity: Newton's constant GN=3/(4ln⁡χ)GN=3/(4lnχ) is derived from entanglement thermodynamics following Jacobson (1995) explicitly. The derivation is clean but relies on a 2D-to-3+1D extrapolation discussed in Section 6.5. • Massive emergent graviton: Yukawa corrections at finite χχ predict an emergent graviton with mass mgrav∼40mgrav∼40 MeV and range ξ∼5ξ∼5 fm. Fifth-force bounds from Eöt-Wash and neutron scattering nominally exclude the naive Yukawa coupling, but spin-2 coupling structure and PPN framework effects may provide an escape—requiring further dedicated calculation. Acknowledged limitations (v10): Anharmonicity is only ~0.3%: the glueball spectrum is essentially harmonic; the model does not explain why lattice mass ratios take their observed values. The 2D PEPS-to-3+1D QCD extrapolation is not rigorously justified—this is the most significant theoretical gap. Full General Relativity is not yet derived—only linearized Einstein equations. Fifth-force bounds challenge the naive Yukawa coupling prediction. ZrenZren parameters are not universal between gauge groups. The collapse field Λ(x)Λ(x) mechanism requires further theoretical development. Future directions: 3D PEPS with SU(3) gauge invariance; numerical verification using TEBD algorithms; PPN parameter calculation for the emergent massive graviton; confinement–deconfinement phase transition from PEPS; connection to AdS/CFT; experimental verification of massive graviton search, GNGN at nuclear scales, and higher glueball spectroscopy. This preprint is a work in progress. Feedback and collaboration are welcome.