Step-by-Step Derivation

How the Yang-Mills Mass Gap Emerges

This page walks through the complete derivation from the paper "Vacuum Strain and the Yang-Mills Mass Gap" in accessible steps. Every parameter is derived from established QCD results — the NSVZ low-energy theorem, large-N scaling, and renormalization group arguments — leaving no free parameters. The final prediction of 1.65 GeV is consistent with lattice QCD results for the lightest scalar glueball.

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Everything begins with the standard Euclidean Yang-Mills action for an SU(3) gauge field. This is the same action used in quantum chromodynamics (QCD) — the theory of the strong force:

SYM=14∫d4x  Fμνa FaμνS_{YM} = \frac{1}{4} \int d^4x \; F_{\mu\nu}^a \, F^{a\mu\nu}

Here FμνaF_{\mu\nu}^a is the gluon field strength tensor. This action describes how gluons interact with each other — a feature unique to non-Abelian gauge theories like QCD, where the force carriers themselves carry color charge.

We also define a gauge-invariant composite operator that measures the local field energy density:

O(x)=Tr ⁣(FμνFμν)O(x) = \text{Tr}\!\left(F_{\mu\nu} F^{\mu\nu}\right)

This operator will become the physical quantity our effective scalar field represents.

To study the dynamics of the composite operator O(x)O(x), we introduce an auxiliary scalar field ϕ\phi using the Hubbard-Stratonovich transformation. This is a well-established technique in statistical field theory — it replaces a complicated interaction with a simpler coupling to an auxiliary field:

1=∫Dϕ  exp⁡ ⁣[−∫d4x 12κ(ϕ−κO)2]1 = \int \mathcal{D}\phi \; \exp\!\left[-\int d^4x \, \frac{1}{2\kappa}\left(\phi - \kappa O\right)^2\right]

Inserting this identity into the path integral gives an equivalent representation:

Z=∫Dϕ  exp⁡ ⁣[−∫d4x(12κϕ2−ϕ O)]Z = \int \mathcal{D}\phi \; \exp\!\left[-\int d^4x \left(\frac{1}{2\kappa}\phi^2 - \phi \, O\right)\right]

After integrating out the gauge fields (the hard part — done non-perturbatively), we obtain an effective action for ϕ\phi alone:

Seff[ϕ]=∫d4x[Zϕ2(∂μϕ)2+V(ϕ)]S_{\text{eff}}[\phi] = \int d^4x \left[\frac{Z_\phi}{2}(\partial_\mu \phi)^2 + V(\phi)\right]

The key insight: the scalar field ϕ\phi now encodes the dynamics of Tr(F2)\text{Tr}(F^2) — the gluon field energy density. Its correlator matches the glueball propagator.

The crucial physical input is the gluon condensate — a nonperturbative property of the QCD vacuum first established by Shifman, Vainshtein, and Zakharov (SVZ) through QCD sum rules, and confirmed by lattice simulations:

⟨αsπGμνaGaμν⟩≈(0.33±0.03  GeV)4\left\langle \frac{\alpha_s}{\pi} G_{\mu\nu}^a G^{a\mu\nu} \right\rangle \approx (0.33 \pm 0.03 \;\text{GeV})^4

This is not a theoretical assumption — it is a measured quantity. It tells us the quantum vacuum has a nonzero energy density. We interpret this as vacuum strain: the vacuum is permanently deformed by quantum fluctuations and cannot relax to the trivial (zero energy) state.

This condensate enters the effective potential as a source term, giving the scalar field a "tilt" that prevents the symmetric vacuum from being the true ground state:

V(ϕ)=λ4ϕ4−Jϕ,J=c⟨αsπG2⟩3/4V(\phi) = \frac{\lambda}{4}\phi^4 - J\phi, \qquad J = c \left\langle \frac{\alpha_s}{\pi} G^2 \right\rangle^{3/4}

The constants cc and λ\lambda are not free parameters — they are derived from established QCD results in the next two steps.

Summary of Key Values

ParameterSymbolValueSource
Gluon Condensate⟨αsπG2⟩\langle \frac{\alpha_s}{\pi} G^2 \rangle(0.33 GeV)4SVZ sum rules / lattice QCD
Quartic Couplingλ\lambda0.8 ± 0.6Large-N + RG + naturalness
Source Coefficientcc5.3 ± 0.5NSVZ low-energy theorem
Leading-Order Mass Gapmgap(0)m_{\text{gap}}^{(0)}~1.0 GeVDerived (EFT)
Radiative Correctionδrad\delta_{\text{rad}}~3.5One-loop + sum rule (Bagan & Steele)
Physical Mass Gapmgapphysm_{\text{gap}}^{\text{phys}}1.65 ± 0.15 GeVFull result (prediction)
Lattice Glueball Massm0++m_{0^{++}}1.71 ± 0.05 GeVLattice QCD (Morningstar & Peardon)
String Tensionσ\sigma(0.44 GeV)2Derived (matches lattice)

Status and Limitations

This derivation demonstrates that the Yang-Mills mass gap can be predicted from the gluon condensate using effective field theory methods, with all parameters derived from established QCD results. It is important to be transparent about what this result does and does not establish:

What this shows

  • The gluon condensate naturally generates a mass gap through vacuum strain
  • No free parameters: λ\lambda from large-N/RG, cc from NSVZ theorem
  • The predicted mass (1.65 GeV) matches lattice results within 3.5%
  • The same mechanism produces both the mass gap and confinement
  • Two consistency checks confirmed by lattice data (SU(2)–SU(12))
  • Three novel predictions provide concrete falsifiability

Open questions

  • The radiative correction factor δrad≈3.5\delta_{\text{rad}} \approx 3.5 is large and warrants further investigation
  • This is an effective field theory argument, not a mathematical proof in the sense of the Clay Millennium Problem
  • The three novel predictions (finite-T scaling, simultaneous gap closure, condensate-tension relation) await dedicated lattice tests
  • Detailed comparison with other approaches (Cornwall, Horak et al.) remains to be published
  • This work is a preprint and has not yet undergone peer review

References

  1. M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Nucl. Phys. B147, 385 (1979). — QCD sum rules and gluon condensate.
  2. V. A. Novikov, M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Nucl. Phys. B 191, 301 (1981). — Low-energy theorem for scalar glueball correlator.
  3. G. 't Hooft, Nucl. Phys. B 72, 461 (1974). — Large-N expansion.
  4. J. M. Cornwall, Phys. Rev. D 26, 1453 (1982). — Dynamical gluon mass generation.
  5. J. Horak et al., SciPost Phys. 13, 042 (2022). — FRG analysis of gluon condensation.
  6. T. Huang, H. Y. Jin, and A. Zhang, Phys. Rev. D 59, 034026 (1999). — Scalar glueball mass from QCD sum rules.
  7. E. Bagan and T. G. Steele, Phys. Lett. B 243, 413 (1990). — One-loop radiative corrections.
  8. C. Morningstar and M. Peardon, Phys. Rev. D 60, 034509 (1999). — Lattice glueball spectrum.
  9. A. Jaffe and E. Witten, "Yang-Mills existence and mass gap," Clay Mathematics Institute Millennium Problem (2000).
  10. B. Lucini and M. Teper, Phys. Rev. D 64, 105019 (2001). — SU(N) glueball masses and string tensions.
  11. A. Athenodorou and M. Teper, JHEP 12, 082 (2021). — SU(N) glueball spectrum for N=2,...,12.
  12. K. Arikawa, H. Sakai, and C. Sasaki, arXiv:2504.09120 (2025). — Finite-temperature glueball mass (revised results).

Connection to the Broader Framework

The vacuum strain mass gap mechanism connects to the other levels of the Ashebo Method. The gluon condensate that generates the mass gap also determines proton internal structure, which feeds into the emergent gravity derivation.

Emergent Gravity Derivation

See how the proton structure set by QCD feeds into G = A(t) × R(t), connecting the mass gap to macroscopic gravity.

Advanced Theory

The retrocausal gravity framework and 6D geometric extension that unifies the mass gap with cosmic-scale predictions.

Alpha-Layered Restoration

How the fine-structure constant α = 1/137 governs nuclear stability and connects to the restoration rate R(t).

Read the Full Paper

The complete derivation with all equations, parameter derivations, confinement mechanism, and scaling tests is available as a PDF formatted in RevTeX4 (American Physical Society template) — 5 pages, 12 references.

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