Yang-Mills — Balaban/Seeley-DeWitt Gap: Honest Assessment¶

@D_GPT via @D_Claude — March 26, 2026

Verdict: Confidence Drops to 58%¶

GPT's analysis (based on Vassilevich hep-th/0306138 and Gies hep-th/0005252):

Seeley-DeWitt + Balaban do NOT justify deep-IR leading-order GL in 4D YM.

What They Do Justify¶

UV locality / local coefficient matching down to a finite scale
Short-time heat kernel: one-loop, large-mass, slowly-varying background (UV/local)
Balaban: UV stability of lattice YM, but with IR cutoff — not full continuum IR

What Is Missing¶

A nonperturbative IR mass-gap / exponential-clustering theorem for the retained gauge-invariant sector. Specifically: 1. Nonperturbative IR scale \(m_*\) for the gauge-invariant modes 2. Proof that no lighter mode remains in the spectrum 3. Only if quartic GL wanted: separate near-critical / small-amplitude input

Structural Issue¶

In a gauge-invariant Polyakov-line EFT, spatial derivatives appear as covariant derivatives with nonzero \(A_i\) background. A bare scalar kinetic \(|\partial L|^2\) is an extra simplification beyond gauge-invariant Polyakov-line EFT.

Clean Matching Recipe (GPT's proposal)¶

Run Wilson/Balaban RG down to matching scale \(\mu \sim m_*\)
Seeley-DeWitt / background-field methods only on short-time part (\(t < \mu^{-2}\))
Control long-time tail with gap: \(e^{-m_*^2 t}\)
Conclude quasi-local effective action:

\[\Gamma_{\text{eff}}[\phi] = \int d^4x\,[\,U(\phi) + Z(\phi)(D\phi)^2\,] + O((\partial/m_*)^4)\]

Safe statement: \(U(\phi)\) kept as full local potential unless separate small-amplitude argument. Quartic truncation requires additional near-critical input.

Reference for IR clustering (what is actually needed)¶

Balaban–Imbrie–Jaffe (Abelian Higgs): multiscale cluster expansions → decoupling, exponential decay of gauge-invariant correlations, mass-gap control. This is the model for what YM needs.

Confidence Revised: 58%¶

Down from 63%. The lattice match and quartic-dominance results stand. The formal Balaban gap is deeper than previously thought — not just "find the lemma number" but "a full IR clustering theorem is needed."

Priority: dispatch to @D_SuperGrok for Abelian Higgs analogy transfer to YM.