Strong coupling for abelian and non-abelian gauge theories: a shared existential gap
Both compact \(U(1)\) and \(SU(3)\) have a volume-uniform strong-coupling Kogut–Susskind gap by the Yarotsky application, with an existential threshold. The continuous-time expansion does not currently provide an explicit uniform threshold for either group. Their weak sides differ: for \(SU(3)\), \[\frac{d\,g^2}{d\log a}=2b_0g^4+O(g^6),\qquad b_0=\frac{11}{16\pi^2}>0,\] whereas the continuum pure photon has no perturbative self-interaction, \(b_0=0\). A massless weak-coupling phase of compact lattice \(U(1)\) is supported by Guth (Phys. Rev. D 21 (1980) 2291) and Fröhlich–Spencer (Commun. Math. Phys. 83 (1982) 411; both metadata level). The perturbative running does not determine the whole lattice phase diagram or establish a constructive arrival in the strong-coupling region.
Correction, 2026-10-02. The earlier uniform \(4\lambda\) bound, an explicit threshold of the same size as for \(SU(3)\), and a target box at \(g^2\sim4\times10^2\) depended on the withdrawn continuous-time count. They are withdrawn here. The finite-volume electric-loop result below survives; the volume-uniform theorem instead uses Yarotsky with an existential threshold. The earlier exact coupling-set and flow equivalences also exceeded the cited perturbative and phase results; the corrected coupling-set note separates them. The original text remains in Git history.
1. The strong-coupling region does not distinguish the groups
For compact \(U(1)\) take the magnetic term \((2/g^2)\sum_p(1-\cos\theta_p)\) and electric term \((g^2/2)\sum_\ell n_\ell^2\). The least nonzero electric eigenvalue per link is \(\varepsilon=g^2/2\). The plaquette operator \(w_p=e^{i\theta_p}+e^{-i\theta_p}\) has norm \(2\) and shifts the four link charges by \(\pm1\). Gauss’s law allows decomposition of a nonzero charge network into closed charged cycles. On a periodic cubic lattice with \(N_s\ge4\), every nonempty support has at least four links. A plaquette loop attains this minimum, so the physical electric gap is exactly \(2g^2\) in units \(\hbar c/a\).
Writing \(P=3N_s^3\), the magnetic operator after removal of its scalar constant has norm at most \(2P/g^2\). The two-eigenvalue min–max argument of the corrected Kogut–Susskind note therefore gives \[\left(2g^2-\frac{4P}{g^2}\right)\frac{\hbar c}{a} \le\Delta^{\rm phys}_{a,L}\le \left(2g^2+\frac{4P}{g^2}\right)\frac{\hbar c}{a},\qquad L=N_sa.\] At fixed \(N_s\), \(\Delta^{\rm phys}_{a,L}/(g^2\hbar c/a)\to2\). The lower bound is extensive and supplies no volume-uniform threshold. A nonpositive lower bound is uninformative.
The former adjacent-growth calculation controls a restricted history class. It misses histories reaching a test link later and does not establish true-vacuum decay; changing the gauge group does not repair those gaps. Uniformity at sufficiently strong coupling follows instead from the group-general Yarotsky application. Its constants depend on the chosen local interaction range.
2. What differs is the weak-side flow
The target-box note now provides an existential strong-coupling region for a specified family of finite-range local corrections. Its edge and tolerance are not the former continuous-time numerical values. An effective Hamiltonian must satisfy its hypotheses before its gap can be transferred.
For \(SU(3)\), perturbative asymptotic freedom gives \(g^2(2a)=g^2(a)+2b_0\log2\,g^4(a)+O(g^6)\) at weak coupling, with \(2b_0\log2\simeq0.0966\). A proposed route is to control all generated corrections until the effective Hamiltonian enters that region. This requires a constructive reduction and spectral transfer; absence of an intermediate fixed point is not an equivalent proof. The position note §6 identifies the region where neither the perturbative expansion nor a strong-coupling theorem applies.
For continuum pure \(U(1)\), perturbative running is absent to all orders. That is not an exact blocking identity for the compact lattice action at every coupling. The weak-coupling massless phase and the existential strong-coupling gap do not alone identify the entire gapped set as one interval or locate its boundary.
3. Non-abelian mechanisms and their limits
The position note §4 isolates the commutator potential of the zero-mode sector, the absence of a gauge-invariant operator linear in the non-abelian electric field, and the one-loop valley potential. These are distinct mechanisms: a small-volume confining potential, a missing photon observable, and a perturbative correction. They motivate the non-abelian route but do not establish the full continuum gap from the beta function alone.
4. What this settles and what it leaves
Both groups have a gap at sufficiently strong coupling by the existential theorem. Compact \(U(1)\) also has a massless weak phase. For \(SU(3)\) the perturbative slope is positive, but controlled arrival, scaling and continuum reconstruction remain open. The strong-coupling edge and tolerance in this Hamiltonian comparison are existential.
5. Consequence for STATE
Use the group-general existential strong-coupling theorem and the explicit finite-volume bound above. Do not reuse the withdrawn uniform \(4\lambda\) rate or numerical box location. A non-abelian continuum gap still requires controlled reduction, scaling and reconstruction; perturbative running alone is insufficient. This written correction supplies no new explicit volume-uniform threshold.