Susceptibility Bound for 4D SU(2) Polyakov Loops — Honest Assessment¶

Date: 2026-03-08
Model: SU(2) lattice gauge theory on \(\Lambda=(\mathbb Z/L\mathbb Z)^3\times (\mathbb Z/N_t\mathbb Z)\) with Wilson action

Verdict¶

The statement

\[ \chi(\beta)=\sum_{\mathbf x\in\Lambda_s}\Big(\langle W^*(\mathbf x)W(\mathbf 0)\rangle-|\langle W\rangle|^2\Big)\le f(\beta)<\infty \]

for all couplings \(\beta\) in the infinite-volume limit \(L\to\infty\) is false as stated for fixed finite \(N_t\).

Reason: fixed \(N_t\) is a finite-temperature problem. Pure 4D SU(2) has a deconfinement transition, in the universality class of the 3D Ising model, and the Polyakov-loop susceptibility diverges at the transition. With the center-symmetric finite-volume definition \(\langle W\rangle=0\), it also diverges throughout the broken/deconfined phase.

Phase-by-phase statement¶

1. Strong coupling / confined phase (\(\beta\) small)¶

Here the strong-coupling expansion gives exponential decay of the Polyakov-loop correlator, hence finite \(\chi\). This is the regime already under control.

2. Critical coupling (\(\beta=\beta_c(N_t)\))¶

For fixed \(N_t\), SU(2) has a second-order deconfinement transition with 3D-Ising critical behavior. Therefore [ \chi_L(\beta_c)\sim L^{\gamma/\nu}, \qquad \chi(\beta)\sim |\beta-\beta_c|^{-\gamma} ] in infinite volume. So \(\chi\) is not finite at \(\beta_c\).

3. Deconfined phase (\(\beta>\beta_c(N_t)\))¶

In finite volume, exact center symmetry forces \(\langle W\rangle_L=0\). But in the broken phase there are pure states with [ \langle W\rangle_\pm = \pm m(\beta),\qquad m(\beta)>0. ] The center-symmetric periodic state is the mixture \(\frac12(\langle\cdot\rangle_+ + \langle\cdot\rangle_-)\), so by clustering in pure phases [ \lim_{|\mathbf x|\to\infty}\langle W^*(\mathbf x)W(\mathbf 0)\rangle = m(\beta)^2. ] Hence, for large \(L\), [ \chi_L(\beta)\ge (m(\beta)^2-\varepsilon)\big(L^3-C_\varepsilon\big)\to\infty. ] So with the user's definition, \(\chi_L\) diverges like the spatial volume in the deconfined phase.

If instead one works in a selected pure phase and subtracts \(m(\beta)^2\), then the connected susceptibility is expected finite away from \(\beta_c\), but that is a different observable.

Where each proposed proof route fails¶

A. Infrared-bound route¶

An infrared bound would have schematic form [ \widehat G(\mathbf p)\le \frac{C}{m(\beta)^2+\widehat p^2}. ] At \(\mathbf p=0\), finiteness of \(\chi=\widehat G(0)\) requires \(m(\beta)>0\). But that is exactly the missing screening-mass / gap input. At the critical point \(m(\beta_c)=0\), so this route cannot prove a uniform all-\(\beta\) bound. It is equivalent to proving you are away from criticality.

B. Reflection positivity + monotonicity¶

Reflection positivity gives positivity and useful inequalities, but it does not prevent a divergent zero-momentum mode. It is compatible with both spontaneous symmetry breaking and critical divergence. So it cannot yield a uniform bound on \(\chi\) across the phase transition.

C. Center symmetry¶

Exact \(\mathbb Z_2\) symmetry only implies \(\langle W\rangle_L=0\) on a finite torus. It does not imply finite susceptibility. In fact, in the broken phase that exact symmetry makes the symmetric susceptibility larger, because the disconnected piece \(m(\beta)^2\) is not subtracted.

The exact place the Ising analogy breaks¶

It does not break. Properly applied, it predicts the opposite of the desired theorem.

The correct analogy is:

confined phase \(\leftrightarrow\) disordered 3D Ising phase: finite \(\chi\),
deconfinement point \(\leftrightarrow\) Ising critical point: divergent \(\chi\),
deconfined symmetric mixture \(\leftrightarrow\) Ising low-temperature mixed state: volume-divergent \(\chi\).

So the desired “finite for all couplings” statement is incompatible with the very analogy being invoked.

Consequence for the Clay Yang–Mills route¶

Even proving finite \(\chi\) for fixed finite \(N_t\) would only control a thermal screening observable. The Clay problem is the zero-temperature 4D continuum mass gap. That means the susceptibility route is, at best, a confined-phase input; it is not the endgame.

The current RTSG/YM route is therefore correctly stated as: - UV side: Balaban multiscale construction, - IR side: Polyakov/GL effective action, - missing theorem: a uniform positive curvature statement such as \(V_L''(0)>0\) in the correct zero-temperature scaling limit.

Bottom line¶

\[ \boxed{\text{No unconditional all-coupling bound exists for the susceptibility as defined at fixed finite }N_t.} \]

The proof fails because the claim itself is false in the thermal setup: - finite in strong coupling, - divergent at the deconfinement point, - and, with the symmetric definition, divergent throughout the broken phase.