2026-10-15 · chapter Yang–Mills and the mass gapPart 13 The crossover bandthe sector bias, the area exponent, Tomboulis–Yaffe · Kiichi, Shiori and Rin

The correct form of the band

We take the notation from 12-03. We write the Wilson action of $SU(2)$ on the lattice $(\mathbb Z/L)^2\times(\mathbb Z/M)^2$ as the ensemble summed over Mack–Petkova twists (links $U$ and closed $\pm1$ variables $z_p$, with weight $\exp(\frac\beta2\sum_pz_p\,\mathrm{tr}\,U_p)$), and fix the background twist $n_{12}=1$ in the two transverse directions 1 and 2. For a disc or square $D$ in the plane (directions 3 and 4), we call $D\times T^2_L$ the container, the $\mathbb Z_2$ flux $\sigma_D=\prod_{p\in D\times\{y_0\}}z_p$ through $D$ its sector, and define the sector bias, conditioned on all the variables outside the container (the frame) $\Phi$, as

We write $\theta(R;T^2_L)$ for its value at a typical frame for a container of size $R$ with transverse $T^2_L$, and in particular $\theta(L,L):=\theta(L;T^2_L)$. Given the frame, the insides of the containers are conditionally independent, and for disjoint containers $D_j$ inside a loop $C$ in the plane

holds (the Mack–Petkova container estimate Cited, 12-03).

The line of the road from the direction of space was this: confinement is visible on a small $T^2_L$ (12-04), and carrying it to $L\to\infty$ gives confinement on $\mathbb R^4$. So the first statement of the band that comes to mind is the following naive form.

It is the statement that "even if the container and the transverse torus are enlarged together, the sector bias does not stick to 1" — disorder remains at every scale.

Proposition 1 Proved here Suppose that, for some $L$, when the plane is tiled with square containers of side $L$, the product on the right-hand side of (2) splits into the product of the individual factors up to a constant $C_{\rm D}\ge1$ (frame decorrelation Assumption), and that each factor is at most $\theta_0$. Then for a square loop $C$ of area $A$ (with side a multiple of $L$)

Proof Place $A/L^2$ square containers of side $L$ inside $C$ and use (2) and the assumption: $|\langle W(C)\rangle|\le C_{\rm D}\prod_j\theta_0=C_{\rm D}\theta_0^{A/L^2}$. Rewriting the exponent gives (4). ∎

The tension $\sigma_L$ in (4) is positive for each $L$. But the naive form (3) gives only one constant $\theta_0$, so the lower bound on the tension obtained by this road is $\ln(1/\theta_0)/L^2$, which goes to 0 in the four-dimensional limit ($L\to\infty$, where the slab $T^2_L\times\mathbb R^2$ approaches $\mathbb R^4$). In physical units, whatever multiple of $1/\Lambda$ we take $L$ to be, the lower bound is $\Lambda^2\ln(1/\theta_0)/(L\Lambda)^2$, and it gives no uniform tension.

What is said here is that "the lower bound that follows from (3) by the container estimate goes to 0". That a theory satisfying (3) does not confine does not follow from Proposition 1. The U(1) estimate in the next section fills that step.

We apply the same language of containers to compact $U(1)$ in four dimensions (the twist enters in the same way, as a centre flux with values $\pm1$). What follows is an estimate by an effective theory Reading (physics).

Weakly coupled $U(1)$ is in the Coulomb phase, and on the slab $T^2_L\times\mathbb R^2$, after integrating over the transverse directions at long distance, it becomes two-dimensional $U(1)$ with coupling $e^2/A_T$ ($A_T=L^2$ is the area of $T^2$). In two-dimensional $U(1)$, the partition function for flux $\Phi$ through a region of area $A_D$ can be written as a Poisson sum over the wrapping $k$:

The sectors $\sigma_D=\pm1$ correspond to $\Phi=0$ and $\Phi=\pi$, so when $q:=e^{-e^2A_D/(2A_T)}$ is small

Taking the container size $R=L$, $A_D/A_T$ is a number independent of $L$ (1 for a square), so $\theta(L,L)$ is a constant independent of $L$ and satisfies the naive form (3). On the other hand, that weakly coupled four-dimensional $U(1)$ does not confine is a theorem (Guth 1980, Fröhlich–Spencer 1982, Villain-type action Cited, 5-06). Together, the two give the conclusion that (3) cannot lead to confinement. But (6) is an estimate, so this conclusion carries the same label, [Reading] (physics).

(6) is consistent with the string tension $\sigma\sim e^2/(2L^2)$ of U(1) reduced to two dimensions. U(1) on the slab confines, but its tension goes to 0 as $1/L^2$. The $\ln(1/\theta_0)/L^2$ of Proposition 1 has exactly the same form as this behaviour.

Note The classical energy count we tried first (the energy $(\beta/2)(\pi/L^2)^2L^4=\beta\pi^2/2$ of the flux $\pi$ spread uniformly over a container of radius $L$ does not depend on $L$) gives the same conclusion, but it drops the sum over the wrapping $k$, so we put the forms (5) and (6) in the text.

To get a uniform tension, the bias must become small with an exponent in the area of the container. In physical units,

This is the correct form of the band. Seen from the side of the container estimate, (7) turns directly into a tension independent of $L$.

Proposition 2 Proved here Under the same frame-decorrelation assumption as Proposition 1, if $\theta\le e^{-cL^2}$ for containers of side $L$, then for square loops with side a multiple of $L$, $|\langle W(C)\rangle|\le C_{\rm D}\,e^{-cA}$.

Proof Replacing $\theta_0$ by $e^{-cL^2}$ in the proof of Proposition 1 gives $C_{\rm D}(e^{-cL^2})^{A/L^2}=C_{\rm D}e^{-cA}$. ∎

(7) is 't Hooft's criterion for confinement — that the free energy of a vortex (a centre flux) becomes small with an exponent in the area of the cross-section — put in a local form conditioned on the frame. The global form can be written with partition functions with twisted boundary conditions. On the torus, define the vortex free energy by $e^{-F_v}:=Z_-/Z_+$ as the ratio of the partition function $Z_-$ with twist flux in the plane $\mu\nu$ of the loop to the one $Z_+$ without; then the global version of the sector bias is

(from the definition, Proved here). Since $\tanh x\le x$, and $\operatorname{artanh}\theta\le\theta/(1-\theta^2)\le2\theta$ for $0\le\theta\le1/\sqrt2$, give $\theta\le F_v/2\le2\theta$, the conditions $\ln(1/\theta)\ge cL^2$ and $F_v\lesssim e^{-cL^2}$ are the same up to constants.

It is the Tomboulis–Yaffe inequality that connects the global form to the area law Cited. In $SU(2)$ lattice gauge theory, for a rectangle $C$ (area $A_C$) in the plane of the directions with periods $L_\mu,L_\nu$,

holds. Here $F_v$ is the free energy of the twist flux in the same plane as $C$; the vortex sheet wraps the remaining two directions and winds once around $C$ (this is a different twist from the background twist $n_{12}$ of the slab). T. Kanazawa extended this inequality to $SU(N)$ and to general classical spin systems, using reflection positivity Cited.

Corollary 3 Proved here Under (9), if $F_v\le e^{-\rho L_\mu L_\nu}$ (with $\rho\gt0$ independent of the size of the torus), then $\langle W(C)\rangle\le2e^{-\rho A_C}$; that is, the string tension is at least $\rho$.

Proof From $1-e^{-F}\le F$, $\tfrac12(1-e^{-F_v})\le\tfrac12e^{-\rho L_\mu L_\nu}$. Raising to the power $A_C/(L_\mu L_\nu)$ gives $2^{-A_C/(L_\mu L_\nu)}e^{-\rho A_C}\le e^{-\rho A_C}$. ∎

$\rho$ is called the 't Hooft string tension. The assumption of Corollary 3 and (7) are conditions of the same type — the former a ratio of global partition functions, the latter the bias of a container conditioned on the frame. The global form does not need frame decorrelation; in exchange, the base point (the estimate on a small $T^2$) must also be shown for the global partition functions (the note in 13-02).

An argument that crosses the band must fail for a theory that does not confine. The naive form (3) does not pass this touchstone; the correct form (7) is expected to pass. All the behaviour in Table 1 is estimated Reading (physics).

Satisfying the naive form is not a touchstone for confinement, because U(1) satisfies it. The right way to impose "failing for U(1)" is as "$\theta$ does not become small with an area exponent as $L$ grows".

(7) is a statement that joins the two sides of the band. At small $L$ (the regime of a dilute gas of fractional instantons), $\ln(1/\theta)$ is expected to grow as the product of the string tension $\propto\Lambda^2(\Lambda L)^{5/3}$ and the area of the container (13-03); at large $L$ (the confinement regime) it is expected to take the form $\sigma L^2$ minus the logarithm of the entropy of the position of the vortex sheet Reading (physics). (7) asks that the region in between — $L\sim1/\Lambda$ — be passed with a single constant $c$.

Two intermediate goals weaker than (7) can also be named.

• (a) $F_v(L,L)\to0$: excluding a non-confining phase. Stronger than the naive form, but gives no uniform lower bound on the tension.
• (b) $\ln(1/\theta(L,L))\ge cL^2/\ln L$: almost an area law. In the form of Proposition 1, the lower bound on the tension is $c/\ln L$, still not uniform.

Both are unsolved.

The conclusion of this chapter also changes the standing of the tools of Part 12. The flow lemma $\theta\le1-a$ (12-06) gives only an estimate of the form $\theta_0\sim\tfrac12$ — the naive form (3) — because the flip probability $a$ of a one-step kernel can be at most of order $\tfrac12$. So that tool cannot be used to test the band; its use is limited to supporting a lower bound on the slab tension at fixed $L$ Proved here.

In the next chapter, 13-02, we reduce (7) to a single inequality that fixes the container and widens only the transverse torus.

The dependency graph (the numbers of premises are smaller than the number of the conclusion):

[01] CIT MP     container estimate |W| ≤ E∏θ                    ensemble summed over twists
[02] ASM DEC    frame decorrelation (constant C_D)
[03] PRV P1     lower bound of the naive form ln(1/θ₀)/L² → 0 ⟸ [01][02]
[04] CIT U1     weakly coupled 4D U(1) does not confine         Villain type
[05] RDG COUL   θ in the Coulomb phase of U(1) is independent of L  estimate
[06] RDG NAIVE  the naive form cannot lead to confinement ⟸ [03][04][05]
[07] PRV TANH   θ = tanh(F_v/2), θ ≤ F_v/2 ≤ 2θ                 global form
[08] CIT TY     the Tomboulis–Yaffe inequality                  SU(2)
[09] CIT KA     extension to SU(N)                              abstract
[10] PRV C3     if F_v ≤ e^(−ρLL), tension ≥ ρ ⟸ [08]
[11] PRV P2     if θ ≤ e^(−cL²), |W| ≤ C_D e^(−cA) ⟸ [01][02]
[12] RDG TEST   the behaviour in Table 1 ⟸ [05]                 estimate
[13] PRV FLOW   the flow lemma gives only naive-form estimates ⟸ [03]

The check is to see by eye that every number to the right of a "⟸" is smaller than the number on the left. There are no cycles. Of the conclusion nodes of this chapter, the ones that stand on paper are [03], [10], [11] and [13], all elementary manipulations on the assumption row [02] or the literature row [08]. The whole of "the naive form is not enough", [06], has the estimate row [05] as a premise, so it is "Reading (physics)". There is no node for the correct form (7) itself. That is the statement of the band, and this chapter does not show it.

What this chapter can say

Sources and reproduction

Terms used in this chapter — container, sector, frame, the sector bias $\theta$, the naive form, the correct form (an area exponent), the vortex free energy $F_v$, the 't Hooft string tension, frame decorrelation, touchstone. Their meanings are in A-02 Glossary and A-06 Index of assumptions.

Next chapter: 13-02 Reduction to a single inequality — the correct form, as a single inequality that widens the transverse directions.

← 13-00 Entrance — confluenceContents13-02 Reduction to a single inequality →