2026-10-15 · chapter Yang–Mills and the mass gapPart 13 The crossover bandthe ledger of exclusions, why they stopped, the selection criteria · Kiichi, Shiori and Rin

A map of the roads tried and stopped

The study in this book tried many roads in search of a tool that crosses the band. Every road stops in the end at the same single point: there is no non-perturbative principle showing that, at intermediate coupling, disorder becomes stronger as the scale is enlarged. Setting the ways of stopping side by side, three lessons come up again and again.

• Tools that work only in a perturbative regime (weak or strong coupling) cannot cross the band.
• Local operations cannot change topological quantities such as the twist sector.
• The sources of monotonicity (correlation inequalities, reflection positivity) work in the opposite direction to the mechanism of confinement (disorder).

The purpose of the ledger is not to repeat the same road. When a new candidate is thought of, we first check whether it is a restatement of something in this ledger (§06 gives examples).

The columns of each row are as follows.

• Road: a one-line summary of the idea tried.
• Why it stopped: where the road became unable to cross the band.
• Label: the label of the reason for stopping. Proved here means the reason has a proof on paper (set out in a chapter of this book); Cited means it rests on literature checked against the original; Computer-assisted is computer assistance; Reading is a numerical reading; "Reading (physics)" is a judgement at the level of a physical estimate. Even when the reason is an estimate, the judgement not to re-examine the road does not change. But a road stopped for a reason that is an estimate can be reopened if there is a new input.
• Origin: the record of this book's study, or the literature. Literature checked against the original is marked [Cited]; literature not checked is marked "not yet checked".

A note on words. In this part, following the convention of 13-00 §02, we write "band", and distinguish it from the "window" of Parts 10 and 11 (the site window, the ħ window, the time window). We also do not use the symbols (P), (R) or T1–T12; we write the names "β-monotonicity", "the renormalisation comparison" and "task".

Numbers 1–28 in Table 1 are roads that this book's study tried in search of a principle that crosses the band; 29 and 30 are roads found to have already stopped in the literature when that study was re-checked by another hand. The labels are labels of "why it stopped".

$N_c$ in Table 1 is the number of colours, a different quantity from the size $N$ of $T^2_N$. Number 11 is kept here as a single line only. Its content was withdrawn, so it is not used elsewhere in this book.

Note that most labels in Table 1 are "Reading (physics)". Most of the reasons for stopping are not counterexamples or negative theorems but estimates of the form "this road comes back to the band itself at this point". The rows with proofs are number 9 (slab confinement is a theorem with $N$ fixed, and is not uniform in $N$) and C1, C2, C5 and C6 of Table 2 (the scope is in the brackets of each row's label). Numbers 29 and 30 of Table 1 are the "Cited" rows that had stopped in the literature. The "Computer-assisted" of number 3 is a result on the hierarchical model and says nothing about the four-dimensional lattice (the hierarchical model is a different thing from four dimensions in that Migdal–Kadanoff decimation is exact there).

Numbers 29 and 30 in Table 1 are roads that, after this book's study had tried them itself, were found to have already stopped in the literature. Both were checked against the text of the original [Cited].

29 The average over local vortex containers — only up to R/ln R in four dimensions

The identity that, with the frame (the outside of the container) fixed, the signs of the sectors inside the containers become independent and the Wilson loop is bounded by the product of the biases of the containers, was Lemma 5 of 12-03 §03. This is the "vortex container" identity of Mack–Petkova (1980), written in eqs. (18)–(22) of Kovács–Tomboulis (hep-lat/0108017). The only differences are that we wrote the sectors with the sum ensemble and took the average over the frame rather than the maximum; this book claims no novelty.

§3 of the same paper states the limit of this road in four dimensions. The length of a container passing through a point inside a rectangular loop with sides $T\times R$ ($T\gt R$) is of order $R^2$, and to keep the cost of the container constant the width of its cross-section must be fattened to order $\sqrt{\ln R}$. Only about $RT/\ln R$ independent containers fit inside the loop, and what comes out is

and no more [Cited]. A linear potential (the area law) needs a single fat container the size of the loop. This is exactly the global form of Tomboulis–Yaffe (13-01 §04), the same condition as crossing the band.

In one line, the reason it stopped is: the road of laying out containers of fixed length independently does not reach the area law on $\mathbb R^4$. On a slab with a small $T^2$, the length of the container (the area of the vortex sheet) stops at the area of $T^2$, so this obstacle does not appear (12-03 §06). The obstacle comes back when $T^2$ is widened — that is, in the band.

30 Monotonicity of the vortex free energy by Migdal–Kadanoff interpolation

Tomboulis (arXiv:0707.2179) claimed confinement at every coupling for four-dimensional $SU(2)$. The flow of the argument is as follows [Cited].

1. By Migdal–Kadanoff-type decimation (lattice spacing $a\to ba$), build upper and lower bounds on the partition function ((3.4) and (3.7) of the original).
2. Interpolate their coefficients with a parameter $t$, and obtain a representation in which decimation becomes an equality at some value $t^*$ ((3.12), (3.16), (3.35)). Build the same representation in the twisted sector ((4.19), (4.26)).
3. The key step: that the same $t^*$ can be taken in the untwisted and the twisted sectors ((5.14)). This is to be derived from the inequality $A\ge A^+$ ((5.15)), and it is stated that showing it at strong coupling is enough.
4. In conclusion, the true vortex free energy is at most the vortex free energy of the decimated system ((5.22)). The decimated system flows to strong coupling, so the strong-coupling expansion gives $F\sim L\,e^{-\hat\sigma A}$, and the Tomboulis–Yaffe inequality gives the area law.

Against this there are two criticisms [Cited].

• Ito–Seiler (arXiv:0711.4930): that $t^*$ can be taken in common for the two sectors reduces to their inequality (3.5), $A(\alpha)\gt A^+(\alpha)$, which has not been shown outside strong coupling. Further, the Migdal–Kadanoff flow goes to strong coupling regardless of whether the group is abelian or non-abelian, so if the argument went through as it stands, $U(1)$ would also confine at every coupling. Four-dimensional $U(1)$ has a non-confining phase, so an input that distinguishes groups is needed somewhere.
• Kanazawa (arXiv:0805.2742, Appendix A): the continuity argument in the interpolation parameter does not hold (a counterexample is given). It is not shown which step of the argument fails for $U(1)$. For any compact connected group the Migdal–Kadanoff flow goes to strong coupling, so the renormalisation side cannot distinguish $U(1)$ from $SU(2)$, and all the non-abelian content goes into the existence of $t^*$.

Footnote 10 of the original itself writes that "for abelian systems, inequalities of type (5.15) follow from the Griffiths inequalities". Together with the remaining steps this would show confinement for $U(1)$, a sign that the argument does not distinguish groups.

In one line, the reason it stopped is: a renormalisation that does not distinguish groups cannot be an argument that closes the band. This is an example in the literature of criterion (m) of 13-06 §08 (show that $SU(2)$ has no transition, distinguishing it from the transition that $U(1)$ has inside the band, at $\beta\approx1.01$). The lateral-widening inequality of this book (13-02 §02) restricts the comparison to the form "change only the geometry, within one theory and one coupling", and has a form that fails for $U(1)$ (13-01 §03, 13-09 §02). But the difficulty of showing it is of the same type, and it is still needed across the band.

Relatedly, Theorem 1 of Kanazawa (arXiv:0805.2742) is an identity relating the ratio of twisted partition functions on lattices of different sizes to a ratio at shifted couplings, but it also holds for $U(1)$, so it says nothing about confinement [Cited]. Another paper of Ito–Seiler (arXiv:0803.3019) is not yet checked.

This is number 27 of Table 1. The comparison principle is the claim that for every planar loop $C$

holds (the right-hand side is the twisted slab, 12-04 §02). If it held, then together with slab confinement (the area law for every $\beta$ with $N$ fixed) it would give the area law for every $\beta$ on $\mathbb Z^4$. That is why it was worth trying.

True at one loop [Reading]. Comparing with the heat kernel of fluctuations around the background, the inequality holds at each time. The colour weight of the sum over images is $c(n)=4\cdot\mathbf 1[n\in(2\mathbb Z)^2]-1$, and we checked this numerically to machine precision (a numerical check at the one-loop level; no proof on paper is set out).

False non-perturbatively [Reading]. By Monte Carlo we compared the twisted slab on the lattice $16\times16\times N_T\times N_T$ (the $\mathbb R^2$ directions periodic with 16, $T^2$ with $N_T$) with $16^4$ ($SU(2)$, Wilson action).

Taking $C$ in (2) to be a single plaquette, the right-hand side is the plaquette of the slab and the left-hand side the plaquette of $16^4$. At $\beta=2.1$, $N_T=2$ the slab is 7σ lower, so (2) fails for small loops. This can be read as the frustration of the twist (the cost of forcing a twist into a small $T^2$) lowering the plaquette [Reading] (physics). The overshoot on the weak-coupling side ($\beta=2.6$) lacks statistics and is not settled.

Further, at $N=1$ it contradicts the scaling behaviour of asymptotic freedom [Reading] (physics). The slab theorem gives $\sigma_{\rm tw}(1)\ge-\ln\tanh\beta\approx2e^{-2\beta}$, but if the scaling behaviour of asymptotic freedom is accepted, $\sigma_\infty a^2\propto e^{-(6\pi^2/11)\beta}=e^{-5.38\beta}$, and at large $\beta$ (2) points the other way.

In one line, the reason it stopped is: an ordering that holds at one loop breaks down non-perturbatively, starting from small loops. Two criteria come out of this (§07): (g) it is a non-perturbative estimate, not a restatement of one loop; (k) it does not fail for small loops (short scales), or it has a form that claims only large loops. The asymptotic form for large loops only ($\sigma_{\rm tw}(L)\le\sigma_\infty$) has not been refuted, but there is no mechanism to show it, and it hits the same wall as number 2 of Table 1 (there is no inequality for $SU(2)$ that fixes the sign of the difference).

One positive observation also remains [Reading] (physics). At weak coupling the twisted $T^2$ is close to infinite volume already from small $N_T$, and approaches it from the opposite side to the untwisted case. This fits the picture of adiabatic continuity, but there are signs that the approach is not monotone, a fact unfavourable to tools based on monotonicity.

In the study of Parts 12 and 13 there are, apart from Table 1, roads that were discarded and roads whose usable scope was limited. Table 2 gathers them. The first five (C1–C5) are "roads discarded", the rest "roads limited in scope".

When thinking of a new candidate, we first check that it is not a restatement of something in the ledger. Table 3 lists three ideas that, when the band candidates were examined, were not made candidates but judged to be restatements of the ledger.

The study of detours in Part 10 (10-10 §04) extracted seven types from failures on the double well in 0+1 dimensions. The individual detour plans do not overlap directly with four-dimensional candidates, but the questions of type apply to four-dimensional candidates as they are. Table 4 lists those questions.

The candidate rules of Part 11 (Table 3 of 11-08 §02) can also be used here, not as exclusions but as a record of "which root they do not work on". No candidate has "works" written in the column of the root that connects the ultraviolet and the infrared ((R2) of 11-07), and the "coupling window" (a word of Part 11, 11-06) that appears in the class of rarity closed by one step of coarse-graining (N1 + N4) was a rewording of that root. Anyone who says a band candidate works on (R2) must name an input different from every candidate.

Turning the reasons for stopping inside out gives candidate conditions that a tool crossing the band should satisfy. Table 6 sets those conditions side by side with the ways of stopping they came from. All are candidates for necessary conditions, not sufficient conditions that guarantee crossing the band if satisfied [Reading] (physics). Especially heavy are the four (d), (g), (j) and (m).

Behind (j) is the fact that even for the two-dimensional $O(3)$ model, exponential decay at all temperatures at the lattice level (Polyakov's conjecture) is unsolved (as stated in the lecture notes of Peled–Spinka, arXiv:1708.00058; not yet checked). A tool that shows a uniform gap in four dimensions must at least have the power to show that.

Calibration against the literature. Where is the current frontier on each side of the band? Only literature whose checking is complete is marked [Cited].

• The strong-coupling side: Cao–Nissim–Sheffield (arXiv:2505.16585) showed, for $U(N)$ lattice Yang–Mills, the area law for all $N$ at $\beta$ below a $\beta_0(d)$ depending only on the dimension $d$ [Cited]. The same authors, in arXiv:2509.04688, check the mass-gap condition of Durhuus–Fröhlich (1980) by methods of stochastic dynamics and obtain the area law in the 't Hooft regime ($\beta\lesssim c_dN$) [Cited]. Both are results on the strong-coupling side and do not reach $\beta$ inside the band.
• The universality of the band: there is a paper that states explicitly that, for every version of the Gross–Neveu model, there is no proof of chiral symmetry breaking in the limit with the ultraviolet cutoff removed (Fabbri–Goller arXiv:2606.13075, not yet checked). The infrared analysis of two-dimensional models at large $N$ (Kopper–Magnen–Rivasseau 1995, Kopper 1999) is with an ultraviolet cutoff (not yet checked).
• Computer-assisted renormalisation: a result showing with computer assistance the flow near the high-temperature fixed point in two-dimensional tensor renormalisation (Ebel–Kennedy–Rychkov arXiv:2506.03247, not yet checked) can be seen as one of the current frontiers. It is far from the band of four-dimensional $SU(2)$.

To sum up, within the range we surveyed, no proof is found of mass generation in an asymptotically free theory in the limit with the ultraviolet cutoff removed. The band is not a hole peculiar to the roads of this book, but the unsolved core common to asymptotically free theories. This sentence, however, is a statement about the range surveyed, and includes literature not yet checked.

The labels of each row of Tables 1–3 are placed in the tables. The rows this page sets out itself are the following fourteen.

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

[01] CIT MP      the container identity is known                            Kovács–Tomboulis (18)–(22)
[02] CIT KT      independent containers reach only R/ln R ⟸ [01]            R⁴
[03] CIT IS      the key step of Tomboulis reduces to (3.5), unproved       outside strong coupling
[04] CIT KZ      the Migdal–Kadanoff flow does not distinguish groups       any compact connected group
[05] RDG CP1     the comparison principle holds at one loop                 one-loop level
[06] RDG CPX     the comparison principle fails at β = 2.1, N_T = 2         7σ, rerun 5.5σ
[07] RDG CPN     the comparison principle at N = 1 contradicts asymptotic freedom  physics
[08] PRV SLAB    slab confinement (N fixed)                                 12-04
[09] PRV ISO     isometric comparison gives β′ ≥ β + 0.023                  12-07, one loop
[10] PRV NOGO    structure alone does not distinguish groups                13-04
[11] RDG REST    why the remaining rows stopped                             physical estimate
[12] CMP HIER    certificate for the hierarchical model                     hierarchical model only
[13] CIT CNS     the frontier on the strong-coupling side                   small β
[14] RDG CRIT    the selection criteria ⟸ [02][04][05][06][07][09][10][11]

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. [14] is the only node on the conclusion side, and its label is "Reading (physics)". No row of this page shows either that the band can be crossed or that it cannot. A stopped road is only recorded as "cannot cross with this set of inputs"; a road that adds something to those inputs lies outside the ledger.

What this chapter can say

Sources and reproduction

Terms used in this chapter — the ledger of exclusions, why it stopped, vortex container, the comparison principle, Migdal–Kadanoff decimation, the isometric comparison, the selection criteria, the crossover band (the band), β-monotonicity, the renormalisation comparison. Their meanings are in A-02 Glossary.

Next chapter: 13-09 The measured record — the numbers this part cites, on one page.

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