2026-10-15 · chapter Yang–Mills and the mass gapPart 13 The crossover bandtwo roads, the ladder of scales, the breakdown of the band · Kiichi, Shiori and Rin

Entrance — confluence

From Part 10 onwards, this book has approached the problem of the mass gap from two directions.

The first is the road from the direction of time (Parts 10 and 11). Around the diffusive-relaxation-time assumption — that the time over which fluctuations are smoothed out grows as the square of the correlation length (symbol: Assumption ξ²; upper side ξ²↑, lower side ξ²↓; formerly Assumption H, 10-01 §05) — we compared the relaxation time on the lattice with the correlation length. The statement aimed at on the four-dimensional lattice is level (T2) of Table 5 in 10-15 §07: the mass in physical units of the lattice, $m_{\rm lat}(\beta)/a(\beta)$, tends to a positive finite value. To get there by combining the one-sided bridge (10-01 §06) with ξ²↑, the constant $K$ of ξ²↑ must be uniform as $\beta\to\infty$.

The second is the road from the direction of space (Part 12). We wrap two directions of space on a small torus $T^2_L$ with the 't Hooft twist $n_{12}=1$, and look at confinement on the remaining plane $\mathbb R^2$. If $T^2_L$ is small enough, the twist removes the zero modes, the theory stays weakly coupled up to the scale $L$, and the vortices can be treated as fractional instantons (12-00, 12-02). The goal of this road is that the string tension $\sigma_{\rm eff}(L)$ of the two-dimensional theory obtained by shrinking the transverse directions to $T^2_L$ is positive uniformly in $L$, that is,

(the uniform slab tension; the definition is in 13-02).

The two goals have the same form. Neither is "it holds at some scale"; both are statements that "it holds at every scale, with the same constant". They ask that the constant does not break down at any intermediate scale, from the lattice spacing $a$ across the physical length $1/\Lambda$ up to large scales. The one line of 10-15 §08 — what is missing is not the number of steps but uniformity, holding at every scale with the same constant — was a statement of this type written from the side of the first road. We also saw at the end of 10-15 §07 that it appears in changed form: uniform as $\hbar\to0$ at the first step (0+1 dimensions), uniform as $\beta\to\infty$ on the four-dimensional lattice.

In four dimensions, between the weak-coupling scales and the strong-coupling scales, there is a scale $L\sim1/\Lambda$ at which the size of the coupling is of order 1. $\Lambda$ is a scale born quantum mechanically through dimensional transmutation (4-07), and measured in lattice units it diverges as $a\to0$. Neither of the two goal statements holds unless it passes through this scale with a uniform constant. In this part we call the interval of scales around $L\sim1/\Lambda$ the crossover band, and we call the statement the two roads reach "a statement that crosses the band".

This confluence also agrees with an arrangement made outside our study. In the division of the continuum tasks (K3 of 9-04), the uniform gap $\inf_L\Delta_L\gt0$ is set as one task, and the two roads attack it in two languages. In the table of the seven stages of 9-06, the band is stage 5.

Parts 10 and 11 used the word "window" in at least three senses. The "band" of this part is different from all of them. If the words were the same, the two roads would read as if they stopped at the same place, so we separate them here.

There are other clashes of symbols, so in this part we write by the following conventions.

• We call by name the property that the vortex free energy increases monotonically in $\beta$ — "β-monotonicity" — and the inequality comparing the vortex free energy of the doubled lattice with that of the coarse lattice — "the renormalisation comparison". We do not use the symbols (P) and (R) (because they are already used for the projection assumption (P) of Part 10 (10-02) and for the ratio $R$ (10-05)).
• We call the assumptions only by name: the small-T² base point, frame decorrelation, the lateral-widening inequality (strong form, weak form), the uniform slab tension, the uniform slab gap. The correspondence with the old symbols is in A-06.
• We write the chessboard density as $\kappa_{\rm cb}$ (to distinguish it from $\kappa$ of Part 10).

Figure 2 sets out, in the language of the road from the direction of space, the scale $L$ (the size of the small torus $T^2_L$, or the size of the box being compared) vertically. The bottom is short distance (the lattice spacing $a$), the top is long distance. Each row carries a label at its right end. Figure 1 (the flow of the logic) is in 9-06. The number and title of Figure 2 are the same as those of Figure 2 in 12-00.

 scale L
   ∞ ┤ confinement regime: widening T² costs only the entropy  [RDG] physics estimate
     │   of the position of the vortex sheet (logarithmic)
     │   strong-coupling end: the area law is known           [CIT] strong-coupling expansion
     │   renormalisation comparison at the strong-coupling end:  [ASM]
     │   constants are crude (bookkeeping sketched)
 1/Λ ┤▓▓▓▓ crossover band ▓▓▓▓  renormalisation comparison:
     │   vortex free energy of (2N, β) ≤ K·(N, β′), β′ < β
     │   one loop: a comparison reshaping the fluctuations    [PRV] one-loop level
     │   gives β′ = β − 0.258 (per doubling)
     │   one loop: a comparison isometric in the fluctuations  [PRV] one-loop level
     │   gives only the direction β′ ≥ β + 0.023
     │   numerics: SU(2) β′ ≈ β − 0.2, U(1) β′ > β            [RDG] conditions in the text
     │   non-perturbative error estimate ← the gap            (open)
     │ dilute gas of fractional instantons:                   [CIT] Tanizaki–Ünsal eq. (2.37)
     │   string tension ∝ Λ²(ΛL)^(5/3)
     │   margin 5/3 = 11/3 − 2 (asymptotic-freedom coefficient  [RDG] physics (one loop, dilute)
     │   minus the transverse two-dimensional entropy)
  L₀ ┤
     │ ultraviolet: Bałaban's multiscale analysis             [ASM] ultraviolet control
   a ┤

Figure 2 The ladder of scales. The weak-coupling tools (bottom) and the strong-coupling tools (top) each work at their own end; only the non-perturbative estimate in the band (L ~ 1/Λ) is missing. Labels are per row.

Just three points on how to read the figure.

(i) The shift of β per doubling. "β′ = β − 0.258" is the value coming from the coefficient $11/3=-1/3+4$ of the one-loop heat kernel of fluctuations in a background magnetic field; in the convention $\beta=4/g^2$ it is $\beta-\beta'=8b_0\ln2$, with $b_0=22/(48\pi^2)$. A comparison that reshapes the fluctuations reproduces this value. In a comparison isometric in the fluctuations (the identity map, translation, gauge transformation, Migdal–Kadanoff-type decimation), the spin term $+4$ does not contribute to the first-order estimate, and only $\beta'\ge\beta+0.023$ on the $-1/3$ side (the same direction as U(1)) comes out. Both are statements at the one-loop level, not non-perturbative comparisons. The breakdown into tables and the computation are in 12-07; the comparison itself is treated in 13-06.

(ii) The conditions of the numerical "β′ ≈ β − 0.2". This is a value read from measurements of the vortex free energy $\Delta F$ for $SU(2)$ with twist $n_{12}=1$, by replica exchange and Bennett's acceptance-ratio method: the whole lattice was doubled ($N_T\times M=4\times8\to8\times16$ and $4^4\to8^4$), and the value on the large lattice was fitted to the $\beta'$ curve of the small lattice. For $\beta=2.5$–$3.0$, with 1–3 seeds, the $\beta-\beta'$ read off is $+0.13$–$+0.25$ (the width of the reading is of order $\pm0.05$–$0.1$; judging from the differences between seeds, the errors are rather underestimated). For $U(1)$ at $\beta=1.5$, $4\times8\to8\times16$, $\beta-\beta'\lt-0.2$ (the opposite direction). The lattices are small and the statistics few, so this is a reading of the direction, not a determination of the value [Reading]. The table is in 13-09.

(iii) The gap. The strong-coupling end and the one-loop end are each held by a different tool. In the band in the middle, there is no tool that estimates the error of the comparison non-perturbatively (at intermediate coupling, large fields included). The "▓" row of the figure is the subject of this whole part.

Table 1 lists, in the language of the road from the direction of space, the components involved in the statement that crosses the band. Labels are per row, and conditions are written in the same row. The labels mean what they mean in the five grades of Part 10: [Proved here] a proof set out on paper; [Cited] known, checked in the original source; [Computer-assisted]; [Reading] a numerical reading (physical estimates are noted "physics"); [Assumption]. The Lean column is outside the labels and lists names and assumptions.

What the Lean column guarantees is only the algebra of finite sums and real numbers. That θ is the sector bias, the Haar measure and the positivity of the transfer matrix lie outside Lean. Names and assumptions are written in the classification table of each chapter.

Of the rows of Table 1, non-abelian dynamics enters only the rows concerned with "the lateral-widening inequality" and "the uniform slab tension" (the third assumption of the reduction, the place of the non-abelian, the renormalisation comparison). These have a form that fails for U(1) and $\mathbb Z_2$. The other rows are structural facts that hold for any group, or arrangements of logic.

The phrase "the two roads reach the same wall" is correct at the level of the goal, and not accurate at the level of the walls at which they now stop.

The correct part is as in §01. The four-dimensional goal of Part 10 (level (T2) of 10-15: ξ²↑ and the one-sided bridge with the same constant as $\beta\to\infty$) and the goal (1) of the road from the direction of space are both statements that must cross the band.

The inaccurate part is the present wall of Part 10. The "15/4 against 4" wall of 10-09 §06 (that the diagonal $15/4$ of one site at the top does not balance the sum of couplings $4$) was the wall of the first step of the staircase, on the double well in 0+1 dimensions. The length at which things cross over there is a classical length fixed by the first conjugate point of the Jacobi equation at the top; it does not depend on $\hbar$ and is finite in lattice units. The length $1/\Lambda$ at which things cross over in the band is born quantum mechanically and diverges in lattice units. Table 2 sets out the differences.

So the wall of the first step is a miniature of the band, but it is not the band itself. At the first step one can think of a road that crosses with a finite block longer than the crossover length and strong mixing (labelled [Assumption]), but in four dimensions there is no "finite block longer than the crossover length". This difference, and what the two have in common in type, are written from the side of time in 13-07. The reason for the difference — that the four-dimensional coupling is dimensionless and has no classical length — is in 11-07 §01.

We recommend reading in the order of the numbers. 13-01 and 13-02 narrow the band down to a single inequality, 13-03 to 13-05 look at what enters where in that inequality, and 13-06 and 13-07 view the band itself from two directions. 13-08 to 13-10 are the record and the candidates. 13-11 is the foundation under the two roads (ultraviolet control), and is the content of ① of the ledger. The whole conditional statement is gathered in Part 14 (14-01).

The labels of the rows of Tables 1 and 3 are fixed in the classification table of each chapter. The rows this page sets out itself are the following twelve.

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

[01] CIT DT     dimensional transmutation: Λ is quantum, diverges in lattice units  4-07
[02] PRV SH     reshaping comparison gives β − β′ = 0.258                     one-loop level
[03] PRV ISO    isometric comparison: direction β′ ≥ β + 0.023                one-loop level
[04] RDG DBL    doubling test: SU(2) β′ < β, U(1) the opposite                small lattices, 1–3 seeds
[05] CIT TU     string tension ∝ Λ²(ΛL)^(5/3)                                 dilute, one loop
[06] RDG MAR    margin 5/3 = 11/3 − 2 ⟸ [05]                                  physics
[07] CIT SC     strong-coupling area law                                      small β
[08] ASM SCR    renormalisation comparison at the strong-coupling end         bookkeeping sketched
[09] ASM UV     ultraviolet control                                           a to L₀
[10] RDG ONE    the goals of the two roads are of one type and cross the band ⟸ [01]  level of the goal
[11] PRV CL     the crossover length of the first step is classical, finite   13-07 §02
[12] RDG MIN    the first-step wall is a miniature of the band, not the band ⟸ [01][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. The rows of Figure 2 are a rearrangement of [02]–[09], and the figure itself creates no new node. The conclusion-side nodes [10] and [12] are "Reading (physics)", and this page has no node for the statement that crosses the band itself (the uniform slab tension, the uniformity of ξ²↑). That is placed as an assumption in the ledger of Part 14.

What this chapter can say

Sources and reproduction

Terms used in this chapter — the crossover band (the band), the ladder of scales, the uniform slab tension, the lateral-widening inequality, frame decorrelation, the small-T² base point, β-monotonicity, the renormalisation comparison, the chessboard density $\kappa_{\rm cb}$, the diffusive-relaxation-time assumption (formerly Assumption H). Their meanings are in A-02 Glossary and A-06 Index of assumptions.

Next chapter: 13-01 The correct form of the band — why "θ moves away from 1" is not enough.

← 12-09 Where this road stops — widening the T² is the bandContents13-01 The correct form of the band →