2026-10-15 · chapter Yang–Mills and the mass gapPart 15 Ultraviolet controlentrance, the number of steps, closing in one step · Kiichi, Shiori and Rin

Entrance — what ultraviolet control is

• Ultraviolet control means showing "one step returns to the same form" at infinitely many steps without the constants degrading (§01).
• If the constant shrinks by a factor $1-\delta_k$ at each step, what remains after $K$ steps is $\prod_k(1-\delta_k)$, and everything is decided by whether the sum of losses $\sum_k\delta_k$ is finite uniformly in the number of steps (§03).
• The misreading of writing a one-step estimate and reading it as closing the induction happens again and again. We divide it into ten patterns and set each beside the correct treatment (§03 Table 1).
• The "step" of this part is one renormalisation step (step $k$); it is different both from the steps of the "staircase" of Part 10 and from the "band" of Part 13 (§02).
• The ten chapters run in this order: the difficulty (15-01), the sources (15-02), the decomposition statement (15-03), numbers for the smallest systems (15-04 to 15-06), the background dependence (15-07), the four rows that do not close (15-08), and the index (15-09) (§04).

The object is Wilson's lattice for $SU(2)$ on a four-dimensional torus of side $L_0$, with lattice spacing $a$. $L_0$ is fixed at a physical scale, and only $a$ is varied. As in 6-05 §01, we repeat blocking with factor $L$ and integrate from the finest scale upwards. The number of steps is

Proved here (from the definition). Example: if $L=2$ and $L_0/a=2^{20}$, then $K=20$, and each halving of $a$ adds one step. Writing $\rho_k$ for the weight (effective density) after step $k$, we have $Z=\int\rho_0=\int\rho_1=\cdots=\int\rho_K$.

The same form. In Bałaban's method, at each step the lattice is divided into a small-field region $\Omega$ and its outside, and one sums over all the ways of dividing it (6-05 §04). The form the effective density must keep is, in abbreviated notation,

Cited (the form follows the exposition for scalar fields, Dimock II §1, and the wording of the three-dimensional theorem; 6-05 §07). $A_\Omega$ is Wilson's action on the small-field region (written in terms of the background field), $E_k(X,\cdot)$ is a correction localised on a connected lump $X$, $R_k$ is the factor carried by the large-field region, and $g_k$ is the effective coupling at step $k$.

The three conditions. Ultraviolet control means keeping the form (2) at every step while satisfying the following three. The constants $r,C,\kappa,c,\gamma$ depend neither on the step $k$ nor on the lattice spacing $a$.

• (U1) Locality and analyticity for small fields. Each $E_k(X,\cdot)$ is analytic in a complex neighbourhood of radius $r$ in the background-field variables, and $|E_k(X)|\le C\,e^{-\kappa\,d(X)}$ ($d(X)$ is the length of a tree joining $X$).
• (U2) Rarity of large fields. $|R_k(\Omega^c)|\le e^{-c\,p(g_k)^2\,|\Omega^c|}$ ($|\Omega^c|$ is the number of large-field blocks, and $p(g)$ is the threshold factor of equation (7) in 6-05).
• (U3) The coupling stays small. The sequence of effective couplings $g_0,g_1,\dots,g_K$ lies entirely in $(0,\gamma]$.

The three put the content of ① into the words of this book, and none of them has been shown in four dimensions Assumption (the row ① of the ledger in 14-02).

The form of closing in one step. Let the class $\mathcal K(g;\,r,\kappa,c)$ be "the collection of effective densities that have the form (2) and satisfy (U1)(U2) at coupling $g$ with constants $r,\kappa,c$". What has to be shown is

The point is that the constants $r,\kappa,c$ are the same on both sides, and only the coupling may move. $b_0=22/(48\pi^2)$ is the one-loop coefficient for $SU(2)$ (6-07 §01), and $r_k$ is the remainder in the flow of the coupling. 15-03 restates (3) by splitting it into "a class that closes with finitely many parameters" and "the one-step remainder".

In this part we use the word "step" only in the sense of one renormalisation step. This book has other words built on "step", "band" and "window", and when words coincide they read as if they referred to the same thing, so Table 0 separates them.

One more caution: in the smallest systems of this part (15-04 to 15-06), we sometimes call "the marginal distribution of one coarse plaquette" a one-step map. This map looks only at the distribution of the holonomy of one coarse plaquette; it is not a true blocking step, which would produce the joint distribution of all the plaquettes of the coarse lattice. Each chapter says this once where it first appears.

Let us see the difference between a one-step estimate and an all-step proof in a single number. If a one-step estimate comes only in the form "constant $r_{k+1}\ge r_k(1-\delta_k)$", then what is left after $K$ steps is

Proved here (the inequality on the right follows from $\ln(1-x)\ge-2x$ for $0\le x\le\frac12$). If the sum of losses $\sum_k\delta_k$ is finite uniformly in $K$, the constant stays positive uniformly in $K$. If the sum grows with $K$, the constant degrades with the number of steps. The content of ultraviolet control is making this sum finite. Bałaban's three-dimensional theorem (equation (10) in 6-05 §07) and the four-dimensional theorem of 1989 (6-07 §03) carry out an induction of this form in finite volume Cited. Why in four dimensions this sum is not naively finite ($\sum_kg_k^2$ grows like $\ln K$) is calculated in 15-01 §01.

In examining ultraviolet control, the same kind of misreading happens again and again. The common pattern is writing a one-step estimate and reading it as closing the induction, which amounts to looking at just one $\delta_k$ without looking at the product $\prod(1-\delta_k)$. Table 1 divides it into ten logical patterns. Every row can be avoided by putting the one-step claim and the all-step claim on separate lines.

The top four rows of the table have the same form. Getting the all-step version from the one-step version needs the assumption that "the constants do not degrade over the steps", and unless this is made explicit on the side of the rules, a one-step proof looks like an all-step proof. The bundle of rules in 15-03 puts this caution into the way the rules are written.

Table 2 lists the ten chapters of Part 15. The level marks are: [2nd year] = readable by second-year students of mathematics or physics; [3rd year+] = third year and above (uses estimates of measures, operators and norms); [research] = research level (follows the construction of the study as it stands).

There are three reading orders. Shortest: 15-00 → 15-03 → 15-09 (what has to be shown, and what remains). The computational line: 15-04 → 15-05 → 15-06 (how the numbers that appear in the decomposition can be measured on the smallest systems). The line of the wall: 15-02 → 15-07 → 15-08 (where the sources stop, and what does not close). 15-01 can be read before any of the lines.

Every statement in this part carries one of the following five labels (the same five grades as the whole book). A label marks "how it was checked", not how weighty the statement is.

Components checked in Lean are written in a separate column rather than as a label, and are collected in the correspondence table of 15-09. The Lean components of this part are all lemmas about finite sums of real numbers or algebra; the lattice, the group and the measure are not formalised.

(a) ① ultraviolet control is not closed. Neither this chapter nor any chapter of this part shows (U1)(U2)(U3), in four dimensions or in two.

(b) (4) is arithmetic for when the one-step losses act multiplicatively; it is not a theorem about an actual sequence of effective couplings or about actual constants.

(c) The ten patterns of Table 1 are logical cautions; they are not a claim that any existing argument is wrong.

(d) The level marks and reading orders of Table 2 are the book's arrangement; they are not a judgement of how hard the contents of the chapters are.

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

[01] PRV STEP   number of steps K → ∞  (L₀ fixed)
[02] CIT FORM   the form (2) of the effective density
[03] ASM UV     (U1)(U2)(U3) and the form (3) that closes in one step ⟸ [01][02]  (from a to L₀)
[04] PRV PROD   lower bound (4) on the product of losses  (0 ≤ δ ≤ 1/2)
[05] CIT BAL    Bałaban's theorems are inductions of type (3) ⟸ [02]  (finite volume, (U3) assumed)
[06] PRV MIS    the ten patterns of misreading ⟸ [04]  (logical cautions)

The way to check it is to see by eye that every number to the right of a "⟸" is smaller than the number on the left. [03] is an assumption row, a restatement of ①. Among the nodes of this chapter there is no node saying that ① has been shown.

What this chapter can say

Sources and reproduction

Words used in this chapter — ultraviolet control (①), the number of steps, the same form, a class that closes in one step, (U1)(U2)(U3), the product of losses, the renormalisation step, the coarse plaquette, the five grades. Their meanings are in the A-02 glossary.

Next chapter: 15-01 The difficulty of the ultraviolet — why in four dimensions the sum of losses is not naively finite.

← 14-04 What is not claimed (a cross-index)Contents15-01 Why the ultraviolet is hard — dimensionless coupling, group-valued fields, gauge →