2026-10-15 · chapter Yang–Mills and the mass gapPart 15 Ultraviolet controldimensionless coupling, group-valued fields, gauge · Kiichi, Shiori and Rin

Why the ultraviolet is hard — dimensionless coupling, group-valued fields, gauge

• In four dimensions the sum of the one-step losses $g_k^2$ diverges like $\ln K$, so that part has to be absorbed exactly into the flow of the coupling, and the rest confined to sizes whose sum is finite (of type $g_k^{2+\delta}$, or a geometric series) (§01).
• In three dimensions $\sum_kg_k^2$ is finite, and the one-step losses can be carried as errors (§01).
• For a group-valued field, among the four properties of the weight, "closes in one step under the expectation of the fluctuations" (property 3) is missing, and the measure of the fluctuations depends on the background (§02).
• At a minimum of a gauge-invariant function the Hessian form in the orbit directions is 0, and the tools of convexity cannot be used unless the gauge is removed. The gauge fixing works only at the size of a block, and the seams where it is re-fixed scale by scale also enter the estimates (§03).
• The core of the wall lies in "group-valued field × dimensionless coupling × large fields"; even for the two-dimensional $O(N)$ σ model without gauge ($N\ge3$), the ultraviolet construction is stuck (§04).

In four dimensions the coupling $g$ has no dimension of length, and the only thing that weakens the coupling at fine scales is asymptotic freedom (11-07 §01). Cutting the one-loop flow into steps, with $b:=2b_0\ln L\gt0$,

Cited (equation (9) of 6-05 §06, keeping only one loop). Below, we treat (1) as an exact equation and count the sum of losses line by line.

(a) The sequence of couplings. Fix the coupling at the coarsest step at $g_K=g_*$, and write in terms of the step $j=K-k$ counted from the coarse end ($j=1,\dots,K$). Using (1) $j$ times,

Proved here. The inequality on the left comes from $1/g_*^2+bj\le(1/g_*^2+b)j$ ($j\ge1$), and the one on the right just drops $1/g_*^2\gt0$. The coupling is smaller on the finer side, but only like $1/j$.

(b) The sum. Comparing with the integral of $1/x$, $\ln(K+1)\le\sum_{j=1}^K\frac1j\le1+\ln K$, so adding up (2),

Proved here. If there is a loss of size $g_k^2$ at each step, its sum keeps growing like the logarithm of the number of steps. Example: for $SU(3)$ and $L=2$, $1/b=1/(2b_0\ln2)\approx10.4$, and each doubling of $K$ raises the upper bound by $(\ln2)/b\approx7.2$. This matches the partial sums in 6-05 Table 4, which grow by about 7.2 with each doubling of $K$ (arithmetic).

(c) The product. If at each step a constant (the radius of analyticity, the rate of decay, the lower bound of the Hessian, and so on) shrinks by a factor $1-cg_k^2$, then from $1-x\le e^{-x}$ and the left of (3),

Proved here (in the range $cg_k^2\le1$). Since $K=\log_L(L_0/a)$, the constant goes to 0 like a power of $\ln(1/a)$. Slowly, but not uniformly.

(d) The size at which the sum is finite. If the loss is of size $g_k^{2+\delta}$ ($\delta\gt0$), then from the right of (2)

and this is finite uniformly in $K$ Proved here. For $\delta=2$ (a loss of $g_k^4$, the size next to one loop) it is $\pi^2/(6b^2)$.

Hence what is needed. Putting (4) and (5) together, the part that moves by size $g_k^2$ after one step cannot be carried as an "error". That part must be made the motion of the parameter of the class itself. Concretely, one defines the class $\mathcal K$ in a form that has the coupling $g$ as its parameter, reads off the coefficient of the leading term (the Wilson term, the direction of $F^2$) from the effective action after one step, and sets that to be $g_{k+1}$. In this way the part of size $g_k^2$ is absorbed exactly into the flow of the coupling (3) of 15-00. The losses left after absorption (such as the degradation of the constants of the localised corrections $E_k$) must be confined to sizes whose sum is finite, of the type (5) — typically $O(g_k^4)$, or irrelevant terms that shrink geometrically in the scale Proved here (a requirement that follows from the arithmetic of (4)(5), in the case where the one-step losses act multiplicatively). Rewriting this requirement in the form of a statement gives the decomposition of 15-03.

On the side of the flow of the coupling things are different. The remainder $r_k$ in (3) of 15-00 may be allowed, in the variable $1/g^2$, to be of size $|r_k|\le Cg_k^2$ (the size of two loops); its sum grows like $\ln K$ as in (3), but since $1/g_k^2$ itself grows like $bK$, the sequence of couplings does not collapse. What is needed is that the sequence stays in $(0,\gamma]$, and that is handled by Lemma F of 15-03.

Contrast with three dimensions. In three dimensions $g^2$ has the dimension of length to the power $-1$, the dimensionless coupling at step $k$ has the form $g^2L^ka$ and is geometrically small on the fine side, and $\sum_kg_k^2$ is finite (it stays within 2 in 6-05 Table 4) Cited. Bałaban 1985 writes that the types that can diverge are finitely many, that the counterterm needed is a single constant, and that by semisimplicity there are no variables that grow in the effective action (relevant variables) (6-06 §05) Cited. In three dimensions the one-step loss $g_k^2$ can be carried as an error, and there is no need to follow the flow of the coupling. In four dimensions the $F^2$ term is exactly marginal and must be followed. This is the precise form, on the ultraviolet side, of root (R1) in 11-07 Table 3.

How "absorption" looks in the smallest setting — that for the one-link heat-kernel class the losses form a telescoping sum — is seen in 15-04 §01.

The edge variables $U_l$ of the lattice are elements of the group $G$. Group elements cannot be scaled, nor added and divided (6-05 §05 (i)(ii)) Cited. Since the group is compact the field is bounded, but a "large field" is the far side of the group (far from the identity), where the curvature of Wilson's one-link weight is negative (Proposition 3 of 11-06 §03) Proved here (proved in 11-06). Large fields can only be paid for with rarity, not with convexity.

The tools for flat fields. In the renormalisation of scalar fields, the rarity of large fields is put into the norm of a weight (regulator) $\tilde G(X,\phi)$, and closing under the expectation of one step is derived from the following four properties Cited (Appendix A.6 of the book by Bauerschmidt–Brydges–Slade (Lemma A.6.1); the four properties are those listed in §A.6.2 as the desired properties of a regulator; 11-06 §07; the chain of the original papers is in 15-02 §04).

1. It factorises into a product over disjoint sets.
2. $\tilde G(X,\phi+\phi')\le\tilde G^2(X,\phi)\,\tilde G^2(X,\phi')$ (for a sum of fields).
3. $\mathbb E_+\tilde G^t(X)\le2^{|B(X)|}$ (closes in one step under the expectation of the fluctuations).
4. $\tilde G^t\le\tilde G_+$ (bounded by the weight of the next step).

From these follows the one-step estimate $\|\mathbb E_+\theta K(X)\|_{\tilde G_+}\le2^{|B(X)|}\|K(X)\|_{\tilde G}$ (Lemma A.6.1). Property 2 uses the addition of fields $\phi+\phi'$, and property 3 uses the fact that the fluctuations are Gaussian variables independent of the background. In the proof of the lemma, property 3 is used as the expectation of a function of the fluctuations alone, and that measure does not depend on the background $\phi$ Reading (from reading the chain of the proof).

What survives on the group, and what falls short. Property 2 holds in the version where the sum of fields $\phi+\phi'$ is replaced by "background × fluctuation" $U_l=W_le^{A_l}$, using only the triangle inequality and conjugation invariance (Lemma R and its corollary; 15-07 §02) Proved here. What falls short is property 3. For a group, the measure of the fluctuations depends on the background (the propagator in a background field, the title of the foundational paper in 6-07 Table 1). So the estimate of $\mathbb E_+G^t$ needs the regularity of the background, and in the region of large fields, where the background is on the far side of the group, the fluctuations are not a small perturbation of the background. Writing the part due to the background dependence of the measure as $\eta_k$, stating the group version of properties 3 and 4 in the form of a statement, and seeing how $\eta_k$ is absorbed is the job of 15-07. For scalar fields the measure does not depend on the background, so $\eta_k=0$, and the problem itself does not exist.

There is no convexity in the orbit directions. On a finite lattice the group of gauge transformations is a product of compact groups, and the action $V$ is gauge invariant, that is, constant along orbits.

Proposition O. Let $V$ be a smooth gauge-invariant function on the configuration space (a compact manifold), and $x_0$ a minimum point of $V$. For a tangent vector $\xi\ne0$ to the orbit through $x_0$, $\operatorname{Hess}V(x_0)(\xi,\xi)=0$. In particular, if there is even one such $\xi$, then $\operatorname{Hess}V\ge m\,g$ ($m\gt0$) does not hold.

Proved here Proof. Let $\gamma(t)=e^{t\eta}\cdot x_0$ ($\eta$ an element of the Lie algebra of gauge transformations with $\gamma'(0)=\xi$). Then $V(\gamma(t))=V(x_0)$ holds for all $t$. Since $dV(x_0)=0$, $\frac{d^2}{dt^2}V(\gamma(t))|_{t=0}=\operatorname{Hess}V(x_0)(\xi,\xi)+dV(x_0)(\nabla_{\gamma'}\gamma')=\operatorname{Hess}V(x_0)(\xi,\xi)$, and the left-hand side is 0. ∎

Tools that rely on convexity (Bakry–Émery, Brascamp–Lieb) cannot be used unless the orbit directions are removed Reading (a physical estimate; the same as the "theory" paragraph of 11-07 §05). To remove them one fixes the gauge, but there is no continuous global gauge fixing that works for all configurations at once (Gribov 1978, Singer 1978) Cited (bibliographic only; 6-05 §05).

A gauge works only at the size of a block. Taking the complete axial gauge on a two-dimensional block of side $\ell$, the plaquette condition $\|U_{\partial p}-1\|\lt\varepsilon$ gives for edges only $\|U_b-1\|\lt\ell\,\varepsilon$ (6-05 §05) Proved here (proved in 6-05). Once $\ell\varepsilon$ exceeds 2 this says nothing, so the gauge is chosen anew at each scale, within the size of a block. Bałaban's three-dimensional theorem introduces the axial gauge block by block by the Faddeev–Popov method (6-06 §05) Cited. We have not checked in the text how the gauge is chosen concretely in the four-dimensional papers (6-07 Table 3).

The seams of re-fixing. The gauge at step $k$ and the gauge at step $k+1$ are different, and the re-fixing is a field-dependent gauge transformation. Its Jacobian (the Faddeev–Popov determinant) and the terms that appear at the seams between neighbouring blocks also enter the effective action, and within the class of (3) of 15-00 they must be estimated as losses whose sum is finite in the sense of §01 Reading (a physical estimate; it states the form of the estimate that is needed). MRS, constructed with a continuum cutoff, also joins the axial gauge of the large-field region with the background-field-dependent gauge of the small-field region, and has seams of the same type (6-07 §06) Cited. The organisation in which, in a tree gauge, the Jacobian of re-fixing becomes a constant and the cost moves elsewhere (to the ghost determinant and the coupling at the seams) is in row 4 of 15-08.

Transcribing the three difficulties into the language of the roots of 11-07 Table 3 gives Table 1. In this part, which deals only with the ultraviolet, the coupling is small at every step, so root (R2) does not appear Reading (a physical estimate).

Models without gauge are also stuck. The two-dimensional $O(N)$ sigma model ($N\ge3$) has no gauge and is a model with a dimensionless coupling and asymptotic freedom, yet the construction of its ultraviolet limit is listed as an open problem Cited (§II.1.1 of the text of Rivasseau's 2000 review, which says that many experts have tried and not succeeded). The same place says that a hierarchical version of this model has been constructed (Gawędzki–Kupiainen, Comm. Math. Phys. 106 (1986) 533) Cited (the review's bibliography). That there is no construction even in finite volume, and that the point is restricted to asymptotically free $N\ge3$, are this book's reading, not the words of the original Reading. The case $O(4)$ is the principal chiral model, whose field takes values in $S^3=SU(2)$. As of 2024, transplanting Bałaban's method has reached the variational problem and the background-field stage (Goswami, a family of two-dimensional nonlinear sigma models; Dybalski–Stottmeister–Tanimoto, two-dimensional $O(4)$: the equation for the critical point is derived by the calculus of variations on the Lie group and solved by a contraction mapping) Cited (both from abstracts). On the other hand, the fermionic two-dimensional Gross–Neveu model (the massive case) has been constructed (11-07 Table 3) Cited. The proposal to use this model as a test bench is in 15-02 §05.

Reading from the contrast Reading (a physical estimate): the core of the wall can be read as lying in the combination "a field that takes values in a group (a nonlinear manifold)", "a dimensionless coupling (root (R1))" and "large fields", with gauge invariance a layer sitting on top of it. We do not say that gauge invariance is not an essential difficulty.

(a) (1)–(5) of §01 are arithmetic that treats the one-loop flow (1) as an exact equation, not theorems about the actual sequence of effective couplings. "A class that absorbs the losses into the flow is needed" is a requirement for the case where the losses act multiplicatively at each step.

(b) §02 only states that, among the four properties of the weight, property 3 falls short for groups; it does not prove the group version of property 3.

(c) "The core of the wall is group-valued field × dimensionless coupling × large fields" is an estimate from the known situation of two-dimensional models; we do not say that gauge invariance is not an essential difficulty.

(d) We do not show that the seams of re-fixing can be estimated as losses whose sum is finite (we only state the form of the estimate that is needed).

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

[01] CIT FLOW   the one-loop flow (1)
[02] PRV RUN    couplings (2), log sum (3) ⟸ [01]  (one loop)
[03] PRV LOSS   product → 0 (4), sum finite (5) ⟸ [02]  (cg² ≤ 1, δ > 0)
[04] CIT 3D     in three dimensions the sum is finite
[05] CIT BBS    the four properties of the weight
[06] PRV CURV   negative curvature on the far side  (one link)
[07] PRV ORB    Proposition O
[08] CIT GRIB   no global gauge fixing
[09] RDG SEAM   the seams of re-fixing ⟸ [03][07][08]
[10] CIT SIGMA  two-dimensional O(N) not constructed
[11] RDG CORE   the core of the wall ⟸ [03][04][06][10]

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. 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 — the dimensionless coupling, the sum of losses, absorption, marginal, the four properties of the weight, property 3, Proposition O, the seams of re-fixing, roots (R1)(R2)(R3). Their meanings are in A-02 Glossary.

Next chapter to read: 15-02 A map of the sources — how far Bałaban's series went, and where it stopped.

← 15-00 Entrance — what ultraviolet control isContents15-02 A map of the sources — Bałaban's programme, the BBS chain, two rows absent from the sources →