2026-10-01 · chapter Yang–Mills and the mass gapPart 6 Constructive field theorythree dimensions, ultraviolet stability

Ultraviolet stability of three-dimensional Yang–Mills — the lattice estimates became uniform, but there is no measure yet

Pure Yang–Mills in two dimensions is solved exactly on the lattice, and the continuum measure has been constructed (6-02). In four dimensions the ultraviolet side has come as far as "the level of control" in finite volume, and it is unfinished (6-07). Three dimensions lies in between, and corresponds to 2+1-dimensional space-time.

The survey says that two and three dimensions are thought to be 'substantially simpler' because they are super-renormalisable (only finitely many diagrams can diverge; §03), and then writes 'a construction of the 3D YM measure and a description of its gauge-invariant observables, even on T3, remains open' (Chevyrev 2022 §1.1). Known

To say it first in one sentence: what has been shown in three dimensions is uniform estimates on the lattice, and what remains is turning the estimates into a limit and deciding what to measure in the limit (observables). The place is stage 3 of the six stages. Saying "it has been done in two and three dimensions" is accurate for scalar fields (6-03), and for abelian groups too the continuum limit has been shown (Table 3). For the non-abelian case, "three dimensions has reached uniform ultraviolet estimates" is accurate.

In this chapter the coupling is put in front of the action, and $A$ does not contain $g$. $A$ is a 1-form with values in anti-Hermitian matrices (matrices satisfying $X^\dagger=-X$; the Lie algebra of $SU(N)$ has this form, 1-03), $F_A=dA+A\wedge A$ and $\lvert F_A\rvert^2=-\sum_{\mu,\nu}\operatorname{tr}F_{\mu\nu}^2\ge0$ (3-03). The relation to the Hermitian $A_\mu$ of the physics chapters (3-02, written with the coupling absorbed) is $A=-iA_\mu dx^\mu$, $F_A=-iF$.

The action $S(A)=\frac{1}{2g^2}\int\lvert F_A\rvert^2\,d^dx$ is a dimensionless number. $A$ has the dimension of length to the power $-1$, the curvature to the power $-2$, and the volume element to the power $d$, so the dimension of the coupling constant, and the dimensionless coupling seen on a lattice of spacing $a$, are fixed.

The formula on the right is the starting point of Bałaban (1985). $g_0^2=g^2a$ is a relation fixed by dimensions, and the constant of proportionality depends on the normalisation of the action. In this book's conventions (5-01, appendix A-01), writing the Wilson action of $SU(N)$ as $\beta\sum_p\big(1-\tfrac1N\operatorname{Re}\operatorname{tr}U_p\big)$ gives $\beta=2N/g_0^2$. The normalisation of tr is not necessarily the same as in the paper's action, so it may differ by a constant factor from the paper's $g_0$. In either convention, the three-dimensional $\beta=2N/(g^2a)$ is proportional to $1/a$. Known

Example: in units with $g^2=1$, $a=2^{-20}$ gives $g_0^2\approx10^{-6}$. The finer the lattice, the weaker the coupling becomes by itself, without waiting for the logarithmic slowness of asymptotic freedom (4-06). At the length $1/g^2$ the coupling is of order 1, and since this is the only length scale, if the theory exists and has a mass, it is a constant multiple of $g^2$ (for four dimensions see 4-07). Physics

We count the diagrams that can diverge in perturbation theory (4-04; for scalar fields 6-03). In perturbation theory one substitutes $A\to gA$. The action takes the form $\tfrac12\int\lvert dA\rvert^2$ with a three-point vertex carrying $g$ and a four-point vertex carrying $g^2$. A one-particle-irreducible diagram (one that does not fall into two pieces when any single line is cut) with $E$ external gluon lines and $L$ loops carries $g^{E-2+2L}$, so subtracting the dimension of the power of $g$ from the dimension of momentum,

is the superficial degree of divergence. In four dimensions $\omega=4-E$, independent of $L$, and divergences appear at every order. In three dimensions $\omega=4-E-L$, which drops by 1 with each additional loop.

ComputedTable 1 $\omega(E,L)$ in three dimensions. Bold is $\omega\ge0$. $E=1$ vanishes by colour symmetry. External lines are gluons only (types with external ghost lines are omitted).

With external lines restricted to gluons, there are seven types with $\omega\ge0$ in three dimensions (three in two dimensions, infinitely many in four).

One constant is the only counterterm

Of the seven, the divergences of the three other than the vacuum (one and two loops with $E=2$ and one loop with $E=3$) need no counterterm under a regularisation that keeps gauge invariance. The reason lies in the form of the counterterms. A counterterm is a gauge-invariant local term (the integral of a polynomial in the field and its derivatives at one point) whose coefficient can be written as a power of $g^2$. In three dimensions $g^2$ has the dimension of length to the power $-1$, so with the conventions of §02 the terms are restricted to dimension 3 or less, and the only such term is a constant. The candidate term linear in $F$, of dimension 2, would have to be a linear form on the Lie algebra invariant under the adjoint action ($X\mapsto hXh^{-1}$), and for a semisimple group there is only 0 (in matrices, $\operatorname{tr}F=0$). The candidate of dimension 3, the Chern–Simons term, changes sign under reversal of orientation (parity), so it does not arise from an action invariant under reversal. Known

This argument belongs to continuum perturbation theory. On the lattice rotation symmetry is broken (only the rotations and reflections of the cubic lattice remain), and it cannot be applied as it is. What corresponds to it on the lattice is Bałaban's "one place semisimplicity is used", seen in §05. Bałaban's starting density also contains, besides the Wilson action, only constants including 'vacuum energy renormalization counterterms'. A vacuum diagram with $L$ loops is of order $g_0^{2(L-1)}$, and the last one, $L=4$, corresponds to $g_0^6$, which matches the paper's expansion up to order 6 in $g_0$ (a cross-check made in this chapter).

The statement of the theorem

A torus $\mathbb T$ of fixed side is approximated by lattices of spacing $a$ (Bałaban's paper writes the lattice spacing as $\varepsilon$). The starting density is

(formula (1) of the paper), where $A$ is the Wilson action and $E_0$ is a constant including the normalisation and the vacuum-energy counterterms ($E$ in the paper; we write $E_0$ to distinguish it from the number of external lines in §03). This corresponds to the single constant being enough as counterterm in §03. The renormalisation transformation $T$ (blocking that coarsens by a factor $L$; 6-05) produces $\rho_{k+1}=T\rho_k$, and one stops at $L^Ka=a_0$ ($a_0$ a positive constant fixed by $g$ alone). Ultraviolet stability means that the upper and lower bounds on the last density do not depend on $a$, and formula (3) of the paper is

Here $U(\partial p)$ is the product of the link variables once round the boundary of the plaquette $p$, $\chi$ is the function (indicator function) that is 1 when $\lvert U(\partial p)-1\rvert$ is small on every plaquette of the last coarse lattice (spacing $a_0$) and 0 otherwise, $\lvert\mathbb T\rvert$ is the volume, and $O(1)$ is a constant fixed by $g$ and $a_0$ alone. Known

The paper notes that this $O(1)$ goes to $\infty$ as $g\to0$, rewrites it in a form with the terms of the action divergent in $g$ written out, and extends it to all steps $k$. That inequality (5) is the main theorem (its form is formula (10) of 6-05): 'The lattice approximations of the three-dimensional pure Yang-Mills theory with a semi-simple compact group Lie G are ultraviolet stable in the sense that the sequence of densities ρk ... satisfies the bounds (5).' The corollary is 'uniform in ε bounds for the partition function'. The author cautions that 'the proof is very sketchy'. Known

The sum over steps

ComputedThe numbers in this figure were computed on this machine (in units of $g^2=1$; $L=2$ and $g^2a_0=1/16$ are values chosen for illustration).

The coupling seen on the lattice at step $k$ is $g_k^2=g^2L^ka$ (the paper's $g_k=g(L^ka)^{1/2}$), shrinking by a factor $1/L$ at each step towards the fine side. A new step is only added at the left end with a smaller coupling than before, and the sum does not exceed $\tfrac{L}{L-1}g^2a_0$ (for $g^2a=2^{-8},2^{-20},2^{-40}$ the steps are 4, 16, 36 and the sums 0.121094, 0.124999, 0.125000). Computed

In four dimensions the same sum grows like the logarithm of the number of steps (§06 of 6-05; about 7.2 more each time the number of steps doubles). Above $a_0$ there is a finite-dimensional integral on a lattice whose number of points does not depend on $a$ (262144 points in the figure), and since it is over a compact group it is bounded. But if the torus is larger than $1/g^2$, scales where the coupling exceeds 1 enter there. Computed

The proof stands on a group of earlier papers (propagators, averaging, spaces of regular configurations, variational problems) and follows the author's own method for the three-dimensional Higgs model. Known

Small fields and large fields. For each plaquette, 1 is partitioned according to whether $\lvert U(\partial p)-1\rvert$ is small or large (1 is written as the sum of two functions, "small" and "large", and each case is estimated). A large plaquette makes $A(U)/g_0^2$ large, so in the end a small factor $\exp(-\tfrac14p^2(g_j))$ appears for each large-field plaquette. $g_j=g(L^ja)^{1/2}$ is the coupling at step $j$, and $p(g)$ is the multiplier for the large-field threshold, which grows like a power of $\log g^{-1}$ (formula (7) of 6-05). This factor becomes small faster than any power of $g_j$.

Inside the small fields. In each block one takes an axial gauge (a gauge setting the link variables in a chosen direction to 1 within the block) by the Faddeev–Popov procedure (a method of fixing the gauge to one and multiplying the integral by a compensating factor; 4-08) (locally, since there is no global gauge fixing; 3-04), and splits into the configuration minimising the action (the background field) and fluctuations. The fluctuations become a Gaussian measure with exponentially decaying covariance plus local interactions, and since $g_0=ga^{1/2}$ is small, an expansion to order 6 in $g_0$ and vacuum diagrams with up to 6 vertices are enough. The paper writes that this integral could properly be expressed by the cluster expansion (6-04), but it uses the simplifications given by super-renormalisability.

Two features. Lattice fields with values in a group cannot be rescaled, so instead of scale transformations one uses 'improvement of regularity properties of typical gauge fields'. Semisimplicity is used in just one place: from the fact that the only element of the Lie algebra invariant under the adjoint action is 0, one derives 'there are no relevant variables in the effective action' (there are no quantities in the effective action that grow as the steps advance and so need to be tracked). This is the lattice-side way of saying the "the only counterterm is a constant" of §03 (this chapter's reading). The problem statement introduces this work as choosing the gauge again for each phase-space cell and each scale, from ultraviolet-oriented to infrared-oriented (JW §6.5). Known

Formula (3) says only that the effective densities and the partition function are uniformly bounded, and says nothing about the limit.

From estimates to a limit. From uniform estimates, compactness (the property that a convergent subsequence can be taken from a bounded sequence) gives the prospect of subsequential limits, but not uniqueness (there is also a usage that calls the existence of subsequences "ultraviolet stability"; Chatterjee 2018 §6). Footnote 2 of the problem statement says that for the four-dimensional problem 'weak-existence (compactness)' alone is not accepted as a solution (unless the properties of the limit are shown separately). Known

What to measure. The problem statement writes 'These results need to be extended to the study of expectations of gauge-invariant functions of the fields' (JW §6.5). In three dimensions this is the problem of defining observables (next section).

Beyond. The constant $O(1)\lvert\mathbb T\rvert$ is proportional to the volume and says nothing about infinite volume (stage 4). Reflection positivity holds on the lattice (Osterwalder–Seiler 1978, 5-03), but recovering rotation symmetry and the gap (stages 5 and 6) are untouched. The paper of Chevyrev–Shen, which treats the dynamics of stochastic quantisation in two dimensions, also writes in §1.2 of its 2026 version 'the construction of the 3D YM measure in the continuum is open, even in finite volume'. Known

Line integrals are not defined

In two dimensions the Wilson loop $W(\gamma)=\operatorname{tr}\mathcal P\exp\oint_\gamma A$ is the natural observable, and the area law holds exactly (6-02). Three-dimensional Yang–Mills is thought to behave at short distances like the Gaussian free field (GFF; 1-09), whose covariance is of order $1/\lvert x-y\rvert$, so the variance of the line integral over a segment is $\int_0^1\!\int_0^1 ds\,dt/\lvert t-s\rvert=\infty$ (Chevyrev 2022 §3.1; the abstract of Cao–Chatterjee also says 'line integrals of the GFF do not exist in such dimensions'). Known

We check this in the simplest case. Making the U(1) free field (a Gaussian measure) on a lattice, with a square loop of side 1 and coupling $e^2=1$, independently of the gauge $\langle W\rangle=\exp(-\tfrac12\operatorname{Var}\oint A)$, and

$G$ is the inverse of the lattice Laplacian, and $G_t$ is the continuum version of the covariance of the field smoothed by the heat flow $A\mapsto e^{t\Delta}A$ (the linearisation of the Yang–Mills flow). $\operatorname{erf}(x)=\tfrac{2}{\sqrt\pi}\int_0^xe^{-s^2}ds$ is the error function, which approaches 1 when $x$ is large. The smoothing width is about $\sqrt{8t}$, the length in the argument of erf; for two points farther apart than that, $G_t$ is almost the same as $1/(4\pi r)$, and for two points closer together it is held to a finite value.

ComputedTable 2 $\langle W\rangle$ for the square of side 1. t is the time of the heat flow. The bottom row is formula (5) integrated in the continuum.

Reading Table 2

In two dimensions, on every lattice $\operatorname{Var}=1$ (the area) and $\langle W\rangle=e^{-1/2}$ comes out exactly. In three dimensions $\operatorname{Var}$ grows in proportion to $\ln(1/a)$, and $\langle W\rangle\to0$. The slope of $\operatorname{Var}$ against $\ln(1/a)$ goes from 0.613 to 0.635 (approaching $2/\pi\approx0.637$). The increment each time $a$ is halved goes from 0.425 to 0.440, approaching $(2/\pi)\ln2\approx0.441$. In four dimensions $\operatorname{Var}\approx1/a$. Computed

With smoothing, the lattice values converge to the continuum values in both three and four dimensions. This is the smallest example of the starting point of Cao–Chatterjee and CCHS, which define observables as "Wilson loops of the smoothed field". Computed

ComputedThe numbers in this figure were computed on this machine (the same computation as Table 2).

The road of solving an equation

Instead of making the measure directly, there is the road of solving a stochastic partial differential equation that should have it as its invariant measure (a measure whose distribution does not change as time passes) (6-09). Chandra–Chevyrev–Hairer–Shen (CCHS 2024), for three-dimensional Yang–Mills–Higgs, extended gauge equivalence to a non-linear metric space of distributions, constructed local solutions (solutions existing only for a short time) of the equation renormalised with regularity structures (the theory of solutions of singular stochastic partial differential equations, treated in 6-09), and showed 'there is a unique choice of renormalisation counterterms such that these solutions are gauge covariant in law'. Known

The Higgs field may be absent, and a statement restricted to pure Yang–Mills is in Chevyrev 2022 Remark 1.2. As a result, a Markov process on the orbit space (a stochastic process whose future behaviour is determined only by the present state) is defined, with the possibility of finite-time blow-up (the solution becoming infinite in finite time).

The state space

How the state space is made is the answer to the previous section. Line integrals cannot be used, so the deterministic Yang–Mills flow (with the DeTurck term; a term in the gauge direction added to put it in the form of a heat equation) is taken as "smoothing compatible with gauge", and gauge equivalence is defined for the fields after smoothing (Chevyrev 2022 §3.1). The space of orbits is a Polish space (a space with a complete metric and a countable dense subset) in two dimensions, but in three dimensions it is only known to be 'completely Hausdorff (and separable)' (any two distinct points can be separated by a continuous function, and there is a countable dense subset) (ibid. Remark 1.5). In the non-abelian case the space of orbits is non-linear, and the difficulty begins with the state space (ibid. §1.1, 3-04).

What remains is the invariant measure, and the survey writes that showing it exists uniquely would give a construction of the three-dimensional Yang–Mills measure in finite volume (ibid. §4). Cao–Chatterjee (2024) showed that, assuming the theory behaves like the GFF at short distances, the theory can be realised on the unit torus as 'random distributional gauge orbits', and smoothed Wilson loops can be defined. This too is a result about the state space. Known

KnownTable 3 What has been reached in three dimensions (stages as in 6-01; categories as in Table 2 of 6-01).

The U(1) column is for reference. What is written in the row for stage 6 is confinement (the force between static charges not weakening with distance), and confinement and the mass gap are different properties (7-02 §02).

The difference from four dimensions (6-07) is that what weakens the coupling in the ultraviolet is the dimension itself. What remains in common is turning estimates into limits, deciding the quantities to measure in the limit, and estimates uniform in the volume. In infinite volume, three dimensions too has the coupling of order 1 at the length $1/g^2$, and enters the same kind of difficulty as the infrared in four dimensions (this chapter's reading; 6-10).

What this chapter can say

Sources and reproduction

Next chapter to read: 6-07 Four dimensions (the four-dimensional version of the same method).

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