2026-10-01 · chapter Yang–Mills and the mass gapPart 0 Entranceoverall map

One picture — the tools go furthest at short distances, and let go at long distances

On 24 May 2000, in Paris, the Clay Mathematics Institute announced seven prize problems. The official problem statement for this problem was written by Jaffe and Witten. Yang–Mills theory is a gauge theory that uses a non-abelian group, and is the framework of the description of the strong and weak forces (3-02). The core sentence is this. Known

Prove that for any compact simple gauge group G, a non-trivial quantum Yang–Mills theory exists on $\mathbb R^4$ and has a mass gap $\Delta>0$ (§4).

"Non-trivial" means that the theory does not reduce to a free field (a field without interaction; its probability weight is Gaussian, the normal distribution extended to infinitely many variables) (0-03).

There are two jobs. Constructing — "existence" includes establishing properties at least as strong as the axioms of quantum field theory (the Wightman axioms, the Osterwalder–Schrader axioms). The axioms are a list, in the language of mathematics, of the conditions a quantum field theory should satisfy. Wightman's are written on the side of spacetime (Minkowski space), and Osterwalder–Schrader's on the side where time is made imaginary (Euclidean space) (4-09, 4-10). Showing — for the energy operator $H$ (the Hamiltonian),

$\sigma(H)$ is the spectrum, which you may think of as the set of values the energy can take (1-10). $0$ is the energy of the vacuum, and formula (1) says that every excitation has energy at least $\Delta$.

Example. The energy of the harmonic oscillator, measured from the vacuum, is $0,\hbar\omega,2\hbar\omega,\dots$, with a gap $\hbar\omega$; the $p^2/2m$ of a free particle on a line comes arbitrarily close to $0$, so there is no gap. In relativity the energy of a particle at rest is its mass ($E=mc^2$), so the gap corresponds to the mass of the lightest excitation. Known To be exact, the gap in the problem statement is defined by the spectrum, not by particles. The object is pure Yang–Mills theory without quarks, and confinement (that things carrying colour charge cannot be taken out on their own) is an extension of the problem (0-03).

According to the problem statement, the classical Yang–Mills equations describe massless waves travelling at the speed of light (§1), and in four dimensions the mass gap is not visible classically (§6.6). Known The only constant in the action is the dimensionless coupling $g$, and a mass cannot be made from the classical theory. Even so, physics holds that in the quantum theory one scale is born (dimensional transmutation). In the one-loop approximation

($\mu$ is a reference scale, $g^2(\mu)$ the coupling there, and $N$ is $N$ for $SU(N)$). The gap is expected to be $\Delta=c\Lambda$ ($c$ a number), but this is not a theorem. Physics

The function $e^{-c/x}$ ($c>0$) vanishes faster than any power as $x\to0$.

So all its Taylor coefficients at $x=0$ are $0$, and perturbation theory, which expands in power series in $g^2$, sees $\Lambda$ as $0$ at every order. Known What is missing is not an infinite value but a quantity that does not appear in the series (Table 1).

The ratio divided by the 20th power exceeds 1 at $g^2=0.3$, but tends to $0$ as $g^2\to0$. Continued in 4-07, 6-11 and 4-12.

Put the length $\ell$ (logarithmic) on the horizontal axis, and the strength of the coupling at that length on the vertical axis. In four-dimensional Yang–Mills the coupling becomes weaker at shorter distances (asymptotic freedom; Gross–Wilczek, Politzer, 1973). Known At one loop, for $\alpha=g^2/4\pi$,

(written with $g^2$, this is formula (2)). Physics As $\ell\to0$, $\alpha$ vanishes only at a logarithmic rate. In two and three dimensions $g^2$ has the dimension of mass to the power $4-d$, and the dimensionless strength $g^2\ell^{4-d}$ vanishes at a power rate, so four dimensions is the boundary (0-04).

A figure that sets the same $4-d$ beside other quantities of the book and moves them with a single slider is 1-12 §10 Figure 4 (opens at d = 4).

The marker on the long side is the correlation length $\xi$. It is the length when the correlation between two distant points decreases at the rate $e^{-r/\xi}$ in the distance $r$, and it is related to the gap by $\xi=1/\Delta$ (in units of $\hbar=c=1$; 4-11).

To construct the theory, cut the short side off with a lattice spacing $a$ and the long side with a box of side $L$, and take $a\to0$ (the ultraviolet limit) and $L\to\infty$ (the infinite-volume limit). Doubling the length is counted as one step (one step of the renormalisation group; 6-05).

ComputedThe numbers in this figure were computed on this machine (formula (4), $N=3$). The guides $\alpha_*=0.2$ and $\xi\Lambda=1$ are choices made for explanation. If JavaScript is disabled, see Table 2.

Each time $a$ is divided by 10 the number of steps grows by about 3.3, and all the added steps are weak-coupling steps. The strong-coupling steps do not depend on $a$ and are about 4.1 (with the guides set to 0.1 and 0.3 they are 8.2 and 2.7, and again do not depend on $a$). Making $L$ larger adds $\log_2(L\Lambda)$ steps on the side longer than $\xi$. Computed

The width of the bands changes with how $\xi$ and $\alpha_*$ are placed. In physics numerics, the mass $m$ of the lightest glueball of pure $SU(3)$ (a bound state made only of gluons, that is, particles of the Yang–Mills field) is estimated as $r_0m\approx4.21$ (Morningstar–Peardon 1999), and $\Lambda$ of the same theory, defined as $\overline{\mathrm{MS}}$, as $r_0\Lambda\approx0.602$ (Capitani–Lüscher–Sommer–Wittig 1999). $r_0$ is a ruler for length fixed from the force between static quarks. Setting $\xi=1/m$ gives $\xi\Lambda\approx0.14$, and $\xi$ is expected to fall on the side shorter than $1/\Lambda$. Physics Its definition differs from the $\Lambda$ of formula (4), so this is a guide, but with this value the strong-coupling band is about 1.3 steps at $\alpha_*=0.2$ and almost 0 steps at $\alpha_*=0.3$. Computed The numbers for the width of the bands are not physical quantities. What is robust in this picture is rather that the width does not depend on $a$.

The weak-coupling band is the side where there is a foothold for expansion, and it is also where mathematics has gone furthest (Bałaban; not complete; §06, 6-07). The strong-coupling band has no small parameter and does not vanish even as $a\to0$. In the band longer than $\xi$, the key is to show exponential decay of correlations in physical units, uniformly in the box (§04). The body of the problem is to join the three bands by estimates that do not break down in either limit. To be exact, formula (4) is a formula of perturbation theory, and this picture is a picture at the level of physics. Physics

What the problem statement asks for is the gap of the theory on $\mathbb R^4$. As a road towards it, the problem's authors consider finite-volume approximations, and name, as places where present methods do not show the way, "a mass gap uniform in the volume" and "the infinite-volume limit $T^4\to\mathbb R^4$" (§6; $T^4$ is a four-dimensional box with opposite faces glued). They write that the former needs new ideas, and that if it were obtained the study of $T^4\to\mathbb R^4$ would become possible (end of §5, §6.5). Known

With the lattice spacing $a$ and the side of the box $L$ fixed, that the lattice gap is positive is the easy side (the detour in 7-13). What the problem's authors see as the key is that the gap of the continuum theory (after $a\to0$, or uniformly in $a$ as well) is bounded from below independently of the size of the box,

(§5, §6.5). $\Delta_{\text{phys}}(L)$ is the gap in a box of side $L$ measured on a physical scale such as $\Lambda$, not measured in lattice units $1/a$. This distinction matters. If $a$ is fixed and one measures in lattice units, exponential decay in infinite volume has already been shown on the strong-coupling side (Table 4). What is needed is that the gap measured in physical units is uniform in both $a\to0$ and $L\to\infty$ (5-09).

Let us see the difference with a field without interaction (a free field). The one-particle levels in a box of length $L$ with the field $0$ at both ends are

and the gap is $\omega_1$ ($\hbar=c=1$). If $m=0$, the gap $\pi/L$ exists in every box but collapses as $L\to\infty$; if $m>0$, it is always larger than $m$ and satisfies a uniform lower bound of the form of formula (5). Known

ComputedThe numbers in this figure were computed on this machine (formula (6)). If JavaScript is disabled, see Table 3.

At $m=1$ the levels with $L^2(\omega_1-1)\to\pi^2/2$, and at $m=0$ those with $E\lt1$, increase at the rate $L/\pi$. Computed To be exact, this $m$ was put in by hand, and the Yang–Mills gap must be born from the interaction. The other face of the gap is the decay of correlations: formula (2) in §5 of the problem statement says that for local quantities with vacuum expectation 0, for any positive $C$ smaller than $\Delta$, correlations at sufficiently large distances are bounded by $e^{-Cr}$ (0-04, 2-09).

The job of constructing can be divided into six stages (6-01). ① regularisation (cutting off short distances, with a lattice or otherwise), ② the finite-volume measure $\mu_{a,L}$ (the probability weight on field configurations; 1-08), ③ $a\to0$, ④ $L\to\infty$, ⑤ the axioms, ⑥ the gap. ①–④ construct

and ⑤⑥ are properties of $\mu$.

Terms in Table 4: "ultraviolet stability" means that quantities do not run wild as the lattice spacing is made small, and convergent subsequences can be taken (uniqueness of the limit is not included). "Gaussian" is the type of the free field in §01. "Axioms" are the Wightman and Osterwalder–Schrader axioms of §01.

How to read Table 4. In the row for four-dimensional Yang–Mills, the part corresponding to ③ stays at the level of assumptions and a sketch, and ④⑤⑥ are untouched. The strong-coupling row shows something like ④ and ⑥, but it is about a lattice with $a$ fixed and does not pass through ③, so it is a different proposition from formula (5) of §04. The $\varphi^4$ row shows that there are theories whose limit becomes a free field in four dimensions. For Yang–Mills a non-trivial limit is expected, but it has not been shown (item 3 of Table 5).

The problem statement says that not a single non-trivial relativistic field theory satisfying the axioms is known in four dimensions. Douglas's report of 2004 places what has been reached at Bałaban and MRS, and Chatterjee's 2018 survey says that a construction of a continuum limit above two dimensions in which Wilson loops (the quantity made by multiplying the rotations by the gauge field along a closed path and taking the trace; 5-01) behave non-trivially does not yet exist. Known "Two and three dimensions can be done" is inaccurate for non-abelian Yang–Mills: in three dimensions the continuum measure has not been constructed even in finite volume ($T^3$) (6-02, 6-06, 9-01).

"It is hard because mathematics does not allow the handling of infinities" is one side of it. Handling divergences is a matter of the left band in Figure 1, and it is the most advanced (unfinished: Bałaban's main theorem assumes that the effective coupling is small, and the theorem removing that assumption is unpublished; 6-07). Where the problem's authors asked for new ideas is the place in §04 that is not about divergence. Known

Terms in Table 5: the "orbit space" is the space made by regarding fields related by gauge transformations as the same. "Reflection positivity" is the positivity condition for recovering the Hilbert space of quantum mechanics from a measure on Euclidean space (4-10). The "interaction picture" is the way of writing time evolution by taking the state space of the free field as it is and adding the interaction to it (4-09).

How to read Table 5. 1 and 2 concern the middle and right bands of Figure 1, that is, the side where the coupling is not small. 3 is the job of showing that the limit constructed has not collapsed to a free field. 4–6 are structures carried by the framework itself of gauge theory and infinitely many degrees of freedom. Each is a different problem from "how to handle infinities", and handling divergences alone does not settle them. Known

"Because it is non-abelian" also, to be exact: for continuous non-abelian groups in three or more dimensions, exponential decay in infinite volume has been shown only on the strong-coupling side, and weak-coupling results are limited to finite groups (Adhikari–Cao; 5-06, 5-10).

Detour: if infinity were given a symbol

The mathematical side starts from "the theory does not yet exist" and constructs in the order existence → axioms → gap (Parts 5 and 6). The physical side starts from "the theory exists" and measures glueball (§03) masses and the string tension (Part 7). As a metaphor, it is a tunnel dug from both ends, and it has not yet met. To be exact, the numbers of physics are checks of consistency on the assumption of existence (7-01, 9-02).

Terms in Table 6: "shore" is a metaphor; exactly, it is a range of parameters where theorems or controlled approximations are available. $\lambda=g^2N$ is the 't Hooft coupling. "Turns into geometry" is also a metaphor; exactly, in holography the gap is computed as an eigenvalue of a wave equation on a geometry that includes extra dimensions (the leading term in $1/N$; at the level of physics; 7-09).

How to read Table 6. All seven are places where a step taken for granted in physics becomes, in mathematics, a proposition that needs a proof. 1 and 4 concern the relation between numerical computation and limits, 2 and 6 what is called a "mass", 3 and 7 the distance between computable models and the theory of the problem statement, and 5 the translation between spacetime and Euclidean space. The rightmost column gives the chapter that treats each in detail.

What the record kept on this machine has touched is only the area around 2: it cut out "one scale" as Assumption H, but has not proved it (10-01; the correspondence with the equivalence principle is an analogy; 7-12). The remaining six are only named.

This book goes through the parts of this chapter's picture in turn, in more detail. The remaining two chapters of Part 0 read the words of the problem statement (0-03) and the two four-dimensional worlds (0-04). Parts 1 and 2 are mathematical and physical preparation, and prepare the tools for reading formulas (1) and (7) and §04 (spectrum, measure, path integral, statistical mechanics). Part 3 is classical Yang–Mills theory and treats §02; Part 4 is quantisation and treats formulas (2) and (4) and the axioms.

Part 5 is lattice gauge theory and treats the $a$ of Figure 1 and the strong-coupling side; Part 6 is constructive field theory and treats Tables 4 and 5. Part 7 looks at the middle and right of Figure 1 from the side of physics. Part 8 is a dictionary that sets out the correspondence between the physical vocabulary and the mathematical vocabulary (8-01 onwards), Part 9 sums up where things stand and the open tasks of §05 and Table 6 (9-01 onwards), and Part 10 is the study of the record kept on this machine (around item 2 of Table 6; 10-01 onwards). The appendices are the table of notation, the glossary, the guide to the literature and the guide to Lean. Chapters and sections marked "Detour" may be skipped. Suggested reading orders are in 0-01.

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