2026-10-01 · chapter Yang–Mills and the mass gapPart 7 From the side of physicsasymptotic freedom, dimensional transmutation, confinement

How physics sees it — freedom at short distances, a single scale, the tube of confinement, and the premise that "the theory exists"

The prize problem asks "does the theory exist, and is there a gap in the spectrum?" (0-03). The questions of physics are "what is the mass of the lightest particle?" and "how does the force change as a pair of quarks is pulled apart?". Physics takes the theory as already existing and asks about its contents.

Jaffe and Witten list three properties that QCD (Yang–Mills theory with quarks) must have at the quantum level Known. (1) The mass gap: 'every excitation of the vacuum has energy at least ∆'. This is why the nuclear force is strong yet reaches only short distances. (2) Quark confinement: physical particle states are invariant under SU(3) transformations, and a single quark is not seen. (3) Chiral symmetry breaking.

Pure Yang–Mills has no quarks, so (3) does not arise. (1) is the body of the prize problem, and (2) is set apart in §5 of the problem statement as a "natural extension". The picture of physics tries to explain (1) and (2) in one sheet, and is made of three parts: the coupling weakens at short distances (§02), a single scale is born (§03), and at long distances the flux is confined to a tube (§04, §05).

Under renormalisation, the coupling $g$ changes its value with the scale $\mu$ at which we look (an energy; as a distance, roughly $1/\mu$) (4-04). The β function that fixes how it changes has the following form in the one-loop computation for pure SU(N) Known.

Because of the minus sign, raising $\mu$ (looking at shorter distances) makes $g$ smaller. This is asymptotic freedom (4-06). The PDG review writes, with $\alpha=g^2/4\pi$, $\mu^2d\alpha/d\mu^2=-b_0'\alpha^2-\cdots$ and $b_0'=11N/(12\pi)$ (the same thing Computed), and says 'The minus sign in Eq. (9.3) is the origin of "asymptotic freedom"'. With the first term alone, $d(1/g^2)/d\ln\mu=2b_0$ is constant, so

With SU(3) and $g^2(\mu_0)=1$, we get $g^2=0.757$ at $\mu=10\mu_0$, and still $0.342$ at $10^6\mu_0$ Computed. The weakening is logarithmic. In the words of the problem statement, 'at short distances the field displays quantum behavior very similar to its classical behavior'. But (1) is a perturbative computation. That the continuum theory really behaves like this at short distances is one of the properties the problem statement requires of a solution.

Reading (2) in the direction of lowering $\mu$, $1/g^2$ becomes 0 at a certain scale $\Lambda$.

Substituting (2), the $\ln(\mu/\mu_0)$ cancels and $\Lambda$ does not depend on $\mu$ (at three points along the flow it agrees up to double-precision rounding Computed). Classical four-dimensional Yang–Mills has no constant with the dimension of mass ($g$ is dimensionless, 1-12). Even so, in the quantum theory a scale $\Lambda$ appears. This is dimensional transmutation (4-07). The PDG explains $\Lambda$ as 'a constant of integration, which corresponds to the scale where the perturbatively-defined coupling would diverge'. What breaks at $\mu=\Lambda$ is the one-loop formula, not the theory. From here physics concludes that every quantity with a dimension is a constant multiple of $\Lambda$ Physics.

$m_{0^{++}}$ is the mass of the lightest glueball (§06), $\sigma$ the string tension (§04), and $c_i$ pure numbers fixed by N. This "single scale" looks obvious in physics as dimensional analysis, and is not proved in mathematics ("Assumption H" in the record kept on this machine; 7-12, 10-01). Also, $\exp(-1/(2b_0g^2))$ has all Taylor coefficients 0 at $g=0$. If (4) is granted, $\Delta$ too appears at no order of the perturbation series. This runs the same way as the problem statement's 'the mass gap is not classically visible' (4-12).

ComputedThe numbers in this figure were computed on this machine. The curves are drawn on the spot from formulas (2)(3), and Table 1 evaluates the same formulas in Python 3 (for SU(3) at $g^2=0.5$, $\Lambda/\mu_0=5.83\times10^{-7}$).

The smaller the coupling, the exponentially smaller $\Lambda$ is, and the more sensitive to changes. SU(2) at $g^2=1.5$ and SU(3) at $g^2=1.0$ give the same value because at one loop only $Ng^2$ (the 't Hooft coupling, 7-04) matters. The numerical value of $\Lambda$ changes by a fixed factor depending on how $g$ is defined, but the form of (4) is kept.

To measure the force in a theory without quarks, we place a pair of static colour charges from outside (a colour charge is a charge that acts as a source of the gauge field; 3-02). In Euclidean spacetime (4-02, One sign is different) take a rectangle of width $R$ and length $T$; the trace of the matrix obtained by parallel transport once around it along the gauge field (the holonomy, 1-07) we call the Wilson loop $W_{R,T}$. The rectangle is the process "create the pair, leave it for a time $T$, annihilate it", and the expectation value decreases roughly as $e^{-V(R)T}$, so we define the potential by the following (Wilson 1974; the notation follows Chatterjee 2018 §4) Known.

If $V(R)\to\infty$ as $R\to\infty$, the pair cannot move apart beyond a fixed distance. This is confinement (the definition in Chatterjee 2018 §4). In particular, if $\langle W_{R,T}\rangle$ decreases as $e^{-\sigma RT}$ in the area $RT$ (the area law), then $V(R)\approx\sigma R$ grows linearly. The coefficient $\sigma$ in this linear case is the string tension. That it grows linearly in four-dimensional non-abelian Yang–Mills stands at the level of 'it is believed that V(R) grows like a multiple of R' (ibid. §4) Physics.

On the lattice, which replaces spacetime by a mesh of points (5-01), the strength of the coupling is expressed by the lattice inverse coupling $\beta$ (for the standard Wilson action $\beta=2N/g^2$; a large $\beta$ is weak coupling; this is not the β function of §02). On the side where $\beta$ is small (strong coupling) the area law is proved, but it holds for every group including U(1), so it is no evidence of non-abelian confinement. The weak-coupling side for non-abelian groups in four dimensions 'remains elusive' (Chatterjee 2018; 5-02, 5-06) Known.

The picture of physics is the flux tube. The lines of force of electromagnetism spread out, and pulling the charges apart the energy approaches a constant value ($V\sim-c/R$). Under confinement the lines of force are squeezed into a tube of fixed thickness, and the energy is proportional to the length. As a metaphor, it is a string whose tension does not change as it is stretched. More precisely, the tube is not a fact derived from the theory but a picture that fits the lattice numbers; the effective theory of the fluctuations of the tube (7-03) takes the existence of the tube as its starting point and gives the correction at long distances as follows Physics.

$-\pi/(12r)$ is the Lüscher term, and its coefficient is held not to depend on the details of the theory ($-\pi(d-2)/24$ in $d$ dimensions). $c_0$ is a non-universal constant. Putting (6) into the lattice length standard $r_0$ ($r_0^2F(r_0)=1.65$ for $F=dV/dr$, Sommer 1994) gives $r_0\sqrt{\sigma}=\sqrt{1.65-\pi/12}=1.178$, which differs by 1.6% from the lattice continuum value $1.160(6)$ (Athenodorou–Teper 2020) Computed. At $r_0$ the tube is still at short distance, so this is a rough guide that extends formula (6) to that distance. It agrees in outline, but the difference is about 3 times the lattice error, and it does not agree in detail.

ComputedThe numbers in this figure were computed on this machine. Confinement is $\sigma r-\pi/(12r)$ of (6) (without $c_0$), electromagnetism is the sample $-0.3/r$, and fm is converted with $\sqrt{\sigma}=485$ MeV (§07). 0.51 is $\sqrt{\pi/12}$, where the two terms balance.

In QCD with quarks, signs that a long tube breaks have been reported on the lattice (SESAM 2005) Physics. Pure Yang–Mills has no quarks to create a pair midway and break the tube.

The particles of pure Yang–Mills are thought to be glueballs, colourless bound states made only of gluons. States are classified by $J^{PC}$ (spin, parity, charge conjugation). $0^{++}$ is a state of spin 0 that changes sign neither under spatial inversion (parity) nor under the exchange of particle and antiparticle (charge conjugation). In physics the mass of the lightest $0^{++}$ is often called "the mass gap" Physics. Chatterjee 2018 explains the correlation length (2-08) as 'the reciprocal of the mass of the lightest glueball', and writes that the existence 'has not yet been experimentally verified'.

The gap of the problem statement is defined as a gap in the spectrum of the Hamiltonian (the problem statement calls its supremum 'the mass'), and does not presuppose the existence of particles. If the lowest excitation is an isolated one-particle state, the gap coincides with the mass of the lightest glueball (4-11). A proof that an isolated one-particle state exists is counted by the problem statement on the side of the "natural extensions".

Lattice computations replace spacetime by a mesh with lattice spacing $a$ (the distance between lattice points) and produce the dimensionless ratio $m/\sqrt{\sigma}$.

ComputedThe errors of the ratios are a naive propagation treating them as independent, a rough guide that ignores correlations. SU(∞) is the value extrapolated in 1/N². On the lattice rotation symmetry is reduced to the rotations of a cube, and spin 2 splits into two sets (E, T₂) that are measured separately. The 2⁺⁺ of SU(∞) is the E value (4.578(11) for T₂; 5-08). 0⁺⁺* is an excited state.

For every N, $0^{++}$ is the lightest, and the ratios change only slowly with N (7-04). On the other hand, $m_{0^{++}}/\sqrt{\sigma}$ for SU(3) is 3.347(68) (Meyer 2005), 3.405(21) (Athenodorou–Teper 2020) and 3.607(87) (Lucini–Teper 2001), scattered more widely than the quoted errors. $\sigma$ is obtained by subtracting the fluctuation correction from the mass of a closed tube. Formula (4) of Lucini–Teper is $am_l=a^2\sigma L-\pi/(3L)$, where $m_l$ is the mass of a flux tube closed once around a periodic box, and $L$ is the side of the box counted in lattice points. The authors note: 'in using eqn(4) we are assuming that we have linear confinement'. The continuum limit is an extrapolation assuming $m(a)/\sqrt{\sigma(a)}=m(0)/\sqrt{\sigma(0)}+c\,a^2\sigma$ (5-08).

The parts and numbers so far rest on the premise that four-dimensional pure Yang–Mills theory really does exist, as one theory. We list the stages of a lattice computation at which the premise enters.

For physics this premise is natural, and the problem statement too writes that experiment and computers 'have given strong encouragement'. But the same paragraph continues 'there does not exist a convincing, whether or not mathematically complete, theoretical computation demonstrating any of the three properties in QCD', and the mathematical side is at the stage of 'nor even a precise definition' Known.

As a metaphor, physics draws a detailed map of a land it is sure exists, and mathematics is asked to show that the land exists. More precisely, the task is, for any compact simple group, to construct a non-trivial quantum Yang–Mills theory on $\mathbb{R}^4$ (with axioms at least as strong as Wightman or Osterwalder–Schrader; 4-09, 4-10) and to show $\Delta\gt0$. A road often taken is to build the Euclidean probability measure with a regulator and take the limit. A footnote of the problem statement says that weak existence, where only a subsequence converges, is not accepted unless the properties of the limit are shown separately. The numbers of physics, however precise, are "predictions made assuming existence" (5-09, 6-10).

In the picture of physics $\sigma$ and $m$ are two faces of the same $\Lambda$ (formula (4)), and confinement and the gap are spoken of together. In mathematics they are separate claims.

There are examples where only one of them holds. For lattice gauge theory with finite groups, exponential decay of correlations at weak coupling is proved, while the absence of confinement is at the level of 'believed' (5-10). For compact U(1) in three dimensions the area law is proved at every coupling, and the continuum limit is a free scalar field (abstract of Göpfert–Mack 1982). In the converse direction, exponential decay under arbitrary boundary conditions on the lattice gives the area law for Wilson loops in representations on which the centre of the group acts non-trivially (Chatterjee 2021) Known. The table and the precise statements are in 7-02 §02 (the master copy in this book).

This converse implication has a form close to "gap ⇒ confinement", but its hypothesis is a strong form of decay on the lattice. There is no theorem deriving confinement from a gap in the sense of the problem statement ($\Delta\gt0$ in the continuum and in infinite volume). The variables physics mainly looks at (the tube and $\sigma$) and the target of the problem statement ($\Delta$) are slightly offset.

Each tool of physics has a "shore" where its computations are under control. More precisely, a shore is a region near some limit ($g\to0$, lattice $\beta\to0$, $N\to\infty$ and so on) in which the errors of the computation can be controlled. The remaining chapters of Part 7 dig into the rows of the table.

The theory of the problem statement — any compact simple group, finite N, no matter, the continuum $\mathbb{R}^4$ — lies on no shore. The picture of physics fills the space between the shores with the premise (§08) that "one theory connects them". Where that premise parts from mathematics is gathered in 7-13 and 9-02.

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Next chapter to read: 7-02 Mechanisms of confinement — on to the explanations physics gives of why the tube of §05 forms (dual superconductivity, monopoles, centre vortices, Gribov–Zwanziger).

← 6-12 Detour: "if 1/0 is given a symbol"Contents7-02 Mechanisms of confinement →