2026-10-01 · chapter Yang–Mills and the mass gapPart 7 From the side of physicsmechanisms of confinement

Mechanisms of confinement — reading four pictures by what each casts in the leading role, and where each is cut off from mathematics

"Confinement" is a word that comes from the observation that quarks cannot be extracted on their own. Pure Yang–Mills theory has no quarks, so we place two infinitely heavy test charges and ask about the energy of pulling them apart (7-01). On the lattice, the expectation of the trace of the matrix obtained by multiplying link variables along a closed path $C$ is the Wilson loop (5-01); for an $R\times T$ rectangle $W\approx e^{-V(R)T}$, where $V(R)$ is the energy of the pair of charges.

There are two ways a large loop can decrease.

Under the area law $V(R)\approx\sigma R$, and the energy grows in proportion to the distance. $\sigma$ is the string tension. Under the perimeter law $V$ is bounded. Example: with $\sigma=c=0.05$ and $R=T=10$, $-\ln W$ is 5 and 2; with sides of 20 it is 20 and 4. To tell them apart we use the Creutz ratio, which is $\sigma$ for a pure area law and 0 for the perimeter lawKnown.

To ask about the "mechanism" is to ask what in the vacuum produces the area law. The area law at small $\beta$ (the lattice inverse coupling; for $SU(N)$, $\beta=2N/g^2$, smaller meaning stronger coupling) is a theorem (5-02), but it holds for every group including U(1), so it is no explanation for the non-abelian caseKnown. The question remains on the side of large $\beta$, heading towards the continuum limit.

The mass gap is exponential decay of correlations (4-11), confinement is the area law, and as logic they are different claims. What the Clay problem statement requires is $\Delta>0$; confinement is set apart as a "natural extension"Known.

On the lattice there is an implication in one direction. Chatterjee (2021) showed the followingKnown. If the representation forming the Wilson loop acts non-trivially on the centre of the group (§05) (the fundamental representation of SU(N) does), and centre symmetry is unbroken (that is, in slab-shaped regions of finite thickness, the measure is invariant under centre transformations whatever boundary condition is taken; Definition 2.1 of that paper), then for any measure on the infinite-volume lattice $\mathbb{Z}^d$ the following holds for some function $V$ with $V(R)\to\infty$ as $R\to\infty$.

This means that $V$ is unbounded, not that $\sigma>0$. Assuming a strong form of the gap, exponential decay under arbitrary boundary conditions, yields even the area law (Theorem 2.4 of the same paper). The decay shown for finite groups is decay of two-point correlations under a specific boundary condition, weaker than the "decay under arbitrary boundary conditions" assumed in Theorem 2.4. So it does not contradict the finite-group row of Table 1 (§2.4 of the same paper). There is no theorem deriving confinement from a gap in the Clay sense.

A superconductor expels magnetic fields (the Meissner effect), and a magnetic field entering from the surface decays over the penetration depth $\lambda_L$. If forced through, it is bundled into thin tubes (Abrikosov vortices), and the flux of one tube is an integer multiple of $h/2e$Known.

A tube with energy density $B^2/2$, cross-section $S$ and flux $\Phi=BS$ carries $E/\ell=\Phi^2/(2S)$ per unit length. In vacuum $S$ can be widened to bring this close to 0, but in a superconductor widening costs something, and $E/\ell$ stays at a positive constant. If a pair of monopoles were embedded, the energy should be proportional to the distance (a schematic estimate).

Maxwell's equations with a magnetic current $j^m$ added are symmetric under exchanging electric and magnetic ($\tilde F$ is the dual; only here do we use real-time notation)Known.

If the vacuum is a condensate of colour-magnetic monopoles, what is expelled is the colour electric field, the colour electric flux forms a tube, and quarks are confined. This is the picture of dual superconductivity, said to have been proposed independently by 't Hooft (a 1975 talk) and Mandelstam (1976)Physics. Nambu (1974) is regarded as its forerunner. More precisely, the monopoles are not fields appearing in the action but objects defined only once a gauge is chosen (§04), and there is as yet no way to write "condensation" as a statement about the measure.

Di Giacomo et al. (2000, SU(2)) measured an order parameter for monopole condensation on the lattice and reported that "confinement appears as a transition to a dual superconductor"Physics. The context is the deconfinement transition at finite temperature, and the mechanism was not shown. Neither step of "condensation ⇒ area law ⇒ gap" is proved for the four-dimensional non-abelian case.

Examples where the mechanism can be computed in an abelian model

In three-dimensional compact U(1) monopoles act as instantons; Polyakov (1977) obtained the area law semiclassically, and Göpfert–Mack proved $\sigma>0$ at every coupling (on the lattice with the Villain action)Known. But the abstract of the same work also states that the continuum limit is a free scalar field and that the string tension in physical units goes to $\infty$ (text not read). From this mechanism "the masses and the string tension are fixed by a single scale" (7-12) does not follow. In four dimensions, for an $\mathcal{N}=2$ supersymmetric theory broken to $\mathcal{N}=1$ by a mass term, Seiberg–Witten (1994) described confinement by monopole condensation (7-11)Physics.

't Hooft (1981) found monopoles, absent from the action, as singularities of the choice of gauge. Choose a quantity $X(x)$ that transforms as $X\mapsto gXg^{-1}$ (for example $F_{12}$) and diagonalise it at each point.

$\varepsilon_i$ are the eigenvalues. The remaining symmetry is $U(1)^{N-1}$, and the diagonal gluons are the "photons". At points where eigenvalues coincide the diagonalisation is not determined, and these are the monopolesPhysics. Example: in SU(2) with $X=\sum_a X^a\sigma^a/2$ the eigenvalues are $\pm|\vec X|/2$, and they coincide at $\vec X=0$, which is 3 real conditions. In three dimensions this is a point, in four a line (a world line) (the same for general $N$, the detour of 1-01). If $\vec X\approx\vec x$ near the zero, then $\vec X/|\vec X|$ covers the sphere surrounding the zero once, and that number of times is the magnetic chargeKnown.

On the lattice, the maximal abelian gauge (MAG, Kronfeld et al. 1987) brings the links close to diagonal.

No method is known for reliably finding the global maximum, and in practice local maxima (Gribov copies) are used. Suzuki–Yotsuyanagi (1990) reported that the string tension of the projected links accounts for most of the original (abelian dominance), and Greensite's review (2003), citing (92 ± 4)% for SU(2) (Bali et al. 1996), writes that "abelian dominance is not yet a proof of confinement by monopoles"Physics.

Detour: Elitzur's theorem

The centre of SU(N) is $Z_N=\{e^{2\pi ik/N}I\}$, which commutes with every element. A centre element $z$ acts in each representation as $z^k$, and this $k\ (\mathrm{mod}\ N)$ is the N-ality. The fundamental representation has 1 and the adjoint 0Known.

The starting point is the lattice observation that the string tension of large loops is fixed by the N-ality alone, and is 0 for the adjoint (screened by the gluons themselves) (Greensite's review §4.3)Physics. A centre vortex is a line in three dimensions and a surface in four, and multiplies a loop linked with it by one centre element ($-1$ for SU(2)). Adjoint loops do not notice the vortex. More precisely, the thinnest vortex on the lattice is a configuration in which the links piercing some region are all multiplied by a centre element, and only the plaquettes on the rim of the region change (inside, the centre elements cancel). This rim is the vortex, and only loops linked with the rim pick up the centre element. Divide the plane into squares of area $\Sigma_{\min}$; if the $Z(C_i)$ picked up by the loops of the squares are nearly independent, a large loop splits into a product and obeys the area law (formulas (94)–(96) of the same review).

Confinement becomes the question of whether the vortex surfaces spread through the volume (percolate). Numerically, Del Debbio–Faber–Greensite–Olejník (1997) reported that projected vortices carry most of the string tension (centre dominance), and de Forcrand–D'Elia (1999) that removing the vortices removes confinementPhysics. As a theorem, Tomboulis–Yaffe (1985, SU(2)) bounded the Wilson loop from above by a function of the vortex free energyKnown. The vortex free energy is $-\ln$ of the ratio of the partition function $\mathcal{Z}_{\mathrm{tw}}$ with a centre twist put into the boundary condition in one direction to the partition function $\mathcal{Z}$ without it, that is $F_v=-\ln(\mathcal{Z}_{\mathrm{tw}}/\mathcal{Z})$, and it is gauge invariant. In the same family of gauge-invariant quantities is the 't Hooft loop (an operator inserting a centre vortex along a loop). Formula (4) is also a theorem bounding Wilson loops from centre symmetry. Tomboulis (2007) claimed a proof at every coupling for SU(2), and Ito–Seiler point out a gap in the argument.

Vortices defined by projection depend on the gauge. The hypotheses of the conditional theorems concentrate on a single point, that vortices condense at weak coupling in four dimensions (gauge-invariantly, decay of the vortex free energy), and this is not proved.

Scatter over a plane the points where SU(2) vortices pierce it, and with $n$ the number of points inside the loop set $W=(-1)^n$. There are two kinds of points. (a) Independent points of density $\rho_f$ (cross-sections of extended vortices). (b) Pairs of points at distance $d$, with point density $\rho_p$ (small vortices piercing the plane twice). A pair changes the sign only when it straddles the edge. For an $I\times J$ rectangle the following holds exactly (averaging over directions the area of the symmetric difference of two translated rectangles)Computed.

The first term comes from the number $n$ of independent points following a Poisson distribution with mean $\mu=\rho_f IJ$, so that $\mathbb{E}[(-1)^n]=\sum_n e^{-\mu}\mu^n(-1)^n/n!=e^{-2\mu}$ (1-08). The first term is the area law and the second the perimeter law, and the Creutz ratio is $2\rho_f$. However many small vortices there are, they contribute nothing to the string tension.

ComputedThe curves are computed on the spot from formula (11), and the points on the left are one random sample. Formula (11) was checked on this machine against a Monte Carlo (MC) of one million samples (Table 2).

Mixed means $\rho_f=0.02$, $\rho_p=0.2$. The MC standard error grows as $-\ln W$ grows, and is at most 0.02 for $-\ln W\le3$. Mixed at $R=6$ has error 0.070, and its difference 0.11 from the exact formula is about 1.6 standard errors. $e^{-6.4}\approx0.0017$ is about the size of the noise $10^{-3}$ of an average of $\pm1$, and cannot be measured. This difficulty is the same as in real lattice computations (5-07). For SU(3), multiplying by $e^{\pm2\pi i/3}$ at each piercing gives the string tension $1.5\rho_f$ for $k=1,2$ and 0 for the adjointComputed.

The fourth picture reads confinement from the shape of the gauge-fixed propagator. The Landau gauge surface $\partial_\mu A_\mu=0$ can meet a single orbit more than once (4-08), so Gribov (1978) restricted the integral to the region $\Omega$ where the Faddeev–Popov operator $\mathcal{M}$ is positive. Its boundary is the Gribov horizon.

"$\mathcal{M}>0$" means that the eigenvalues of $\mathcal{M}$ (the spectrum of a self-adjoint operator, 1-10) are all positive. At $A=0$, $\mathcal{M}=-\partial^2$, which is positive apart from constants. As $A$ grows the smallest eigenvalue goes down, and where it reaches 0 is the horizon. On the horizon an infinitesimal gauge transformation in the direction of the zero eigenvalue preserves the gauge condition, so copies arbitrarily close appear.

What is certain as mathematics goes only as far as Dell'Antonio–Zwanziger (1991): "every orbit passes inside the horizon"Known. Copies remain inside $\Omega$ too, and no method is known for handling a copy-free region analytically (Zwanziger claims that the copies inside do not affect expectation valuesPhysics).

Zwanziger (1989) wrote the restriction to $\Omega$ as a local action. The propagator of its tree approximation (lowest order, without loop contributions, 4-03) is on the left below, and the version with condensates added (RGZ, Dudal et al. 2008) on the rightPhysics.

$\lambda^4=2g^2N\gamma^4$, and $\gamma$ is fixed by the horizon condition. Here $\lambda$ is a mass scale, different both from the penetration depth $\lambda_L$ of §03 and from the large-$N$ $\lambda=g^2N$. $M^2$ and $m^2$ are mass parameters added in RGZ. The GZ form is 0 at $p=0$ and its poles are complex, $p^2=\pm i\lambda^2$, quite unlike $1/(p^2+m_0^2)$ for a free particle of mass $m_0$.

If the gluon were a physical particle, the propagator could be written as a superposition with positive density $\rho$ (the Källén–Lehmann representation), and the correlation in the time direction $C(t)$ could not be negative (formulas (159)(160) of Dudal et al. 2008, 4-10). $\omega$ is the mass being superposed.

At $m=0$, computing by complex integration (residues) gives a closed form (it can also be checked by numerical integration; bottom of Figure 3). For $M^2\lt 2\lambda^2$, setting $\cos\varphi=M^2/(2\lambda^2)$Computed

and it first becomes negative at $t_*=(\pi-3\varphi/2)/(\lambda\sin\frac{\varphi}{2})$. For GZ, $t_*=\pi/(2\sqrt2\,\lambda)\approx1.111/\lambda$. For $M^2\gt2\lambda^2$ it splits into real poles, $D=A/(p^2+r_1)+B/(p^2+r_2)$ ($0\lt r_1\lt r_2$), and although $B$ is negative ($\rho$ is not positive), $C(t)\ge e^{-\sqrt{r_2}\,t}C(0)\gt0$ and it does not become negative. This test works in one direction only.

ComputedThe curves are computed on the spot from formulas (13)(15) and the partial fractions for real roots. The closed formula was checked on this machine against numerical integration, with differences at most $2\times10^{-11}$.

The lattice did not agree with the original prediction $D(0)=0$. Cucchieri–Mendes (SU(2), $128^4$, about $(27\ \mathrm{fm})^4$; 1 fm = $10^{-15}$ m) report in their abstract that at zero momentum it "seems to show a finite non-zero value", and Bogolubsky et al. (SU(3)) point the same wayPhysics. RGZ is built to reproduce thisPhysics. BRST symmetry (the symmetry that remains after gauge fixing, 4-08) is softly broken in GZ. Soft breaking means that the breaking term has low dimension and has no effect at short distances. This breaking is not a price paid by RGZ but a property of the horizon term of GZ itselfPhysics.

Every leading variable becomes visible only once a gauge is chosen, and the gauge-invariant core is only the vortex free energy and the inequalities around it. What every picture explains is $\sigma$; $\Delta$ is merely expected as a consequence (§02). None of the pictures yields "a single scale". The seams are sorted out in 9-02, obstacles other than divergence in 6-11.

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Next chapter to read: 7-03 The string picture — treating the squeezed tube as a single string and looking at its fluctuations.

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