2026-10-01 · chapter Yang–Mills and the mass gapPart 5 Lattice gauge theorythe strong-coupling expansion

The strong-coupling expansion — at small β the Wilson loop becomes a sum over surfaces, and the area law comes out

A small coefficient $\beta=2N/g^2$ in the Wilson action (5-01 §04) means a large coupling $g$, so the expansion on this side is called the strong-coupling expansion. $\beta$ plays the role of inverse temperature, so it is also called the high-temperature expansion (2-08). At $\beta=0$ the links scatter independently with Haar measure, and since $\int_GU\,dU=0$, the expectation value of a simple loop is 0. The abstract of Wilson 1974 writes “The lattice gauge theory has a computable strong-coupling limit; in this limit the binding mechanism applies and there are no free quarks. There is unfortunately no Lorentz (or Euclidean) invariance in the strong-coupling limit.” Known.

Example: $\mathbb Z_2$. For the group of links $\pm1$ (2-09) with action $\beta\sum_P(1-U_P)$, since $U_P=\pm1$, the weight of one plaquette splits into two terms, up to a constant.

Opening the product gives a term $t^{|X|}\prod_{P\in X}U_P$ for each set $X$ of plaquettes. The average of a uniform $\pm1$ is 1 for even powers and 0 for odd ones, so in the partition function only sets in which every link is surrounded by an even number of plaquettes (closed surfaces) survive, and in the numerator of the Wilson loop $\prod_{\ell\in C}U_\ell$ only surfaces with boundary $C$. The smallest surface is a flat rectangle, so

and the expectation value decreases exponentially in the area (the area law). That the gap for small $\beta$ in the $\mathbb Z_2$ of 5-03 §08 is close to $-4\log\tanh\beta$ is the same count.

For a general group too, the weight of one plaquette $e^{\frac\beta N\mathrm{Re\,tr}\,U}$ is a function unchanged by conjugation $VUV^{-1}$ (a class function). To split it by representation, we prepare just two tools.

Characters. For a representation $r$ of the group (1-04), $\chi_r(U)=\mathrm{tr}\,r(U)$ is called its character. The trace is unchanged by conjugation, so characters are class functions. The characters of irreducible representations are orthonormal with respect to Haar measure $dU$ (the uniform probability measure on the group; 1-08 §03) (Schur's orthogonality relations; proof omitted) Known.

Class functions can be expanded in these characters. It is the generalisation of Fourier series to non-commutative groups Known.

Integrals over classes of $SU(2)$. Elements of $SU(2)$ can be written as $U=V\,\mathrm{diag}(e^{i\theta},e^{-i\theta})\,V^{-1}$ ($0\le\theta\le\pi$), so a class function is a function of $\theta$ alone. Its integral takes the following form (the $SU(2)$ version of Weyl's integration formula) Known.

Then $\mathrm{tr}\,U=2\cos\theta$, the character of the spin-$j$ representation is $\chi_j=\sin((2j+1)\theta)/\sin\theta$, and its dimension is $2j+1$.

Character coefficients

For a general group, the weight is expanded in characters as follows.

$d_r$ is the dimension. We call $u_r$ the character coefficients, and write the one for the fundamental representation simply as $u$ (for $\mathbb Z_2$, $u=t$). For $SU(2)$, the class integral above can be written with modified Bessel functions $I_n$ (Detour: the derivation of (4)):

Known (agreeing with numerical integration to $2\times10^{-13}$ Computed). It is the same number as the ratio $\lambda_j/\lambda_0$ of eigenvalues of the transfer matrix of a single link in 5-03 §06.

If $\beta\gt0$, then $0\lt u_r\lt1$. They are positive because each term of the opened exponential is a non-negative sum of characters of tensor products (5-03 §06). They are below 1 because $\mathrm{Re}\,\chi_r(U)\le d_r$, with equality only when $r(U)$ is the identity matrix.

At small $\beta$, $u$ approaches $\beta$ for $\mathbb Z_2$, $\beta/2$ for $U(1)$, $\beta/4$ for $SU(2)$, and $\beta/(2N^2)$ for $SU(N)\ (N\ge3)$ Computed. The reason is the first-order term. Opening the exponential of (3) to first order gives $u\approx\frac{\beta}{N^2}\int(\mathrm{Re\,tr}\,U)\,\overline{\chi_F(U)}\,dU$ ($F$ is the fundamental representation), and this integral is $1/2$ for $N\ge3$, since $\int(\mathrm{tr}\,U)^2dU=0$, and 1 for $\mathbb Z_2$ and $SU(2)$, since $\mathrm{tr}\,U$ is real Known.

For $SU(2)$, the $\beta$ of 6-02 is $\beta/2$ of this chapter (compare the $w$ of (3.3) in 6-02 with the $u$ of (4) in this chapter; for $U(1)$ it is the same $\beta$).

Detour: the derivation of (4)

Multiplying (3) over all plaquettes and opening the product, each term is an assignment of "which representation to put on which plaquette". There are two rules for integrating out the links one at a time (Schur's orthogonality relations Known; (3.1) of 6-02 is a lemma of the same idea).

The left says "if an edge lies on only one plaquette, the result is 0", and the right says "gluing two plaquettes with the same representation along a common edge makes a larger loop, with a factor $1/d_r$". What survives are terms in which the plaquettes carrying representations form a closed surface, or a surface whose boundary is the observed loop.

Example: the $1\times2$ loop. The observable is $\frac1d\chi(U_C)$ ($d$ is the dimension of the fundamental representation). The leading term is the one with representations on the two plaquettes inside the loop, $(du)^2\chi(U_{P_1})\chi(U_{P_2})$. For $SU(N)$, the surface carries the conjugate representation $\bar r$ so as to pair with the $\chi$ of the observable (for $SU(2)$, $\bar r=r$). We integrate in three steps.

1. Integrate the edge shared by the two plaquettes with (5) right. The two loops join into the boundary loop $\chi(U_{\partial})$ of the $1\times2$, with a factor $1/d$.
2. Integrate one of the six boundary edges with (5) right. The loop of the observable and the boundary loop are glued with opposite orientations, the remaining five links cancel in the form $VV^{-1}$, and $\chi(1)/d=d/d=1$ remains.
3. Normalisation: the $\frac1d$ of the observable remains (the partition function is 1 in the leading term apart from powers of the constant $c_0$, so dividing by it changes nothing).

Altogether $(du)^2\cdot\frac1d\cdot1\cdot\frac1d=u^2$, that is, $\langle W_{1\times2}\rangle=u^2$. Tiling an $R\times T$ too, the powers of $d$ cancel, and what remains is

The energy of a static pair of charges $V(R)=-\lim_{T\to\infty}\frac1T\ln\langle W_{R\times T}\rangle$ (7-02 §01) is $\sigma_0R$ in the leading term: the confining form that grows in proportion to the distance Known. Example: for $SU(2)$ at $\beta=1$, $\sigma_0=1.426$, and the $2\times2$ loop is $u^4=0.0033$ Computed.

ComputedThe numbers in this figure were computed on this machine ($u$ from the series of (4), computed on the spot). If JavaScript is disabled: the surface has $RT$ plaquettes with weight $u^{RT}$, and in four dimensions there are $4RT$ bumps, each with $u^4$.

Detour: sums over surfaces and strings

The next surfaces that matter replace one plaquette of the flat surface by the remaining five faces of the unit cube with that plaquette as its base (Figure 1). Paying five plaquettes instead of one multiplies the weight by $u^4$, and there are $2(d-2)$ such cubes, on both sides along the $d-2$ axes perpendicular to the surface: 4 in four dimensions. Separated bumps may be counted independently at this order, so together they make an exponential (the count for $SU(2)$).

The next corrections come from overlapping bumps, two-storey bumps and surfaces carrying higher representations (higher coefficients are in the review of Drouffe–Zuber 1983, and the higher series for the string tension of $SU(2)$ in four dimensions in Münster 1981; see the table of sources). Justifying the exponential form uniformly in the volume is the cluster expansion (5-04, 6-04) Known.

Two dimensions have no bumps. For $d=2$ the surface cannot leave the plane, and the leading term $u^{RT}$ is exact for every $\beta$ (6-02 §03) Known. The difficulty in three or more dimensions begins with the surface being able to fluctuate out of the plane. Example: for $SU(2)$ at $\beta=2$, $4u^4=0.141$ reaches 17% of $\sigma_0=0.837$, and the truncation becomes doubtful Computed.

Mass is the rate at which correlations decrease (4-11). The smallest closed surface joining a spatial plaquette $P=\frac12\mathrm{tr}\,U_P$ and the same shifted by $t$ in the time direction, $P(t)$, is the $1\times1\times t$ box with the two as lids. The lids are observables and carry no weight, and the $4t$ side plaquettes pay $u^{4t}$.

Computed (lowest-order counting). The lowest-order form is said to be standard (Drouffe–Zuber 1983, bibliographic data only). That the gap in the $\mathbb Z_2$ of 5-03 §08 is close to $-4\log\tanh\beta$ has the same form. The correlation length $\xi_0$ is 0.175 at $\beta=1$ and still 0.38 lattice units at $\beta=2.6$: the correlation vanishes within one lattice unit Computed. The continuum limit needs $\xi$ much longer than the lattice spacing ($\xi(\beta)\to\infty$ in Problem 5.1 of Chatterjee 2018; 5-09), and the leading term says the opposite.

Extending the ratio $m_0/\sqrt{\sigma_0}=4\sqrt{\sigma_0}$ of (8) outside the region of convergence, the ratio of leading terms falls to 0 as $\beta$ grows. The ratio from lattice numbers rises from 3.3 ($\beta=2.25$) to 3.8 ($\beta=2.6$) and approaches a constant value (7-12 §08) Physics. This concerns the form of the leading terms, not a statement about the theory. The leading terms at $\beta=2.6$, $a\sqrt{\sigma_0}=0.808$ and $am_0=2.61$, are 6.0 and 5.1 times the $0.13395(62)$ and $0.514(8)$ of Table 1 of Lucini–Teper (2001) Computed (the literature values are Physics). From $\beta=2.4\to2.6$ the literature $a\sqrt\sigma$ halves, while the leading term decreases by only 4%.

We integrated $SU(2)$ on four-dimensional periodic lattices (6 and 8 points per side) with the heat-bath method (an update that redraws the links one at a time from their conditional distributions; 5-07), and measured the string tension with Creutz ratios ((3) of 7-02).

The perimeter and corner terms cancel in the ratio, and with the form (7), $\chi(R)=\sigma_1$.

ComputedThe numbers in this figure were computed on this machine (curves computed on the spot, points the values of Table 2, ■ literature values). If JavaScript is disabled, Table 2 gives the content.

There are three readings Computed. (i) For $\beta\le1$ the plaquette agrees with $u+4u^5$, and $\chi(2)$ with $\sigma_1$, within errors. (ii) From 1.8, $\chi(2)$ falls below $\sigma_1$, and at 2.4 it is nearly half (that the plaquette is close to $u+4u^5$ at 2.6 is a coincidence of crossing curves). (iii) $\chi(3)$ is smaller still, yet 2 to 5 times the literature $(a\sqrt\sigma)^2$ (small loops mix in short-distance contributions Physics). The plaquette differs from the literature values by about 0.0003, four times the statistical error at $\beta=2.6$. The difference may include finite-volume differences between the lattice of 8 per side and the literature's $12^4$ lattice ($\beta=2.4$) and $24^4$ lattice ($\beta=2.6$) Computed.

The abstract of Creutz (1980), who did the same kind of calculation, reports that when the lattice is made finer with the string tension held fixed, the bare coupling decreases (logarithmically) as predicted by the perturbative renormalisation group, and concludes “This supports the coexistence of confinement and asymptotic freedom for quantized non-Abelian gauge fields.” It can be read as the area law on the strong-coupling side and asymptotic freedom on the weak-coupling side appearing connected without being cut by a phase transition. That $\beta_c=\infty$ (the continuum limit only on the $\beta\to\infty$ side) for many non-abelian lattice models of interest in four dimensions is, in the words of Chatterjee (2018) §5, at the level of “It is believed” Physics.

On a lattice with $L$ points per side, $Z_L(\beta)=\int e^{-S_W}\prod dU_\ell$ is an integral over a finite-dimensional compact space and an entire function of $\beta$ (a function expressible as a power series converging on the whole complex plane). $Z_L$ itself is entire, but expectation values and $\ln Z_L/|\Lambda_L|$ ($|\Lambda_L|$ is the number of lattice points; the free energy) involve dividing by $Z_L$, so zeros of $Z_L$ in the complex $\beta$ plane shrink the radius of convergence of their expansions. The question is whether that radius of convergence survives independently of $L$ (phase transitions in 2-08) Known. The number of connected surfaces grows with the number $n$ of plaquettes as $K^n$ ($K$ fixed by the dimension), and each plaquette weighs $u$, so the sum converges for roughly $Ku\lt1$ (a crude estimate; the precise condition is in 5-04): that is the core. The introduction of Shen–Zhu–Zhu (2023) summarises Osterwalder–Seiler (1978) as “when the coupling is sufficiently strong, the cluster expansion ... for the expectation values of local observables ... is convergent, uniformly in volume”, and writes that the mass gap (exponential clustering: correlations of distant observables decreasing exponentially in the distance; the mass gap of 4-11) and the existence and analyticity of the infinite-volume limit follow from it Known. SZZ themselves showed “a strictly positive mass gap” with Langevin dynamics (6-09) for $SU(N)$ with $|\beta_{\mathrm{SZZ}}|\lt1/(16(d-1))$. From their formula (1.2), $\beta=N^2\beta_{\mathrm{SZZ}}$, so for four-dimensional $SU(2)$, $\beta\lt1/12$ (the conversion is this book's arithmetic). It is a sufficient condition, not the place where the expansion stops.

Where it stops

(a) Roughening: physics considers that at some $\beta$ the fluctuations of the surface win, and the series for $\sigma$ cannot be extended beyond it (7-03 §07; about 2 for SU(2), with a large uncertainty in the estimate) Physics. (b) For four-dimensional $U(1)$ the area law breaks down at large $\beta$ (§08). (c) Formulas truncated at finite order cannot express exponential smallness like $e^{-c\beta}$ as $\beta\to\infty$ (the leading term $-\ln u$ decreases only as a power, like $3/(2\beta)$). On the other hand, in the continuum limit of asymptotic freedom (4-06, 4-07), at one loop (the lowest order of the β function of 4-06) $a^2\sigma\propto e^{-(6\pi^2/11)\beta}$ should become exponentially small Physics. Taking $\beta$ from 2 to 4 merely halves $-\ln u$, from 0.837 to 0.418, while $e^{-(6\pi^2/11)\beta}$ becomes about $5\times10^4$ times smaller Computed. The continuum-limit side cannot be reached with finite-order expansions (this book's organisation; 5-05).

The $u$ of Table 1 is below 1 for every group, so the area law (6) comes out for $\mathbb Z_2$ and for $U(1)$ too. Chatterjee (2018) §4 writes “the area law holds for any lattice gauge theory at sufficiently small β”, and goes on to say that therefore “the area law at small β cannot be evidence for quark confinement”. The proof is Osterwalder–Seiler (1978), whose abstract writes “We then prove existence and analyticity of the infinite volume limit of strongly coupled Yang-Mills theories on the lattice and we verify Wilson's confinement bound.” Known. The four-dimensional $U(1)$ lattice does not satisfy the area law at large $\beta$ (Guth 1980; Guth showed it with the Villain form of the action, different from the Wilson form; a fully rigorous proof is Fröhlich–Spencer 1982; 5-06) Known. The strong-coupling area law is a property of the lattice being coarse, and does not use non-commutativity. Problem 4.1 of the same paper asks to show, in any infinite-volume limit of a four-dimensional non-abelian compact group, for all $\beta\gt0$,

and writes that at large $\beta$ it “remains elusive”. On the strong-coupling side, Cao–Nissim–Sheffield extended the range of the area law for $U(N)$ at large $N$ (arXiv:2505.16585, a preprint, publication not checked; abstract: “This improves on a classical result of Osterwalder-Seiler from 1978”), and in the 't Hooft regime (letting $N$ grow with $\beta$ proportional to $N$; for converting conventions, Table 4 of 5-01) they checked, with the SZZ method, a mass gap condition that the abstract cites as [DF80], and derived the area law from it (arXiv:2509.04688, a preprint, publication not checked) Known. Both are results at fixed lattice spacing. The mass gap and confinement are different statements (7-02 §02; finite groups in 5-10), and this chapter shows neither for the continuum theory.

What this chapter can say

Sources and reproduction

Next chapter to read: 5-03 Reflection positivity and the transfer matrix (extracting the Hamiltonian from the same lattice).

← 5-01 Wilson's latticeContents5-03 Reflection positivity and the transfer matrix →