2026-10-01 · chapter Yang–Mills and the mass gapPart 7 From the side of physicslarge N, planar diagrams

Large N — sending the number of colours to infinity sorts Feynman diagrams by surfaces

Four-dimensional pure Yang–Mills has no small number to expand in. The mass scale $\Lambda$ has the form $e^{-1/(2b_0g^2)}$ ($b_0=11N/(48\pi^2)$, formula (1) of 7-01) and shows up at no power of $g$ (7-01 §03). 't Hooft (1974) treated the size $N$ of the group as a knob and considered the following limit (the group in the original paper is U(N)) Known.

According to the abstract, 'only planar diagrams with the quarks at the edges dominate', and 1/N corresponds to the coupling of the string models of the time. As a metaphor, the idea is to see $N=3$ as "$N=\infty$ with a correction of order $1/9$"; more precisely, how small the correction is has to be checked with lattice numbers (§08). The problem statement calls the duality conjecture, that through the 1/N expansion a gauge theory might be equivalent to a string theory with coupling $1/N$, 'A prominent speculation', and writes that such a description could be 'perhaps a good starting point for a rigorous proof (for sufficiently large N)' Known. But the requirement is finite $N$ (0-03).

The gauge field $A_\mu$ is an $N\times N$ Hermitian matrix at each point (3-02), and we regard the two indices of a component $A_\mu{}^{i}{}_{j}$ as the two ends of colour lines. The action is (Euclidean)

The trace of a product joins indices into a ring, as in $\operatorname{tr}(ABC)=A^{i}{}_{j}B^{j}{}_{k}C^{k}{}_{i}$, and the three-point and four-point vertices of (2) are all of this form. The colour part of the propagator (4-03) for U(N) is

One delta is one index line, so a gluon line is two lines of opposite orientation, a thin ribbon. This is the double-line notation. In the continuum action (at the level of the Lie algebra) the direction of the identity matrix drops out of the commutators, and its cross terms with the traceless part vanish too, so the U(N) of pure gauge theory is SU(N) plus one free U(1) field that does not interact Known. As groups, U(N) does not split into SU(N)×U(1) (the identity matrix times $e^{2\pi i k/N}$ lies in both), and on the lattice U(N) (§07) this separation does not hold.

From (2)(3), each vertex carries $N/\lambda$, each propagator $\lambda/N$, and each closed index ring $\sum_i\delta^i{}_i=N$. A vacuum diagram with $V$ vertices, $E$ propagators and $F$ index rings is proportional to

Gluing a disc to each index ring gives an oriented closed surface, and $V-E+F$ becomes the Euler characteristic $\chi=2-2h$, fixed by the surface alone ($h$ is the number of handles = the genus; for a tetrahedron on a sphere $4-6+4=2$). Therefore

where $f_0$ is the sum of diagrams that can be drawn in the plane (planar diagrams), and the corrections go in steps of $1/N^2$ Known. A quark line is a single index line, and opens a boundary in the surface, multiplying by $1/N$. (5) is a rearrangement of the perturbation series; convergence is a separate question (4-12). The reason for fixing $\lambda$ lies in the β function (the formula describing how the coupling constant changes with the scale $\mu$ at which we look; 7-01 §02). Writing it in terms of $\lambda$ with the $b_0$ of §01, $N$ disappears Computed.

The left-hand side is how $\lambda$ changes when the scale $\mu$ is changed slightly, and since the right-hand side is negative, $\lambda$ is smaller at shorter distances (larger $\mu$). Fixing $\lambda$ fixes $\Lambda$ too, and on the lattice this choice gives a smooth limit (Lucini–Teper) Physics.

The counting (4) can be checked in a model where spacetime is shrunk to a single point. Take one Hermitian matrix $M$ distributed according to a Gaussian (1-08) ($dM$ is the Lebesgue measure on the real components).

The expectation of a product of Gaussian variables is the sum over ways of pairing them (Wick's theorem, 2-07). The pairings of $\frac1N\operatorname{tr}M^4$ are the three of Figure 0: (a)(b) give $\frac1N\cdot N^3\cdot N^{-2}=1$ and (c) gives $N^{-2}$.

In general

where $\varepsilon_g(k)$ is the number, among the $(2k-1)!!$ pairings, of those of genus $g$ (with $k+1-2g$ index rings). The coefficients can be obtained from the recursion of Harer–Zagier (1986).

ComputedThe numbers of Table 1 and Figure 1 were computed on this machine (for $k\le7$, full enumeration checked against the recursion). If JavaScript is disabled, see Table 1.

The total $(2k-1)!!$ grows much faster than the number of planar diagrams, and at $k=9$, $N=1$ planar diagrams are under 0.02%. The large-N limit is the limit that keeps only the few diagrams.

Detour: Catalan numbers and the semicircle

We read the ribbon diagrams inside out. Replace four-point vertices by squares, three-point vertices by triangles, and propagators by the gluing of two edges, and each diagram becomes one closed surface glued from polygons. (5) is

a sum over random surfaces, where $N$ counts the topology and $\lambda$ the number of polygons ($w_\lambda$ is a weight that includes the momentum integrals). As a metaphor it is a sum over shapes of a membrane; more precisely, (10) is a rearrangement of a series, and convergence as a sum over surfaces has to be shown separately.

Physics reads this membrane as the world sheet of the string swept out by the tube of confinement (7-03). $N^{2-2h}$ has the form of the perturbative expansion of a string with string coupling $1/N$; in the problem statement, 'may be equivalent to a string theory with 1/N as the string coupling constant' Physics. In two-dimensional QCD, the coefficients of the $1/N$ expansion are fixed by sums over maps of surfaces, and it can be read as a theory of closed strings (Gross–Taylor 1993) Known.

One more thing follows from the counting (4): the covariance of quantities normalised with $\frac1N\operatorname{tr}$ is smaller by $1/N^2$ Known.

This is factorisation, and for $A=B$ the standard deviation shrinks as $1/N$.

At $N=\infty$, quantities made from traces (for the matrix model (7), quantities unchanged under conjugation $M\to UMU^\dagger$; for gauge theory, gauge-invariant quantities) stop being random, and the measure behaves as though concentrated on a single configuration. By custom, where it concentrates is called the master field. As a metaphor it is the picture of a heaving sea freezing; more precisely, what freezes is the distribution of quantities made from traces, while the components of the matrix keep fluctuating.

The representative case in gauge theory where factorisation works is the Wilson loop. It is the $\frac1N\operatorname{tr}$ of the matrix (an $N\times N$ unitary matrix) obtained by parallel transport with the gauge field along a closed path $C$, defined in 7-01 §04. The Schwinger–Dyson equations are relations among expectation values that come from the integral not changing when the integration variable is shifted. Putting factorisation into these equations for Wilson loops gives equations closed in the expectations of loops alone (Makeenko–Migdal 1979) Physics. In the plane $\mathbb{R}^2$, Lévy constructed the large-N limit of the Yang–Mills measure (6-02) and showed that each Wilson loop converges in probability to a deterministic limit, and that these equations hold both at finite N and in the limit Known.

We write as much lattice vocabulary as needed (for details see 5-01). Replace spacetime by a mesh of lattice points, and on each edge $\ell$ joining two neighbouring points place a group element $U_\ell\in U(N)$. Write $U_p$ for the product of the elements on the four sides of the smallest square of the lattice (a plaquette) $p$, taken in order around it; the action is made from $\operatorname{tr}U_p$. The $\frac1N\operatorname{tr}$ of the product of edge elements along a closed path is the lattice Wilson loop. Each $U_\ell$ follows the Haar measure (the uniform probability measure on the group, 1-08 §03) multiplied by the weight of the action.

On the two-dimensional lattice, the tree gauge, which sets some of the edge elements to 1, makes the plaquettes independent, and a Wilson loop enclosing $n$ faces is exactly $w^n$ (6-02 §02, §03). Large N can be studied with the integral over a single plaquette. Weighting the Haar measure of U(N), set

(in terms of the $\beta$ of 6-02, $\beta=2N/\lambda$). At finite N, Weyl's integration formula (which turns the Haar integral of a function depending only on traces into an integral over the eigenvalues $e^{i\theta_1},\dots,e^{i\theta_N}$) and Heine's identity (which turns that eigenvalue integral into a determinant) give a determinant whose entries are the modified Bessel functions

Known.

Gross–Witten (1980) evaluated $N\to\infty$ by the saddle-point method (a method that approximates $\int e^{Nf(x)}dx$, containing a large parameter $N$, around the point where $f$ is maximal) and found a third-order phase transition at $\lambda=2$. The same transition appears in Wadia's 1979 preprint Known. In this normalisation the limit has the following form, and the finite-N values approach it as in Table 3 Computed.

At $\lambda=2$ both the value 1/2 and the slope −1/4 join up, and the second derivative jumps from 0 to 1/4 (the third derivative, for the free energy). The string tension is $\sigma=-\ln w$.

ComputedThe numbers in this figure and table were computed on this machine ((13) evaluated to 300 digits; N=1,2 agree with direct integration to 10 digits; the second derivative in the figure is a difference with step 0.1). If JavaScript is disabled, see Table 3.

Three things can be read off Computed. (i) At finite N, $Z_N$ is an entire function of $N/\lambda$ and positive, so $w_N$ is smooth, and the kink first appears at $N=\infty$. (ii) The manner of approach differs between the phases. The deviation is nearly $-1/(16N^2)$ at $\lambda=1$; at $\lambda=4$ it is exponentially small, $2\times10^{-12}$ at $N=20$; at $\lambda=2$ doubling N only reduces it by about 1/2.5. (iii) Extending the strong-coupling formula $1/\lambda$ to $\lambda=1$ gives 1, away from the correct 0.75. This is an example of the strong-coupling expansion not reaching beyond the transition (5-02).

Wadia's preprint (reprinted on arXiv) writes of this transition that it 'seems to be a pathology of the infinite N limit', and holds that in this model it is unrelated to the physical continuum limit (the $\lambda\to0$ side). Still, $N\to\infty$ can create singularities that finite N does not have, so exchanging $N\to\infty$ and $a\to0$ needs a reason.

In four dimensions we rely on lattice numbers. Lucini–Teper (2001) fitted the continuum limits of $m_{0^{++}}/\sqrt\sigma$ (the lightest glueball and the string tension, 7-01) for SU(2) to SU(5) with a linear function of $1/N^2$, and summarised: 'all SU(N) gauge theories are close to SU(∞), at least as far as the low-lying mass spectrum is concerned' Physics.

Refitting (15) to the four points of the paper with error weights gives $a=3.366\pm0.106$, $b=1.93\pm0.54$, $\chi^2=0.26$ (2 degrees of freedom), with central values the same as the paper's $3.37(15)+1.93(85)/N^2$. A constant alone gives $\chi^2=12.9$ (3 degrees of freedom) and does not fit Computed.

PhysicsExtrapolated values from primary sources. Computed"Line (15)" is the fit on this machine (correlations and systematic errors ignored).

From SU(2) to SU(∞) the ratio changes by about 12.5% on the line (15), and by about 19% in the Athenodorou–Teper values Computed. The SU(∞) values of the two primary sources differ by about 2.0σ Computed. All are extrapolations presupposing that the continuum limit exists, and $\sigma$ is extracted through the string picture. "N=3 is close to N=∞" is an observation within that premise.

Large-N results that have become theorems lie on the following shores Known.

• Theories of planar diagrams: 't Hooft (1983) treated a theory of planar diagrams alone in four-dimensional Euclidean space, and showed that 'if all particles involved have non-vanishing masses' and the couplings are small enough, an absolutely convergent procedure gives Green functions (expectations of products of fields; the propagator is one of them) that uniquely solve the Dyson–Schwinger equations (the same as the Schwinger–Dyson equations of §06) (abstract: 'an absolutely convergent procedure exists to obtain Green functions that uniquely solve the Dyson-Schwinger equations'). The gap is an assumption.
• The strong-coupling lattice: Chatterjee (2015) computed the expectations of large-N Wilson loops for SO(N) at strong coupling as absolutely convergent sums over trajectories of strings on the lattice, and Chatterjee–Jafarov, for the same strong-coupling SO(N), proved the $1/N$ expansion of Wilson loops as an asymptotic expansion (the first term is genus 0, higher orders surfaces of higher genus).
• All N and β: Cao–Park–Sheffield expressed the expectations of Wilson loops, for any matrix size $N\ge1$, $\beta>0$ and dimension $d\ge2$, as sums over embedded maps (graphs drawn on a surface, mapped onto the lattice). For U(N) the weight of one map is proportional to the product of $N^{\chi}$ and the weights of the lattice edges (Weingarten functions, which themselves depend on $N$).
• Two dimensions: the planar master field (§06) and the area law at all couplings (6-02).

Large N is used again for strings (7-05) and for holography, which also takes $\lambda\to\infty$ (7-08). The distance to finite N is in 7-10.

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Next chapter to read: 7-05 String theory and extra dimensions — what is needed to read the sum over surfaces as a theory of strings.

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