2026-10-01 · chapter Yang–Mills and the mass gapPart 5 Lattice gauge theoryU(1) and non-abelian

How U(1) differs from the non-abelian case — on the same lattice, an abelian group changes phase on the way

Wilson's lattice (5-01) has the same form for every compact group. Group elements are placed on the edges, and the product around the four edges of a plaquette is weighted. For $U(1)$ (2-03 §08) each edge carries an angle $\theta_\ell$, and the product becomes a sum of angles $\theta_P$. For $SU(2)$ each edge carries a $2\times2$ matrix $U_\ell$, and the product is the matrix product $U_P$.

The convention for $\beta$ is chosen so that the action returns to the Maxwell or Yang–Mills action in the classical continuum limit Known. The $\beta=1/g^2$ of $U(1)$ is not the $2/g^2$ obtained by putting $N=1$ into the $\beta=2N/g^2$ of $SU(N)$. This is because the generators are normalised differently (for $U(1)$ the edge carries $e^{-iagA_\mu}$ (the sign is the same as in (2) of 5-01), for $SU(N)$ $\mathrm{tr}\,T^aT^b=\delta^{ab}/2$). For $U(1)$, $\theta_P\approx -a^2gF_{\mu\nu}$ and $1-\cos\theta_P\approx\theta_P^2/2$ make the action approximately $\frac{\beta g^2}4\sum_x a^4\sum_{\mu,\nu}F_{\mu\nu}^2$, and matching it to Maxwell's $\frac14\int F^2$ gives $\beta=1/g^2$. The difference shows in the form of the formulas. $\theta_P$ is a linear expression in the edge angles (the lattice curl), and the only non-linearity of the $U(1)$ action is the periodicity of $\cos$, that is, the compactness of the group. For $SU(2)$, writing $U_\ell=e^{i\vartheta_\ell\cdot\sigma}$, the exponent of $U_P$ contains terms with the commutator $[\vartheta,\vartheta]$ (1-03). This is the self-interaction on the lattice (3-02). Example: if the angles on the four edges are $0.1,0.2,0.3,0.4$, then $\theta_P=-0.4$, and the order of addition does not matter. For $SU(2)$ with $A=e^{0.3i\sigma_1}$ and $B=e^{0.3i\sigma_3}$, $\tfrac12\mathrm{tr}(ABA^{-1}B^{-1})=0.9847$, while the reordered $\tfrac12\mathrm{tr}(ABB^{-1}A^{-1})=1$ Computed. For $U(1)$ the sum of angles is 0 in both cases. Cyclic permutations (such as $BA^{-1}B^{-1}A$), however, do not change the value, because $\mathrm{tr}\,XY=\mathrm{tr}\,YX$.

On the side where $\beta$ is small the two groups behave alike. The variable $u$ of the character expansion of 5-02 is

($I_n$ are modified Bessel functions), and since $u\lt1$ both have the area law. Example: for $U(1)$ at $\beta=0.5$, $u=0.2425$ and $\sigma\approx1.417$; at $\beta=1$, $u=0.4464$ and $\sigma\approx0.807$ Computed. The area law and exponential decay at small $\beta$ are theorems for every compact group (Osterwalder–Seiler 1978) Known. They are a property of the lattice being coarse and do not use non-commutativity (5-02 §08).

The difference shows only in numerical coefficients. As $\beta\to0$, $u_{U(1)}\approx\beta/2$ and $u_{SU(2)}\approx\beta/4$, so $\sigma\approx-\log u$ grows in both cases in the form $\log(1/\beta)$, and the difference is the constant $\log2$. At $\beta=1$, $u_{SU(2)}=0.2402$ and $\sigma\approx1.426$, larger than for $U(1)$ at the same $\beta$ ($\sigma\approx0.807$) Computed. The convention (1) differs from group to group, so comparing sizes at the same $\beta$ means little. What matters is that both have an area law of the same form.

When $\beta$ is large the weight demands $\theta_P\approx0$, and with $1-\cos\theta_P\approx\theta_P^2/2$ the action is quadratic. Since $\theta_P$ is linear, this is a lattice Maxwell field without interaction (photons).

The right-hand formula of (3) is the form for one fixed gauge (Feynman gauge, 4-08) Known. Gauge-invariant quantities (Wilson loops) do not depend on this choice.

That $1/\hat k^2$ diverges as $\hat k\to0$ is what "massless" means, and correlations of the field strength decrease as the power $|x|^{-4}$ Known.

Per point there are 3 physical degrees of freedom (4 edges minus 1 for the gauge), and 6 plaquettes. From the Gaussian integral $\int x^2e^{-x^2/2}dx\big/\int e^{-x^2/2}dx=1$, the mean action of one mode with quadratic action $\frac\lambda2x^2$ is $1/2$ (equipartition). Hence $6\cdot\tfrac\beta2\langle\theta_P^2\rangle=\tfrac32$, that is,

($SU(2)$ has 3 generators). Example: for $U(1)$ at $\beta=2$, $0.875$, and Monte Carlo gives $0.865$ (Table 2) Computed.

The force between static charges

An $R\times T$ Wilson loop is the amplitude of a pair of static charges held for time $T$. For a Gaussian field it can be written in closed form, and as $T\to\infty$

remains Known. $G_3$ is the Green function of the three-dimensional lattice, $1/(4\pi R)$ at large distance. So $V$ has the Coulomb form and stays at the finite value $G_3(0)/\beta$. It does not confine. Computing with $G_3(r,0,0)=\int_0^\infty e^{-6t}I_r(2t)I_0(2t)^2dt$ gives $G_3(0)=0.252731$ Computed. The $1/6$ at $R=1$ is a check from $6G_3(0)-6G_3(1)=1$.

ComputedThe numbers in this figure were computed on this machine (the solid line is the calculation of Table 1; the dashed line computes the series for $I_1/I_0$ on the spot). Near $\beta\approx1$ neither approximation can be used (§04).

For finite squares $R\times R$ too, we obtained $-\beta\log\langle W\rangle$ from a Fourier sum on a periodic $64^4$ lattice. At $R=1,4,8,16$ it is $0.250,\ 1.631,\ 3.584,\ 7.557$: divided by the area $R^2$ it goes towards 0 as $0.250\to0.030$, and divided by the perimeter $4R$ it settles to almost a constant, $0.063,\ 0.102,\ 0.112,\ 0.118$ (the perimeter law). The value per perimeter approaches $G_3(0)/2\approx0.126$ as $R\to\infty$ (the same as the limit of long rectangles). It approaches with corrections of $1/R$ and logarithms from the corners Computed.

That the same picture remains when the approximations are dropped is a theorem for $U(1)$. The theorem, however, is about actions of Villain type (Guth's abstract). The Villain type is the action whose weight is the periodic Gaussian $\sum_{n}e^{-\frac\beta2(\theta+2\pi n)^2}$ in place of $e^{\beta\cos\theta}$. The abstract of Fröhlich–Spencer does not state the type of action (the text was not obtained). (2), (4), Table 2, Figure 2 and the $\beta_T$ below are numbers for the Wilson action (1).

The logic of the theorem is short: if the string tension is positive at small $\beta$ and 0 at large $\beta$, then $\sigma(\beta)$ cannot be an analytic function. An analytic function that is 0 on an interval is 0 everywhere. A point where a thermodynamic quantity is not analytic is a phase transition (2-08 §04).

The second half was shown by Guth (1980): for an action of Villain type with $g^2\lt0.168$, a lower bound on Wilson loops “has the form of a Coulomb interaction and guarantees that electric charges are not confined”, and together with the strong-coupling result of Osterwalder–Seiler he concludes “this model has at least one phase transition” (abstract) Known.

The abstract of Fröhlich–Spencer (1982) is “We give a new, elementary proof for the existence of a deconfining transition to a massless (QED) phase in the four-dimensional U(1) lattice gauge theory” Known. Chatterjee (2018) §4 introduces it as “A fully rigorous proof of Guth's theorem”.

Convergence to the continuum limit — Driver

The abstract of Driver (1987) deals with the continuum limit. “compact U(1) lattice gauge theory with general energy function converges to a renormalized free electromagnetic field on the current sector as the lattice spacing approaches zero” comes with the condition “provided the coupling constant is sufficiently large”. For the Wilson action it says “it is possible, by judicious choice of the Gibbs state, to get convergence for arbitrary coupling strengths” Known. The "current sector" is the range of observables that smear the field with smooth currents. Whether the “coupling constant” of the abstract is on the side of $\beta$ or of $g$ has not been checked, because we have not been able to read the text.

The jump seen in numbers

For the Wilson action the transition point is reported numerically as $\beta_T=1.011128(11)$, with evidence of a first-order transition (Arnold et al. 2001, conference proceedings; the abstract of the follow-up proceedings hep-lat/0210010 also writes “strong evidence for a first-order phase transition”) Physics. A first-order transition is one in which the mean plaquette jumps at $\beta_T$, and there the two phases can coexist. Because they can coexist, a run started in one phase stays in that phase for a while, and runs that raise $\beta$ and runs that lower it disagree. This is hysteresis.

On this machine we compared, with the Metropolis method on a $6^4$ lattice, a run lowering $\beta$ from above (cold start) with a run raising it from below (hot start) (at each $\beta$, 100 sweeps discarded and 100 sweeps measured).

ComputedThe numbers in this figure were computed on this machine (the points are Monte Carlo, of which Table 2 is an excerpt; the curves compute (2) and (4) on the spot).

For $U(1)$ the two runs disagree for $\beta=0.98$–$1.02$, and at $\beta=1$ the difference is $0.074$. It can be read as each run staying in its original phase (the hysteresis above). The $U(1)$ transition was also seen in the numerics of Creutz–Jacobs–Rebbi (1979) Physics. This table comes from short runs on a small lattice and has no power to decide the order (5-07).

For $SU(2)$ in Table 2, the difference between the two runs is at most about $0.007$, about a tenth of that for $U(1)$. The difference $0.0069$ at $\beta=1.8$ is about 3 times the error in the table, but it is a difference between runs thermalised for only 100 sweeps at each $\beta$, and as $\beta$ varies the cold and hot runs do not split into two systematic branches (how to tell insufficient thermalisation from a jump is in 5-07). The plaquette follows the leading strong-coupling term for $\beta\lesssim1.8$, approaches the leading weak-coupling term for $\beta\gtrsim3$, and joins the two smoothly around $\beta\approx2.2$ Computed.

The abstract of Creutz (1980) fixes the string tension, makes the lattice finer, sees the bare coupling decrease logarithmically as the renormalisation group predicts, and writes “This supports the coexistence of confinement and asymptotic freedom” Physics.

The peak of the specific heat around the crossover is said to be the “shadow” of a nearby critical end point in a family of actions with an added term in the adjoint representation (Bhanot–Creutz 1981) Physics. Whether there is a phase transition can depend on the choice of action, so we consider it separately from the continuum limit. That for four-dimensional $SU(2)$ there is no point at finite $\beta$ where the correlation length diverges is a believed statement, not a theorem. Chatterjee (2018) §5 writes “It is believed … βc = ∞”, that is, $\beta_c=\infty$ (5-05 §01).

What separates $U(1)$ and $SU(2)$ is whether the effective coupling changes with distance (4-06). At one loop

$C_A$ is the Casimir number of the adjoint representation, $N$ for $SU(N)$ and 0 for $U(1)$ Known. The right-hand formula is the value of the coupling $g_0$ at spacing $a$ seen at length $L$.

With no matter fields, $b_0$ is 0 for $U(1)$, 0.04644 for $SU(2)$ and 0.06966 for $SU(3)$ Computed.

For $U(1)$, $b_0=0$, so if $\beta$ is large the Gaussian approximation is self-consistent at all distances, and the Coulomb force of §03 continues to large distances. For $SU(2)$, however large $\beta$ is, taking $L$ large enough brings the right-hand side of (8) towards 0, and the coupling becomes strong. The rough guide is $L\sim a\,e^{\beta/(4Nb_0)}$, the same form as the $a(\beta)$ of 5-05 §03. That there is no point at finite $\beta$ where the correlation length becomes infinite is what "$\beta_c=\infty$" means Physics. Example: for $SU(2)$ at one loop, raising $\beta$ by 1 multiplies the correlation length by about $e^{1/0.3715}\approx15$ ($4Nb_0=11/(3\pi^2)$) Computed. This is a one-loop guide only; corrections from two loops and beyond, and the breaking of asymptotic scaling in the range of $\beta$ actually used, are treated in 5-05.

Using known examples with groups and dimensions exchanged, we list which phrasings fail (see also 9-05 §04).

The "screened" of Table 3 is the “screened” of Elitzur–Pearson–Shigemitsu, the phase in which Wilson loops follow the perimeter law and excitations are massive. It is a different meaning from the Debye screening of §05 (which produces confinement). The same paper derives this sequence of phases from the existence of the phases of the O(2) model, using self-duality and correlation inequalities (abstract). Numerically, the abstract of Creutz–Jacobs–Rebbi (1979) reports two transitions for $N\ge5$ Physics.

What matters is not only "non-abelian" but that the group is continuous. Continuity matters, however, for whether massless waves remain at the end of sufficiently large $\beta$. For a finite group the edge values are discrete, and at sufficiently large $\beta$ the edges freeze to the identity, so no massless waves like photons can be made Physics. On the other hand $\mathbb Z_N$ (with $N$ large) has a massless phase at intermediate $\beta$ (row 4 of Table 3). Exponential decay at large $\beta$ for finite groups is a different thing from the gap of the continuum theory (5-10). $U(1)$ is continuous, so it has massless waves, and since $C_A=0$ they survive.

(ii) is the form of Problem 5.1 in 5-09. Neither (i) nor (ii) has been proved. It is more accurate to say that the role of non-commutativity is not "to make a gap" but "to break the massless weak-coupling picture at large distances".

On the side of theorems are the strong-coupling results of §02, Guth and Fröhlich–Spencer (actions of Villain type) and Driver (with the conditions of the abstract) in §04, Göpfert–Mack in §05, rows 1 and 3 of Table 3, and Guth and Fröhlich–Spencer in row 4 Known. (7) and Polyakov in §05, and row 2 of Table 3, are at the level of physics. On the side of belief: that for four-dimensional $SU(N)$ there is no point at finite $\beta$ where the correlation length diverges ($\beta_c=\infty$, Chatterjee §5), the area law for all $\beta$, and that the gap remains in physical units in the continuum limit Physics. Depending on the action, there can be bulk (lattice-scale) phase transitions not accompanied by a divergence of the correlation length. Lattice calculations of $SU(N)$ with large $N$ report such a transition on the strong-coupling side, limiting the range that can be computed (abstract of Lucini–Teper–Wenger 2005) Physics. This does not contradict $\beta_c=\infty$. Problem 4.1 of Chatterjee (2018) asks for the area law at all $\beta\gt0$ for four-dimensional non-abelian compact groups, and writes that at large $\beta$ it “remains elusive” (5-02 §08).

What the $U(1)$ theorem teaches is that an argument extending strong-coupling results to weak coupling without distinguishing groups is wrong. What must be shown for $SU(N)$ includes a negative statement, "the divergence of the correlation length and the breaking of the area law do not occur at finite $\beta$", and the tools to say that over the whole range of $\beta$ exist at present only on the strong-coupling side (6-10, 9-04).

What this chapter can say

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Next chapter to read: 5-07 The Monte Carlo method (the method that made Table 2, and how to read its errors).

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