2026-10-01 · chapter Yang–Mills and the mass gapPart 5 Lattice gauge theorycontinuum limit

Weak coupling and the continuum limit — raising β makes the lattice finer, and asymptotic freedom sets how fast

A lattice theory has no unit of length; what comes out are numbers of the form "so many lattice spacings". For example, the correlation length $\xi_{\mathrm{lat}}(\beta)$ when the correlation of two plaquettes decreases as $e^{-|x|/\xi_{\mathrm{lat}}}$ (4-11). A mass in physical units is obtained by dividing by the lattice spacing.

To hold $\Delta$ fixed and take $a\to0$ on the lattice with $\beta=2N/g_0^2$ (5-01), we need $\xi_{\mathrm{lat}}\to\infty$, that is, to approach a critical point. Chatterjee (2018) §5 summarises the physical usage as “the model has a continuum limit if there is a critical point βc ∈ [0,∞] such that as β → βc, the correlation length tends to infinity”, and for four-dimensional non-abelian models writes “It is believed … βc = ∞” Known. $\beta=\infty$ is $g_0\to0$, the same direction as asymptotic freedom (4-06).

There are two limits: first hold $\beta$ fixed and make the box infinite (5-04), and afterwards take $\beta\to\infty$.

How many fm $a$ is, is decided afterwards by matching the lattice value $\widehat Q=a^kQ$ of a quantity $Q$ of mass dimension $k$ against its physical value.

As standards one uses the string tension $\sigma$ (Creutz 1980 fixed $a$ by “hold fixed the string tension”), the length $r_0$ defined from the force, and $t_0$ of the gradient flow (Lüscher 2010) Known. $r_0$ is the length fixed by $r_0^2F(r_0)=1.65$ from the force $F(r)$ acting between a pair of static charges (Sommer 1994; 4-06 §07). $t_0$ is a length fixed by smoothing the field with a diffusion equation (the gradient flow) (8-04 §08 calls the same thing the Wilson flow).

Example: for pure SU(3), $r_0/a$ is 2.922 at $\beta=5.7$ and 9.74 at 6.4 (Table 1 of Necco–Sommer 2002; the values for $\beta\le6.4$ are reproduced from earlier calculations). Setting $r_0\approx0.5$ fm (a reference value used in QCD; the introduction of Necco–Sommer) gives $a\approx0.171$ fm and $0.051$ fm (1 fm = $10^{-15}$ m) Computed. The physical value of $r_0$ is an input from outside the lattice.

Changing the standard changes $a(\beta)$ slightly at finite $\beta$, and the ratio of the two $a$ is fixed by the ratio $\widehat Q_1/\widehat Q_2$ measured on the lattice. That this ratio settles down is the scaling of §04. In the mathematical formulation (§08) too the choice of calibration is an issue, and the existence of $\varepsilon(\beta)$ in either convention is unproved (5-09).

To keep the same physics as the lattice is made finer, the bare coupling $g_0$ is weakened. How fast begins with the scheme-independent $b_0,b_1$ (4-06 §05) Known. Reading $\mu=1/a$ and rewriting in terms of $\beta=2N/g_0^2$ gives

and the lower formula is formula (8) of 4-06, obtained by integrating the upper one. The constant of integration $\Lambda_{\mathrm L}$ (4-07) does not come out, but the ratio $a(\beta)/a(\beta')$ does. At one loop $a\propto e^{-\beta/(4Nb_0)}$, and the step that halves $a$ is $\Delta\beta=4Nb_0\ln2$: 0.2575 for SU(2) from $4Nb_0=11/(3\pi^2)$, and 0.5794 for SU(3) Computed.

ComputedThe numbers in this figure were computed on this machine (formula (3) computed on the spot; agrees with Table 1 to 4 digits. If JavaScript is disabled, Table 1).

That lattice numbers follow (3) is called asymptotic scaling. Creutz (1980) saw for SU(2) “a logarithmic decrease of the bare coupling constant in a manner consistent with the perturbative renormalization-group prediction” Known. The step in pure SU(3) data is 0.354 at $\beta=5.70$ and 0.524 at 6.20, short of the two-loop 0.61 (Table 4 of 4-06) Physics. The abstract of Lepage–Mackenzie (1993) says that expanding in a coupling defined by a physical quantity makes perturbation theory and numerics agree well. The abstract also says that if that coupling is used instead of the bare coupling, scaling already follows perturbation theory from about $\beta=5.7$. Much of the deviation seen with the bare coupling comes from the choice of coupling.

Detour: the price of the continuum limit

What the continuum limit needs is that ratios of physical quantities stop depending on $\beta$.

being constant within errors is called scaling, and that range of $\beta$ the scaling window. Example: Necco–Sommer (2002) write, for the ratio $r_c/r_0$ of two lengths defined by the force, “No dependence on the resolution is seen within the errors of below 1%”, and obtain 0.5133(24) by extrapolation Known. $m/\sqrt\sigma$ is looked at in 5-08.

Scaling is ratios standing still, and is necessary for the continuum limit. Asymptotic scaling is $a(\beta)$ itself following (3), an additional claim tying the lattice to the Λ of perturbation theory Physics. If two quantities deviate from (3) by the same amount, the deviation cancels in the ratio. Lattice results such as $m/\sqrt\sigma$ rely on scaling (8-04 §05).

We see why the correction in (4) starts at $a^2$ in a free field, which can be solved exactly. In the symmetric difference $f(x+a)-2f(x)+f(x-a)$, odd powers of $a$ cancel. For a free scalar field of mass $m$, the transfer matrix (5-03) fixes the mass $E$ seen on the lattice Known. Concretely, looking for the pole of the lattice propagator $1/(\hat p^2+m^2)$ ($\hat p^2$ is formula (6) below) at zero spatial momentum and imaginary momentum $p=iE$ gives $\tfrac4{a^2}\sinh^2\tfrac{aE}2=m^2$, and rewriting this gives the upper formula below.

The second column approaches $-1/24=-0.04167$. The same holds in the spatial directions: applying the lattice Laplacian to a wave of wave number $p$ gives $\hat p^2$ instead of $p^2$. Building a difference that uses points two steps away, with weights $\tfrac43,\,-\tfrac13$, removes the $a^2$ term. This is the simplest example of improvement.

ComputedThe numbers in this figure were computed on this machine ((6) computed on the spot). If JavaScript is disabled: $\hat p^2/p^2-1$ at $n=4,8,16,32$ is −0.189, −0.0504, −0.0128, −0.00321 for the standard difference and −0.0543, −0.00400, −0.000261, −0.0000165 for the improved one. Doubling $n$ makes the standard error about 1/4 and the improved one about 1/16.

Symanzik (1983) read a lattice theory at small $a$ as "the continuum theory plus terms with powers of $a$", and says in the abstract “Corrections to continuum theory results stemming from finite lattice spacing can be diminished systematically by use of lattice actions that also include suitable irrelevant terms” Known. For pure gauge theory it is thought to take the following form Physics.

The first term has the same normalisation as 5-01 ($g_0$ put into the covariant derivative). $\mathcal O_i$ are gauge-invariant terms of mass dimension 6 ($\mathrm{tr}F^2$ has dimension 4, so $a^2$ multiplies them; 4-04). According to the abstract of Lüscher–Weisz (1985), if only spectral quantities (masses of stable particles, the static potential and so on) are to be improved, one of the three coefficients can be chosen freely Known. Why there is no $a^1$: a term of dimension 5 has two field strengths and one derivative, so 5 indices. The lattice is invariant under reversing each axis, so the indices of each direction must appear an even number of times, and an odd number of indices cannot be placed Physics. This is an argument about the action and spectral quantities (masses). For quantities that involve operators, such as the force between static charges, it has to be shown separately that the correction starts at $a^2$, and Necco–Sommer (2002) §4 gives that argument.

Adding rectangles

We add $1\times2$ rectangles to the plaquette with weights $c_0,c_1$ ($c_0+8c_1=1$). A rectangle has two orientations in each plane, and has twice the area so acts 4 times as strongly on the leading term; hence 8. The lowest-order (classical) coefficients are attributed to Lüscher–Weisz (1985), and for the action that adds only rectangles they are $c_1=-\tfrac1{12}$ ($c_0=\tfrac53$) Known. Table 3 reproduces this number. We check it with a smooth U(1) field on a two-dimensional torus (a sum of six wave numbers; the flux computed exactly with an antiderivative).

The ratio of errors on halving $a$ is 3.95→3.99 (→4) for Wilson and 15.2→15.8 (→16) for the improved action. The Wilson coefficient approaches the expected value $-\tfrac1{12}\int|\nabla F|^2/\int F^2=-5.522$. At $L=16$, taking $c_1=0,\,-\tfrac1{24},\,-\tfrac1{12},\,-\tfrac18$ makes the error divided by $a^2$ move linearly as $-5.43,\,-2.85,\,-0.27,\,+2.31$ Computed. For a four-dimensional non-abelian field, see Table 3 of 5-01.

Coefficients and logarithmic corrections in the quantum theory

In the quantum theory $c_i$ depend on $g_0$ (one loop: Lüscher–Weisz 1985). Because of asymptotic freedom, the leading error comes with a logarithmic correction of the form $a^2[\bar g^2(a^{-1})]^{\hat\gamma_1}$, and Husung–Marquard–Sommer (2020) conclude for Yang–Mills that “the log-corrections are a rather weak modification” Physics. Here $\bar g^2(a^{-1})$ is the running coupling at the scale $1/a$ (4-06), which approaches 0 like $1/\ln(1/(a\Lambda))$. $\hat\gamma_1$ is an exponent fixed for each correction term (a number that plays the role of an anomalous dimension). The error therefore takes the form $a^2$ times a power of $\ln(1/a)$.

Detour: the price of improvement

A computer cannot reach $a=0$, so one measures a ratio $R$ at several $\beta$, draws a straight line against $a^2$ on the horizontal axis, and takes the intercept.

The third column of Table 2 is the error of this two-point extrapolation, and the error drops to $a^4$. Even if the slope $c$ differs from action to action, the intercept $R_0$ should be common (universality; §08). For the Wilson case of Table 3 too, building it from $L=16,32$ takes the relative error from 0.54 % to 0.009 % Computed. In actual numerics, Necco–Sommer (2002) fit the force in the form $(1+s(a/r_c)^2)$ and report that the slope $s$ is not significant Known.

The pitfalls are points that are too coarse (at $L=4$ in Table 3, $/a^2$ is off by 20 %), the logarithmic corrections of §06, and boxes that are too small. All are procedures resting on the premise that $R(a)$ has the form (8), and the premise is not a theorem Physics.

Chatterjee (2018) §3 writes “the general belief in the physics community is that β should scale like log(1/ε) in dimension four, although there are doubts about this belief and the question remains an open mathematical problem” Known. Integrating the upper formula of (3) gives $\beta\approx4Nb_0\log(1/a)$, and this is that $\log(1/\varepsilon)$. Problem 5.2 in §5 of the same paper asks whether, in the infinite-volume theory as $\beta\to\infty$, there exist sequences $\varepsilon(\beta)\to0$, $c(\beta)\to\infty$ and a nonzero $d$ such that, for a rectangular loop of height $R$ and width $T$,

holds. $d$ is the continuum string tension, and about the perimeter term it says “it is not clear to me whether the renormalization term c(R+T) is indeed necessary” Known.

In two dimensions it has been shown that the Wilson, Villain and Manton lattice actions go to the same Yang–Mills measure (Chevyrev–Shen, the trivial bundle on the torus; 6-02) Known. Four-dimensional U(1) behaves differently (there is a massless photon phase on the weak-coupling side, and the critical point is at finite $\beta$; 5-06), and the renormalisation steps of the non-abelian case are treated in 6-07. In the four-dimensional non-abelian case, neither the convergence of lattice correlation functions, nor their independence of the action, nor the limit of the ratios (4) is a theorem. The tables and figures of this chapter are procedures that assume the limit, not evidence of its existence.

What this chapter can say

Sources and reproduction

Next chapter to read: 5-06 How U(1) differs from the non-abelian case.

← 5-04 The infinite-volume limit and the cluster expansionContents5-06 How U(1) differs from the non-abelian case →