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

What to hold fixed when taking the limit — what gives lattice numbers a unit is a scale we choose

Pure Yang–Mills lattice theory (5-01 §action) has only one parameter: the $\beta$ multiplying the action. The lattice sits on the integer points $\mathbb Z^4$, and nowhere is a "metre" or an "MeV" written.

So every number a lattice calculation returns is a number in units of one lattice step. A mass comes out only as $a m$ (the product of the lattice spacing $a$ and the mass $m$), a string tension as $a^2\sigma$, and a correlation length as $\xi=1/(am)$ (how many lattice spacings). In this book we write $m_{\rm lat}$ for the mass in lattice units and $\xi$ for the correlation length in lattice units.

Each left-hand side is a function of $\beta$ alone, and has no dimension. The $a$ and $m$ on the right are not yet fixed separately at this stage. What is fixed on the lattice is only the product.

As a metaphor, it is like measuring things with a ruler on which the spacing of the marks is not written. One can say "4 marks", but not "so many centimetres". Precisely, given $\beta$, lattice theory determines a set of dimensionless functions such as $m_{\rm lat}$ and $\sigma_{\rm lat}$, and $a$ is defined only once a physical value is assigned to one of them.

Example. We copy the numbers of a lattice calculation from a primary source (Lucini–Teper 2001, $SU(2)$) into Table 1. At $\beta=2.60$, $a\sqrt\sigma=0.13395$ means "the square root of the string tension is 0.134 times the inverse lattice spacing", and $m_{\rm lat}=0.514$ means "the mass of the lightest scalar glueball is 0.514 times" (we write this glueball $0^{++}$: the state with spin 0, positive parity and positive charge conjugation; notation in 5-08); both are unitless numbers Physics.

ComputedTable 1 The numbers used in this chapter ($a\sqrt\sigma$, $m_{\rm lat}=a\,m(0^{++})$ and lattice sizes are from Table 1 of Lucini–Teper. The six $m_{\rm lat}$ are the same as in 9-02 Table 3 (the master copy). $\xi=1/m_{\rm lat}$, and $L\,a\sqrt\sigma$, the side of the box measured in units of $1/\sqrt\sigma$, are computed on this machine).

To attach units, one chooses a quantity with a dimension and fixes its value by convention. This is called setting the scale. If the chosen quantity is $Q$ (of mass dimension) and the agreed value $Q_*$, the lattice spacing is fixed as follows.

The numerator is a lattice number, the denominator a conventional number. Other quantities can be converted to physical units as $m=m_{\rm lat}(\beta)/a_Q(\beta)$. There are many ways to choose.

• The string tension $\sigma$: Creutz 1980, with 'hold fixed the string tension', saw numerically that holding the string tension fixed makes the bare coupling decrease logarithmically (abstract) Physics.
• The Sommer scale $r_0$: the length fixed by $F(r_0)\,r_0^2=1.65$ for the force $F$ between static charges (Sommer 1994) Physics.
• The correlation length itself: Chatterjee 2018 §5 explains sending the lattice spacing $\epsilon$ to 0 so that the correlation length approaches a finite positive value Known.
• The $\Lambda$ of asymptotic freedom: the two-loop formula gives $a\Lambda_{\rm L}$ as a function of $\beta$, and $\Lambda_{\rm L}$ is held fixed (5-05, 4-06 §two loops).

The ratio of the lattice spacings made with two choices $Q$ and $Q'$ can be written with lattice numbers alone.

The first factor on the right is the lattice value of the dimensionless ratio $Q/Q'$. If it settles to a constant as $\beta\to\infty$, the two scales differ only by a constant factor in the limit. That it settles is not a theorem but a matter supported by numerics (§03).

Example. We compare the scale fixed by the string tension with the scale fixed by the two-loop $\Lambda_{\rm L}$. For $SU(2)$, with $g^2=4/\beta$, $b_0=11N/(48\pi^2)$ and $b_1=\tfrac{34}{3}\bigl(\tfrac{N}{16\pi^2}\bigr)^2$,

Dividing the $a\sqrt\sigma$ of Table 1 by (4) gives Table 2 (higher-order corrections to (4) are not included).

ComputedTable 2 The ratio $\sqrt\sigma/\Lambda_{\rm L}$ of the two scales for $SU(2)$.

At $\beta=2.25$ and $2.30$ the lattice is coarse ($a\sqrt\sigma>0.3$) and the ratio goes up and down, so comparisons are made at $2.40$ and above. Between $\beta=2.40$ and $2.60$ the ratio moves from 57 to 48, by about 16%. In this range of $\beta$, "holding $\sigma$ fixed" and "holding the $\Lambda_{\rm L}$ of the bare coupling fixed" do not give the same scale. This distinction (scaling and asymptotic scaling) is treated in 8-04 §the continuum limit and a(β) and in 5-05 Physics.

Detour: MeV is borrowed

Dividing two lattice numbers cancels the lattice spacing.

This ratio has the same value whichever scale is chosen, since the $Q_*$ of formula (2) appears nowhere. Pure Yang–Mills theory has no dimensionless parameter left to choose freely after renormalisation (dimensional transmutation, 4-07 §one parameter disappears). So what has meaning as a prediction of the theory are only the limiting values of ratios such as (5) Physics.

At finite $\beta$ lattice corrections remain in the ratios too. For the Wilson action their leading term is thought to be proportional to $a^2$, and Lucini–Teper extrapolate in the following form (formula (5) of the paper; the coefficient $c$ is really a power series in $g^2$, but they state that they treat it as a constant) Physics.

The procedure of the paper is to fit all points and, if the fit is poor, to drop points one at a time starting from the coarsest. Tables 3 and 4 redo the fits on this machine from the numbers of Table 1.

The fit is least squares with weights $1/\delta_i^2$ from the error $\delta_i$ of each point. The goodness of fit is measured by the following quantity.

Here $y_i$ is the $m/\sqrt\sigma$ of each point. If $\chi^2$ divided by the degrees of freedom (number of points − 2) is around 1, the fit matches the size of the errors. The confidence level is the probability that a $\chi^2$ larger than this would occur by chance if the line were correct; the smaller it is, the worse the fit.

ComputedThe numbers in this figure were computed on this machine (points from Table 3, lines from the fits of Table 4).

ComputedTable 3 The ratio at each $\beta$ (errors propagated treating the errors of $m_{\rm lat}$ and $a\sqrt\sigma$ as independent). The rightmost column is the "misreading" value used in §04 (with $m_{\rm lat}$ held at 1.390).

ComputedTable 4 Weighted fits of formula (6).

How to read it. The lattice ratio moves from 3.29 to 3.84, but put on a straight line in $a^2\sigma$ the intercepts line up at 3.84–3.86. The central value of the 6-point intercept agrees with the 3.844 of Table 5 of the paper (the paper's error assignment is larger). The ratio mostly increases, approaching from below, but drops once, within errors, at $\beta=2.475\to2.55$ Physics. A more recent calculation (Athenodorou–Teper 2021, 5-08 §numbers) gives 3.781(23) Physics.

What can be said here goes only as far as "assuming $a^2$ corrections, the numbers in this range line up at one value". It is not a proof that the limit exists. How the quantities are measured (the variational method, extracting the string tension) is treated in 5-08.

Here we come to an easily misread point. One is tempted to read the "mass gap" on the lattice as "the gap in lattice units $m_{\rm lat}(\beta)$ is at least some fixed positive number for all $\beta$". That the gap on a finite lattice is positive is nearly obvious (4-11 §finite boxes and uniformity), and it seems enough to make it uniform in $\beta$.

But a mass in physical units is the mass in lattice units divided by the lattice spacing.

In the continuum limit $a\to0$, so if the gap in lattice units is uniformly positive, the physical mass goes to infinity. Correlations shrink to zero width in physical distance, and the correlation between two separated points becomes 0. This is not a limit in which "the mass is positive" but one in which "nothing is transmitted".

For the mass to remain at a finite positive value in the continuum limit, $m_{\rm lat}$ must on the contrary go to 0 as fast as $a$.

As a metaphor, when the marks on a ruler are made finer and finer, an object of fixed length should span more and more marks. If it always spans 2 marks, the object is shrinking. Precisely, (8) says that the correlation length in lattice units $\xi$ diverges while the correlation length in physical units $a\xi$ is kept constant (the form seen exactly for the Gaussian chain of 2-08 §continuum limit).

ComputedThe numbers in this figure were computed on this machine ($a\sqrt\sigma$ and $m_{\rm lat}$ from Table 1; the curves are exponentials drawn on the spot).

From $\beta=2.25\to2.60$ the lattice spacing shrinks by a factor of about 3.2 and $m_{\rm lat}$ by about 2.7. The two shrink at almost the same rate, so the ratio is almost constant. The misreading column of Table 3 keeps growing in proportion to $1/a$. Note that $\xi$ in this range does not reach 2 lattice spacings, and the numbers are not looking at the limit itself Physics.

There is a statement of the direction of the previous section, written clearly as a mathematical question: Problem 5.1 (Mass gap) in §5 of Chatterjee's survey "Yang–Mills for probabilists" (arXiv:1803.01950, 2018) Known. We translate the gist.

Consider any compact non-abelian Lie group $G\subseteq U(N)$ ($N\ge2$) and any infinite-volume limit of four-dimensional lattice gauge theory. For each $x\in\mathbb R^4$ take the nearest plaquette $p_x$, and let $f_\beta(x)$ be the correlation of $W_{p_0}$ and $W_{p_x}$. Show that for every $\beta>0$ there is some $\xi(\beta)\in(0,\infty)$ such that

Show further that $\lim_{\beta\to\infty}\xi(\beta)=\infty$.

The $\beta$ of Problem 5.1 corresponds to $\beta'=\beta/N$ in 5-01 §action, but as far as $\beta\to\infty$ is concerned there is no difference.

Here the correlation is the connected correlation $\langle W_{\gamma_1}W_{\gamma_2}\rangle-\langle W_{\gamma_1}\rangle\langle W_{\gamma_2}\rangle$, and the same section writes that the logarithm is taken because there may be a power correction in front of the exponential. It continues 'The number ξ represents the reciprocal of the mass of the lightest glueball in the theory.'

This question has two parts.

1. Finiteness at each $\beta$: $\xi(\beta)\lt\infty$, that is, correlations decrease exponentially in lattice units. This corresponds to the gap being positive in lattice units (4-11 §correlation length).
2. Divergence as $\beta\to\infty$: $\xi(\beta)\to\infty$. This is the direction of (8) in §04. The gap in lattice units must go to 0.

The first part alone does not contradict "uniformly positive in lattice units". The second part does. Problem 5.1 asks for the first part at each $\beta$ separately, and the second part explicitly denies uniformity.

What this question does not ask

We also note what this question does not ask. It does not ask for the rate of divergence. What asymptotic freedom predicts is a form like $\xi\propto e^{\beta/(4Nb_0)}$ ($\beta$ in the Wilson normalisation of 5-01; the reciprocal of the one-loop formula $a\propto e^{-\beta/(4Nb_0)}$ of 5-05, which in Chatterjee's $\beta'$ reads $e^{N\beta'/(4Nb_0)}=e^{\beta'/(4b_0)}$). The ratio $m/\sqrt\sigma$ does not appear in the wording either. That the limiting theory exists, and that it satisfies the Osterwalder–Schrader axioms (4-10), are separate questions. Even if Problem 5.1 were shown, the mass gap of the Clay problem statement (0-03 §mass gap) would not follow directly.

Example. The $\xi$ of Table 1 grows from 0.72 to 1.95 with $\beta$. These numbers agree with the direction of the second part, but $\xi$ growing over finitely many $\beta$ is not a proof of divergence in the limit Physics.

Problem 5.2 (Continuum limit) in the same §5 gives another way to fix the scale Known. For the rectangular Wilson loop $W_{R,T}$ (5-01 §plaquettes and Wilson loops), it asks to show that there are sequences with $\epsilon(\beta)\to0$ and $c(\beta)\to\infty$ as $\beta\to\infty$, and a nonzero constant $d$, such that for every $R,T$

$o(1)$ is a term that goes to 0 as $\beta\to\infty$. $RT$ is the area of the rescaled loop, and the same section writes that $d$ corresponds to the string tension of the continuum theory. $R+T$ is a quantity proportional to the perimeter of the rescaled loop (half the perimeter; the factor 2 is absorbed into $c$). The $d$ here is the coefficient of the area, not the spacetime dimension.

In (10), all that is asked is that there exist an $\epsilon(\beta)$ "in whose scale the coefficient of the area of Wilson loops is constant". That is, it is a formulation that fixes the scale with the string tension. The explanatory paragraph of §5 fixes the scale with the correlation length instead. Problem 5.2 does not presuppose Problem 5.1. For neither $\epsilon(\beta)$ has existence been shown.

One more caution. About the perimeter term $c(R+T)$, Chatterjee writes in the same section 'it is not clear to me whether the renormalization term c(R + T) is indeed necessary. It seems entirely possible that the limit in (5.2) holds without the renormalization term.' Subtracting the perimeter term is one proposal for how to pose this question.

Putting the two questions together with the same scale

The ratio $m/\sqrt\sigma$ appears when the two questions are put together with the same $\epsilon(\beta)$. Suppose Problem 5.2 holds and moreover $\epsilon(\beta)\,\xi(\beta)\to\ell\in(0,\infty)$. The physical correlation length is $\epsilon\xi\to\ell$, the physical mass $1/\ell$ and the string tension $d$, so

and $\epsilon$ cancels and disappears. This is (5) of §03 written in the limit. This construction is not in Chatterjee's text; it reads the two questions together. Note that the two limits existing simultaneously with the same $\epsilon$ is itself an assumption.

Example. The lattice values of the right-hand side of (11) are the $m/\sqrt\sigma$ row of Table 3. At $\beta=2.60$ it is $(1/1.95)/0.134\approx3.84$.

The phrase "a uniform gap" has no definite meaning until one says uniform in what. At least three kinds of "uniform" appear in discussions of the mass gap.

The Clay problem statement §6.5 writes 'New ideas are needed to prove the existence of a mass gap that is uniform in the volume of space-time.' Known. The next sentence, 'Such a result presumably would enable the study of the limit as T4 → R4', has in mind the limit from compact spacetime $T^4$ to $\mathbb R^4$. The uniformity here is that of the first row, uniformity in the volume. It is a requirement separate from the third row, and both are needed.

Theorems of the form of the second row exist. Shen–Zhu–Zhu show, for $SU(N)$ with $\beta$ small enough (in four dimensions $\beta\lt1/48$ in the normalisation of the paper, where the inverse coupling is $N\beta$; in the Wilson $\beta$ of this book $\beta\lt N^2/48$: 5-01 Table 4), uniqueness of the infinite-volume measure and exponential decay of correlations (abstract) Known. Earlier, Osterwalder–Seiler 1978 proved the area law rigorously at small $\beta$ (according to the text of Chatterjee 2018 §4) Known. The abstract of that paper says it also showed that at strong coupling (small $\beta$) the infinite-volume limit exists and is analytic in $\beta$ (text not read). These are results with $\beta$ held in a small range, theorems about a world with the lattice spacing held fixed. They sit on the side opposite the continuum limit ($\beta\to\infty$) (5-02 §tubes, 5-04).

The constants of these estimates do not extend to $\beta\to\infty$, and do not reach the "behaviour as the scale is made finer" asked about in the continuum limit.

For gauge theories with finite groups, exponential decay of correlations of gauge-invariant quantities at large $\beta$ has been proved (Adhikari–Cao, abstract) Known. This is an example in which the form of the first part of Problem 5.1 holds at large $\beta$, but since the group is not continuous it is a different setting from the question asking for $\xi\to\infty$ in the second part (5-10).

Fixing one scale and taking $\beta\to\infty$ looks as if it would give some limit. But there is no guarantee that the limit obtained is the theory sought. We list three types that fail.

Type 1: the correlation length does not grow

If $\xi(\beta)$ is bounded, then whatever $\epsilon(\beta)\to0$ is chosen, $\epsilon\xi\to0$. The physical correlation length is 0 and the mass infinite: exactly the misread world of §04. What remains in the limit is a field in which no two points are correlated (like white noise).

Type 2: the correlation length grows, but a ratio goes to 0 or infinity

If two quantities diverge at different rates, then once the scale is fixed with one, the other escapes to 0 or infinity. For the three-dimensional compact $U(1)$ lattice theory, Göpfert–Mack proved the area law at every $\beta$ (according to the text of Chatterjee 2018 §4) Known. Even so, the abstract of that paper says, the continuum limit is a free scalar field, and the string tension in units of the mass goes to infinity (Villain action; text not read) Known. It is an example that confines at every lattice $\beta$, yet the ratio $m/\sqrt\sigma$ goes to 0.

Type 3: the ratio is finite, but it collapses to a Gaussian

Even if a limit can be taken with the correlation length fixed, if it is a free field (a Gaussian measure) it is not a non-trivial theory. For four-dimensional Ising-type models and $\varphi^4$ (a scalar field with a quartic self-interaction added; 6-03), the scaling limits near the critical point have been shown to be Gaussian (Aizenman–Duminil-Copin 2021) Known. This is why the problem statement asks for a "non-trivial" theory (0-03 §non-trivial, 6-08).

Putting the three together, the conditions on the limit sought are the following set.

The second uses confinement, while the problem statement itself does not require confinement (4-11 §definition). Even if another scale such as $r_0$ is used instead of the string tension, the second of (12) takes the form "the ratio of two dimensionful quantities is finite and positive". On the numerical side, Tables 3 and 4 support the second for $SU(2)$ Physics.

The theory of the problem statement needs two limits: volume $L\to\infty$, and lattice spacing $a\to0$ (in four dimensions, tied to $\beta\to\infty$) (5-01 §the map of Part 5). Problem 5.1 is written in the order of taking the infinite-volume limit first at each $\beta$, and considering $\beta\to\infty$ afterwards.

Even for the first limit, Chatterjee 2018 §3 writes 'the uniqueness (or non-uniqueness) is in general unknown for lattice gauge theories in dimensions higher than two when β is large.' Known. This is why Problem 5.1 says "any infinite-volume limit".

Numerical calculations take another road. The rightmost column of Table 1, $L\,a\sqrt\sigma$, is the side of the box, the lattice side $L$ (in lattice points) measured in physical units.

Each time the lattice is made finer, the number of lattice points is increased, keeping the physical box at about $3.2/\sqrt\sigma$. That is, numerical extrapolation takes the road "hold the physical volume fixed and take $a\to0$". The paper uses only lattices with $aL\sqrt\sigma\ge3$, as the lengths at which the formula that obtains the string tension by subtracting the string correction $\pi/(3L)$ from the mass of a closed flux tube (formula (4) of the paper) can be used. The paper writes that finite-volume effects on glueballs also become invisible at this size Physics.

That this road and the road of (13) reach the same value, that is, exchanging the two limits, is something checked numerically, not a theorem. The exchange needs an estimate uniform in the size of the box, which is the requirement of the first row of the table in §07.

Part 10 of this book uses the direction set here (the gap in lattice units should go to 0) as its starting point. The correspondence with the definitions declared there is in 10-01 The line of construction and Assumption H.

What this chapter can say

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Next chapter to read: 5-10 Lattice gauge theory with finite groups (an example where correlations decrease at large $\beta$, yet there is no progress in the direction of the continuum limit).

← 5-08 Measuring glueballs and the string tensionContents5-10 Lattice gauge theory with finite groups →