2026-10-01 · chapter Yang–Mills and the mass gapPart 5 Lattice gauge theoryglueballs and the string tension

Measuring glueballs and the string tension — read masses from how correlations decrease, and form the ratio $m/\sqrt\sigma$

Advancing time by one step on the lattice is applying the transfer matrix $T=e^{-aH}$ once (5-03). The eigenvalues of $H$ are read from how the correlation of an operator $O(t)$, built from quantities on one time slice $t$, decreases. $O$ is summed over space to give zero momentum, and its vacuum expectation value is subtracted (lattice units $a=1$).

$|\Omega\rangle$ is the vacuum, and $E_n$ are the eigenvalues measured from the vacuum. The weights $|\langle n|O|\Omega\rangle|^2$ are squared absolute values, so they are 0 or more. The eigenvalues $e^{-E_n}$ are positive reals because link-reflection positivity ($\beta\ge0$) gives $T\ge0$, and for $\beta\gt0$, $T$ does not have 0 as an eigenvalue (5-03 §chain, §transfer matrix). A sum of positive exponentials with weights of 0 or more is log-convex (by Cauchy–Schwarz, $C(t)^2\le C(t-1)C(t+1)$), so the effective mass $m_{\rm eff}(t)$ decreases monotonically in $t$ and approaches from above the energy of the lightest state that $O$ touches Known. A value read at finite $t$ is an upper bound, and no lower bound comes out (for an example with a finite window, 4-11 §window). On a lattice periodic in the time direction (length $N_t$), each term gets a backward $e^{-E_n(N_t-t)}$. For $N_t=32$, $t\le7$ of §07 it is negligible.

Example. For $C(t)=0.8\,e^{-0.77t}+0.2\,e^{-1.8t}$, $m_{\rm eff}$ goes down as 0.908, 0.824, 0.790, 0.777, 0.773, 0.771 for $t=0,\dots,5$ Computed.

Gauge-invariant states are called glueballs and are classified by spin $J$, parity $P$ and charge conjugation $C$. $P$ is the operation reversing the spatial coordinates $\vec x\to-\vec x$ (with the links mapped accordingly), and $C$ is the operation mapping links to their complex conjugates $U\to U^*$. For the trace of a loop, $\operatorname{tr}U^*=\overline{\operatorname{tr}U}$, which is the trace of the loop with the opposite orientation, so $C$ reverses the orientation of loops. The lightest is thought to be $0^{++}$, and the next $2^{++}$ Physics.

The rotation group of the cube and its characters

The only rotations left on the lattice are the 24 that carry a cube onto itself, and the irreducible representations of this group $O$ are $A_1,A_2,E,T_1,T_2$ (dimensions 1, 1, 2, 3, 3). "Name tag" is a metaphor; precisely, it is the pair of the irreducible representation of $O$ to which a state belongs and the eigenvalues ($\pm1$) of $P$ and $C$.

The representation of spin $J$ splits on the lattice into a sum of these. Characters decide how it splits. The character $\chi(g)$ of a representation is the trace of the matrix representing the element $g$. The 24 elements of $O$ fall into five sets on which characters take the same value (conjugacy classes): the identity (1), 90° rotations about axes through face centres (6), 180° rotations about the same axes (3), 120° rotations about diagonal axes (8), and 180° rotations about axes through edge midpoints (6). The number of times a representation $\Gamma$ appears is the following sum, which comes from a theorem, the orthogonality of characters of finite groups (here we only use it) Known.

$\theta$ is the rotation angle of $g$. For example, for $J=2$, $\chi_2=(5,-1,1,-1,1)$, and $n_E=\frac1{24}(5\cdot2+3\cdot1\cdot2+8\cdot(-1)\cdot(-1))=1$.

KnownCharacters of $O$ The columns are the five conjugacy classes above (number of elements).

ComputedKnownTable 1 How the representation of spin $J$ splits under the rotation group $O$ of the cube (computed from the characters of the five conjugacy classes; we checked that the dimensions add up to $2J+1$).

$0^{++}$ is measured in $A_1^{++}$, but $A_1$ also contains $J=4$. Lucini–Teper 2001 themselves raise the possibility that the excitation $0^{++*}$ is the lightest $4^{++}$. $2^{++}$ appears in $E^{++}$ and $T_2^{++}$, and the two agreeing is a sign that rotational symmetry is restored (4.582(14) and 4.578(11) for $SU(\infty)$ in Athenodorou–Teper 2021) Physics.

Example. The sum of spatial plaquettes in the three orientations $P_{12}+P_{23}+P_{31}$ is $A_1^{++}$, and the difference $P_{12}-P_{23}$ is $E^{++}$. For $SU(2)$, $U^*=gUg^{-1}$ ($g=i\sigma_2\in SU(2)$, where $\sigma_2$ is the Pauli matrix of 1-04), so $C$ is the same as making the same gauge transformation at every point. Hence every gauge-invariant state has $C=+$.

$C(t)$ is estimated by an average over $N$ configurations (how configurations are made and their errors: 5-07). The variance of the product $O(t)O(0)$, which sets the error, approaches a constant of about $\langle O^2\rangle^2$ as $t$ grows, and does not decrease. The signal falls as $e^{-E_{\min}t}$, so

This argument is standard in lattice spectroscopy and is said to go back to Parisi 1984 Known. Glueballs have a large $E_{\min}$, and $0^{++}$ also needs the vacuum subtracted, so it bites especially hard.

Example. In the demonstration of §07 ($E_{\min}\approx0.77$, about 35,000 configurations), the relative errors of the plaquette $C(t)$ were 0.004, 0.011, 0.025, 0.050, 0.121, 0.214, 0.379, 0.657 for $t=0,\dots,7$ Computed. The factor 1.8–2.9 per step is of the same order as $e^{0.77}\approx2.2$.

A plaquette has size $a$, but the physical length $1/\sqrt\sigma$ is about $7.5a$ for $SU(2)$ at $\beta=2.60$ (Table 2). So we "fatten" the operators. Precisely, we replace links by averages over nearby paths, so that the operator overlaps strongly with extended states.

$i,j$ run over spatial directions only, and $\operatorname{Proj}$ is the operation that brings the sum back to the group. There is also blocking, which builds links of length $2a$. Lucini–Teper 2001 cite Teper 1987 and Albanese et al. 1987 (APE smoothing) for these as standard methods Known. Averaging only in space is the key: the operator stays on one time slice, so the positive weights of (1) and "approaching from above" are kept.

Quality is measured by the fraction $c_0^2$ of the weight in the ground state. For the best operators of Lucini–Teper 2001, for $0^{++}$ it was from about 0.95 ($SU(2)$) to about 0.90 ($SU(5)$) on coarse lattices, and from about 0.92 to about 0.85 on fine lattices. For periodic strings it is from about 0.90 to about 0.75 on both. The larger $N$, the worse the overlap Physics.

For the three-dimensional $\mathbb Z_2$ of §07, the rough fraction $C(2)e^{2m}/C(0)$ ($m=1/1.296$; it comes out on the large side) was 0.57 for the plaquette and 0.85 for the $3\times3$ loop. With a smoothing in which $\operatorname{Proj}$ is a majority vote of the link and two staples, 0.59 rose to 0.66 in one step Computed.

We line up candidates $O_1,\dots,O_N$ and let the correlations themselves choose the best linear combination. For the correlation matrix $C_{ij}(t)=\langle O_i(t)O_j(0)\rangle-\langle O_i\rangle\langle O_j\rangle$ we solve

The variational idea began with Wilson's 1981 talk, and is said to have been used for glueballs by Berg 1982 and Michael–Teasdale 1983. The form of the generalised eigenvalue problem with $t_0$ and the estimate of its corrections are attributed to Lüscher–Wolff 1990 (according to the introduction of Blossier et al. 2009) Known. Blossier et al. 2009 show that taking $t_0\ge t/2$ makes the corrections vanish as $\exp\bigl(-(E_{N+1}-E_n)t\bigr)$. The $t_0=0$ of Figure 1 is outside this condition, and is used only in the sense of the upper bound of formula (5).

For $t_0=0$, $t=1$ it is the Rayleigh–Ritz variational principle itself (1-10, 2-05). For $\psi=\sum_iv_iO_i|\Omega\rangle$ (with the vacuum component subtracted)

so $-\ln\lambda_{\max}$ is the best upper bound within the space spanned by the candidates Known. The variational method improves the upper bound, but does not change the fact that it is an upper bound.

ComputedThe numbers in this figure were computed on this machine.

ComputedThe numbers of Figure 1 (jackknife errors, 39 bins. The variational basis is $1\times1$, $1\times2+2\times1$, $2\times2$, $3\times3$, with $t_0=0$).

How to read it. At $t=0$ the variational value is lowest (formula (5)); every column goes down to 0.80–0.85 at $t=2,3$ and approaches the reference value 0.772 from above. For $t\ge4$ the errors exceed 0.1 (Figure 1). The rightmost column is the rough fraction (§04).

The string tension $\sigma$ is the slope of the energy $V(R)$ of two separated static charges. An $R\times T$ Wilson loop (5-01) decreases like $e^{-V(R)T}$, and at long distances the following form is used (7-03) Physics.

$d$ is the spacetime dimension. In the Creutz ratio $\chi$ (attributed to Creutz 1980) the perimeter and corner contributions cancel. Assuming the form (6) and ignoring finite-$T$ corrections, $\chi(R,R)\approx V'(R-\tfrac12)=\sigma+\pi(d-2)/\bigl(24(R-\tfrac12)^2\bigr)$ approaches $\sigma$ from above Physics.

Periodic strings

The other way is to read, by the same method as §01, the mass $m_l$ of a closed flux tube going once around the spatial period $L$ (a periodic string). Lucini–Teper 2001 extracted $\sigma$ in four dimensions from the following formula (the correction is in general $\pi(d-2)/(6L)$; in three dimensions $\pi/(6L)$; formula (7) of 7-03 §closed strings).

The authors write 'in using eqn(4) we are assuming that we have linear confinement ... Nambu-Gotto universality class' (Nambu–Goto is the approximation that treats the string as a membrane with action proportional to its area; 7-03), and caution that with two values of $L$ 'we cannot claim to have evidence for both the 1/L functional form and for its coefficient'. Only lattices with $aL\sqrt\sigma\ge3$ are used. The authors write that to study the string correction itself, periodic strings are much better than Wilson loops: the coefficient of the correction is larger ($\pi/3$ against $\pi/12$), and the Coulomb term of static charges (which has the same $1/r$ form and dominates at short distances) does not mix in.

Periodic strings do not mix with glueballs because of centre symmetry. Multiply all links in direction $i$ crossing the plane $x_i=c$ of spatial direction $i$ (at all times) by an element of the centre of the group (an element commuting with every element; $-1$ for $SU(2)$; 1-04). Every plaquette either does not contain these links or contains them as a there-and-back pair, so the action does not change. A loop going once around in direction $i$ crosses the plane once and picks up one centre element (for $SU(2)$ its sign changes), while contractible closed loops do not change. In the confining phase this symmetry is not broken, so the correlation between the two is 0 Known.

ComputedThe numbers in this figure were computed on this machine (formula (7)).

ComputedTable 2 $a\sqrt\sigma$ recomputed by formula (7) from the $a m_l$ of Table 1 of Lucini–Teper 2001 ($SU(2)$). The master values of the six $\beta$ and masses are in 9-02 Table 3.

How to read it. The $a\sqrt\sigma$ of the paper are reproduced by formula (7) within rounding (within 1 in the error of the last digit). $L\,a\sqrt\sigma$ is 3.1–3.4 at every $\beta$, so the box is almost constant in physical length. The correction is about 10% of $\sigma L$, and leaving it out underestimates $\sqrt\sigma$ by about 5% Computed.

We check the same way of measuring in the simplest gauge theory: three-dimensional $\mathbb Z_2$ gauge theory, with $\pm1$ on the links and the Wilson action $S=-\beta\sum_pU_p$. It is known to map by duality to the three-dimensional Ising model (attributed to Wegner 1971; for how the correspondence is used, Agostini et al. 1997). The mass of $0^+$ corresponds to the inverse correlation length of the Ising model, and the string tension to the interface tension (the free energy per unit area carried by the boundary between up and down regions; 2-09 §low temperature) Known.

Agostini et al. 1997 give, at $\beta=0.72484$, $\xi_{0^+}=1.296(3)$ (stable for $N_s\ge16$, Table 2 of that paper), $\sigma=0.05558(10)$ and $\sigma\xi^2=0.0934(6)$ (Table 6, from the literature on interface tension), so $m/\sqrt\sigma=3.27$ Physics.

On this machine we made about 35,000 heat-bath updates on a $16^2\times32$ lattice (three independent runs), measuring the operators of §04 and §05 and the periodic string every time, and Wilson loops once every 5 (majority-vote smoothing in two other runs).

ComputedTable 3 Demonstration and reference values for three-dimensional $\mathbb Z_2$ at $\beta=0.72484$. $\sigma$ is the $m_{\rm eff}(2)$ of the periodic string put into $m_l=\sigma L-\pi/(6L)$. The rough guide for Creutz ratios is the crude approximation $\chi\approx V'(R-\tfrac12)$, without corrections for finite $T$ (the loops being square).

How to read it. Both the mass and the string tension agree with the reference values within errors. The central value of the mass lies above the reference value, but the difference is about 0.5 times the error and cannot be distinguished. Being an upper bound is a property of the exact expectation value $C(t)$; an estimate from finitely many configurations can also come out below because of noise ($t\ge4$ in Figure 1). The value at $t=3$ with $t_0=0$ is a ratio of two largest eigenvalues, and is not a strict upper bound like formula (5) either. The Creutz ratios are 16–19% larger than $\sigma$ even at $R=5,6$, slightly above the guide. The periodic string gave $\sigma$ to 4% accuracy from the single point $L=16$.

The ratio differs from theory to theory. The continuum value of Agostini et al. 1997 is $m_{0^+}/\sqrt\sigma=3.08(3)$, and the value for three-dimensional $SU(2)$ that Table 9 of that paper cites from other literature is 4.763(31) Physics.

These are continuum-limit values from primary sources for four-dimensional pure Yang–Mills theory (for the extrapolation procedure, 5-09 §ratios).

PhysicsComputedTable 4 Continuum limits of $m/\sqrt\sigma$, and $2^{++}/0^{++}$ divided on this machine (errors combined as independent. Only the Meyer row is the ratio from the source's table; divided on this machine it is 1.47(4)). LT = Lucini–Teper 2001, AT = Athenodorou–Teper 2020 ($SU(3)$) and 2021 ($SU(2)$, $SU(\infty)$; $2^{++}$ is $E^{++}$). $SU(\infty)$ is the $1/N^2$ extrapolation.

Morningstar–Peardon 1999 and Chen et al. 2006 report for $SU(3)$, in units of the Sommer scale $r_0$ (the distance at which the force $F$ between static charges satisfies $r_0^2F(r_0)=1.65$), $r_0m(0^{++})=4.21(11)(4)$ and $4.16(11)(4)$; dividing by the $r_0\sqrt\sigma=1.160(6)$ of AT 2020 gives 3.63(10) and 3.59(10) Computed. It is a conversion combining numbers from different sources and is only a rough guide.

How to read it. The $0^{++}$ ratio goes down with $N$, and $2^{++}/0^{++}$ is 1.4–1.5 for every $N$. Even for the same group the sources differ: the difference 0.063 between LT and AT for $SU(2)$ is about 1 times the combined error, and the difference 0.20 for $SU(3)$ about 2.3 times Computed. This chapter does not identify the cause of the differences. Conversion to GeV is treated in 5-09 §detour, and large $N$ in 7-04.

Table 4 is numerical support for the physical view that the positive gap is about 3 to 4 times $\sqrt\sigma$. To make it a mathematical statement, four things remain.

1. Upper bounds. The $a\,m$ at each $\beta$ is an upper bound at finite $t$ or a fit to a plateau (§01, §05). No lower bound comes out of a finite window (4-11 §window).
2. Boxes and quantum numbers. In a finite box, two glueballs or a pair of periodic strings also appear with the same quantum numbers (Morningstar–Peardon 1999 say in their abstract that they distinguished them). Spin is identified through the representations of the cube (§02).
3. Extrapolation. Continuum-limit values are extrapolations that assume the form of the $a^2$ corrections (5-09, 5-05). That the limit exists has not been shown.
4. The string assumption. $\sigma$ is extracted by assuming linear confinement and corrections of Nambu–Goto type (§06). Confinement at large $\beta$ for four-dimensional non-abelian groups has not been proved, so using this $\sigma$ as evidence of confinement is circular (7-03).

What the mass gap problem (4-11) asks for is the continuum theory and a lower bound on the gap, while every method in this chapter approaches from above. What to hold fixed when taking the limit is treated in 5-09.

What this chapter can say

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