2026-10-01 · chapter Yang–Mills and the mass gapPart 8 A dictionary of two vocabulariesmass and length

Dictionary B: mass and length — pairs that overlap on the lattice and come apart on the side of the limit

In physics, mass and length are two sides of one coin. More precisely, in units with $\hbar=c=1$ the inverse of a mass has the dimension of length; the force carried by a particle of mass $m$ stops reaching beyond a distance of about $1/m$ (Yukawa type, 3-01), and correlations fade as $e^{-r/\xi}$ with $\xi=1/m$. In mathematics this one number splits into separate quantities: the spectral gap, the decay rate of correlations, and the rate of relaxation of a stochastic dynamics.

The pairs of Table 0 fall into two layers. The inner layer is pairs that become the same thing by the spectral theorem alone (1-10) once the lattice spacing $a$ and the dynamics are fixed (the gap of the transfer matrix and the decay rate of correlations, for example). The outer layer is pairs that appear where $\xi\to\infty$ and $a\to0$ are joined, and the conditions and the Unsettled grades gather there.

The two columns of the table are the grades of 8-01. (a) is "if both objects existed, would they be the same?": to the three grades Settled, Conditional and Unsettled we add "Does not correspond" for pairs known to be different quantities (the "false friend" of 8-01). (b) is column (b) of 8-01 given at the level "lattice ($a$ fixed)", read as: Constructed = exists at every $\beta$ in finite volume; Conditional = only versions of limited range, such as strong coupling; None = nothing shown at this level (including things known only numerically). At the continuum four-dimensional level every row is "None". Notation: $\beta=2N/g^2$, lengths in lattice units, $m_{\rm lat}=1/\xi$, $\Delta=m_{\rm lat}/a$. $A\asymp B$ means that the ratio $A/B$ is bounded above and below by positive constants (A-01).

Note: (c) is column (c) of 8-01 §02 (the direction reachable by finite measurement). Relatives of the gap (B1, B4, B10, B11) reach an upper bound from a single correlation or test function, and the lengths and times that are their inverses (B2, B13, B14, B16, B20) swap direction and reach a lower bound. Statements about uniformity, families or limits (B3, B9, B12, B15, B17) are out of reach. The ground for B20 is (14) (choosing one set gives a lower bound on $C_P$; §08), and for B14 the definition of mixing time (§06; if the total variation distance exceeds $\varepsilon$ on a single event, that time is before the mixing time). B5 (string tension) gets "—", since the main text of this book does not treat a form that pins down one side from finite measurements. The (b) of B10 has the form that for each momentum there is a dispersion curve isolated from the energy spectrum, since the lattice has no Lorentz symmetry and the mass operator $M$ is not defined as it is (Schor 1984; §04). Both sides of B12 and B15 are defined on a fixed lattice, but $\asymp$ is a statement about a family with the lattice spacing sent to 0, so on a single lattice it has no content. "None" in (b) means that nothing has been shown about that family.

The problem statement (Jaffe–Witten) defines the mass gap as the Hamiltonian $H$ having no spectrum in $(0,\Delta)$ (§4), and lists equal-time clustering as a consequence (§5) Known. After OS reconstruction (4-10), if the vacuum $\Omega$ is unique, the spectral theorem gives

($\rho_O$ is the spectral measure of $O\Omega$ with 0 removed). The converse of (1) holds only when a dense family of observables decreases at a uniform rate (B1). On the lattice, reflection positivity lets one build a positive self-adjoint transfer matrix $T=e^{-H_T}$ (Lüscher 1977, 5-03), and from its eigenvalues $\lambda_0\gt\lambda_1\ge\cdots$,

The correlation length of the plaquette correlation of Chatterjee (2018) is in general $\xi_{\rm plaq}\le1/m_{\rm lat}$ (equality needs overlap with the lowest state). Example: for the free field $\xi=1/(2\,\mathrm{arsinh}(m/2))$ (Table 1).

Two quantities under one word

Many of the results called "the mass gap is positive" in mathematics are uniform lower bounds in lattice units, $m_{\rm lat}\ge c\gt0$. An example that can be checked is Shen–Zhu–Zhu (2023), whose abstract states a 'strictly positive mass gap' for strongly coupled lattice Yang–Mills Known. Then $\Delta=m_{\rm lat}/a\ge c/a\to\infty$, which in physical units is a property of the side where the mass goes to infinity, the opposite side from the continuum limit. A gap in physical units needs $\xi\to\infty$ to cancel $a\to0$, and Problem 5.1 also asks for $\xi(\beta)\to\infty$ as $\beta\to\infty$ (5-09, 9-05). This is why B3 in Table 0 is "Does not correspond".

The static potential $V(R)$ is the energy of an infinitely heavy quark pair placed at distance $R$, read, following Wilson (1974), from the $R\times T$ Wilson loop (5-01) as

Taking the left as the definition it is Settled, but identifying it with the ground-state energy of static charges needs the $T\to\infty$ limit and overlap with the ground state (B4). From the monotonicity and concavity of $V$ shown by Bachas (1986), $\sigma\in[0,\infty)$ exists Known. The area law holds at strong coupling for every group (Osterwalder–Seiler 1978, 5-02). The continuum string tension is a proposal to define it as the $d$ of Chatterjee's Problem 5.2.

He himself writes that it is not clear whether the perimeter term is needed. The physical rule of thumb is $\sqrt\sigma\approx485(6)$ MeV. This is a conversion that tentatively applies to the $r_0\sqrt\sigma=1.160(6)$ of SU(3) pure gauge theory the length scale $r_0=0.472$ fm fixed by lattice calculations of QCD, and the authors themselves note that it is a conversion that cannot strictly be made, since pure gauge theory and QCD are different theories (Athenodorou–Teper 2020) Physics.

Three ways of saying confinement

The three are (a) $V(R)\to\infty$, (b) $\sigma\gt0$ and (c) unbroken centre symmetry. The physical definition of (c) is at finite temperature. On a lattice whose time direction is a ring of length $1/T$ ($T$ is the temperature), the trace of the product of links once round the time direction from a point is called the Polyakov loop, and its expectation being 0 is taken as the sign of confinement (Polyakov 1978, Susskind 1979; the universality argument through an order parameter transforming under the centre is Svetitsky–Yaffe 1982) Physics. A centre transformation multiplies all the time-direction links crossing the surface at some time by an element of the centre of the gauge group (for $SU(N)$, $e^{2\pi ik/N}$ times the identity matrix). The action does not change and only the Polyakov loop is multiplied by $e^{2\pi ik/N}$, so if this symmetry is unbroken the expectation is 0.

What has been shown in mathematics is only the one-directional implications of Table A (Chatterjee 2021) Known. Definition 2.1 is that for some $N\ge1$ the theory restricted to the slab $\{0,\dots,N\}\times\mathbb Z^{d-1}$ is invariant under centre transformations for every boundary condition, and Definition 2.3 is uniform exponential decay for every cube and boundary condition (strong decay). The Wilson loops of Theorem 2.2 are those in representations acting non-trivially on the centre.

The gap and confinement are separate claims (7-02, 9-05, 5-10). On centre symmetry, Chatterjee proposes Definition 2.1 on the grounds that there is "to my knowledge" no rigorous definition, and cites Borgs–Seiler (1983) for the finite-temperature analogue. Whether the two definitions separate the same phases is unproved (B7).

The value of the dimensionless coupling $g$ changes with the scale $\mu$ at which it is measured. Asymptotic freedom means that $g$ becomes smaller the larger $\mu$ is (the shorter the distance) (4-06), and the one-loop coefficient that fixes the rate is $b_0$. Solving the one-loop formula,

becomes a mass scale independent of $\mu$ (dimensional transmutation, 4-07). In physics every mass is a constant multiple of $\Lambda$, and $m/\sqrt\sigma$ is read as a pure number Physics. The side of mathematics has no definition of the four-dimensional $\Lambda$ (the lattice $\Lambda_L$ is in 8-04). The record kept on this machine writes "a single scale" as chain (ii) $m^2\asymp\sigma$, and calls it, by analogy, the sentence that plays the role of an equivalence principle (7-12). There is no proof. If Problem 5.2 holds with $\varepsilon(\beta)$ and $\varepsilon\xi\to\ell\in(0,\infty)$, then

and this pairing is a reading not in Chatterjee's text. The numbers are 3.781(23) for SU(2), 3.405(21) for SU(3) and 3.072(14) for SU(∞) ($0^{++}$, Athenodorou–Teper 2020, 2021) Physics. There are also opposite examples. In three-dimensional compact U(1) (Villain action) the string tension is positive at every coupling, the continuum limit is a free field of mass $m_D$, and $\sigma/m_D^2$ in physical units diverges to infinity (Göpfert–Mack, abstract) Known. It is an example where there is confinement but the scale does not become one (7-12).

Glueballs are measured as the lowest decay rate in each $J^{PC}$ (spin, parity, charge conjugation) channel (5-08; the SU(3) $0^{++}$ is $r_0m=4.21(11)(4)$, Morningstar–Peardon 1999) Physics. The rotational symmetry of the lattice is only the 48 rotations and reflections that carry the cube onto itself (the group $O_h$), so the channels are divided by the irreducible representations of $O_h$. The correspondence with continuum spin presupposes the recovery of rotational symmetry. A one-particle state in mathematics is an isolated eigenvalue of the mass operator

($P$ is the momentum; §6 of the problem statement), and in §5 the problem statement counts "an isolated one-particle state (the upper gap)" among the extensions Known. A gap and a particle are separate claims. The lattice has no Lorentz symmetry and $M$ is not defined as it is, so what has been shown at strong coupling on the lattice has the form that for each momentum there is a dispersion curve isolated from the energy spectrum (Schor 1983, 1984) Known.

There are two times. One is the imaginary time of the path integral: correlations are carried by $T=e^{-H_T}$, and the gap is $1/\xi$. The other is the "fictitious fifth time" $s$ of stochastic quantisation (Parisi–Wu 1981, 6-09) (the abstract of Zwanziger 1981): the whole configuration moves by the following stochastic differential equation and settles to $e^{-S}/Z$ (the time of Monte Carlo methods is of the same kind).

$W_x(s)$ is a Brownian motion independent at each point $x$ (1-09), and $dW_x$ is the Gaussian increment over time $ds$, with mean 0 and variance $ds$. The first term moves in the direction that lowers the action, and the second adds fluctuations. On a computer one advances with step $h$ as $\phi_x\leftarrow\phi_x-h\,\partial S/\partial\phi_x+\sqrt{2h}\,Z_x$ ($Z_x$ independent standard normals), which is called the Euler–Maruyama method. The operator that generates the time change of the expectation of a function $f$ (the generator) is

($\mu=e^{-S}/Z$, and $\mathcal E$ is the Dirichlet form; the $\Delta$ in (9) is the Laplacian, not the $\Delta$ of the mass gap), and the inverse of the gap above 0 is the Poincaré constant $C_P$ (1-10). On a finite lattice both are defined (B11), but $H_T$ acts on the Hilbert space of a time slice, while $L$ acts on the space $L^2(\mu)$ of functions of the whole configuration (functions square-integrable with respect to $\mu$), and no theorem joins the two in general. For the free field $\mathrm{gap}(H_T)=2\,\mathrm{arsinh}(m/2)\approx m$ and $\mathrm{gap}(L)=m^2$, so $C_P\approx\xi^2$ (1-10).

Assumption H and the unit of time

Assumption H of the record kept on this machine is the assumption that $\mathrm{gap}(L)\asymp\mathrm{gap}(H_T)^2$ holds uniformly in the lattice spacing (10-01, 8-05). However, the unit of time of $L$ is fixed by the normalisation of $S$ and of the noise. Replacing $S$ by $\beta S$ multiplies the gap of $L$, in the Gaussian case, by exactly $\beta$. In gauge theory with $S=\beta\times$ (Wilson action), in the free approximation $C_P\approx\xi^2/\beta$; that is, $C_P$ is proportional to $g^2(a)\,\xi^2$, and this factor moves with the lattice spacing. Written at the level of a band (constant factors), Assumption H is a statement made after fixing this normalisation once (10-01). By the convention of the record, H is put at the level of powers, and the band level is split off as a separate assumption that names the normalisation (8-05 §03).

ComputedThe numbers in this figure were computed on this machine. The fluctuations are Euler–Maruyama with step $h=0.05$. At this step the mean $M$ is multiplied by $1-hm^2$ each step and noise is added, a first-order recurrence whose integrated autocorrelation time $1/m^2-h/2$ agrees with $1/m^2$ to within 2.5% over the whole range of the knob ($0.1\le m\le1$). Measured on the histories run, it was checked at $m=0.3,\ 0.5,\ 1$ and agrees to within 10%. Without JavaScript it can be read from Table 1.

The gap agrees to 6 digits with the slope of the correlation on a ring of length 8192, and $C_P$ with the smallest eigenvalue of $-\Delta_{\rm lat}+m^2$ on a ring of 256 points Computed. $C_P/\xi^2\to1$ is the $z=2$ of the free field.

When the dynamics is run from equilibrium, the correlation between values of an observable $f$ separated by time $t$, divided by the variance, is called the autocorrelation $\rho_f(t)$. The time of its slowest decay is $\tau_{\exp}$, $\tau_{{\rm int},f}=\tfrac12+\sum_{t\ge1}\rho_f(t)$ is the integrated autocorrelation time, and the variance of the sample mean is about $2\tau_{{\rm int},f}\mathrm{Var}(f)/n$ (Wolff 2004) Known. In continuous time the sum is replaced by an integral, $\tau_{{\rm int},f}=\int_0^\infty\rho_f(s)\,ds$. For reversible dynamics in continuous time $\rho_f(s)=\int e^{-s\lambda}d\nu_f(\lambda)$, with $\nu_f$ supported on $[\mathrm{gap},\infty)$, so

For a reversible chain in discrete time (eigenvalues $1\gt\lambda_2\ge\cdots\gt-1$) they split into three (Levin–Peres–Wilmer §12.2) Known.

Example: for the AR(1) chain $X_{n+1}=\rho X_n+\sqrt{1-\rho^2}\,Z$ (eigenvalues $\rho^k$) with $\rho=-0.9$, $C_P=5.263$, $t_{\rm rel}=10$ and $\tau_{\exp}=9.491$, all three different, and $\tau_{\rm int}$ is 0.026 for $x$ and 4.763 for $x^2$. In continuous time (Ornstein–Uhlenbeck) all three line up at 1 Computed. For a lazy chain (one that stays put with probability 1/2 each time, so all eigenvalues are at least 0), $\lambda_\star=\lambda_2$ and $C_P=t_{\rm rel}$ (LPW §12.2, Exercise 12.3) Known.

Differences of dynamics, and mixing time

The heat-bath method (redrawing variables one at a time from their conditional distributions), Langevin and HMC (hybrid Monte Carlo, Duane et al. 1987: add fictitious momenta, move by classical mechanics for a fixed time, and correct by acceptance or rejection) have different generators, and regarding the time of HMC as the $C_P$ of Langevin is an assumption (5-07). Thermalisation time and mixing time are also a Conditional pair. The mixing time $t_{\rm mix}(\varepsilon)$ is the number of steps until, from any initial state, the total variation distance (the largest difference between the probabilities two distributions give to the same event) between the distribution and the equilibrium distribution $\pi$ is at most $\varepsilon$. The $\log(1/\pi_{\min})$ in the upper bound $t_{\rm mix}(\varepsilon)\le t_{\rm rel}\log(1/(\varepsilon\pi_{\min}))$ (LPW Theorem 12.4; $\pi_{\min}$ is the smallest probability of $\pi$) grows with the volume. This is a theorem for reversible chains with finitely many states; for gauge fields taking values in a continuous group $\pi_{\min}$ has no meaning, and other tools are needed, such as the log-Sobolev constant (§07).

As $\xi$ grows, the relaxation of the dynamics also becomes slower (critical slowing down). Physics writes this with the dynamic scaling hypothesis $\tau\propto\xi^{z}$ and calls $z$ the dynamical exponent (Halperin–Hohenberg 1969, Hohenberg–Halperin 1977). $z$ changes with the dynamics and the observable. (8) is the relaxational dynamics without conserved quantities called model A in the classification of Hohenberg–Halperin. In mathematics it is $C_P\asymp\xi^z$, but the existence of a power is itself a hypothesis, and there is no theorem fixing as a power the $\xi$-dependence of $C_P$ in Yang–Mills (B15).

The conventional theory and RG

The static exponents are those of 2-08: $\xi\sim|t|^{-\nu}$, $\chi\sim|t|^{-\gamma}$ ($t$ is the distance in temperature from the critical point), and $\eta_s$ is the amount by which the critical correlation $G(x)\sim|x|^{-(d-2+\eta_s)}$ deviates from the power of the free field (the anomalous dimension). From the scaling relation $\gamma=\nu(2-\eta_s)$, $\chi\propto\xi^{\gamma/\nu}=\xi^{2-\eta_s}$ Physics. The conventional theory (called by van Hove's name) is summarised as making the relaxation time proportional to $\chi$, giving $z=2-\eta_s$ Physics. The abstract of Halperin–Hohenberg–Ma (1972), who applied Wilson's expansion method to a continuum version of the kinetic Ising model, writes that the results 'disagree with the mode-mode coupling approach and the conventional (Van Hove) theory'. RG (the $\varepsilon$ expansion) and numerics both suggest $z\gt2-\eta_s$ (Table 2) Physics.

Computed$2-\eta_s$ ($\eta_s=1/4$ in two dimensions, $\eta_s=2\Delta_\sigma-1$ in three, with the scaling dimension of the spin $\Delta_\sigma=0.5181489(10)$) and $\eta_{\rm dyn}$ computed on this machine. $z$ and $\Delta_\sigma$ are values from primary sources Physics. Glauber is a dynamics that updates spins one at a time at random.

The variational lower bound

The counterpart on the side of mathematics is the variational lower bound. Putting $f=M=\sum_x\phi_x$ into (9), $|\nabla M|^2=|\Lambda|$ (the number of points) and $\mathrm{Var}(M)=|\Lambda|\chi_\Lambda$, so

which is a theorem holding on every lattice of finite volume Known. For the free field $\chi=1/m^2=C_P$, with equality (Table 1). Moving from here to a lower bound on the exponent needs two hypotheses: dynamic scaling $C_P\asymp\xi^z$ and static scaling $\chi\asymp\xi^{\gamma/\nu}$, under which $z\ge\gamma/\nu=2-\eta_s$ Physics (a lower bound regarded as standard in the Monte Carlo literature). An example of a rigorous lower bound from an inequality of the same type is $z_{\rm SW}\ge\alpha/\nu$ for the Swendsen–Wang method ($\alpha$ is the specific-heat exponent; the abstract of Li–Sokal 1989) Known. The value of the conventional theory sits exactly at this lower bound, and the $z$ of Table 2 all lie above it.

In gauge theory this form does not reach $\xi^2$ Physics. Measuring the dimension $\Delta$ of an operator by the power when correlations behave at short distances like $|x|^{-2\Delta}$, the lowest dimension of a gauge-invariant local operator is 4. The two-point correlation of a dimension-4 operator decreases as $|x|^{-8}$, so its sum $\chi$ in four dimensions is decided at short distances and does not grow with $\xi$ (10-01).

Two $\eta$

We split $\eta$ into two. The static $\eta_s$ is the anomalous dimension above, the dynamic one is $\eta_{\rm dyn}:=2-z$, and they are equal only within the conventional theory (in Table 2 they differ even in sign). This book always attaches the subscript, and which one the record's $\eta$ is is fixed in 8-05.

No slowing down away from criticality

Conversely, the premise of physics is that if $\xi$ is finite there is no slowing down (B17). For the mathematical form we use two tools. The log-Sobolev inequality bounds the entropy, instead of the variance, by $\mathcal E$:

It is stronger than the Poincaré inequality and gives $C_P\le C_{LS}$ Known. The Dobrushin–Shlosman mixing condition is that, whatever boundary condition is imposed on a finite box, correlations inside the box decrease exponentially with distance at a uniform rate. The mathematical form of "no slowing down away from criticality" is, for discrete spin systems (finite spin space, finite range), the equivalence theorem "uniform mixing independent of boundary conditions (Dobrushin–Shlosman) ⟺ uniform log-Sobolev" (Stroock–Zegarlinski 1992; bibliographic data only) Known. A finite correlation length alone is not enough of a hypothesis. For gauge fields taking values in a continuous group the equivalence is not found within the material of this chapter. For lattice Yang–Mills, Shen–Zhu–Zhu (2023) show Poincaré, log-Sobolev and exponential decay at strong coupling. Neither result fixes the power on the side $\xi\to\infty$.

Detour: for the free field the relaxation time equals the susceptibility

Effective strings and the Lüscher term

A long flux tube spontaneously breaks transverse translation, and Goldstone bosons live in 1+1 dimensions (Lüscher 1981). A Goldstone boson is a massless fluctuation that appears when a continuous symmetry is spontaneously broken; here it is the fluctuation of the transverse position of the tube. These fluctuations add a universal $1/R$ term to $V(R)$ (Lüscher–Symanzik–Weisz 1980, 7-03).

A two-dimensional conformal field theory (a two-dimensional field theory with a symmetry that includes scaling of distances) has a number called the central charge $c$, and one free boson has $c=1$. The part of the ground-state energy of fluctuations confined to a strip of width $R$ that depends on the width (the Casimir energy) is proportional to $c$ (Blöte–Cardy–Nightingale 1986, Affleck 1986): $-\pi c/(24R)$ for an open strip and $-\pi c/(6L)$ for a periodic strip of circumference $L$. Setting $c=d-2$, the former is $\gamma$ and the latter at $d=4$ is $-\pi/(3L)$, which agrees with the correction subtracted when extracting the string tension on the lattice (Lucini–Teper 2001) Computed. The lattice $\sigma$ contains the physical assumption "the central charge of the tube is $d-2$" (B18).

Rigid tubes and roughening

At strong coupling, where the area law has been shown, the tube is said to be rigid. More precisely, each shift of the surface by one lattice step costs a small weight, so transverse fluctuations are exponentially suppressed (7-03). The shape of the tube at strong coupling can be read by analogy with the Wulff construction from a direction-dependent tension. The Wulff construction is a drawing that gives, under a tension $\tau(\mathbf n)$ depending on direction, the shape that minimises the total tension of the boundary for a fixed enclosed area (7-03). It is said to be separated from the region of fluctuating tubes by the roughening transition. This is a picture of physics found in the title (abstract) of Lüscher 1981, and no theorem showing its existence in lattice gauge theory is found in the material of this chapter Physics (B19).

Topological freezing and bottlenecks

In HMC for pure gauge theory, the effective dynamical exponent of the square of the topological charge is about 5, and Wilson loops decouple from this slow mode (the abstract of Schaefer–Sommer–Virotta 2011) Physics. The mathematical word is a bottleneck: putting into the variational principle a smoothed version $f$ of the indicator function of a set $A$ of configurations (configurations with topological charge close to 1, say),

and the narrower the passage joining $A$ and its outside, the larger $C_P$ Known (the two-valley model, 1-10). More precisely, the configurations near the boundary of $A$ have small measure, so $\mathcal E(f)$ becomes small. (14) is a theorem, but that it is the cause of freezing, and carrying the phenomenon of HMC over to the $C_P$ of Langevin, have not been checked (B20, 10-02).

Detour: sectors of topological charge and narrow passages

What this chapter can say

Sources and reproduction

Next chapter to read: 8-04 Dictionary C: scale and limits. For the correspondence with the assumptions of the record, see 8-05.

← 8-02 Dictionary AContents8-04 Dictionary C →