2026-10-01 · chapter Yang–Mills and the mass gapPart 6 Constructive field theoryobstacles other than divergence

Obstacles other than divergence — problems remain even on a lattice that contains not a single infinity

The explanation that "the difficulty is handling infinities" is one side of it. This chapter puts each obstacle to the following test. Does the obstacle remain even with the lattice spacing $a$ kept fixed (with the ultraviolet cutoff left in place)? If it remains, no handling of divergences will dispose of it.

Divergences were quantities that go to infinity when the cutoff is removed, like the variance of the free field at a point, $G_a(0)\approx0.1549/a^2$ (in four dimensions) (6-01 §02). Lattice gauge theory in a finite box with fixed lattice spacing is a finite-dimensional integral.

$\Lambda$ is the set of edges of the box, $\mathrm dU_\ell$ the Haar measure of total weight 1, $S$ the Wilson action, and $\beta$ the inverse lattice coupling (not an inverse temperature; large $\beta$ is weak coupling) (formula (3) of 6-10). There is not a single infinity.

Example: even so, for the theory with only the box extended to $\mathbb Z^4$, Chatterjee's talk slides (no date given; the PDF was created in September 2020) write 'Even showing any kind of correlation decay at large β for non-Abelian theories in 3D and 4D is open.' Known Since then, Adhikari–Cao have shown decay of correlations at weak coupling for finite groups. Known Within the literature this chapter has read, what is open now is the case of continuous non-abelian Lie groups ($SU(N)$ and so on). The results are summarised in Table 4.

Many rigorous estimates are expansions in a small number, but the coupling of four-dimensional Yang–Mills is weak at short distances (asymptotic freedom) and strong at long distances, and no foothold for an expansion is found on the infrared side (6-10 §05). The small $\beta$ (strong coupling) where the expansion converges is the shore opposite to the continuum limit $\beta\to\infty$. More precisely, the range of $\beta$ where the strong-coupling expansion converges is limited to near 0, and does not overlap with $\beta\to\infty$, which the continuum limit needs.

The target on the lattice is Problem 5.1 of Chatterjee (2018): for the length $\xi(\beta)$ (in lattice units) over which correlations of gauge-invariant local quantities decrease like $e^{-|x-y|/\xi}$,

is to be shown (four dimensions, compact non-abelian Lie group). Known The first half is that there is a gap at every lattice spacing; the second half is the prospect of a continuum limit (5-09).

Example: changing the group breaks (2). $U(1)$ enters a massless phase at large $\beta$, and the area law fails (Guth 1980; rigorously Fröhlich–Spencer 1982). Known For finite groups correlations decrease exponentially at weak coupling (Adhikari–Cao) Known, but they are thought not to confine and to have no continuum limit. Physics $SU(N)$ is believed to be in the confining phase at every $\beta$. Physics

An argument that works equally for every group goes wrong for $U(1)$, so what ought to work is "for a continuous non-abelian group, the gap survives even with the transition point at $\beta=\infty$" (5-06, 5-10). What is missing is an estimate that crosses, independently of the volume, the scales where the coupling grows to $O(1)$.

In the physics picture, the mass gap is a constant multiple of the single scale $\Lambda$ of the theory (4-07), and at one loop in the renormalisation group it is related to the coupling $g$ at scale $\mu$ as follows. Physics

$b_0$ is the coefficient of the running of the coupling $\mu\,\mathrm dg/\mathrm d\mu=-b_0g^3+\cdots$ (4-06). The right-hand side approaches 0 faster than any power.

So all Taylor coefficients at $\alpha=0$ are 0, and perturbation theory answers $\Lambda=0$ at every order (for quantum mechanics, 2-06). Known

Example: the order $n_*=c_N/\big(\alpha\ln(1/\alpha)\big)$ at which $\alpha^n$ equals $e^{-c_N/\alpha}$ grows as the coupling gets smaller.

ComputedTable 1 $e^{-c_N/\alpha}$ and the order $n_*$ ($c_2=0.8568$, $c_3=0.5712$).

ComputedThe numbers in this figure were computed on this machine (formula (3) and bisection for the crossing points).

What is missing is not the value of an infinity but information that does not appear in the series. Divergence of the series is a separate phenomenon. The problem statement (Jaffe–Witten 2000, JW below) writes in §6.1 that perturbation series are in many examples 'divergent expansions'. Beneke (1999) writes 'The sum of a convergent series may still differ from R by exponentially small terms exp(−1/α)'. Known Even a convergent series can miss $e^{-c/\alpha}$ (4-12, 6-12).

Even if divergences are handled and a limit is taken, a free field (a Gaussian measure) is not an answer (JW §4 asks for a non-trivial theory). From the two- and four-point correlations $S_2$, $S_4$ of a field with mean 0, form

and for a Gaussian $u_4\equiv0$ (Wick's rule; 6-08 §01).

Example: for four-dimensional $\phi^4$, the infinite-volume scaling limit with the ultraviolet cut on a lattice and approaching the critical point is Gaussian (Aizenman–Duminil-Copin 2021). Known The divergences have been handled, and still the limit is a free field. A special scaling limit of $SU(2)$ Yang–Mills–Higgs is also Gaussian, and a non-Gaussian limit is said to 'remain open' (Chatterjee 2026). Known

What to hold fixed is a problem of the same kind. The mass in physical units, $m_{\mathrm{lat}}(\beta)/a(\beta)$, has the form 0/0, and what has meaning is the limit of dimensionless ratios such as $m/\sqrt\sigma$, that is, ratios of rates of approach to 0 (5-09). The requirement is in 6-08 §10.

The place to put the measure is the orbit space, the space of connections divided by gauge transformations, which is not a linear space (3-04). An orbit is a set of connections carried into one another by gauge transformations. Singer (1978) showed that on $S^4$ ($\mathbb R^4$ with one point at infinity added), for non-abelian compact groups, one cannot continuously choose exactly one connection from each orbit ('No gauge fixing is possible.'). Known "Curved" in the title is a metaphor; precisely, it refers to these two things (not being a linear space, and representatives not being choosable globally and continuously).

The part reducible to divergence

With the formal $\mathrm dA$ (meant as Lebesgue measure in infinitely many dimensions), $Z=\infty$ (Chatterjee 2018), and the volume of the orbits is also expected to be infinite (Singer 1978). The lattice disposes of this. Formula (1) is finite without fixing the gauge.

The 0/0 that remains

What remains is when gauge fixing is needed in weak-coupling analysis. Bałaban uses local gauges that change from cell to cell (6-05 §05). Fixing a single covariant (Landau-type) gauge on the whole lattice BRST-symmetrically produces the following 0/0 (Neuberger 1987; the formula is (2) of Mehta–Daleo–Hauenstein–Seaton 2014). Known We define the words used, one line each (details in 4-08).

• Landau-type gauge: a fixing that builds a function $F$ (a function of the gauge transformation) measuring how far the transformed edge variables are from 1, and chooses its critical points as representatives of the orbit. The $F$ of the example below has this form.
• Gribov copies: when there are several critical points of $F$ on one orbit, each one of them.
• Faddeev–Popov matrix $M_{\mathrm{FP}}$: the matrix of second derivatives of $F$ at a critical point. The gauge-fixed integral has $\det M_{\mathrm{FP}}$ as its weight.
• BRST-symmetric fixing: a way of fixing so that a formal symmetry (BRST) remains after gauge fixing. Then the contribution of one orbit is the sum of $\operatorname{sign}\det M_{\mathrm{FP}}$ over all the copies.
• Euler characteristic $\chi$: the number obtained by adding up the critical points of a smooth function (with non-degenerate critical points) on a compact manifold without boundary (1-05), each with the sign of the determinant of the matrix of second derivatives ($+1$ if the number of negative eigenvalues is even, $-1$ if odd). It does not depend on the choice of function (the Poincaré–Hopf theorem). For a surface it is "the number of minima and maxima − the number of saddles", and for the sphere it is 2. On a circle a minimum counts $+1$ and a maximum $-1$, so it is 0.

$N$ is the number of lattice points, and a gauge transformation places one element of the group $G$ at each point, so the space moved over is the product of $N$ copies of $G$. The sum of signs is its Euler characteristic $\chi(G)^N$, and for compact connected Lie groups of positive dimension $\chi(G)=0$. Gauge-invariant quantities are constant on orbits, so for gauge-fixed expectation values both numerator and denominator become 0: 0/0.

Example: for a smooth function on the circle $S^1=U(1)$, maxima and minima alternate and the sum of signs is $\chi(S^1)=0$. On a ring of $N$ points we counted all the critical points of $F=\sum_j\big(1-\cos(\phi_j+\theta_{j+1}-\theta_j)\big)$ in formula (3) of Mehta et al. ($\theta_j$ is the angle of the gauge transformation at point $j$, with $\theta_0=0$ to remove the freedom of rotating everything uniformly; $\phi_j$ are the edge angles, uniform random numbers, 300 orbits).

ComputedTable 2 Gauge fixing on a $U(1)$ ring.

The number of copies was the same on every orbit examined, growing exponentially as $\frac N2\binom{N}{N/2}$ for even $N$, and the sum of signs was always 0. Computed Restricting to minima avoids 0/0, but that the number of minima can vary from orbit to orbit ($N=3$) is another difficulty (Mehta et al. §I). The 0 comes from topology. Singer himself named as a way round it localisation, gluing local covers together with a partition of unity (4-08, 6-12).

The road back from Euclidean to space-time depends on reflection positivity $\langle\overline{\Theta F}\,F\rangle\ge0$ ($\Theta$ is time reflection, $F$ a quantity on the positive-time side) (4-10). The Wilson action keeps it (Osterwalder–Seiler 1978) and breaks rotation symmetry. Continuum regularisations are the reverse: the problem statement writes that there are few ways to recover reflection positivity lost in regularisation, giving dimensional regularisation and Pauli–Villars as examples (JW §6.5; quoted in 0-03 §03). Known

Example: fixing the spatial momentum $p$ of a free field, the time dependence of the two-point function is $G(t)=e^{-\omega|t|}/(2\omega)$ ($\omega=\sqrt{p^2+m^2}$). For a Gaussian field, reflection positivity is, for positive times $t_i$ and real numbers $f_i$,

If $G$ is a sum of exponentials with positive weights $\rho(E)$, then $Q=\sum_E\rho(E)\big(\sum_if_ie^{-Et_i}\big)^2\ge0$. Pauli–Villars subtracts a term with a heavy mass $M$, giving $G_{\mathrm{PV}}(t)=e^{-\omega t}/(2\omega)-e^{-\Omega t}/(2\Omega)$ ($\Omega=\sqrt{p^2+M^2}$), so for $f$ with $\sum_if_ie^{-\omega t_i}=0$, $Q=-\frac{1}{2\Omega}\big(\sum_if_ie^{-\Omega t_i}\big)^2\lt0$. With $p=0$, $m=1$, $M=5$, $t=(0.1,0.2)$ and $f=(1,-e^{0.1})$, $Q=-4.0\times10^{-3}$. Computed As $M\to\infty$ the violation shrinks, but it cannot be used at the stage where the cutoff is kept.

That Wilson loops must be smoothed by the heat flow above two dimensions (Cao–Chatterjee) is an ultraviolet problem, but regularisations that keep positivity, recovery of rotation symmetry, and putting observables valued on curves rather than points (Chatterjee 2018) into the axioms are problems of structure. Known

With finitely many oscillators the irreducible representation of the canonical commutation relations is essentially unique (Stone–von Neumann; 2-06), but with infinitely many, countless inequivalent representations appear. Known One consequence is Haag's theorem, by which an interacting theory cannot be put on the Fock space of the free field (in the interaction picture) (Haag 1955, Hall–Wightman 1957; 4-09).

The mechanism can be seen without infinities. The overlap of the ground states of oscillators with frequencies $\omega\ne\omega'$ is $\big(2\sqrt{\omega\omega'}/(\omega+\omega')\big)^{1/2}\lt1$, and the overlap of the vacua of free fields of masses $m$ and $m'$ on a three-dimensional periodic lattice with $L$ points on a side and spacing $a$ is the product over modes.

The product runs over the $L^3$ modes $k=(n_1,n_2,n_3)$, $n_j=0,\dots,L-1$ ($\omega'_k$ is the same with $m'$). The logarithm for each mode is negative.

Example: we compare two limits. A keeps the lattice spacing fixed ($am=0.5$) and enlarges the box; B keeps the box fixed (side $2/m$) and refines the lattice. Both increase the number of points on a side, $L$.

ComputedTable 3 The vacuum overlap $\langle\Omega_m,\Omega_{m'}\rangle$ for $m'=1.2m$.

In A, $-\ln\langle\Omega_m,\Omega_{m'}\rangle$ is proportional to $L^3$ ($1.44\times10^{-5}$ per point), and the overlap goes to 0. Computed

The reason the overlap settles to a positive value in B lies in the high-momentum modes. When $\omega_k$ is large, the contribution of one mode to the logarithm is about $-(m'^2-m^2)^2/(64\omega_k^4)$, and $\omega_k$ grows nearly in proportion to $|k|$, so the whole is a sum of the form $\sum_k|k|^{-4}$. In three dimensions the shell $|k|\approx R$ has about $R^2$ modes, and the contribution of each shell decreases like $R^{-2}$, so this sum converges. Computed

ComputedThe numbers in this figure were computed on this machine (formula (8); m′/m = 1.1, 1.2, 1.5, 2).

The overlap goes to 0 because of the volume, not the ultraviolet (that free fields of different masses give inequivalent representations is said to be a generalisation of Haag's theorem; Reed–Simon). Constructive field theory makes the measure as a density in a finite box on a lattice, and reconstructs the state space from its limit (4-10). The state space is an output, not an input (Chatterjee 2018's 'we do not even know for certain the right spaces').

We contrast this with a related phenomenon. The vacua of this section (at a fixed time, on a lattice in three space dimensions) have positive overlap in finite volume (B of Table 3). Comparing two free fields of different masses as measures on Euclidean four-dimensional space-time, they are mutually singular even in a finite box (torus), because of the ultraviolet (the detour in 1-09, formula (11)). How it acts divides between space dimension 3 and space-time dimension 4.

Table 4 The six obstacles and their divergent part (the part that only arises when the ultraviolet cutoff is removed; the judgements are this book's organisation).

The ultraviolet side is not complete either (Bałaban's four-dimensional result goes only as far as estimates with a hypothesis, and subsequential limits are a prospect; 6-07 §05). What can be said goes as far as this: the ultraviolet side is the most advanced, and what the setters asked new ideas for was the side of a gap uniform in the volume (6-10 §01). Outside the six are the correspondence with local fields (JW footnote 1) and topological sectors (detour). The seams are in 9-02 (a glossary of names in the bilingual table of 9-02), the tasks in 9-04.

What this chapter can say

Sources and reproduction

Next chapter to read: 6-12 Detour: "if 1/0 is given a symbol", which looks again at the $e^{-c/\alpha}$ of §03 and the 0/0 of §05 from the side of symbols. From the physics side, 7-01.

← 6-10 Where four dimensions stopsContents6-12 Detour: "if 1/0 is given a symbol" →