2026-10-01 · chapter Yang–Mills and the mass gapPart 7 From the side of physicsthe seams between physics and mathematics

Where the way physics digs parts from mathematics — translating a sentence of physics into a sentence of mathematics brings the assumptions into the open

Table 5 of Lucini–Teper 2001 gives 3.844(61) as the continuum value of the ratio of the mass $m_{0^{++}}$ of the lightest glueball of $SU(2)$ to the string tension $\sigma$ Physics. Translated into a mathematical statement, this sentence is

($\beta=2N/g^2$, and $m$ and $\sigma$ are quantities in units of the lattice spacing $a$; 5-01). (1) brings in four premises.

1. For each $\beta$ the infinite-volume limit exists and is unique. Subsequential limits always exist by compactness, so the question is uniqueness.
2. For each $\beta$, $m(\beta)>0$ and $\sigma(\beta)>0$.
3. The limit $\beta\to\infty$ exists.
4. The limit is a ratio of quantities of a continuum theory satisfying the axioms (4-10).

For four-dimensional $SU(2)$ at large $\beta$, none of (a)–(d) is a theorem (§05) Known. "(61)" is an estimated numerical error, not an error bound given by inequalities. What this chapter calls "parting" is a place where some stage of the procedure of physics has no corresponding theorem and can only be put as an assumption.

The calculations of physics presuppose that a four-dimensional pure Yang–Mills theory exists, and is one. In a finite box $B$ (a finite cube of lattice points) and at finite $\beta$, the probability measure on the link variables $U_\ell\in G$,

is properly defined ($U_P$ is the product of the link variables once round the plaquette $P$, and $dU_\ell$ is Haar measure) Known. The premises enter in the limits. The problem statement does not accept weak existence by compactness (convergence of subsequences) alone as a solution, unless the properties of the limit (the gap and the axioms) are shown by other means (footnote 2 of §6.5 of the problem statement, 7-01). Further, in §5 the problem statement names a uniform gap: 'the existence of a uniform gap for finite-volume approximations may play a fundamental role in the proof of existence of the infinite-volume limit' Known. The numbers of physics are "predictions made after assuming that it exists".

Detour: why the gap in a finite box is easy

The $\sigma$ in the denominator of (1) cannot be measured directly. What is measured is the mass $m_l$ of a flux tube closed once round a periodic box (side $L$ lattice points), and Lucini–Teper regard the tube as a string and subtract the Lüscher term (7-03 §03) Physics.

The authors write 'in using eqn(4) we are assuming that we have linear confinement …, that the effective string theory … is in the Nambu-Gotto universality class, and that our string is in fact long enough', and also note 'with only two values of L … we cannot claim to have evidence for both the 1/L functional form and for its coefficient'. The full form of the Nambu–Goto ground state (equation (23) of Athenodorou–Bringoltz–Teper) is

($l=aL$ is the length of the tube and $d$ the dimension of spacetime; the primary source writes $D$). Taking the square root and expanding in $1/l$, the first two terms are (4) Known. That is, (4) is a long-string approximation, and the shorter the tube, the more the later terms matter. On the lattices of the primary source the tube lengths are $L\sqrt\sigma\approx3.1$ to $3.4$ (lengths in units of $1/\sqrt\sigma$) Computed, and that (4) is enough at these lengths is, as in the quotation above, put as an assumption by the authors too.

From the tube masses in Table 1 of the primary source we extracted $\sigma$ in three ways (no correction $a^2\sigma=am_l/L$, (4) and (5)) and recomputed the ratio Computed. No correction is the ruler that measures the contribution of the string picture.

How to read the fits — The ratio at each $\beta$ is placed against the horizontal axis $a^2\sigma$ (the coarseness of the lattice), each point is weighted by the inverse square of its error, and a straight line is drawn (weighted least squares; the form of the line is (6) of §05). The value at the left end $a^2\sigma=0$ is the continuum value. $\chi^2$ is the sum of the squares of the deviations between points and line, each divided by the error of the point; if it is of about the same size as the degrees of freedom (number of points − 2), the line fits the points within the errors. The errors of the ratios in Table 1 are the square root of the sum of the squared relative errors of $am_{0^{++}}$ and $a\sqrt\sigma$, treated as independent.

ComputedThe numbers in this figure were computed on this machine (the results of the same calculation as Table 1 embedded).

The central value 3.844 for (4) with all points agrees with Table 5 of the primary source. With or without the Lüscher term the continuum value moves by 5.6% (about 3.5 times the primary source's error 0.061), and the difference between (4) and (5) is 0.3% Computed. On the lattices of Table 1 the correction is 9–11% of the leading term $\sigma L$. The precision of the ratio rests on the leading correction of the string picture being right (Athenodorou–Bringoltz–Teper report that many of the low states are well described by Nambu–Goto) Physics.

Translated into mathematics, the starting point of the string picture is a long tube of confinement, that is, confinement. Chatterjee 2018 writes that for four-dimensional non-abelian groups the area law at large $\beta$ 'remains elusive', and that the area law at small $\beta$ holds for all groups including $U(1)$, so it is no evidence Known.

The primary source expects the lattice corrections of the plaquette action to start at $a^2$, and

fits this (taking $c$ as a constant), dropping the coarsest point if the fit is poor Physics. The knobs of Figure 1 reproduce this procedure; with (4) the continuum value stays within 3.84–3.86 Computed (for movement due to the form of the fit, see 7-12).

On the side of mathematics, Problem 5.1 of Chatterjee 2018 asks for the following to be shown. Set $W_P=\operatorname{tr}U_P$, and let $f_\beta(x)=\langle W_{P_0}W_{P_x}\rangle-\langle W_{P_0}\rangle\langle W_{P_x}\rangle$ be the correlation (covariance) between the plaquette $P_0$ at the origin and the plaquette $P_x$ nearest to the point $x\in\mathbb R^4$. For any infinite-volume limit of four-dimensional non-abelian lattice gauge theory, at every $\beta>0$,

holds ($\xi$ is the correlation length in lattice units; Problem 5.2 is the continuum limit of Wilson loops). In words: correlations decrease far away like $e^{-|x|/\xi}$, the length $\xi$ is finite and positive at every $\beta$, and moreover grows without bound compared with the lattice spacing as the continuum limit is approached. The author writes 'Since none of the above has been proved', and states that at large $\beta$ in dimensions higher than 2 even the uniqueness of the infinite-volume limit is unknown, and that $\beta_c=\infty$ in four dimensions is at the level of 'It is believed' Known. Example: even at the finest point ($\beta=2.6$, $24^4$), $\xi=1/(am_{0^{++}})=1.95$ lattice spacings, and the $\xi$ of the six points are 0.72–1.95 Computed. Extrapolation is the assumption that extends the slope over this range all the way to $\xi\to\infty$.

Perturbation theory and semiclassics are both expansions around $g\to0$, but the scale of dimensional transmutation (4-07) at one loop is

$e^{-1/x}$ approaches 0 as $x\to0^+$ faster than any power, and all its Taylor coefficients are 0 Known. If $\Delta=c\Lambda$, the gap appears at no order of the series in $g^2$ (an expectation) Physics (6-12, 4-12). Example: for $SU(3)$ at $g^2=0.5$, $\Lambda/\mu=5.8\times10^{-7}$, while $g^{20}$ is still $9.8\times10^{-4}$ Computed.

The examples where semiclassics reaches the gap are none of them the theory of the problem statement. In three-dimensional compact $U(1)$, monopoles cause confinement (Polyakov 1977, semiclassical), and Göpfert–Mack proved the area law at all couplings on the lattice with the Villain action Known (the abstract of the same paper says that in the continuum limit the string tension in physical units goes to infinity; main text not read). On $\mathbb R^3\times S^1$, Ünsal–Yaffe 2008 derived a gap and linear confinement for a deformed $SU(N)$ theory from a semiclassical analysis in the regime of small $N$ where the circumference $L$ is small enough (at the level of physics) Physics. That there is no phase transition on the way to $L\to\infty$ is a conjecture, based on the argument that there is no order parameter distinguishing small $L$ from large $L$. What mathematics needs is an estimate non-perturbative in $g$, and the constructive methods too stop where they cross over to strong coupling (6-10).

In the move of going to a computable neighbouring theory and turning a knob to come back, the assumption is to put that nothing happens on the way. Translated into mathematics this is an exchange of limits, and uniformity is needed (7-10 §06). For large N, for a dimensionless ratio $F$,

and what the problem statement needs is the limit at fixed $N$.

Figure 2 is a picture of the content of Table 3, and has no numbers.

For large N, 't Hooft 1983 is said to have constructed planar field theory under the condition "all particle masses non-zero and the coupling small enough" (bibliographic data and search summaries only), and the gap is an input. In §6.6 the problem statement writes of 1/N as 'perhaps a good starting point for a rigorous proof (for sufficiently large N)' Known. Example: the lattice ratio is 3.405(21) for $SU(3)$ (Athenodorou–Teper 2020, 2021) and 3.072(14) for $SU(\infty)$ (same, 2021) Physics. Physics reads this as "large N is close", but what mathematics needs is a proof of the uniformity in (9).

Many mechanisms of confinement are told in terms of variables defined after choosing a gauge (monopoles, centre vortices, the Landau-gauge propagator) (7-02) Physics. On the side of mathematics, there is in general no globally continuous gauge fixing (Singer, 3-04), and even local conditions leave several solutions (Gribov, 4-08) Known. On the lattice, the expectation of a local quantity equals the expectation of that quantity averaged over gauge transformations (its gauge-invariant part), and components that change under gauge transformations have expectation 0 (Elitzur; for example $\langle U_\ell\rangle=0$) Known.

The target of the problem statement is gauge-invariant local fields. For example, for $O=\operatorname{tr}F_{\mu\nu}F_{\mu\nu}$ (summed over $\mu,\nu$),

is one way the gap $\Delta$ shows itself (4-11). For the propagator, $D(0)$ is reported to look finite and non-zero on large lattices ('seems to') (Cucchieri–Mendes), and Zwanziger argues that copies inside the region do not affect expectations Physics. There is no theorem deriving (10) from $D(p)$.

There are gauge-invariant bridges too. The centre of a group $G$ is the set of elements that commute with every element; for $SU(N)$ it is the identity matrix times an $N$-th root of 1, $e^{2\pi ik/N}I$. Centre symmetry is the symmetry under the transformation that multiplies all links of one layer by the same element of the centre (which does not change the values of the plaquettes), and we say it is "unbroken" when no measure changes under this transformation (precisely, on a slab of finite thickness, for every boundary condition; 7-02). Tomboulis–Yaffe 1985 are said to have bounded the Wilson loop from above by the free energy of centre vortices for $SU(2)$ (search summaries only). Chatterjee 2021 proved two things on the lattice for Wilson loops in representations acting non-trivially on the centre (representations in which some element of the centre maps to something other than the identity matrix; for example the fundamental representation of $SU(N)$) Known: that if centre symmetry is unbroken there is confinement (in the sense $V(R)\to\infty$, weaker than the area law), and that if exponential decay of correlations for any boundary condition is assumed, centre symmetry is unbroken and the area law follows. What is missing is that this assumption holds at weak coupling in four dimensions (the claim of Tomboulis 2007 has been pointed out to have 'missing links'; Ito–Seiler).

Physics regards the scale after dimensional transmutation as one, and treats separate quantities as faces of the one scale (7-01 §03). Translated into mathematics, the sentence splits.

In every row, what is missing is not the line of argument of physics but the corresponding theorem.

Most of the places where they part lie not in divergences but in limits and uniformity (6-11), and for mathematics the numbers of physics are not steps of a proof but targets and checks. Where the translation succeeded, the result is conditional theorems on the lattice. The overall picture set beside the side of mathematics is in 9-02, and the parallel list of names is the parallel table in 9-02.

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