2026-10-01 · chapter Yang–Mills and the mass gapPart 9 Where things stand, and the tasksthe seven seams

Where the two ways of digging part — seeing the mismatch of two tunnels in seven seams

There are two directions in which to dig at this problem (0-02). Mathematics starts from "there is no theory yet" and passes through the six stages (6-01) in order; physics starts from "the theory exists" and computes quantities inside it. As a metaphor, they are two tunnels dug into the same mountain from both sides; more precisely, there are places where a sentence on one side cannot be handed over as it is to the other. The goal of mathematics has the following form (0-03) Known.

$G$ is a compact simple group, $\mathcal T_G$ the quantum Yang–Mills theory on $\mathbb R^4$, and $H$ its Hamiltonian. The physical sentence "computing the quantity $Q$ in $\mathcal T_G$ gives $q$" includes $\exists\,\mathcal T_G$ among its premises. We translate a sentence of physics $P$ into a mathematical statement $M$, and call a place where the translation does not become a theorem a seam.

Example: the $m_{0^{++}}/\sqrt\sigma=3.844(61)$ of $SU(2)$ (Lucini–Teper 2001) Physics is a prediction in physics, and in mathematics a target that the constructed theory must hit (7-13 §01).

A parallel table of walls — the six obstacles of mathematics and the seven moves of physics

The seams of Table 1 were made by pairing rows of the lists on the two sides. Before that, we give a parallel table of the rows by name. On the left are the six obstacles of 6-11 Table 4, and on the right the seven moves of 7-13 Table 5. The stock phrases are the expressions each side uses when pointing at that wall. Some rows exist on one side only.

Parallel table Names of the walls (the correspondence is this book's arrangement)

The four rows on both sides call the same wall by different words. The rows on one side only are places that, in that side's language, are not seen as walls. Positivity and equivalence of representations pass unasked in the procedures of physics, and the area law passes unasked in the mathematical problem statement. Rearranging this parallel table into seams gives Table 1, and lining it up with the partial bridges gives Table 4 of §07.

Table 1 lines up the seven seams. The order of mathematics (existence → axioms → gap) and the order of physics (values → mechanisms) run in opposite directions, so stages at the same position do not correspond. As a metaphor this is the mismatch of the tunnels; more precisely, a premise on one side becomes the goal of the last stage on the other.

Table 1 The seven seams (the division and the types are this book's arrangement)

Type A is a seam where a premise of physics becomes a goal of mathematics; type B is a seam where physics computes at a particular value (a shore) of a parameter $p$, and mathematics needs the limit that moves $p$ to the value $p_*$ of the problem statement; type C is a seam where the same word points at different objects. Type B has the following form.

Example: in seam 4, $p=L$ (the side of the box) with $p_*=\infty$; in seam 7, $p=R$ (the size of the extra dimensions) with $p_*=0$.

The following sections are ordered by type (type A is 1 and 2, type B is 3, 4 and 7, type C is 5 and 6).

Seam 1: the premise of existence

The calculations of physics stand on the existence of the theory (a measure and a Hilbert space), and mathematics has to make that existence. Partway through the making, neither mass nor string tension is defined. The problem statement writes 'one does not yet have a mathematically complete example of a quantum gauge theory in four-dimensional space-time, nor even a precise definition' Known. Mathematics shows $\exists\,\mathcal T$ as the limit of approximations $\mu_{a,L}$ with lattice spacing $a$ and side $L$.

The problem statement writes in footnote 2 of §6.5 that subsequential limits by compactness (weak existence) alone are not accepted as a solution (unless the properties of the limit are established by another method) Known. Example: two dimensions has been made up to the measure on $\mathbb R^2$ (6-02); for three dimensions 'a construction of the 3D YM measure and a description of its gauge-invariant observables, even on T³, remains open' (Chevyrev 2022) Known; four dimensions is at the stage where ultraviolet stability in finite volume has been obtained with a hypothesis (6-07).

Seam 2: a single scale

In physics every quantity with a dimension is taken to be a constant multiple of a single scale $\Lambda$ (4-07) Physics. Translated into mathematics, for example for the mass $m(\beta)$ and string tension $\sigma(\beta)$ in lattice units ($\beta=2N/g^2$ is the lattice inverse coupling, not an inverse temperature), it is that the following holds.

This is peculiar to four dimensions because $[g^2]=(\text{mass})^{4-d}$ Known. For $d\lt4$, if the theory exists, dimensional analysis gives a single scale; at $d=4$, $\Lambda=\mu\,e^{-1/(2b_0g^2(\mu))}$ is born as a quantum effect from the running of the coupling Physics. But dimensional analysis fixes only the form. For example at $d=3$, $g^2$ has the dimension of mass, so one can write $m=c_1g^2$ and $\sigma=c_2g^4$, but this does not forbid the constants $c_1,c_2$ from going to 0 or ∞ in the limit. Example: examples where one scale breaks are three-dimensional compact $U(1)$ (said to have $\sigma/m^2\to\infty$ in the continuum limit with the mass fixed in physical units; the abstract of Göpfert–Mack writes that in the continuum limit with $m_D$ fixed the string tension in physical units is infinite; main text not read) and the supergravity regime ($m^2/\sigma\sim1/\lambda$) Physics (7-12 §08). The record kept on this machine cut out a sentence close to (5) as Assumption H and split it into three chains (7-12, 10-01), but it has not become a theorem.

Seam 3: different shores

Rigorous results and calculations work at three shores: perturbation theory at $g\to0$, the strong-coupling expansion at small $\beta$, and holography at $N\to\infty$ together with $\lambda=g^2N\to\infty$. The theory of the problem statement is on none of these shores. Converting the lattice coupling to the 't Hooft coupling gives the following.

On the strong-coupling side, the infinite-volume limit and exponential clustering (the gap) at sufficiently small $\beta$ go back to Osterwalder–Seiler 1978 (range not made explicit; according to the introduction of Shen–Zhu–Zhu) Known. The condition of the theorem with an explicit range (Shen–Zhu–Zhu; arXiv 2022, CMP 2023) is $\beta\lt N^2/(16(d-1))$ in Wilson's normalisation (9-01) Known. Converted by (6), it is $\lambda\gt32(d-1)$, or $\lambda\gt96$ in four dimensions, independent of $N$ Computed. The continuum limit is in the direction $\lambda\to0$, and the lattice numbers are at $\beta=2.25$–$2.60$ for $SU(2)$, which is $\lambda=3.6$–$3.1$ Computed (a rough guide, since the conventions differ). Example: both strong shores are places where the cutoff and the physical scale are not separated (on the strongly coupled lattice the correlation length is shorter than the lattice spacing, and in supergravity the KK tower stays at the same height as the glueballs; 7-10).

Seam 4: finite and infinite

In a finite box at finite $\beta$ the entries of the transfer matrix are positive, so the gap is always positive (7-13) Known. What is asked is that it does not collapse as the box is enlarged.

On the latter, the problem statement writes 'no present ideas point the direction to establish the existence of a mass gap that is uniform in the volume' Known, and the form on the lattice is Problem 5.1 of Chatterjee 2018 (9-05 §08). Physics measures in finite boxes, checks volume effects with rules of thumb such as $L\sqrt\sigma\ge3$ ($L$ is the physical length of the side of the box), and extrapolates in the lattice spacing with $a^2$ (7-13 §05) Physics. Example: for a strip of the two-dimensional Ising model, the gap closes as $1/L$ at the critical point, but it is positive for every $L$ (2-09). $\inf_L$ cannot be read from numbers in finite boxes.

Seam 7: extra dimensions

In holography the gap comes from the geometry of the extra dimensions (the cigar where a circle shrinks) (7-09) Physics. To return to four dimensions one sends the size of the extra dimensions $R=1/M_{\mathrm{KK}}$ to 0 with $\Lambda$ fixed, but from the one-loop estimate

$R\to0$ is the same direction as $\lambda\to0$ (7-10) Physics. Supergravity can be used for $\lambda\gg1$, and Csáki et al. write 'This is the opposite of the limit that is required for the supergravity description to be valid' Physics. Example: at $\lambda\approx16.6$ converted from the Sakai–Sugimoto fit (a fitted value from a model with quarks, not a value for pure Yang–Mills; $N=3$; can be off by a factor of about 2 by convention), (8) is 0.27, and the scales are not separated Computed. Within the range we searched, we found no justification for exchanging the limits.

How far the shores are depends on the quantity. For the four-dimensional $SU(2)$ Wilson action $S=\sum_P\bigl(1-\tfrac12\operatorname{tr}U_P\bigr)$ with weight $e^{-\beta S}$ ($\beta=4/g^2$; 5-01), we compare the leading terms of the two shores with the numbers, for a local quantity (the plaquette) and a long-distance quantity (the mass). Both leading terms are the standard lowest-order expansions (the strong-coupling expansion is treated in 5-02).

What is inside the two leading terms

$I_n$ is the modified Bessel function, defined by the following integral.

The leading strong-coupling term $u$ is the average of $\tfrac12\operatorname{tr}U$ with weight $e^{\beta\cdot\frac12\operatorname{tr}U}$, treating one plaquette as a $SU(2)$ variable unrelated to its surroundings. Under the Haar measure on the group (5-01 §06) the density of $\tfrac12\operatorname{tr}U=\cos\theta$ is proportional to $\sin^2\theta$ ($0\le\theta\le\pi$), and this average is exactly $I_2/I_1$ Known (agreeing to 10 digits by numerical integration Computed). When $\beta$ is small the ties with the surrounding plaquettes are weak, and this single-plaquette average is the leading term Known.

The leading weak-coupling term is a count by equipartition. Equipartition means that under the weight $e^{-\beta S}$, each degree of freedom entering $S$ quadratically contributes $\tfrac12$ to $\langle\beta S\rangle$ (from $\int x^2e^{-\beta x^2}dx\big/\!\int e^{-\beta x^2}dx=1/(2\beta)$). When $\beta$ is large the plaquettes are pressed close to 1, and $S$ can be approximated by a quadratic form in small fluctuations of the links. Per lattice point there are 4 links, and $SU(2)$ has 3 directions, making 12. Of these, the 3 gauge-transformation directions do not change the action and are not counted, leaving 9. There are 6 plaquettes per lattice point, so per plaquette $\langle1-\tfrac12\operatorname{tr}U_P\rangle\approx\frac{9/2}{6\beta}=\frac{3}{4\beta}$ Known.

The mass is the rate at which correlations decrease. At strong coupling, joining two distant plaquettes needs the space between them filled with plaquettes in the shape of a tube, and each lattice step the tube grows picks up $u^4$ for the 4 plaquettes of its side. Correlations decrease with distance $t$ like $u^{4t}=e^{-(-4\ln u)t}$, so $am\approx-4\ln u$ Known. On the weak-coupling side, expanding $am$ as a power series in $g^2$ gives coefficient 0 at every order (because of the form $e^{-1/g^2}$), and only the form comes from asymptotic freedom Physics.

This is two-loop asymptotic scaling (4-06). The constant $C$ is not fixed by asymptotic freedom, and it is the very information of the gap.

Setting them beside the numbers

The numbers are, for the plaquette, the heat-bath method on a $6^4$ lattice on this machine Computed, and for the mass, Table 1 of Lucini–Teper 2001 Physics. The heat-bath method is a Monte Carlo method that repeats updates redrawing one link at a time directly from its conditional distribution with the surrounding links fixed (5-07).

ComputedThe plaquette points (Table 2) and the curves ((9) and (10) computed on the spot) were computed on this machine. PhysicsThe mass points are Table 1 of Lucini–Teper 2001.

ComputedTable 2 $\langle\tfrac12\operatorname{tr}U_P\rangle$: heat-bath method on $6^4$ (in parentheses a rough guide to the statistical error; extracted from the 15 points measured) and the two leading terms. Deviations are percentages of the numerical value.

Table 3 Mass $am$ (lattice units). PhysicsThe numerical $am$ column is Table 1 of Lucini–Teper (the master copy when this book cites the six SU(2) points; $a\sqrt\sigma$ and lattice sizes are in 5-09 Table 1). ComputedThe columns $\xi=1/(am)$, strong coupling and (10) are calculations on this machine, with (10) fitted to the point $\beta=2.6$ ($C=185$).

Example: for the plaquette, the leading strong-coupling term is within 5% of the numbers for $\beta\le1.5$, and the leading weak-coupling term for $\beta\ge2.75$ (the agreement at $\beta=1$ is only where the curves cross); at the points with $1.75\le\beta\le2.6$ neither is within 5% Computed. The $\beta=2.25$–$2.60$ used to extrapolate the continuum value of the mass is inside this band. For the mass, the leading strong-coupling term is 2.2–5.1 times the numbers, and (10) reproduces the other five points to within 11%, but $C$ is a fitted number Computed. For a local quantity the numbers fill the space between the two leading terms; for a long-distance quantity the weak-coupling shore has no value (6-11 §03), and the strong-coupling shore does not reach.

The physical continuum value 3.844(61) is the ratio $m/\sqrt\sigma$ of the six points in this band ($\xi\le2$) extended to $a^2\sigma\to0$ (Lucini–Teper Table 5; for how it is extended, see 7-13 §05) Physics. This is an extrapolation to the continuum limit ($a\to0$), and volume effects are checked separately with rules of thumb for finite boxes (seam 4).

Seam 5: spacetime and space

The intuition of physics lives on spacetime (Minkowski), where particles fly and there is causality. The construction of mathematics is a probability measure on Euclidean $\mathbb R^4$, with neither light cones nor causality (0-04). The way back is reflection positivity (4-10). With $\Theta$ the reflection of time, $F,G$ functions on the positive-time side, and $\tau_t$ translation in time,

the inequality in the middle is reflection positivity, and only with it can $\mathcal H$ and $H\ge0$ be built (the Osterwalder–Schrader reconstruction) Known. "Time", "energy" and "particle" are words that acquire meaning through (11). That the Wilson action preserves reflection positivity is due to Osterwalder–Seiler 1978 Known. On the other hand, the lattice itself breaks rotational symmetry. The problem statement writes that when reflection positivity is lost through regularisation, there are few ways to recover it (quotation in 0-03 §03) Known. Example: for the Pauli–Villars two-point function an $F$ with $\langle F,F\rangle_{\mathcal H}\lt0$ can be built (6-11 §06).

Seam 6: the word "particle"

Physics counts glueball masses Physics. What the problem statement asks for is a spectral gap, and the existence of an isolated one-particle state at the bottom is set aside among the 'natural extensions' (0-03 §09) Known.

($\mathbf P$ is the momentum.) Example: the way correlations measured on the lattice decrease tells the position of the bottom, but not whether it is an isolated point or the edge of a continuous spectrum (9-05 §05). The unit GeV does not come out of pure Yang–Mills either, and Athenodorou–Teper note 'Strictly speaking this is of course not possible: the pure gauge theory and QCD ... are two different theories' Physics.

6-11 listed the obstacles on the side of mathematics, and 7-13 the moves of physics. The seams pair their rows. The partial bridges that already exist all have the following form.

Example: in Shen–Zhu–Zhu, $A$ is $\lambda\gt96$ (an explicit range; below $N^2/48$ in Wilson's $\beta$, with the conversion in Table 4 of 5-01), the conclusion is a gap in infinite volume, and $A$ does not hold in the direction of the continuum limit $\lambda\to0$.

KnownTable 4 The seams with the obstacles of 6-11, the moves of 7-13 and partial bridges (the correspondence is this book's arrangement)

The partial bridges all lie on the side of the shores (strong coupling, low dimensions, finite lattices), and within the range we searched none extends towards the problem statement. Gauge is not put among the seven because local gauge fixing is a wall common to both mathematics (6-05) and physics. Chatterjee's 2021 theorem is one road that goes on in front of this wall without choosing a gauge. Topological sectors and $\theta$ (6-11) could also be counted as a seam, in that the problem statement does not touch $\theta$. What would need to be known to move which rows is lined up in 9-04.

What this chapter can say

Sources and reproduction

Next chapter to read: 9-03 Candidates for new concepts, which looks from the side of concepts at what seems to be needed to cross the seams. The rearrangement as tasks is in 9-04.

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