2026-10-01 · chapter Yang–Mills and the mass gapPart 7 From the side of physicsa single scale, Assumption H

The sentence that plays the role of an equivalence principle — cutting out "a single scale" as one assumption

General relativity is sometimes explained as a combination of the equivalence principle, the constancy of the speed of light, general covariance (the form of the laws is the same in any coordinates) and Riemannian geometry. Riemannian geometry is a tool of mathematics, the constancy of the speed of light is a premise inherited from special relativity (setting $c=1$ is a convention of units), and general covariance is a requirement on how laws are written. On top of these, the one sentence of physics newly added about gravity is the equivalence principle. We write falling in Newtonian mechanics.

$m_{\rm i}$ is the inertial mass (how hard a body is to move), and $m_{\rm g}$ is the gravitational mass (how strongly gravity pulls on it). They are defined by separate experiments, and the theory gives no reason for them to be equal. The equivalence principle says that the ratio $m_{\rm g}/m_{\rm i}$ is the same for every body; that is, there is no second kind of mass. Then gravity disappears locally inside a freely falling box, and gravity can be written as the geometry of spacetime (said to begin with Einstein 1907; the spacetime article) Physics. Example: if the ratio differs between materials by $\eta$, the difference in acceleration is about $\eta g$ (§02).

The question is: does the path to the mass gap also contain "one sentence that is obvious physically and unproved mathematically"? It does: "the tube of confinement has no length parameter other than the string tension". Its role, however, differs from that of the equivalence principle (§10).

Classical four-dimensional Yang–Mills has no constant with the dimension of mass, and in the quantum theory a single scale $\Lambda$ appears (4-07). Physics takes a quantity whose mass dimension is $k$ to be a constant multiple of $\Lambda^k$ (for example the mass of the lightest glueball $m=c_1\Lambda$ and the string tension $\sigma=c_2\Lambda^2$; 7-01 §03) Physics. Then the ratio $m/\sqrt\sigma=c_1/\sqrt{c_2}$ is a pure number fixed by the group alone. On the lattice (5-01) what is measured is $am$ and $a\sqrt\sigma$ in lattice units; as $\beta\to\infty$ both go to 0, and in the ratio $a$ cancels. Below, $m,\sigma$ are numbers in lattice units. Making "a single scale" into an inequality uniform in $\beta$ gives the following sentence (as the record kept on this machine writes it).

The constants $c,C>0$ do not depend on $\beta$. It assumes neither $m>0$ nor $\sigma>0$; it only says that if they go to 0, they do so at the same rate. In terms of length, we call the length $\xi=1/m$ of the lightest glueball (glue plus ball) the ball length below (the correlation length: the distance over which correlations decay as $e^{-|x|/\xi}$; 2-08 §03). $\xi_\sigma=1/\sqrt\sigma$ is the string length. $A\asymp B$ means that the ratio $A/B$ is bounded above and below by positive constants independent of $\beta$. (3) is the same as $\xi\asymp\xi_\sigma$. It is a claim of the same form as "two kinds of mass", and in the language of physics it is the statement that no second dimensionless parameter appears. If there were a second parameter $z$, the ratio would be $F(z)$, and could go to 0 or to infinity.

Example: for SU(2) at $\beta=2.60$ ($24^4$), $a\sqrt\sigma=0.13395$ and $am=0.514$ (Lucini–Teper), so in lattice units $\xi_\sigma=7.47$ and $\xi=1.95$, and the ratio is 3.837 Computed. That this ratio stays bounded is the content of (3).

These are the six SU(2) points of Lucini–Teper (2001). $\sigma$ is the mass $am_l$ of a tube of glue wound once round the periodic direction, plus the string-fluctuation correction $\pi/(3L)$, divided by the length $L$ (in lattice steps) (equation (4) of the original paper: $am_l=a^2\sigma L-\pi/(3L)$). In using this formula the authors state 'we are assuming that we have linear confinement' (7-01 §06).

Table A $m/\sqrt\sigma$ for SU(2) (from Lucini–Teper Table 1. The master copy of the six values of $am$ is 9-02 Table 3, and of $a\sqrt\sigma$ 5-09 Table 1)

ComputedThe numbers in this figure and table were computed on this machine (ratios of values from primary sources; the errors are a rough guide, treated as independent).

The string length in lattice units, $1/(a\sqrt\sigma)$, grows from 2.36 to 7.47, but the ratio stays within 3.29–3.84, and broadly increases, approaching from below (it falls once from 2.475 to 2.55, within the errors). A fit with a constant independent of $a$ is rejected with $\chi^2=38.6$ (5 degrees of freedom), so the ratio depends on $a$ Computed. The continuum value is extrapolated after fixing the form of this dependence.

Table B Fits of (4) (least squares weighted by the errors, all 6 points)

$p=2$ agrees with the primary source in the central value, but $p=1$ gives a $\chi^2$ just as good, with a continuum value of 4.05 Computed. The data alone cannot choose the form of the extrapolation. The authors write 'the functional form of such corrections is known' and use $O(a^2)$, that is, $p=2$ (5-05). 3.844 is a number resting on the premise that the limit exists and is approached in this form (7-01 §08) Physics.

There are four lengths, each counted in lattice steps: the bookkeeping length $\xi_{\rm RG}$, the string length $\xi_\sigma$, the ball length $\xi$, and the length over which fluctuations are smoothed out $\sqrt{C_P}$.

The bookkeeping length is the length fixed by writing down, for each step that coarsens the lattice by a factor $b$ (one step of the renormalisation group, 6-05), how much the coupling $\beta$ falls, and piling up the steps. After $k$ steps lengths are multiplied by $b^k$, so the number of steps needed to fall to a coupling of a fixed strength decides "how many lattice steps make one physical unit". We call that $\xi_{\rm RG}$, and in this unit set the lattice spacing to $a:=1/\xi_{\rm RG}$ (the definition of the record kept on this machine; 10-02). Counting by the one-loop running alone (4-06), for SU(2), up to a constant factor, $\xi_{\rm RG}\approx e^{3\pi^2\beta/11}$ ($\beta=4/g^2$ = $2N/g^2$ of the table of notation; the coefficient in the exponent is a rewriting of the one-loop coefficient $b_0$) Known.

$C_P$ is, for the probability distribution $\mu$ of the lattice gauge field (SU(2), Wilson action, infinite volume, no gauge fixing),

the best constant for which this holds for every $f$ (the Poincaré constant). $\operatorname{Var}_\mu$ is the variance under $\mu$, and $\nabla_\ell$ is the derivative in the SU(2) directions of the link $\ell$. Depending on whether the derivative is taken in the angle $\vartheta$ of the link ($U=e^{i\vartheta\cdot\sigma}$) or in the field $A=\sqrt\beta\,\vartheta$, $\mathcal E$ changes as $\mathcal E_A=\mathcal E/\beta$, and $C_P$ differs by a factor $\beta$. In the free approximation $C_P=(g^2/4)\xi^2$ for the former and $C_P=\xi^2$ for the latter. Whether the band (6) below holds depends on this choice (with the angle normalisation the lower side breaks logarithmically even in the free approximation: 8-05 §03). The best constant is the inverse of the gap (1-10 §07). The dynamics at work is Langevin dynamics, a stochastic time evolution that moves each link little by little in the direction that lowers the action while adding a small random shake. Run for long, the distribution settles to $\mu$. $\lambda_1=1/C_P$ is the gap of this dynamics, and $C_P$ is the time of the slowest return. More precisely, $C_P$ equals the return time only for reversible dynamics in continuous time (Langevin); for discrete-time updates or HMC it is a different quantity (8-03 §06). The assumption the record kept on this machine makes is that the length of fluctuations is on the same scale as the bookkeeping length, which in band form is (6) below. The names follow A-06 Index of assumptions. Assumption H is the statement at the level of powers $\lim\ln C_P/\ln\xi=2$, and (6) is the band statement $H_{\text{band}}$ (naming the normalisation, $H_{\text{band}}[\mathcal E_\vartheta]$ or $H_{\text{band}}[\mathcal E_A]$) combined with the chains (ii)(iii).

We split this into a product of three ratios.

That each factor is $\asymp1$ we call one chain. (i) $\lambda_1\asymp m^2$: the fluctuation length and the ball length line up (the relaxation chain). Langevin dynamics is diffusive, so smoothing out fluctuations of length $\xi$ takes a time of order $\xi^2$ (dynamical exponent $z=2$) Physics. That is why $\lambda_1$ is compared not with $m$ but with $m^2$. (ii) $m^2\asymp\sigma$: the ball length and the string length line up (one scale, (3)). (iii) $\sigma\asymp a^2$: the string length and the bookkeeping length line up (confinement happens at the bookkeeping scale). If all three hold, (6) follows. Under (ii)(iii), (6) is the same as the band statement $C_P\asymp\xi^2$ ($H_{\text{band}}$), and the power form of H follows from it (A-06). The converse does not follow: if the factors cancel each other, the product is constant even when chains are broken (Figure 2). The algebra in the direction from (7) to (6) is machine-checked Lean (chain_band, sq_times_poincare_band, LatticeGaugeOneScale.lean. The hypotheses are three band inequalities, and the conclusion is that $a^2C_P$ lies in a positive band. The list is in 10-13), but none of the three chains themselves has been proved.

ComputedA schematic drawing of the sum of the logarithms in (7), drawn on the spot (not a measurement).

(i) is a relation between the dynamics chosen as a tool and the gap of the theory (the gap of the transfer matrix, 5-03); change the dynamics and the statement changes too. (iii) depends on how the steps are booked. If we call the small error dropped at each coarsening step the shortfall, (iii) is governed by the one-loop coefficient and by the shortfalls summed over the steps (10-02). In the record's arrangement, only (ii) is a sentence of the theory that depends neither on the tool nor on the bookkeeping.

In the same pattern as "there is no second kind of mass", the record made the following table of correspondences.

As a metaphor, just as general relativity made "weight" one kind, this sentence makes "length" one kind. More precisely, the table is an analogy and no theorem joins its left and right; the equivalence principle is an equality at each point, while (3) is a uniform inequality.

(a) Dual superconductivity and (b) centre vortices (7-02) are statements about configurations after fixing a gauge and projecting. In raw form they depend on how the gauge is fixed (Gribov copies, 7-02 §04), and do not become a sentence about the probability measure. The core that can be written gauge-invariantly ('t Hooft's vortex free energy) is a statement about $\sigma$, and merges into chain (iii). (c) Non-perturbative asymptotic scaling is the sentence that lengths in lattice units grow as perturbation theory predicts. The two-loop prediction is the following $f_2$ (SU(N), $g^2=2N/\beta$) Known.

The coefficients agree with the values in the PDG review (without quarks) (checked at $N=3$ Computed). (c) says "the inverse of every length in lattice units, for example $am/f_2$ or $a\sqrt\sigma/f_2$, converges to a positive constant". If $f_2$ is read as the scale marking of physical units, the convergence of $am/f_2$ is itself the statement that the mass gap in physical units is positive and finite. Since it contains the conclusion we want to show, the record set it aside as circular.

It differs numerically too. For SU(2) with the bare coupling $g^2=4/\beta$, as $\beta=2.40\to2.60$, $a\sqrt\sigma/f_2$ goes 57.26→48.23 (−15.8 %), while the ratio goes 3.618→3.837 (+6.1 %) Computed. The ratio moves less, and does not use perturbation theory either.

The bare coupling $\beta$ carries large corrections peculiar to the lattice. Lucini–Teper use the mean-field improved coupling $\beta_I=\beta\cdot\frac1N\langle\operatorname{Re}\operatorname{Tr}U_p\rangle$ ($U_p$ is the matrix of the plaquette) and write 'Transforming this to $\beta_I$ is known to remove a large part of the lattice corrections at small $a$.' (§5 of the original paper, the sentence after equation (11); as the origin of this coupling the original paper cites earlier literature) Physics. The values obtained by finding the two-loop $\Lambda$ with this coupling and extrapolating to the continuum limit are in Table 6 of the original paper, but for the SU(2) extrapolation it says 'the best continuum fit is very poor'.

That (3) is not obvious is seen from examples where it breaks. Three-dimensional compact U(1): Göpfert–Mack proved $\sigma>0$ at all couplings Known, and also state that in the continuum limit, with the mass fixed in physical units, the string tension goes to infinity ($\sigma/m^2\to\infty$) (from the abstract of the same paper; Villain action; the main text not read). If that description is right, there is confinement and yet the ratio $m/\sqrt\sigma$ goes to 0 (7-02 §03). The supergravity regime of holography: the ratio goes to 0 as $\lambda^{-1/2}$, and $\lambda=g^2N$ is the second parameter (7-09 §08) Physics.

The leading term of the strong-coupling expansion is not a counterexample but an approximation with the opposite slope. With the action $S=\beta\sum_p\big(1-\tfrac12\operatorname{Tr}U_p\big)$ (the sum over all plaquettes, 5-01), the leading term for SU(2) gives, with $u=I_2(\beta)/I_1(\beta)$ (a ratio of modified Bessel functions), $\sigma=-\ln u$, and the lightest state is $m=-4\ln u=4\sigma$ (5-02) Known, so

which is the dashed line in Figure 1 (a straight line that would pass through the origin if extended). Extended to $a\sqrt\sigma\to0$ the ratio falls to 0, the opposite direction to the lattice numbers. The leading term at $\beta=2.40$ gives $a\sqrt\sigma=0.840$ (measured 0.2634) Computed, outside the region where the expansion converges.

An example where it holds: the three-dimensional $\mathbb{Z}_2$ gauge model. The group is abelian, but by a duality transformation it maps to the three-dimensional Ising model; the glueball mass becomes the inverse of the correlation length, and the string tension becomes the interface tension. Agostini–Carlino–Caselle–Hasenbusch (1997), taking it that $\sigma\xi^2$ should be constant in the scaling limit, found the continuum value $0.1056(19)$, checked that it agrees with the prediction $0.1024(88)$ of $\phi^4$ theory, and obtained $m/\sqrt\sigma=3.08(3)$ Physics.

The ratio $1/\sqrt{\sigma\xi^2}$ is 3.272(11) at $a\sqrt\sigma=0.236$, 3.117(52) at 0.069, and 3.077(28) in the continuum, approaching from above (from Table 6 of the original paper Computed). The physical explanation is this. A relevant direction is a direction of couplings whose effect grows as the lattice is coarsened. Within the range that keeps the $\mathbb{Z}_2$ symmetry (in the Ising model, with no magnetic field), there is only one relevant direction near the critical point, so $m$ and $\sqrt\sigma$ are fixed by one and the same $\xi$ (2-08 §07) Physics. It is not a theorem.

If the description of the continuum limit of three-dimensional U(1) is right, one scale breaks for the abelian U(1) and holds for the abelian $\mathbb{Z}_2$, so it is not decided by whether the group is non-abelian. What is needed is "an input that forbids a second small parameter". The record takes this input in four-dimensional SU(2) to be the positivity of the one-loop coefficient (asymptotic freedom, 4-06), and rewrote it in two forms. I: there is one relevant direction, and $m$ and $\sqrt\sigma$ are fixed by the same $\xi$. R: the rotational symmetry broken by the lattice comes back at long distances (the shape of the tube matters; 7-03 §08). Whether these two are enough is not known.

In general relativity the equivalence principle is an axiom. The prize problem asks us to prove existence and the gap from the definitions, so (ii) cannot be added as an axiom; it has to be made a theorem or bypassed.

The problem statement defines the gap $\Delta$ as there being no spectrum in $(0,\Delta)$, and continues 'The supremum of such ∆ is the mass m, and we require m < ∞.' Known. Splitting the form $C_P\asymp\xi^2$ of (6) under (ii)(iii) (the band statement $H_{\text{band}}$) into upper and lower halves, in the record's reading $C_P\lesssim\xi^2$ (H↑) corresponds to the side of $\Delta>0$, and $C_P\gtrsim\xi^2$ (H↓) to the side where the mass does not collapse to infinity (the form $m\lt\infty$). The precise treatment that splits H itself into upper and lower halves is in 10-01. H↓ is a different thing from "non-trivial" in the Clay sense (that there are interactions).

The lower half of (ii), $m^2\gtrsim\sigma$, reduces to an operator inequality saying that the deficit of the tube is at least $c/\ell$ for perimeter $\ell$ (called $T(c,1)$ in the record) Lean (mass_sq_ge_of_tube, LatticeGaugeTubeEntropy.lean. Hypothesis: the norm of the transfer operator is at most $e^{-2\sqrt{c\sigma}}$, that is, a consequence of $T(c,1)$. Conclusion: $m^2\ge4c\sigma$. The list is in 10-13). There is no pattern of proof for $T(c,1)$ itself (10-02). From finitely many measurements only an upper bound on $m$ follows (a restatement of a known relation; 10-02). The one sentence can be named, but there are not yet tools to make it a theorem.

What this chapter can say

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Next chapter to read: 7-13 Where the way physics digs parts from mathematics — picking up, from the whole of Part 7, the places where physics and mathematics part. For the detailed form of Assumption H, go to 10-01.

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