2026-10-01 · chapter Yang–Mills and the mass gapPart 8 A dictionary of two vocabulariesthe physical side of the assumptions

The physical counterparts of our assumptions — are the record's assumptions stronger than the hypotheses of physics?

The record kept on this machine wrote the line towards the continuum limit as a sequence of "steps of bookkeeping" (10-01), and put assumptions in the language of mathematics along the way. Set against old hypotheses of physics, the answers are of three kinds: the same (reducing to the same inequality or definition), a conditional restatement (the same once a power form or a choice of dynamics is added), and an analogy (similar in form, but no theorem joins them). These correspond to Settled, Conditional and Unsettled in column (a) of 8-01 (column (b) is "None" in every row). The central Assumption H splits into an upper and a lower half.

The record's $\eta$ pointed at two quantities (8-03 §07). The static $\eta_s$ is the power of correlations fixed by the scaling dimension $\Delta$ of an operator $O$ (not the $\Delta$ of the mass gap) as $\eta_s(O)=2\Delta-(d-2)$, and the record's Lean theorems that deal with static exponents (theorem names and hypotheses in 10-13) are about this one. What enters the power form of $C_P$ and the correlation length $\xi$ is the dynamic exponent.

This book fixes the $\eta$ of the power form and of the equivalence (S4) (§07) to $\eta_{\rm dyn}$, and uses $\eta_s$ only for the test-function lower bound (§05). The two are equal only within the conventional theory $z=2-\eta_s$.

There is one price. The record had put the premise $\eta\ge0$ of the equivalence as an assumption, grounded on Källén–Lehmann positivity (8-02). What follows from that positivity is $\eta_s\ge0$ (the unitarity bound) Physics. For the Glauber dynamics of the two-dimensional Ising model, $\eta_s=1/4$ yet $\eta_{\rm dyn}=-0.1665$ (8-03 Table 2). The premise $\eta_{\rm dyn}\ge0$ is $z\le2$, that is, it includes the power form of H↑ (§04) among the premises.

Assumption H is the assumption that the slowest relaxation time $C_P$ of Langevin dynamics lines up with the square of the correlation length up to constant factors independent of the lattice spacing (8-03 §05, 10-01).

(2) is the way of writing at the level of a band, and depends on the normalisation of $\mathcal E$ ((3) below). By the record's convention H is put at the level of powers, $\lim\ln C_P/\ln\xi=2$, and the band level is split off as a separate assumption naming the normalisation (end of §03, 10-01).

The closest thing in physics is the dynamic scaling hypothesis $\tau\propto\xi^z$ (Halperin–Hohenberg 1969, Hohenberg–Halperin 1977), with the $z$ of Langevin dynamics (model A in that classification) set to 2 Physics. For the free field in the normalisation of the field $A$ ($s^2=1$ below), (2) holds exactly (8-03 Table 1).

Question: is H the same as "$z=2$"? Answer: H is stronger. The hypothesis says only the power, not uniformity up to constant factors. Moreover, (2) includes the normalisation of the Dirichlet form $\mathcal E$. For the Gaussian measure $N(0,s^2(K+m^2)^{-1})$ ($K$ is the lattice Laplace operator $-\Delta$ [here $\Delta$ is the Laplacian], $s^2$ the amplitude, $m$ the mass), taking $\mathcal E(f)=\mathbb E|\nabla f|^2$ makes $C_P$ the largest eigenvalue of the covariance, $s^2/m^2$, while the correlation length does not depend on $s$ (1-10) Known. Written in the link angle $\vartheta$ ($U=e^{i\vartheta\cdot\sigma}$), $s^2\propto g^2$; written in the field $A=\sqrt\beta\,\vartheta$ ($A=2\vartheta/g$ in this chapter's convention $\beta=4/g^2$), $s^2=1$. The $\beta$ of this chapter is the SU(2) lattice inverse coupling $\beta=4/g^2$ (the $2N/g^2$ of the table of notation), not an inverse temperature.

The free approximation here is a model that substitutes a Gaussian field with the mass $1/\xi_{\rm RG}$ put in by hand, and does not include the gauge directions (zero modes). $\xi_{\rm RG}$ is the correlation length in lattice units fixed by the one-loop running (the record sets the lattice spacing to $a:=1/\xi_{\rm RG}$, §07).

The record's $\mathcal E$ is written in link angles. Putting in the one-loop $\ln\xi_{\rm RG}=3\pi^2\beta/11+\text{const.}$ (SU(2), 4-06), $C_P/\xi^2\propto4/\beta$ falls to 0 like $1/\ln\xi$, and the effective exponent is $z_{\rm eff}=2-\kappa/\beta$ ($\kappa=4Nb_0=11/(3\pi^2)$).

In powers $z=2$, but in the link-angle normalisation the lower side of (2) breaks logarithmically. To set up H one must first decide either to write $\mathcal E$ in the normalisation of the field $A$ ($\mathcal E_A=\mathcal E/\beta$, that is, multiply the angle $\mathcal E$ by $g^2/4$; $C_P$ is conversely multiplied by $\beta$), or to put H at the level of powers ($\lim\ln C_P/\ln\xi=2$) and split the band-level statement off as a separate assumption naming the normalisation. In interacting theories the scale of stochastic time is also renormalised Known (at the level of perturbation theory; 10-00 Entrance — the staircase, and the random walk, the ruler and the clock, 10-01 The line of construction and Assumption H; there is also an example checked at one loop in supersymmetric Yang–Mills Physics), so there is no guarantee that the logarithm disappears in either normalisation.

In terms of column (c) of the dictionary (the direction reachable by finite measurement; 8-01 §02), H↓ is on the "Lower bound only" side: exhibiting one test function gives a lower bound on $C_P$ (§05). H↑ is on the "Out of reach" side. An upper bound on $C_P$ is an inequality about every $f$, so no number of added test functions, and no amount of autocorrelation measurement up to finite times, reaches it (the box "The asymmetry of upper and lower bounds" in 1-10). H↑ and the mass gap are not the same statement, but they are of the same type in that both stand on the side of lower bounds.

Detour: why diffusion takes $\xi^2$

H↑ ($C_P\lesssim\xi^2$) is, in the record's reading, the side of "the mass gap is positive" (7-12 §10). The record restates it as $z\le2$. Question: is this a restatement under the dynamic scaling hypothesis? Answer: under three conditions. (i) The power form $C_P\asymp\xi^z$ exists. (ii) The dynamics is reversible Langevin in continuous time ($C_P=\tau_{\exp}$ is an equality only there, 8-03 §06). (iii) The normalisation of $\mathcal E$ has been decided (§03). Writing $q$ for the power of a logarithmic factor, the judgement is as follows.

The hypothesis of Hohenberg–Halperin only assumes the existence of $z$, and does not claim $z\le2$. The Glauber dynamics of the Ising model in 8-03 Table 2 has $z\gt2$ in both two and three dimensions Physics. Reading H↑ as a conjecture of physics needs the condition "$\eta_s=0$ at the relevant fixed point", and we have not checked any primary source stating it for four-dimensional Yang–Mills.

ComputedA schematic drawing of equation (5), drawn on the spot. Without JavaScript it can be read from the next paragraph.

Examples ($\xi\to\infty$): $\xi^2$ satisfies both; $\xi^2/\ln\xi$ (the angle normalisation) and $\xi^{1.9}$ satisfy H↑ only; $\xi^2\ln\xi$ and $\xi^{2.1665}$ satisfy H↓ only. For $\xi^2\ln\xi$ the power is 2 and yet H↑ breaks, so dropping (i) leaves only the one direction "H↑ ⟹ $z\le2$".

H↓ ($C_P\gtrsim\xi^2$) is the side that can be shown with test functions: for any $f$, $C_P\ge\operatorname{Var}(f)/\mathcal E(f)$ (1-10). Physics says "local dynamics cannot relax faster than the susceptibility". Taking the magnetisation as $f$ gives $C_P\ge\chi$ (equation (12) of 8-03) Known. Under the two hypotheses of dynamic and static scaling, $z\ge\gamma/\nu=2-\eta_s$ ($\gamma$ is the susceptibility exponent and $\nu$ the correlation-length exponent; 8-03) Physics. It is a lower bound regarded as standard in the Monte Carlo literature, and we have not checked its authority.

Question: is H↓ the same as this lower bound? Answer: the inequality is the same, but where it reaches differs. Taking $f$ to be a sum of local operators of scaling dimension $\Delta$, the lower bound is fixed by the susceptibility $\chi_O$.

It reaches $\xi^2$ only when there is an operator with $\eta_s(O)=0$, that is, of canonical dimension $\Delta=(d-2)/2$. We check this with the four-dimensional free field.

Dimension 2 grows with slope 0.0408 in $\ln\xi$ (continuum estimate $1/(16\pi^2G(0))=0.0409$), and dimension 4 stops at $1/16$ Computed. The infinite-volume integrals were reduced to one dimension and computed, and matched at $m=0.5$ to 5 significant figures with sums on a $24^4$ lattice.

The gauge-invariant local operators of Yang–Mills are of the $\mathrm{tr}\,F^2$ type, of dimension 4, so they fall into the third line of (6) (10-01). The link angle has dimension 1, but under the measure without gauge fixing the correlations between angles of links not sharing an endpoint are exactly 0 after averaging over gauge transformations (the same mechanism as Elitzur's theorem, 8-02). H↓ is the demand to push the same inequality up to $\xi^2$ in a theory with no local order parameter, and is stronger by that much. The record checked that three families of test functions (some by U(1) calculations) do not reach it, but it does not say that H↓ is false (10-02).

H↓ is, in the record's reading, the side of "the mass in physical units is finite", a condition different from "non-trivial" (there are interactions) in the Clay problem statement. It is satisfied even by a free massive field.

The record writes one step that coarsens by a factor $b$ as $\beta\mapsto\beta-\kappa(\ln b-\delta(b))$, and calls $\delta(b)$ the shortfall (10-02, 8-04 §03). $\delta$ is a number measured in the one-loop kernel model, and in the record's measurements in four dimensions it has the form $c_2/b^2+c_3/b^3+\cdots$ Computed (that there is no $\ln b/b^2$ term is within the detection limit). Advancing $k$ steps with factors $b_1,b_2,\dots$,

For an exact step within the same scheme the third term is absent (equation (3) of 8-04), and a change of scheme shifts $\beta$ once only by $c$ (Dashen–Gross, 8-04 Table 1). If the third term converges, the step-by-step shifts add up to a one-off shift $c=\kappa\sum\delta$, indistinguishable from a single change of scheme.

Rough guide: for SU(2), $\Lambda_{\overline{\rm MS}}/\Lambda_{\rm L}=19.82$ corresponds to $\ln19.82=2.987$ in $\sum\delta$ Computed. With $b$ fixed, it is bounded only when $\delta(b)=0$ Lean (fixed_step_bounded_iff. Hypothesis: the error of a step is the same at every step).

Question: which thing in physics is (8)? There are three candidates, and all are analogies. (i) It is not Symanzik's $O(a^2)$. The variables of expansion differ, $a$ against $b$, and the former does not accumulate step by step (8-04 §06). (ii) The first SU(3) result measuring the $\Delta\beta$ of gauge theory by the method of MCRG (the Monte Carlo renormalisation group, introduced by Swendsen 1979 for spin systems) was 'no asymptotic scaling' for $K_F\lt6.1$ ($K_F$ is the coupling of the Wilson action) (Gupta et al. 1984) Known. This $\Delta\beta$ and the record's decrease per step are, with the same projection, quantities that correspond by definition (the identification is this dictionary's reading). (iii) The error of truncating to the Wilson form: for untruncated perfect actions Hasenfratz–Niedermayer (1994) write 'There exist lattice actions which give cut-off independent physical predictions even on coarse grained lattices'. One reading is that $\delta$ is close to this error.

Kernels with negative weights and reflection positivity

The blocking kernel that gave $\delta(2)=0$ in the record's calculation had a negative weight on the neighbours Computed. If the weights are non-negative, $c_2\gt0$ Lean (c2_ge_of_nonneg_weights. Hypothesis: the measurement that $c_2$ is an increasing affine function of the amount of weight, positive at weight 0). The abstract of Lüscher–Weisz (1984) writes 'When operators of dimension 6 are added to the standard Wilson action in lattice gauge theories, physical positivity is lost in general.' Known. The pattern, a negative coefficient that removes the leading term taking away positivity, is the same. The difference is the object: theirs is the action (5-03), the record's is the blocking kernel. Whether a kernel with negative weights breaks reflection positivity has not been checked.

The record treats (S4) (the mass gap in physical units) in the form "$a^2C_P$ lies in a positive band independent of the lattice spacing $a=1/\xi_{\rm RG}$", and puts two powers.

$\theta$ is the exponent of accumulation of the shortfall; in the model with $b$ fixed, $\theta=\delta(b)/\ln b$, and if $\sum\delta\lt\infty$, then $\theta=0$. If $\eta_{\rm dyn}\ge0$ and $\theta\ge0$, then $p=0$ is equivalent to both being 0, and this is the content of the record's "(S4) ⟺ $\sum\delta\lt\infty$ ∧ $\eta=0$" (the record's $\eta$ is $\eta_{\rm dyn}$, §02) Lean (drain_exponent, drainExp_eq_zero_iff. Hypotheses: the power form of (9) and the sign conditions). That the error of the steps is additive and bounded, and that $\delta$ is the value in the one-loop kernel model, are also premises.

Question: is this the product of two statements of physics? Answer: within the algebra of the model, it can be read so. $\theta=0$ is the statement "physical mass is invariant under the renormalisation group, and $m_{\rm lat}(\beta)\propto a(\beta)\Lambda$". Physics treats $\theta=0$ as following, if a continuum theory is presupposed, from the Callan–Symanzik equation (4-04) for quantities with no anomalous dimension (dimensional transmutation is in 4-07) Physics. Physics has not made this into a named hypothesis. In mathematics, Problem 5.1 of Chatterjee (2018) ($\xi(\beta)\in(0,\infty)$, and $\xi\to\infty$ as $\beta\to\infty$) asks for the same direction Known. $\eta_{\rm dyn}=0$ is $z=2$, the expectation of §03 and §04.

Dropping $\eta_{\rm dyn}\ge0$, $\theta^*=-\eta_{\rm dyn}/(2-\eta_{\rm dyn})$ gives $p=0$, and $a^2C_P$ lies in a constant band, yet neither H nor $\theta=0$ holds (the same type as Figure 2 of 7-12).

ComputedA schematic drawing of equation (9), drawn on the spot. The light shading is the region of the premises, and the dashed lines are $p=\pm0.2$. Without JavaScript it can be read from Table 3.

Repeating the step that halves $a$ 20 times, $a^2C_P$ grows by about 10 times at $p=-0.1665$, and stays at 1 times at the point of cancellation Computed. The numerical tables alone cannot tell a breaking of H from an accumulation of shortfall.

This section is easier to follow after reading 7-09 and 7-10 (the cigar, $M_{\rm KK}$, supergravity). The record has the sentence "a single scale ⟺ extra dimensions are visible (radius $R\sim1/\Lambda$)". Question: is it the same thing as the non-separation of KK? In the holographic model (Witten's 1998 cigar, 7-09) the background has a single length $R=1/M_{\rm KK}$, and multiplying the metric by $s^2$ multiplies all eigenvalues of the fluctuations by $s^{-2}$. So $m_n=c_nM_{\rm KK}$. Since there is only one length scale $R$, every quantity with the dimension of length squared (both $\xi^2$ and $C_P$ measured in Langevin time) is proportional to $R^2$, and H becomes an identity by dimensional analysis inside the model Physics. Precisely, this is counting dimensions after defining $C_P$ in the model. For the same reason the KK modes do not separate from the glueballs. Sakai–Sugimoto (2005), a model of QCD with quarks, also writes about mesons: 'our D4/D8 model deviates from QCD at the energy scale of M_KK, which is, unfortunately, the same energy scale of the mass of the vector mesons'.

At $\lambda=30,\,16.6,\,10,\,5$, $\Lambda/M_{\rm KK}=0.49,\,0.27,\,0.12,\,0.013$ Computed (16.6 is the Sakai–Sugimoto fit converted to $N_c=3$, the $\lambda$ in that convention. Putting it into the one-loop formula (10) is a rough guide, and can be off by a factor of 2 by convention). Supergravity can be used on the side of large $\lambda$, and Witten writes of the small side 'This looks like a tall order' (7-10).

Answer: an analogy. "One length ⟹ one scale" is obvious inside the model, but there is neither the converse nor a passage to $R\to0$. The physical name for one scale in pure Yang–Mills is dimensional transmutation (4-07), and the ⟺ is the formulation of the record kept on this machine.

The following rows are treated as known, and are not listed as results in the later part (the investigation on this machine). (1) The correction of direction, that a uniform gap in lattice units is on the trivial side $m_{\rm phys}\to\infty$, is just the definition that takes the continuum limit where the correlation length diverges, and the direction of Problem 5.1 (Settled, 5-09). (2) $\kappa=11/(3\pi^2)$ is $4Nb_0$ (Gross–Wilczek, Politzer 1973), and the 5-digit agreement with measurement is agreement within the one-loop model (Settled, 4-06). (3) In the record's one step of bookkeeping, the strong-coupling $\beta'=\beta^4/64$ corresponds only to the decimation step of Migdal–Kadanoff (one MK step including the bond-moving factor $b^{d-2}$ is $\beta'\approx4\beta^4$, 8-04 §04), and the weak-coupling $\beta'=\beta\,b^{d-4}$ is dimension counting (Conditional). (4) Reflection positivity and the transfer operator of (S2) and the strong-coupling area law are Osterwalder–Seiler 1978, Lüscher 1977 and Wilson 1974 (Settled, 5-03). (5) (S1) is the lattice version of uniqueness of the vacuum, and $R\le\tau_{\rm int}\le C_P$ is error analysis within the same reversible dynamics (Conditional, 8-02, 8-03).

What could not be asserted to be the same are the uniformity of H up to constant factors (§03) and that step-by-step accumulation fits into a single change of scheme (§06); both go only as far as analogy.

What this chapter can say

Sources and reproduction

Next chapter to read: 9-01 How far the world has come (reading where things stand from a timeline and tables by dimension). The conventions for $\eta_s$ and $\eta_{\rm dyn}$ are in A-01 Table of notation.

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