2026-10-09 · chapter Yang–Mills and the mass gapPart 11 Further considerations — beyond Assumption H, and other viewsnearly conserved quantities, five examples, τ_int ≤ C_P · Kiichi, Shiori and Rin

One form that obstructs H — a nearly conserved global quantity

Assumption H is the power statement $\lim\ln C_P/\ln\xi=2$ ((13) of 10-01), and its upper half H↑ is "for every test function $f$, $\operatorname{Var}(f)\le C\,\xi_{\rm RG}^2\,\mathcal E(f)$" ((15) there). Since it is a statement about every $f$, it collapses if there is even one function whose ratio $\operatorname{Var}/\mathcal E$ grows faster than $\xi^2$. Because Part 10 set up H as a sharp sentence, we can name what such a function looks like. The answer of this chapter is the following pattern.

(iii) is the condition that separates this pattern from the mass gap. Since the quantity is not reflected in the gap, the gap may stay positive however slow the relaxation is. The smallest example of this separation was Proposition C of Part 10 (a free particle on a circle; the gap is positive and $C_P=\infty$).

The one line by which H↑ fails. If the dynamics is reversible (the generator $L$ is self-adjoint on $L^2(\mu)$, $-L\ge0$), then the integrated autocorrelation time of any observable $f$ does not exceed $C_P$. Writing $\nu_f$ for the spectral measure of $-L$ for $\tilde f=f-\mu(f)$ (supported on $[\lambda_1,\infty)$, $\lambda_1=1/C_P$), we have

Proved here (one only integrates $\lambda^{-1}\le\lambda_1^{-1}$ over $\lambda\ge\lambda_1$; equality holds when $\tilde f$ lies in the slowest eigenspace). In Lean the version with a finite spectral sum is checked (LatticeGauge.FourDTrial.tau_int_le_CP, assumptions: weights $\ge0$, each eigenvalue $\ge\lambda_1\gt0$, sum of weights $\gt0$). The version that does not use the dynamics is the definition of the supremum itself:

Proved here. Hence, if the relaxation time of $Q$ ($\tau_{\rm int}(Q)$ for (1), the static ratio for (2)) grows at least as $c\,\xi^p$ ($p\gt2$) along a sequence with $\beta\to\infty$, then $C_P\le K\xi^2$ fails for every constant $K$. In the range where the correlation length $\xi$ and the bookkeeping length $\xi_{\rm RG}$ are comparable up to constant factors, this is the negation of H↑.

Proved here Lean LatticeGauge.FourDTrial.not_hup_of_fast_tau (assumptions: $0\lt c$, $2\lt p$, $\xi_n\to\infty$, $\xi_n\gt0$, and for each $n$, $\tau_n\le C_n$ and $c\,\xi_n^p\le\tau_n$). Exponential growth $\tau\ge c\,e^{\alpha\xi}$ includes $p=3$ ($e^x\ge x^3/6$). The caution in the converse direction is also one line. That $\tau$ is small for a fast quantity does not support H↑. For an observable whose weight lies on fast modes, $\tau_{\rm int}$ stays small however large $C_P$ is (negative control tau_small_but_CP_large).

Let us check the level of this one line. Of the two assumptions on the left of (3), Lean takes only inequalities between sequences of real numbers; that $\tau_n$ is the autocorrelation time of a reversible dynamics and that $C_n$ is the Poincaré constant of the Wilson measure lie outside Lean. What supports the second assumption of (3) in four dimensions is the Monte Carlo numbers of the original sources Cited, reinterpreted across the difference of algorithms as Langevin autocorrelations, and that reinterpretation is an assumption (§06 (a)).

Table 1 sets out, in the same columns, five examples that satisfy the three conditions of the pattern. The columns are: the quantity outside the family (the $Q$ of the pattern), the event that moves it (the breaking of conservation), the relation between the scale of slowness and the correlation length, the strength, the numbers measured and their labels, and the chapter where the body text is. The label on a row is the label of that row's numbers; the classification "falls into the same pattern" itself is Estimate on every row.

(a) The number of kinks in the double well. In the path measure in 0+1 dimensions (10-04, $V=(x^2-1)^2/8$, $S_0=2/3$), which well the path is in changes only when a kink (a path moving from valley to valley) enters. The weight of a kink $e^{-S_0/\hbar}$ fixes the correlation length $\xi\propto\hbar^{1/2}e^{S_0/\hbar}$, and the diffusion of kinks fixes the scale $\xi^2S_0/\hbar$ on the upper side of $C_P$, so the slow quantity and the correlation length are born from the same event. The ratio $C_P/\xi^2$ is of order $S_0/\hbar\approx\ln\xi$, so the band form of H↑ fails only by a logarithm while the power form of H survives ((14) of 10-04, the accounting for a dilute gas of kinks) Assumption. The same pattern appears in local ratios with limited windows. In a window of a single time, the Poincaré constant of the marginal $\psi_0^2$ moves by the ground-state transformation to the gap of $(H-E_0)$, and its ratio to the harmonic oscillator with the curvature of the bottom is exactly $\omega_b/m$ (when $\psi_0$ is the Schrödinger ground state) Proved here. With the transfer on the lattice $\varepsilon=\tfrac12$, at $\hbar=0.3,\ 0.15,\ 0.08$ it is $4.61,\ 21.6,\ 638$, $1.04$ to $1.05$ times $\omega_b/m$ Reading. The detailed behaviour of this ratio (lengthening the window moves it to $(\omega_b/m)^2$) is treated in 11-04.

(b) The winding number of a particle on a circle. This is Proposition C of Part 10. For the path lifted to the real line, $\tilde x(\tau)$, the integral of its increments with weight $w$, $f=\int w\,d\tilde x$, has $\operatorname{Var}(f)=\hbar\int w^2$ and $\mathcal E(f)=\hbar\int w'^2$, and lengthening the window makes the ratio as large as one likes. The winding number never breaks for continuous paths, so the strength is infinite. This $f$ is a functional supported in a window of length $L$; with the window fixed the ratio is finite, but lengthening the window makes it unbounded Proved here. The winding number is not a time-integrated single-time function, so it satisfies condition (iii) of the pattern.

(c) The flux and the topological charge of the two-dimensional $U(1)$ strip. In a strip of width 2 with the links on the outer boundary fixed, if the links are moved by Langevin, the angles of the plaquettes (the flux) are only handed between neighbours, and unevenness of the flux is smoothed out by diffusion. Because the group is compact, a "wrapping", in which the angle of a single plaquette crosses $\pi$, changes the total flux (the topological charge $Q$ of the strip) by one, and conservation is broken. The length at which conservation breaks, $\ell_c$, is $3.29,\ 9.88,\ 24.7,\ 53.6,\ 139$ at $\beta=2,\dots,6$; the slope of $\ln\ell_c$ is $0.916(21)$, and the ratio to the static length $\xi_{XY}=-1/\ln(I_1(\beta)/I_0(\beta))$ ($2.78$ to $10.9$) opens from $1.2$ to $12.8$ Reading. When the strip is longer than $\ell_c$, relaxation is fixed by the time for $Q$ to move by one, and $\tau(Q)/\xi_{XY}^2$ is about $0.4$ at $\beta=4$ and about $24$ at $\beta=8$ (strip length 32) Reading. Conditions: two-dimensional $U(1)$, width 2, outer boundary frozen, gauge quotient of link Langevin, strip length $\le256$, $\tau_{\rm int}$ as a reading of an upper bound on the gap, fit of the form $A/M^2+B$. The length at which conservation breaks is discussed in detail in 11-02.

(d) The topological charge in the four-dimensional periodic box. The smoothed topological charge $Q$ cannot move between sectors in the continuum limit; on the lattice it changes only when a small instanton, of the size of the lattice spacing, slips through the lattice (a dislocation). In the periodic box, the autocorrelation time of $Q$ is reported to grow clearly faster than $\xi^2$ (Del Debbio–Panagopoulos–Vicari 2002 with local heat bath and overrelaxation, Schaefer–Sommer–Virotta 2011 with HMC, Bonanno–Bonati–D'Elia 2021 with local algorithms; if a power, the exponent is about 5, or else exponential in $1/a$, and all three write that power and exponential cannot be distinguished by the data) Cited. The groups in all three are $SU(3)$ and above. For $SU(2)$, the primary sources measuring the dependence of $\tau_{\rm int}(Q)$ on the lattice spacing number one in the range searched (Abe–Morikawa–Suzuki 2025; HMC, periodic, a small box with side about $2.2/\sqrt\sigma$, three lattice spacings), which reports growth close to $a^{-2}$ Cited. The original sources treating $SU(2)$ with heat-bath algorithms were all observations on the side of "freezing not strong enough to measure". The conditions for using this, through (3), in a reading of H↑ are collected in §06.

(e) Flux through a box. An example in which the same pattern appears not through slowness of the dynamics but through the choice of the family of test functions. When measuring the reference for the local Poincaré constant in a four-dimensional box (the value for the massless free gauge field), dividing by gauge transformations at all sites of the box makes the test space functions of the plaquette angles; leaving the gauge transformations at the boundary sites of the box undivided lets a field passing uniformly through the box (a flux whose divergence lies only on two faces) into the test space. The latter value is $2.22$ (side 4) to $2.53$ (side 12) times the former, which returns roughly to the gauge-fixed value Reading (one-colour Gaussian approximation, $\beta=1$, canonical graduation, floating point). In a contractible box, the space divided at all sites coincides with the space of functions of the plaquettes Proved here (11-03 §03). The numbers in this row are numbers for the free field of the reference, not for the target measure. Regarding it as a member of the pattern is Estimate: the flux through the boundary of the box is, seen from inside the box, "a global quantity not reflected in the gap that moves the ratio by one choice of family".

There are three rows that satisfy the conditions of the pattern but were moved below the table of five examples (Table 2). Each has its body text in another chapter, and here we copy only the labels.

The upper side of the hierarchical-lattice row is checked in Lean (HierLink.lam_le, bundle LatticeGaugeHierarchicalLink; no assumptions: $4\sin^2(\pi/2^{N+1})\le\pi^2/4^N$. That the smallest eigenvalue equals this expression is outside Lean).

Rearranging the five examples by how the ratio of slowness $C_P/\xi^2$ (or $\tau(Q)/\xi^2$) grows with the correlation length, the pattern is one but the strengths divide into three.

The middle row of Table 3 is the row that matters in four dimensions. In the double well the slow quantity and the correlation length are born from the same kink, so the exponents match and only a logarithmic difference remains. In the two-dimensional strip, the correlation length (the inverse string tension) grows only linearly in $\beta$, whereas the wrapping that breaks conservation is an event crossing a barrier of height $2\beta$, and its length grows exponentially in $\beta$. Whether the same split occurs in four dimensions, and if so why, is the question of 11-07.

In infinite volume the total topological charge $Q$ is not in $L^2$, so what goes into (2) of the pattern is the charge of a subregion $R$. Writing its ratio $\operatorname{Var}/\mathcal E$ split into two coefficients makes clear what the local H of 11-03 asks.

Setting. Take a translation-invariant lattice measure $\mu$ in four-dimensional infinite volume, and the Dirichlet form of link Langevin $\mathcal E(f)=\sum_\ell\mathbb E|\nabla_\ell f|^2$. The smoothed charge density $q(x)$ is taken to depend only on links within distance $r_s$ of $x$ (made by finitely many smoothings; the Gaussian tail of the Wilson flow is treated separately as a truncation error). The gradient field $J_\ell(x):=\nabla_\ell q(x)$ is 0 for $|x-x_\ell|\gt r_s$. Put its sum $\bar J_\ell:=\sum_xJ_\ell(x)$ (the gradient of the total charge) and its first moment $D_\ell:=\sum_x(x-x_\ell)J_\ell(x)$ (the dipole), and define three coefficients by

Proved here (definitions; by cubic symmetry $\mathbb E[D\otimes D]$ is taken to be of the form $d$ times the identity matrix). $\chi_t$ is the topological susceptibility in lattice units, $h$ is the density of breaking of conservation (in the continuum $\bar J=0$ by topological invariance; only on the lattice is it non-zero, on dislocation configurations), and $d$ is the coefficient of transport. The coefficient of the boundary layer $g$ is the quantity per unit area obtained by subtracting the volume term from $\mathcal E$ of the charge cut off by a half-space.

A sharp box. For a box of side $R$, assuming cluster decay $\sum_z|z|\,|\operatorname{Cov}(q(0),q(z))|\lt\infty$ Assumption (an assumption equivalent to a gap, so beware of circularity when bringing this back to arguments about the gap),

($b$ is the one-sided first moment of the correlation). Note the form of the denominator. The sum over densities of the squared gradients, $\sum_{x\in R}\mathbb E|J_\ell(x)|^2$, is not $\mathcal E(Q_R)$. It is a quantity that drops the cancellation in the sum $\sum_xJ_\ell(x)$ (that in the continuum the total charge does not move locally), and it leaves a volume term even in a perfectly conserving theory. The correct volume term is the density $h$ of $|\sum_xJ_\ell(x)|^2$. Inserting $|R|=R^4$ and $|\partial R|=8R^3$ gives

Proved here (algebra) Lean LatticeGauge.LocalH.subregion_gap_identity (assumptions: $0\lt h$, $0\lt hR+8g$). As $R\to\infty$ the ratio returns to the overall ratio of the periodic box, $\operatorname{Var}(Q)/\mathcal E(Q)=\chi_t/h$, and the leading ratio $\chi_tR^4/(hR^4+gS)$ does not exceed $\chi_t/h$ (subregion_ratio_le, assumptions: $0\le\chi_t$, $0\lt h$, $0\le g$, $0\lt R^4$, $0\le S$).

A smooth window. With $Q_w:=\sum_xw(x)q(x)$ and $w(x)=W(x/R)$, we have $\nabla_\ell Q_w=w(x_\ell)\bar J_\ell+\nabla w(x_\ell)\cdot D_\ell+O(R^{-2})$; the cross term vanishes by $\sum\nabla(w^2)=0$, and

Assumption (an estimate on top of cluster decay and the definitions of $h,d$; $\lambda_W$ is $4\pi^2$ for the Dirichlet window on the unit four-dimensional cube). The two regimes divide at the crossing length $R_*$. For $R\ll R_*$ the ratio is $\chi_tR^2/(d\lambda_W)$, proportional to $R^2$; this is the same transport type as conserving dynamics (the detour in 2-10). For $R\gg R_*$ the ratio is $\chi_t/h$, the barrier type (breaking of conservation). Reading $D_{\rm eff}:=d/\chi_t$ and $r\propto h/\chi_t$ gives $R_*^2\approx D_{\rm eff}/r$, a four-dimensional candidate for the "length at which conservation breaks" of 11-02 Estimate.

The two coefficients bear on different questions. The necessary condition for the global H↑ is $\chi_t/h\lesssim\xi^2$, which asks about the coefficient of breaking of conservation $h$. The local H↑ of 11-03 is the condition obtained by putting (7) into a window of side $c\xi$, and in the regime where $h$ is small it asks about the transport coefficient $d$ (11-03 §05). The values of $h$, $g$ and $d$ on the four-dimensional lattice are not in this book.

The first step of Part 10 stopped at the common wall: the curvature at one site at the top falls short of convexity by $\tfrac14$ ((11) of 10-09: the diagonal $4-\tfrac14=\tfrac{15}4$ does not reach the sum of couplings $4$). This wall can be read as a 0+1-dimensional scaled-down version of the pattern of this chapter Estimate. The reason is the following correspondence.

Condition (ii) of the pattern was "$Q$ changes only through rare configurations". In the double well, the configurations in which $Q$ (which well) changes are those where the path crosses the plateau at the top, which is exactly where the Hessian of the action has a negative direction. Tools that rely on convexity (Bakry–Émery, Brascamp–Lieb, diagonal dominance) produce a shortfall on these configurations, and the shortfall can be paid only by the rarity (entropy) of the plateau (10-10 type 4). That is, $\tfrac14$ is the local face of "on the configurations of the event that breaks a nearly conserved quantity, convexity falls short". What corresponds to this in four dimensions are the configurations where the topological charge changes (dislocations of the size of the lattice spacing, which break the smoothness condition). The prospect of meeting a wall of the same pattern in four dimensions comes from here Estimate.

(a) Extending the conjecture. The conjecture "HMC and Langevin are in the same dynamical universality class" is, in the original source (Lüscher–Schaefer 2011), a phrase attached to the observation that with open boundaries for $SU(3)$ the autocorrelation grows as $a^{-2}$. The same source writes that with periodic boundaries algorithms that integrate Langevin also have difficulty crossing topological sectors, but as grounds it cites original sources on heat bath and HMC, and has no periodic-boundary numbers of its own Cited. Putting row (d) of Table 1 into (3) to say "H↑ reads as false in the periodic box" is a reading that extends the conjecture to the slowness of topology with periodic boundaries Assumption. Between the Dirichlet form of heat bath and that of Langevin there is only the one-directional one-link comparison $\mathcal E_{\rm HB}\le c_1\mathcal E_L$, and a lower bound on the Langevin $C_P$ does not follow directly from the slowness of heat bath. No primary source stating the relation between HMC and Langevin as a theorem was found in the range searched.

(b) $SU(2)$ is undecided. The groups in the original sources are $SU(3)$ and above. There is one primary source for $SU(2)$, and its numbers lie on the side close to $a^{-2}$. In a rough semiclassical accounting (taking the action of a dislocation as a constant multiple of the one-loop instanton action), with the constant fitted to the numbers for $SU(3)$, $SU(2)$ falls right near the boundary $z=2$ (11-07). For $SU(2)$, whether freezing of the topological charge breaks Assumption H is not settled.

(c) Two dimensions and four. Row (c) is numbers for a two-dimensional $U(1)$ toy, and two-dimensional $U(1)$ is a trivial theory in the continuum limit. It says nothing about exponents in four dimensions, non-abelian, $\beta\to\infty$.

(d) Power or exponential. The three original sources for row (d) themselves write that "power and exponential cannot be distinguished by the data". The "power or exponential" of Table 3 copies that phrasing.

(e) The pattern is a classification. That the five examples fall into one pattern is not a theorem. Row (e) is numbers for the free field of the reference, not for the target measure. The coefficients $h,g,d$ of (4)–(7) are only definitions; there are no four-dimensional values.

(f) The mass gap. None of the examples breaks the gap. Condition (iii) of the pattern is what guarantees this, and what this chapter says goes only as far as "there is slowness that is not caught in the form of H↑".

We put the rows of this chapter into one table in the format of the five grades (10-00 §01), with the estimate label Estimate added. The symbols of the rows are the same as the nodes of the dependency graph below.

[01] ASM CL  cluster decay                                  infinite volume
[02] ASM HL  conjecture extended to slow topology           periodic box
[03] CIT Q3  τ(Q) faster than ξ² for SU(3) and above        periodic box
[04] CIT Q2  one primary source for SU(2) (near a⁻²)        HMC, small box
[05] ASM KR  overall ratio of the double well ≈ S₀/ħ        ħ → 0
[06] RDG ST  ℓ_c and τ(Q)/ξ_XY² of the strip                two-dim. U(1)
[07] RDG FO  factor, reference with frozen boundary gauge   one-colour Gaussian
[08] PRV TI  τ_int ≤ C_P                                    reversible semigroup
             Lean tau_int_le_CP
[09] PRV RC  Proposition C (10-01)                          circle
[10] PRV OS  one-site ratio = ω_b/m                         0+1
[11] PRV SB  subregion ratio (6) ⟸ [01]                     infinite volume
             Lean subregion_gap_identity, subregion_ratio_le
[12] PRV NH  if τ ≥ cξ^p (p>2), H↑ fails ⟸ [08]
             Lean not_hup_of_fast_tau
[13] ASM R4  H↑ reads false in periodic box ⟸ [02][03][12]  SU(3) and up
[14] EST TY  one form ⟸ [05][06][07][09][10][13]

How to check: on each "⟸" row, see by eye that the numbers on the right are smaller than the one on the left ([11] has [01], [12] has [08], [13] has [02][03][12], [14] has [05][06][07][09][10][13]). There are no circles. [04] ($SU(2)$) is a premise of no row. That is how the unsettled reading for $SU(2)$ appears on the graph. Among the leaves of [13] is the assumption [02], and this is why the label of [13] is "Assumption".

Strictly, the statement of [13], "H↑ reads as false", contains the quantity of the conclusion $C_P$, but the premises [02][03] do not contain $C_P$. [03] is numbers for Monte Carlo autocorrelations and [02] is a reading about the universality class of dynamics; the only passage that reaches $C_P$ is the single one through [08].

What this chapter can say

Sources and reproduction

Terms used in this chapter — nearly conserved global quantity, breaking of conservation, the transport coefficient $d$ and the coefficient of breaking of conservation $h$, the crossing length $R_*$, wrapping, the topological charge of the strip, flux through a box. The notation is in 11-A1.

Next chapter to read: 11-02 Choosing the ruler and the coordinates again — is the length to compare with the static correlation length, or the length at which conservation breaks? The two forms that restate H to avoid the pattern are in 11-03.

← 11-00 Further considerations — entranceContents11-02 Choosing the ruler and the coordinates again — the length at which conservation breaks →