2026-10-15 · chapter Yang–Mills and the mass gapPart 15 Ultraviolet controlheat-kernel class, Migdal–Kadanoff, the invariant curve · Kiichi, Shiori and Rin

The one-link heat-kernel class and the hierarchical model — the class that closes with one parameter, and the attracting curve

• The one-link heat-kernel class is a semigroup and closes with the single parameter $\tau$; the remainder of one step is 0, and the change of a quantity over the steps is a telescoping sum in which only the difference of the end values remains (§01).
• In one Migdal–Kadanoff step the image is attracted at $1/16$ per step to the curve of the form of the heat kernel $\times\exp(\theta^4/3600+\cdots)$, and on the curve the coupling flows as $\beta\mapsto\beta-\frac14-c/\beta$ (§02, §03).
• The contraction of the coefficient of $\theta^{2m}$ is $b^{d-4m}$ (with $d=4$, $b=2$, $m=2$ is $1/16$), the same form as dimensional counting (§03, the leading saddle-point term).
• In this model it has been confirmed by interval arithmetic that, for $\beta\in[2.46,64]$, the tube around the heat-kernel class is invariant under the map, and that, for $\beta\in[15.26,64]$, all orbits in the tube approach a single curve (§05). $\beta\gt64$ (the limit of lattice spacing 0) is not closed.

We look at how "absorbing the loss into the flow of the coupling" (15-01 §01) appears in the smallest setting, taking as the weight a one-link class function of $SU(2)$ (a function only of the angle $\theta$ from the identity) and following the flow that convolves it with the heat kernel one step at a time (the candidate of 11-06 §07, and Question F7 of 11-08). Every statement in this section is a statement about one link, the heat-kernel class, and the range written below; it says nothing about one four-dimensional step, which contains many links and couplings between plaquettes.

The heat kernel $p_\tau$ is a semigroup,

so within the family of heat kernels a step yields a heat kernel again, and the parameter $\tau$ (the inverse of the effective coupling) merely moves by $t$ Proved here (the semigroup property alone). Hence every quantity computed from a heat-kernel weight is a function of the parameter at that moment only, and when the changes over the steps are added up only the difference of the end values remains (the telescoping sum). It does not become a product Proved here. In terms of the identity of 15-03 §04, this is the case where the deviation from the class $\delta_k$ is identically 0.

In the coordinates divided by the centre $\mathbb Z_2$ ($SO(3)$, $\theta\in[0,\pi/2]$, the branch set $\operatorname{tr}U=0$ is $\theta=\pi/2$), write $\kappa$ for the minimum of the Bakry–Émery curvature outside the layer $|\theta-\pi/2|\lt h$. For weights obtained from the Wilson weight by convolving with the heat kernel, the curvature outside the layer was positive at every point of $\beta\in\{2,3,4,6,8\}$, $t\in\{0,0.05,0.1,0.2,0.5\}$, half-width of the layer $h=\beta'^{-1/2}$ (without dividing, $SU(2)$, it was negative at every point of the same range) Reading (two independent implementations, in the character series and in position space, agree). The curvature outside the layer takes the form

and for the Wilson weight $A=\sin h$, $B=2$ Reading (for the heat-kernel weight it agrees with the numerics).

Applying this form to one four-dimensional step $\beta'=\beta-8b_0\ln L$ (the one-loop flow written in $\beta=4/g^2$), the leading term $A\beta$ moves with the coupling (the part that is absorbed), and the one-step change of $\kappa/\beta$ is $B\,(1/\beta'-1/\beta)=O(1/\beta^2)=O(g^4)$, a loss with a finite sum Reading (a reading obtained by fitting the form above). The $\kappa_{j+1}\ge\kappa_j(1-c/\beta_j)$ that 11-06 §07 put forward as a candidate is of product type and goes to 0 as a power of the number of steps, but within this class it is merely pessimistic.

There are, however, three reservations. (i) The rarity rate of the layer is proportional to $\beta$, falls by $O(1)$ per step and runs out at $\beta=O(1)$ Reading. This is the one-link form of what 11-06 §07 described as the limit of the range of couplings that can be reached, not a new obstacle on the ultraviolet side. (ii) The convexity of the divided coordinates can pay only for the part invariant under the centre (the adjoint part); the sign of $\mathbb Z_2$ on each link needs another currency. With many links this sign part corresponds to centre vortices Reading (a physical estimate). (iii) One step of four-dimensional renormalisation is not a convolution of the one-link heat kernel; multi-link couplings between plaquettes enter. Whether the form of $A+B/\beta$ is preserved there has not been examined.

This class has no direction leading off the curve, so neither the transverse contraction nor the source appears. In the language of the decomposition, the heat-kernel class is a class with "remainder 0", the case in which (Rem-shape) of 15-03 is empty.

The map. To the weight of one plaquette of $SU(2)$ (a class function) $w=\sum_nc_n\chi_n$ ($\chi_n$ is the character of spin $n/2$, $a_n=c_n/(n+1)$) we apply one Migdal–Kadanoff step ($d=4$, $b=2$). The conventions are the same as in 8-04:

Bond moving is the operation that bundles four parallel plaquettes into one (the product of the weights), and decimation is the operation that convolves four plaquettes in series. On the hierarchical lattice this map is not an approximation but the map itself, yet it is not one step of renormalisation on the four-dimensional hypercubic lattice Cited (the definition of the Migdal–Kadanoff approximation; 8-04).

The invariant curve. The heat-kernel class does not close exactly under this step, but the image is attracted, step by step at the rate $1/16$, to a one-parameter curve (the heat kernel $\times\exp(\theta^4/3600+\cdots)$). The ratio of the differences (total-variation distance) between weights that arrive at the same $\beta'$ after different numbers of steps was $16.06$, $17.00$ at $j=1:2:3$ (the prediction of a contraction by $1/16$ each: $16.0$, $17.0$), and it was the same for every starting $\beta_T\in\{3.5,7,15,30\}$ (starting from Wilson, $15.8$, $16.7$) Reading (numerical values from this book's study; one class function, the map on the hierarchical lattice, $\beta\le32$, half-width of the layer $h\in\{\pi/4,\pi/8\}$). On the curve, $\kappa/\beta$ outside the layer is a function only of $\beta$ at that moment, and the telescoping sum holds.

The flow of the coupling. In one step starting from the heat kernel,

was found Reading (coefficients 0.022, 0.022, 0.021 at the three points $\beta=4,8,16$). $\frac14$ agrees with $(1-b^{-2})/3$ of 8-04. Written in the form of (S) of Lemma F (15-03 §03), in this model $b=1/16$ (in units of $1/g^2$), $|r_k|\approx0.0054/\beta_k\approx0.0013\,g_k^2$ Reading ($\beta\in[2,32]$). The remainder relative to the heat-kernel class is $O(1/\beta)$ at $\kappa/\beta$ (a deviation of shape that does not depend on $\beta$), but it does not accumulate as the steps are repeated Reading. As a negative control, replacing the coefficients by Wilson's at each step keeps the deviation at $\delta\approx-0.04$ ($\beta\approx16$) without decreasing it (the test has detecting power) Reading.

Fitting into the ledger. In the language of 15-03 §04, transverse stability holds with $\lambda\approx1/16$, and the remainder is a correction of the form $O(1)$ relative to the heat-kernel class. It is therefore uniform in the form of "the transversely contracting case", and the deviation from the heat-kernel class was $|\delta|\le1.6\times10^{-3}$ ($\beta\ge2$, $h=\pi/4$), and $\le6.0\times10^{-5}$ at $\beta\ge8$ Reading (over the range measured).

At the saddle point for large $\beta$ we estimate one step of the map as a map of the coefficients of the action Proved here (the leading saddle-point term; it drops the higher orders of the fluctuations and the other terms of the sum over images, of order $e^{-\pi\beta h}$; the constants of the error are not given).

• Heat kernel (taking only one image): $\mathrm{HK}_\beta(\theta)\propto(\theta/\sin\theta)\,e^{-\beta\theta^2/2}$. $\log(\theta/\sin\theta)=\theta^2/6+\theta^4/180+\theta^6/2835+\cdots$.
• Put the weight $w=\mathrm{HK}_\beta\cdot e^{R(\theta)}$, $R=\sigma\theta^4+\tau\theta^6+\cdots$. Bond moving gives $$w^4=\mathrm{HK}_{4\beta}\Bigl(\frac\theta{\sin\theta}\Bigr)^3e^{4R}=\mathrm{HK}_{4\beta-1}\,e^{\rho+4R},\qquad \rho:=3\log\frac\theta{\sin\theta}-\frac{\theta^2}2=\frac{\theta^4}{60}+\frac{\theta^6}{945}+\cdots \tag{5}$$ where $\beta\mapsto4\beta-1$ is $b^2\beta-(b^2-1)/3$ of 8-04. $\rho$ is the source: what is left over from the prefactor $\theta/\sin\theta$ (the remnant of the Haar weight) when four plaquettes are bundled.
• Decimation (the convolution of four plaquettes): for large $\beta$ the dominant configuration is the one in which the four group elements are spaced equally along the geodesic from the identity to $U$ (each $\theta/4$), and the correction $S$ is mapped to $4S(\theta/4)$. The heat-kernel part is a semigroup: $\mathrm{HK}_{(4\beta-1)/4}=\mathrm{HK}_{\beta-1/4}$.

As a consequence, the map of the coefficients of $\theta^4$ and $\theta^6$ is

The contraction is $1/16$, the source from the heat kernel in one step is $1/3840$, and the fixed point is fixed by the balance of source and contraction,

Proved here (the arithmetic of the geometric series on (6)). In general the contraction of the coefficient of $\theta^{2m}$ is $b^{d-2}\cdot b^2\cdot b^{-4m}=b^{d-4m}$ (with $d=4$, $b=2$, $4^{2-2m}$): $m=1$ gives 1 (marginal, $\beta$ itself), $m=2$ gives $1/16$, and $m=3$ gives $1/256$ Proved here (the leading saddle-point term). Reading $\theta\sim a^2F$, $\theta^{2m}$ corresponds to an operator of dimension $4m$, and $b^{d-4m}$ is the same as the dimensional counting for that dimension Reading (an estimate).

Check. The coefficient of $\theta^4$ in one step is $2.615\times10^{-4}$ ($\beta=60$) against $1/3840=2.604\times10^{-4}$, and on the attracting curve $2.789\times10^{-4}$ against $1/3600=2.778\times10^{-4}$. The difference has the form $6.5\times10^{-5}/\beta$ (the correction from the fluctuations). $\theta^6$ was $1.06\times10^{-6}$ against $1.04\times10^{-6}$ Reading (numerics for the same system). The ratios $16.06$, $17.00$ of §02 are a direct measurement of the contraction $1/16$.

Effect on the curvature. Since $-\sigma_*\theta^4$ enters $V=-\log w$, the curvature in the tangential direction at the edge of the layer $\theta=\pi/2-h$ moves by $-4\sigma_*\theta^3\cot\theta$. With $h=\pi/4$, $-4(\pi/4)^3/3600=-5.4\times10^{-4}$ Proved here (arithmetic). The measured deviation $\delta\cdot\beta$ of $\kappa/\beta$ approaches this value: $-2.6\times10^{-4}$ ($\beta=16$), $-3.6\times10^{-4}$ (24), $-4.1\times10^{-4}$ (32) Reading.

Reading Reading: in this model the source and the contraction come from separate mechanisms. The source is the $\theta^4/60$ that bond moving (the operation of bundling in parallel) pours in from the prefactor, and it exists only at $d\gt2$. The contraction is what decimation (the operation of combining in series) does in dividing the coefficient of $\theta^4$ by $1/64$, the contraction of the scale of the coupling, to give $1/16$. In the language of 15-03 §04, the two numbers to ask about (the size of the source and $\lambda$) are $1/3840$ and $1/16$ in this model. Values of the same two numbers measured in a small four-dimensional box are in 15-05 ($\lambda$ in the band $0.026$–$0.049$, the source in the band $4$–$6\times10^{-4}$).

Consider on the two-dimensional open strip (a narrow strip-shaped lattice) one step that blocks plaquettes two at a time. In axial gauge the plaquette variables on the open strip are independent, so the step combining two plaquettes is exactly the one-link convolution $w*w$, and the two coarse plaquettes are also independent Proved here (a known structure; with 200,000 independent samples the averages of the characters agree to around $1\sigma$, correlation $0.003\pm0.002$ Reading). So this setting is no test of many links (no new couplings are born).

On that basis, decimation combining $m$ plaquettes multiplies the coefficient of $\theta^4$ by $1/m^3$ and the deviation measured at $\beta'=\beta/m$ by $1/m^2$. For two plaquettes this is $\lambda=1/4$; since there is no bond moving there is no source term either, and the fixed form is the heat kernel itself Proved here (the leading saddle-point term). The path dependence starting from Wilson has an order that differs with the observable: $O(1/\beta^3)$ for the coefficients of low representations, $O(1/\beta)$ for the total-variation distance, and $O(1)$ for $\kappa/\beta'$ outside the layer (all $1/m^2$ in $m$) Reading ($\beta_T=8$–$64$, $m=1,2,4$). The heat kernel and the Villain weight at $U(1)$ do not depend on the path; the deviation is at most $10^{-8}$ Reading.

The smallest example with many links is the three-dimensional $2\times2\times1$ slab, in which the plaquettes are not independent because of the cube constraint. The numbers beyond that are in 15-05.

§02 and §03 were numerics and a saddle-point estimate. Within this model, we list, together with their conditions, the results of checking each row of the ultraviolet decomposition (15-03 §02) over a finite range of couplings. All are statements about the MK map $T$ ((3)), the weight of one plaquette of $SU(2)$, $d=4$, $b=2$.

Coordinates and the tube. Let the reading of the coupling be $s(w):=1/b(w):=-\frac23\log a_1$ (for the heat kernel $\mathrm{HK}_\beta$, $a_n=e^{-q_n/(2\beta)}$ and $q_n:=n(n+2)$, so $b(\mathrm{HK}_\beta)=\beta$). Put the transverse coordinate

Then in the heat-kernel class $e\equiv0$, and decimation gives $e\mapsto4e$, $s\mapsto4s$ exactly Proved here (from the definition). The tube $U$ is the set of weights such that "$b(w)\in[\beta_{\min},\beta_{\max}]$, $|e_n(w)|\le R\,(q_n^2s^4+10\,q_ns^3)$ ($2\le n\le N$, $R=10^{-3}$), and the tail of high $n$ is bounded by a weighted sum of the coefficients". $q_n^2s^4$ corresponds to the size of a shape of type $\theta^4$ (for $w=\mathrm{HK}\cdot e^{\sigma\theta^4}$, $e_n\approx\sigma q_n(q_n-3)s^4$).

$\sigma\mapsto\sigma/16+1/3840$ in (6) is the case $k=2$ of the flow of the shape coefficient in Table 1 ($\ell_2=1/180$, $(16\sigma+12/180)/256=\sigma/16+1/3840$).

What ultraviolet means in this model. Even if the starting $\beta$ is made large within a finite range and the flow is run for $K$ steps, the weight at the last scale lies within $\Lambda^K$ × (the starting distance) of a point on the curve, and the error is uniform in $K$ Computer-assisted (a statement on the range $\beta\in[15.26,64]$; $\beta$ falls by about $1/4$ per step, so for example from the final $\beta=16$ up to 64 is $K\approx190$ steps). That is, the form of (4) of 15-03 §04 stands, over a finite range in this model, as a computer-assisted inequality.

The range it does not reach (in the same rows). $\beta\gt64$ — the limit of lattice spacing 0 itself — has only the leading saddle-point term, with no estimate carrying the constants of the error Assumption. Interval arithmetic for the curvature outside the layer and the weight of the layer, separating $c_*=1/60$ from $1/48$ by interval arithmetic (the width of $\beta'$ over the whole tube is $\pm0.0007$ at $\beta=64$, which is coarse), and $\beta\lt2.46$ are not reached either. Above all, it closes here because this is a model in which the weight of one plaquette returns to the weight of one plaquette in one step. Structurally it has no multi-link couplings, no re-fixing of the gauge and no large-field remainder, and the "two rows absent from the sources" of 15-02 (the one-step remainder including large fields, and the source of observables) are empty in this model. The hierarchical model confines even at $U(1)$, so it is no test of non-commutativity either. $\frac14$ is close to the four-dimensional one-loop value ($8b_0\ln2\approx0.258$ for factor 2), but there is no basis for comparing $c_*=1/60$ with the four-dimensional two-loop coefficient Reading.

Questions. We set down two questions that come out of this chapter (numbered consecutively through Part 15; the index is in 15-09).

Table 3 lines up how each component of the decomposition looks in three smallest systems.

What to read from Table 3 Reading. The (Cls) class of 15-03 is either the heat-kernel class itself (remainder 0) or the invariant curve that deviates from the heat kernel only by the form $O(1)$ (the remainder is bounded above by the transverse contraction); in both the only parameter is the coupling. The source comes from the operation of bundling in parallel, and the contraction from the operation of combining in series. The two-dimensional open strip has no bundling in parallel and so no source; Migdal–Kadanoff has one. One step on the four-dimensional lattice contains both operations, and moreover multi-link couplings are born. The smallest system for that is the flux network of 15-05.

(a) All results of this chapter are statements about maps applied to the weight of one plaquette (the convolution of the one-link heat kernel, the two-dimensional open strip, the map on the hierarchical lattice), and imply nothing about one four-dimensional step containing many links, couplings between plaquettes, re-fixing of the gauge and large fields.

(b) The form (2) of the curvature for the one-link heat-kernel class is a reading of numerics over the registered range of $\beta$ and $t$.

(c) (5)–(7) and $b^{d-4m}$ are estimates from the leading saddle-point term, dropping the higher orders of the fluctuations and the other terms of the sum over images. The constants of the error are not given.

(d) The tube and the attracting curve of the hierarchical model are computer-assisted statements over a finite range of $\beta\le64$; the limit of lattice spacing 0 ($\beta\to\infty$) is not closed. The curvature outside the layer, the weight of the layer and the separation of $c_*=1/60$ have not been confirmed by interval arithmetic.

(e) The hierarchical model confines even at $U(1)$, so what closed here is no test of non-commutativity. $c_*=1/60$ has not been compared with the four-dimensional two-loop coefficient.

Dependency graph (the numbers of premises are smaller than the number of the conclusion):

[01] PRV SEMI   telescoping sum in the heat-kernel class          (one link)
[02] RDG HK1    curvature outside the layer A + B/β ⟸ [01]          (one link, registered range)
[03] RDG MK     invariant curve, 1/16, flow of β ⟸ [01]             (hierarchical lattice, β ≤ 32)
[04] PRV SADDLE source 1/3840, contraction 1/16, 1/3600 ⟸ [01]      (leading saddle-point term)
[05] PRV STRIP  open strip is a one-link convolution ⟸ [01][04]
[06] CMP TUBE   tube invariance, contraction, flow of β ⟸ [03][04]  (β ∈ [2.46, 64])
[07] CMP CURVE  attracting curve ⟸ [06]                             (β ∈ [15.26, 64])
[08] PRV FLOW   flow of the shape coefficient, c* = 1/60 ⟸ [04]     (leading term)
[09] ASM MKUV   β > 64, curvature, weight of the layer ⟸ [02][07][08]  (question)

To check it, see by eye that in every "⟸" the numbers on the right are smaller than the number on the left. There are no cycles. [09] is an assumption row, in the form of a question. Among the nodes of this chapter there is no node saying that the four-dimensional ① has been shown.

What this chapter can say

Sources and reproduction

Terms used in this chapter — the heat-kernel class, semigroup, the telescoping sum, one Migdal–Kadanoff step (bond moving, decimation), the invariant curve, the transverse contraction $\lambda$, the source, balance, the tube, the attracting curve. Their meanings are in the A-02 Glossary.

Next chapter: 15-05 The flux-network theorem — how the two numbers, source and contraction, are fixed in the smallest system with many links.

← 15-03 The decomposition statement — a class that closes with finitely many parameters, and three remaindersContents15-05 The flux-network theorem — the transverse contraction and the parameter flow in one projection →