2026-10-15 · chapter Yang–Mills and the mass gapPart 15 Ultraviolet controlparameters, loop shapes, the attracting class · Kiichi, Shiori and Rin

Counting the parameters — couplings by loop shape and the attracting class

• The coordinates for counting the parameters of (Cls) are not "the coefficient of $F^2$ and the commutator coefficient" but the couplings by loop shape. The commutator coefficient follows afterwards as the BCH image of the coupling of the composite loop (§02, §03).
• For four plaquettes in a row the second coordinate (the ratio $\kappa_{ab}/\kappa_a$) moves with $O(1/\beta)$, and part of it, $2/(3\beta)$, is due to the curvature of the group and does not vanish even if the action is replaced by the heat kernel (§04, §05).
• Keeping the ratio at 2 for all $\beta$ requires an action whose quartic term in the exponent is steeper than Gaussian ($q/B=-1/12$); the family of powers of the prefactor has no root, and the fundamental + adjoint family has a root at $\beta\ge12$ (§06).
• That root curve does not close under the one-step flow, and the deviation stays at $O(1)$ ($9/89$), because the one-step map contracts all the coordinates of the shape of the action (§07).
• The power of the prefactor follows a linear contraction map and is drawn to the unique fixed point $\gamma^*=37/76$ ($\psi^4$), where the ratio is $2+(2/3)/B$. (Cls) can be stated as "the one-parameter attracting class + a contracting shape" (§08, §09).

Conventions: $SU(2)$ is written with unit quaternions, $\frac12\operatorname{tr}U=\operatorname{Re}U$, background $W=\exp(i\,a\cdot\sigma/2)$ ($a\in\mathbb R^3$, angle $\theta=|a|$). The numbers were measured at $\beta=4$–$32$, $\theta\le0.3$.

On a single plaquette (four links), consider the measure $\mu_W$ of the fluctuations around the background holonomy $W$. It is the smallest model of the $\eta_k$ of 15-07. Of the 12 dimensions of the four links, the conditional law of the 9 gauge directions does not depend on $W$, and the background dependence gathers in the 3 dimensions of the holonomy Proved here (invariance of the Haar measure). Placing on the fluctuation $X=i\,x\cdot\sigma/2$ ($r=|x|$) the constraint $x_3=0$ that removes the direction of the generator of the background (the smallest substitute for the background-field constraint), we have $\frac12\operatorname{tr}(We^X)=\cos\frac\theta2\cos\frac r2$, so

holds exactly Proved here. The background dependence goes entirely into a single replacement of the coupling $\beta\mapsto\beta\cos(\theta/2)$. The $\eta$ measured by the ratio of covariances is $\theta^2/8$, and the score is $\theta/4$; neither depends on $\beta$ (main term Proved here; by two routes, quadrature and Monte Carlo, the ratio to the main term is 0.98–1.00 Reading).

• Without the constraint it is not small. If the background is not taken at a critical point, the ratio of covariances grows with $\beta$ as $\beta\theta^2/12$ Reading. From this one plaquette we read that the minimal condition for the background dependence to be small uniformly in the background is "the background is a constrained critical point".
• The one-step map of the coupling. The weight of the background with the fluctuations integrated out is, under the constraint, $\beta_{\rm eff}=\beta-d/2+O(1/\beta)$ ($d$ is the dimension of the fluctuations; $\beta-\beta_{\rm eff}=1.005$ at $d=2$, $\beta=64$) Reading (Proved here as a count of the dimensions of the fluctuations, at the one-loop level).
• The source of $\eta$ is not non-commutativity. In the $U(1)$ control it is $\eta=0$ only for the quadratic (Gaussian) action, and for the $\cos$ action a $\eta$ of the same $\theta^2$ type appears Reading. The source is the non-quadratic nature of the action (the compactness of the group). Measuring $\eta$ by a single number changes the type (the ratio of covariances is $\theta^2$, the Kullback–Leibler quantity is $\theta^4$, the score is $\theta$), so the statement must specify "which $\eta$" Reading.

On a $2\times1$ rectangle with one shared link, fix the outer loop $W=P_1P_2$ ($P_1,P_2$ are the holonomies of the two plaquettes, independent Haar after the tree gauge). The constrained minimum is $P_1=P_2=W^{1/2}$, in which the two plaquettes tilt by half each in the same direction; setting $P_1=W^{1/2}e^X$, $P_2=e^{-X}W^{1/2}$,

holds exactly Proved here. The action is even in the fluctuations, and the odd cubic term that appeared when the substitute of "subtracting the first-order term" was used for one plaquette vanishes at the true constrained minimum (numerically the odd part of the score is $\le3\times10^{-12}$) Reading. The weight with the fluctuations integrated out is a function $Z_1(\beta|1+W|)$ of the class of $W$ alone, $Z_1(b)=2I_1(b)/b$ (the same integral as the one-link distribution of the heat-bath method) Proved here, and the coupling after one step is $\beta_{\rm eff}=\beta-\frac34+O(1/\beta)$ (Proved here, at the one-loop level; $\beta-\beta_{\rm eff}=0.723\to0.747$ at $\beta=4\to32$ Reading). The commutator of the field strengths $a,b$ of the two background plaquettes enters only through the BCH of the group product $e^{F_1}e^{F_2}$:

Proved here (algebra under truncation of BCH at third order). So the coefficient $c$ of $|a\times b|^2$ in the weight is tied to $1/12$ of the coefficient $k$ of $F^2$, and $c/k=1/12$ holds to $10^{-8}$ for $\beta=4$–$32$ Reading. The commutator is not a new parameter at this step.

With three in a row, fix the two overlapping $2\times1$ loops $W_a=P_1P_2$, $W_b=P_2P_3$ and integrate over the central $P_2$; the weight is then, with $Z_1(\beta|Q|)$, $Q=1+W_a+W_b$,

Proved here (agrees with direct quadrature to $3.5\times10^{-15}$ Reading). It contains the two single loops and the composite loop $W_aW_b^{-1}$ in the ratio 1:1:1. The invariants of simultaneous conjugation of a pair of elements of $SU(2)$ are determined by the three traces $\operatorname{Re}W_a,\operatorname{Re}W_b,\operatorname{Re}(W_aW_b^{-1})$ Proved here (two imaginary-part vectors are determined up to rotation by their lengths and inner product), so every gauge-invariant two-loop weight is a function of the three class angles. Taking the class angle of the composite loop as the reference, the coefficient of the commutator is necessarily $1/12$ of the quadratic coupling $\kappa_{ab}$ of the composite loop Proved here ((3) enters as it is). The coupling after one step is $12k=\beta-\frac12+O(1/\beta)$ Reading. The "commutator coefficient" measured in the basis of monomials appears to move with $\beta$, but it appears with the same size even for a composite function with no commutator, so it is not a quantity of the commutator Reading.

With four in a row, fix the two overlapping $3\times1$ loops $W_a=P_1P_2P_3$, $W_b=P_2P_3P_4$ and integrate over the two plaquettes inside. Setting $U=P_2P_3$,

holds Proved here. $w$ is a function of the two variables $\operatorname{Re}W_a+\operatorname{Re}W_b$ and $\operatorname{Re}(W_aW_b^{-1})$ and does not collapse to the single variable $|Q|$ of the three-plaquette case (on the same $|Q|$ background, $\log w$ moves by $1.8\times10^{-2}$, $\beta=4$) Reading. The quadratic couplings are defined by

($\theta_{ab}$ is the class angle of the composite loop $W_aW_b^{-1}$). The tree (the commuting Gaussian minimisation) gives the ratio $\kappa_{ab}/\kappa_a=2$ Proved here, and the main term at one loop is $-\frac1{40}$ for both $\kappa_a,\kappa_{ab}$, with ratio 1. Together,

(the first two are $+O(1/\beta)$) Proved here, at the one-loop level. Numerically $\kappa_{ab}/\kappa_a=2.3211,\ 2.1409,\ 2.0662,\ 2.0322$ ($\beta=4,8,16,32$, the two routes agree to $7.8\times10^{-12}$) Reading. The reason the ratio moves is "the tree shape ≠ the one-loop shape". The quadratic form of the tree is 1:2, fixed by the geometry of the lattice, and the quadratic form at one loop is 1:1, fixed by the measure; since the two are not parallel, the direction of the sum rotates with $\beta$. For the $\cos$ action of $U(1)$ too the ratio moves with $2+0.2/\beta$, while for the Villain action of $U(1)$ it is 2 for all $\beta$ Reading. That one step of blocking produces terms of larger loops whose coefficients depend on the coupling is, we think, a known type in the same direction as the idea of improved actions (the original sources have not been checked).

If the weight of each plaquette is replaced by the heat kernel $p_t$ of $SU(2)$ (character coefficients $(n+1)e^{-tn(n+2)/2}$), the two inner plaquettes combine by the semigroup property $p_t*p_t=p_{2t}$ into

Proved here. Even so the ratio is $2.2090,\ 2.0927,\ 2.0439,\ 2.0214$ ($t=1/4,\dots,1/32$, the two routes agree to $7.6\times10^{-12}$) Reading, and at one loop $2+\frac{2t}3+O(t^2)$ Proved here, at the one-loop level. For the heat kernel of $U(1)$ (Villain) it is 2 for all $t$ Proved here.

The reason can be written as a one-line criterion. Let the outer kernel be $e^{F(\psi)}$ and the inner kernel $G(\psi)$ (both class functions; $\psi$ is the angle of a unit quaternion), and write the expectation with respect to the measure $\sin^2\psi\,G\,e^{2F}d\psi$ as $E$; then integration by parts gives

Proved here (under the conditions that $F,G$ is a smooth class function for $[0,\pi]$ and that the boundary terms vanish). A pointwise sufficient condition is that the inner kernel is the square root of the outer one, $G\propto e^{F/2}$. For a flat Gaussian the semigroup $p_{2t}$ and the square root $p_t^{1/2}$ are the same thing, so the ratio stays at 2; for a curved group the two split. The heat kernel of $SU(2)$ is a semigroup, but the Van Vleck prefactor $P(\psi):=\psi/\sin\psi$ enters $p_t$ and $p_{2t}$ with the same power, so it is not a square root. The Gaussian in the geodesic distance $e^{-\psi^2/(2t)}$ satisfies the square-root relation but is not a semigroup Proved here (numerically too the ratio is 2 for all $t$, $3\times10^{-12}$ Reading). Matching $t$ and $\beta$ by the coefficient of the fundamental representation, the main term of the deviation of the ratio is $1/\beta$ for Wilson and $2/(3\beta)$ for the heat kernel. The $2/(3\beta)$ that does not vanish even for the heat kernel is the part due to the curvature of the group, and the remaining $1/(3\beta)$ is the part due to the non-quadratic nature of the action, we read Reading (main term, this system, quadratic order).

Is there an action that satisfies the criterion (9) for all $\beta$ (an action that closes with one parameter for four plaquettes in a row)? Writing an action concentrated at the identity as $\log f=\text{const}-B\psi^2/2+q\psi^4+\cdots$ ($B$ is the tree coupling, $q=O(B)$) and counting the correction of the inner kernel by the Brownian-bridge average (the flat approximation), we get

Proved here, at the one-loop level. $\frac25$ comes from the curvature of the group (the Van Vleck prefactor $\log P=\psi^2/6+\cdots$), and $\frac{24}5q/B$ from the quartic term of the exponent (non-Gaussianity). A multiplicative correction of $\psi^2$ is absorbed into a shift of the tree coupling $B$ and does not enter $D$ Proved here. Calibration: for Wilson ($q=\beta/24$) $D=\frac35$ and ratio $2+1/\beta$; for the heat kernel ($q=0$) $D=\frac25$ and ratio $2+2t/3$; for $U(1)$, $D=\frac{24}{25}q/B$ ($\frac1{25}$ at $\cos$, 0 for Villain). All the values of §04 and §05 are reproduced Reading.

• The family of powers of the prefactor has no root. For $f=e^{\beta\cos\psi}(\psi/\sin\psi)^\gamma$, in the range $\gamma\lt\frac12$ where the quadratic coupling exists, $D\in[0.534,0.600]$ at every point of the measured grid $\gamma\in[-2,0.45]$, $\beta\in\{4,8,16,32\}$, and $D\ge0.343$ also on the extension $\gamma\ge-128$; it is monotonically increasing in $\gamma$, with no change of sign Reading (the two routes agree to $\le6.5\times10^{-9}$). At one loop too, $D\gt8/25$ for all $\gamma=c\beta$ ($c\lt3$) Proved here, at the one-loop level. At $\gamma\ge\frac12$ the singularities of $\psi=\pi$ overlap and the quadratic coupling itself does not exist, and at $\gamma\ge\frac32$ the weight itself diverges Proved here. The reason is that the power of the prefactor is absorbed into $B$ by the correction of $\psi^2$, and the curvature term $\frac25$ does not move.
• The fundamental + adjoint family has a root. From (10), $D=0$ requires $q/B=-\frac1{12}$ (a quartic term steeper than Gaussian). $F=\beta\cos\psi+\rho\beta\cos^2\psi$ is $B=\beta(1+2\rho)$, and with $q=\beta(\frac1{24}+\frac\rho3)$ $$D=\frac25+\frac15\,\frac{1+8\rho}{1+2\rho}+O\Bigl(\frac1B\Bigr),\qquad \text{one-loop root}\ \rho=-\tfrac14 \tag{11}$$ Proved here, at the one-loop level. The numerical root is $\rho^*(\beta)=-0.318,\,-0.279,\,-0.264,\dots,-0.2516$ ($\beta=16,32,64,\dots,512$; ratio 2 to $3\times10^{-10}$ by bisection and two routes), $\rho^*=-\frac14-0.84/\beta+O(\beta^{-2})$ Reading. For $\beta\le10$ there is no root on the measured grid (minimum $D=+0.029$), and the root appears between $\beta=10$ and 12 Reading.
• One parameter for one step only. In neither family does $f*f$ belong to the same family (the residual of the fit is at least $10^5$ times that of the heat-kernel control) Reading. Even if the root action is taken, the inner kernel after one step leaves the family, so the same criterion cannot be used at the next step. The pointwise square-root relation $f*f\propto f^{1/2}$ cannot be satisfied at the one-loop level by a concentrated class function Proved here (the coefficient $-\frac13$ of $\psi$ remains).

We check whether the root curve $\rho^*(\beta)$ returns to itself under the one-step map. The outer kernel after one step is defined on the diagonal as $f'(\psi):=w(W,W)^{1/2}=[(f*f)*f^2]^{1/2}(W)$ (the diagonal definition: at $W_a=W_b$ the angle of the composite loop is 0), refitted to the fundamental + adjoint family by the Taylor coefficients of $\psi^2$ and $\psi^4$, and $(\beta',\rho')$ is read off.

The tree map. With the outer kernel $F=-Bx^2/2+qx^4+\cdots$, at the flat, collinear saddle point (on the geodesic joining $1$ and $W$, $u_0=4w/5$),

holds Proved here, tree (main term of the saddle point). Because the saddle point is collinear, the commutator and the curvature of the group do not enter the tree, and enter only the one-loop $O(1/B)$. Combined with (10), the image of the ratio-2 curve $q/B=-\frac1{12}$ is $q/B=-\frac{17}{1500}$, and for fundamental + adjoint

holds Proved here, tree (algebra of substitution). Numerically, $\rho'-\rho^*(\beta')=0.160,\ 0.127,\ 0.113$ (starting from $\beta=128,256,512$, $\beta'=17.8,36.1,72.5$) approaches $9/89$ as $O(1/\beta')$ Reading (the two routes agree to $\le6\times10^{-9}$ at $q'$). Adding a third coordinate $\sigma\beta\cos^3\psi$ does not bring $D$ back to 0 ($\sigma$ enters $D$ only at $O(1/B')$) Reading. One step contracts the shape coordinate $q/B$ and draws the flow towards the Gaussian side. For $\mathrm{SU}(2)$ the ratio-2 condition lies at a point $O(1)$ away along the contracting direction, so the curve does not close, we read. For $U(1)$ the ratio-2 line $q/B=0$ is itself the fixed point of the tree map (Villain), so it closes Reading.

Then where does the flow draw to? Write the class function as $f=\exp(-B\psi^2/2+h(\psi))$ ($h=O(1)$) and follow one step by the main term of the Laplace method with the argument held at $B\to\infty$. Regarding $SU(2)$ as the $S^3$ of radius 1, the saddle point of the convolution of two kernels lies on the geodesic joining 1 and $W$, and the determinant of the Hessian in the two transverse directions gives the ratio of the prefactors $P$:

($a,b$ is the split at the saddle point) Proved here, main term of Laplace. Writing $g:=h-\log P$, $g$ is carried by convolution in the same way as the flat tree. This is why the heat kernel ($g=0$) is preserved; it is a restatement of the semigroup property. The product (the square at the same point) leaves one prefactor over, and the square root halves it. Substituting one step $f'^2=(f*f)*f^2$, the power of the prefactor $\gamma_k:=h_k/p_k$ read from $\psi^{2k}$ ($\log P=\sum p_k\psi^{2k}$, $p_2=\frac1{180}$) follows

a linear contraction map Proved here, main term of Laplace (algebra of the coefficients). The unique fixed point is

Proved here. The heat-kernel power 1 moves to $313/625$ in one step, which explains why the power of the prefactor appears to change from 1 to about $\frac12$. The contraction rate is $\frac{17}{625}$ ($\psi^4$), not $\frac12$, and $\frac12$ is the value at the fixed point. Numerically: the extrapolation of $\gamma'_2,\gamma'_3$ from $B=32$–$256$ agrees with (15) to $\le10^{-7}$; in the iteration holding $B$, four starts ($\gamma=0,\frac12,1,2$) and the start of $q/B=-\frac1{12}$ gather to the same point with ratio $17/625$; and the one-step extrapolation including the $h^*$ of (16) gives $37/76$ and $1.4\times10^{-8}$ Reading (two routes: the character series and a one-dimensional multiple integral).

The ratio at the attracting class. Putting $(q/B)^*=O(1/B)$ into (10), on the attracting class

$\kappa_{ab}/\kappa_a=2+c/B+O(B^{-2}),\qquad c=\dfrac{2/5}{3/5}=\dfrac23$

Proved here, at the one-loop level (numerical extrapolation 0.666667, two routes $1.8\times10^{-9}$ Reading). $c$ does not depend on $h^*$ and is the same as for the heat kernel; in the main term of the quadratic ratio the attracting class cannot be distinguished from the heat kernel. For $U(1)$ there is no inhomogeneous term, and the fixed point is Villain ($h=0$), $c=0$ Proved here.

Putting §01–§08 together Reading (limited to four plaquettes in a row, two loops, the diagonal definition and the main term): the coordinates of the parameters of (Cls) are the couplings by loop shape, and the commutator is their BCH image. One step contracts all the shape coordinates (Table 2), and the family is drawn to "one $B$ + the shape $h^*$ of $O(1/B)$ fixed by the curvature of the group". The ratio 2 is not a coordinate that can be tuned by a parameter, but a one-loop quantity that becomes $2+2/(3B)$ at the attracting class. So stating (Cls) not as "the class that keeps the ratio 2" but as "the one-parameter attracting class + a contracting shape carried by (Rem-shape)" is what is consistent with this model. If we assume that the shape deviation $r_n$ satisfies $r_{n+1}\le\lambda r_n+C/B_n$ ($B_n=B_0/5^n$) in one step, then $r_n\le\lambda^nr_0+C/((5-\lambda)B_n)$, and it closes in the geometric-series form of 15-03 §04 Proved here (the one-step inequality is an assumption; the existence of the constant $C$ has only been seen numerically).

(a) The systems in this chapter are one plaquette up to four in a row of $SU(2)$, $\theta\le0.3$, quadratic order in $F$ (§07 and §08 go up to the quartic Taylor coefficients); these are not the numbers for one step including blocks embedded in the four-dimensional lattice, gauge re-fixing, or large fields.

(b) "Two coordinates are needed at the quadratic level" is a reading for four in a row with two loops fixed; we have not looked at the quartic order (the direction of the commutator) or at systems with three or more block variables.

(c) That the family of powers of the prefactor has no root is a statement of numbers on the measured grid and of an inequality at the one-loop level, not a statement about all actions. The fundamental + adjoint root is a number at $\beta\ge12$, and the coefficient of $-0.84/\beta$ is an extrapolation.

(d) The maps of §07 and §08 are maps for the diagonal definition $f'=[(f*f)*f^2]^{1/2}$ and the main term ($B\to\infty$); mixed quartic terms do not enter, and they cannot be used in the strong-coupling region of $B=O(1)$. The one-step inequality of (Rem-shape) is an assumption.

(e) We say neither "there is no action that closes with one parameter" nor "(Cls) closes with finitely many parameters". What we say is a main-term statement: in this model all the shape coordinates contract, and the attracting point is determined as a single one.

(f) This chapter does not show ultraviolet control ①.

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

[01] PRV ONE    one plaquette: (1)                           constraint x₃ = 0
[02] RDG BEFF   β_eff = β − d/2 ⟸ [01]
[03] PRV BCH    two, three plaquettes: commutator is the image of the composite loop ⟸ [01]
[04] PRV FOUR   four plaquettes: the ratio moves with β ⟸ [03]   (one-loop level)
[05] PRV CRIT   heat kernel 2 + 2t/3, criterion (9) ⟸ [04]
[06] PRV D1     general one-loop formula (10)(11) ⟸ [05]   (concentrated actions)
[07] RDG ROOT   no root in prefactor family; fund. + adj. root ⟸ [06]
[08] PRV TREE   tree map (12), deviation 9/89 (13) ⟸ [06]   (diagonal definition)
[09] RDG GAP    numbers for the deviation ⟸ [07][08]
[10] PRV FIX    fixed point of the prefactor power 37/76, c = 2/3 ⟸ [06][08]   (main term of Laplace)
[11] RDG FIXN   numbers for the fixed point ⟸ [10]
[12] ASM CLS    statement of the attracting class, F-UV8, F-UV9 ⟸ [09][10][11]

Every number to the right of "⟸" is smaller than the one on the left, and there are no cycles. The only assumption row is [12], the form of the statement and the questions.

What this chapter can say

Sources and reproduction

Terms used in this chapter — couplings by loop shape, the composite loop, the criterion, the curvature of the group, the Van Vleck prefactor, one parameter for one step only, the diagonal definition, the tree map, the power of the prefactor, the attracting class, (Cls). Their meanings are in the A-02 glossary.

Next chapter: 15-07 Background-dependent fluctuations — the $\eta$ of one plaquette in §01, as a problem of absorption at each step and of induction over many steps.

← 15-05 The flux-network theorem — the transverse contraction and the parameter flow in one projectionContents15-07 Background-dependent fluctuations η — absorption step by step →