2026-10-15 · chapter Yang–Mills and the mass gapPart 13 The crossover bandβ-monotonicity, the renormalisation comparison, the two ends and the band · Kiichi, Shiori and Rin

Renormalisation comparison — a comparison that distinguishes groups

The object is the slab of 12-01. On the lattice $(\mathbb Z/N)^2\times(\mathbb Z/M)^2$ we put the $SU(2)$ Wilson action and insert the twist $n_{12}=1$ in the $(1,2)$ plane. We take the flux $n_{34}$ of the $(3,4)$ plane as the sector, and write the vortex free energy as

$\theta$ is the quantity used in 't Hooft's criterion, and the correct form of the band was "$\ln(1/\theta(L,L))\ge cL^2$ uniformly in $L$" (13-01). The larger the vortex free energy, the smaller $\theta$, which is the confining side.

We split the statement that crosses the band into the following two Assumption (both set as unproved statements).

The left-hand side of (2) can be written as an equality, as an average of the difference of actions: $\partial_\beta\Delta F=\langle\Delta S\rangle$ (the difference of the average actions in the sectors with and without a vortex) Proved here (just differentiate the partition function in $\beta$). (3) is the comparison that a vortex on the lattice coarsened by a factor 2 costs no more than a vortex on the lattice of the original size at a stronger coupling.

Together, the two give a line that crosses the band. Iterating (3) as $\beta_0\gt\beta_1\gt\beta_2\gt\cdots$, the vortex free energy on the lattice enlarged $2^k$ times is bounded by $K^k$ times the value on the original lattice at coupling $\beta_k$. Once $\beta_k$ enters the strong-coupling range, the vortex free energy there is small with an area exponent, and it becomes doubly exponentially small with each step, so it beats the loss $K^k$. (2) plays the role of an adjustment: nothing is lost if, in the course of the iteration, $\beta'$ is chosen smaller than necessary.

Asymptotic freedom enters only through $\beta'\lt\beta$ in (3). (2) is a statement of a form that contains no mechanism that distinguishes groups. For $\mathbb Z_2$ and $U(1)$, it follows from Ginibre's inequality when there is no twisted background, but the slab of this chapter has the twist $n_{12}=1$, and in that case it does not follow from Ginibre's inequality even for $\mathbb Z_2$ Assumption (the application of Ginibre's inequality has not been checked against the text of the original source). The direction of (3), on the other hand, changes with the group. For $U(1)$ one loop gives $\beta'\ge\beta$, and the iteration does not head for strong coupling: it fails, correctly. That $\beta'\lt\beta$ for the non-abelian $SU(2)$ is a restatement of asymptotic freedom (4-06), that is, of the effective coupling becoming stronger when the lattice is coarsened. In this sense (3) has to be "a comparison that distinguishes groups".

Example. At the strong-coupling end, (3) holds with $K\approx4$ (§03). Doubling the lattice makes the area of the vortex sheet 4 times larger, and it balances at the $\beta'$ that makes the rate $\sigma(\beta)$ of the area exponent 4 times larger, that is, $\sigma(\beta')=4\sigma(\beta)$. For $\beta=1.0$, $\beta'\approx0.013$ (Table 1). At strong coupling the coupling moves a long way in one step.

Tomboulis (2007) tried to build a comparison of type (3) with Migdal–Kadanoff decimation, and Ito–Seiler and Kanazawa pointed out that the flow heads for strong coupling also for $U(1)$ Cited (Ito–Seiler reduce the remaining step to the inequality $A(\alpha)\gt A^+(\alpha)$ of their eq. (3.5), and write that it has not been shown). The history and the checking are in 13-08. What this chapter needs is just one point: a comparison that does not distinguish groups cannot cross the band.

(Mon) looks like "a structural component that can probably be shown". In fact it is not easy. Rewriting it with the electric flux makes this clear.

Splitting the partition function of the sector with flux $n_{34}$, by the Fourier transform on $\mathbb Z_2$, into the partition functions $Y_0,Y_1$ with electric flux $e\in\{0,1\}$ gives $Z_{n_{34}=0}=Y_0+Y_1$, $Z_{n_{34}=1}=Y_0-Y_1$. Setting $E_e=\frac{d}{d\beta}\ln Y_e$,

Proved here Proof. Differentiate $\Delta F=\ln(Y_0+Y_1)-\ln(Y_0-Y_1)$, insert $Y_e'=E_eY_e$ and bring to a common denominator; the numerator is $(E_0Y_0+E_1Y_1)(Y_0-Y_1)-(E_0Y_0-E_1Y_1)(Y_0+Y_1)=2Y_0Y_1(E_1-E_0)$. The expression for $\theta$ is $\tanh(\Delta F/2)=(e^{\Delta F}-1)/(e^{\Delta F}+1)$ with $e^{\Delta F}=(Y_0+Y_1)/(Y_0-Y_1)$ inserted. ∎

If $Y_1\gt0$ (the positive case of $\theta=Y(1)/Y(0)\ge0$ in 13-04), then from (4)

Proved here. In the confining phase, the free energy of the electric flux is of the order of the product of the string tension and the length the flux passes through. So it can be read that, if (Mon) holds on every lattice, it follows that the string tension in lattice units $\sigma a^2$ decreases in $\beta$ Assumption (basis: the step connecting the electric-flux free energy with the string tension has not been written out). The monotonicity in $\beta$ of the string tension in lattice units has not been proved for $SU(2)$. Even the monotonicity in $\beta$ of the expectation value of the Wilson loop has not been shown, as far as we searched. (Mon) is a problem at least of the same weight as this Assumption. The initial estimate that "(Mon) is structural and can probably be shown" is corrected here.

The partial results to hand are the following Assumption (the proofs of none of them are set out in this book). In two dimensions with 1 or 2 sheets (by the monotonicity of the conditional energy of the $O(4)$ chain); near $\beta\to0$ on any fixed lattice; near $\beta\to\infty$ in the twisted $n_{12}$ background ($\partial_\beta\Delta F$ approaches a positive constant because $SU(2)$ has no flat connection carrying both $n_{12}$ and $n_{34}$). No counterexample has been found in any of the closed two-dimensional formulas ($N\le100$, $\beta\le20$), the exhaustive enumeration of $2\times2\times2\times1$ for $\mathbb Z_2$, or the four-dimensional Monte Carlo on $1^4$ and $2^4$ Reading.

Two reasons why it gets stuck are recorded. In the Mack–Petkova representation, when the $SO(3)$ configuration carries a twist class, the conditional Ginibre inequality goes the wrong way, and $\partial_\beta\Delta F$ becomes exactly "the difference of the two contributions we want to measure". In the character expansion, in three or more dimensions the sign of the spin-network integrals is not fixed Reading (physical estimate).

Consequence. The chain across the band does not need (Mon) on every lattice. If (3) is shown directly in the form $\Delta F(2N,2M,\beta)\le K\Delta F(N,M,\beta')$, the freedom in choosing $\beta'$ absorbs the role of (Mon). So below we concentrate on (RG-cmp).

The strong-coupling end

We write the coefficient of the character expansion of the Wilson action in the fundamental representation as $u(\beta)=I_2(\beta)/I_1(\beta)$ ($u\approx\beta/4$ as $\beta\to0$). Expanding the vortex free energy of the slab by the strong-coupling expansion (5-02), the leading terms are the $N^2$ flat sheets wrapping the $(3,4)$ plane (area $M^2$, weight $u^{M^2}$). The record contains a proposition and a line of proof of the following form.

Proposition S (the strong-coupling end). There are absolute constants $u_0,K_0,C$ such that, for $u(\beta)\le u_0$ and all $N,M\ge2$,

holds. $\sigma_N$ depends on $\beta$, $N$ and $n_{12}$ but not on $M$, and $\sigma_\infty=-\ln u-4u^4+O(u^5)$. Assumption (line: in the polymer expansion the sign of the twist is fixed by the $\mathbb Z_2$ homology class; integrating the flat parts of the sheet by gluing two-dimensional characters gives exactly $u^{M^2}$; a Peierls estimate for contours on the sheet. The bookkeeping of the interaction between volume polymers and sheet contours (step 5 of the proof) is sketched, and the proof is not set out in this book. The form of the expansion is of the Münster (1981) type, but the text of the original source has not been checked)

Corollary (the comparison at the strong-coupling end). Defining $\beta'$ by $\sigma_N(\beta')=4\sigma_{2N}(\beta)$, for all $N,M\ge2$, $\Delta F(2N,2M,\beta)/\Delta F(N,M,\beta')=4(1+O(u))$, $\beta'\lt\beta$, $u'=u^4(1+O(u^4))$ Assumption (algebra on top of Proposition S).

The constants are extremely crude: $u_0\approx10^{-80}$, which in $\beta$ is about $4\times10^{-80}$. On paper, the strong-coupling end is closed only in the range of extremely strong coupling. This is a matter of constants, and there is a prospect of large improvement by tightening the counting, but the expansion in flat sheets stops at roughening ($\beta_R\approx1.9$), so the strong-coupling side alone cannot cover the band Reading (physical estimate). $U(1)$ also confines at strong coupling and satisfies (3). The difference between the groups appears on the weak-coupling side.

The $\beta$ of Table 1 lie outside the range of Proposition S ($u\le u_0$). The table is a rough guide to the direction and size of the comparison from the leading terms only, not a claim that the comparison holds.

The one-loop end

On the weak-coupling side we build (3) as a variational inequality. For a measurable bijection $T$ carrying the measure $\mu_0$ of the sector $n_{34}=0$ to the sector $1$, the Gibbs variational principle gives

Cited (the standard form of the Gibbs variational principle; $J_T$ is the Jacobian of $T$). The right-hand side is written only with averages in $\mu_0$, so the estimate needs only one side. We consider the following construction: block the configuration of the fine lattice $(2N,\beta)$ and map it to the coarse lattice $(N,\beta')$, use a nearly optimal insertion map on the coarse lattice, and carry the fluctuations when going back to the fine lattice. Depending on how the fluctuations are carried, the one-loop answer splits in two.

The one-loop count is done with the trace of the heat kernel of the $SU(2)$ fluctuations (Feynman gauge) in a constant background magnetic field $B$. Per volume, apart from $1/(16\pi^2s^2)$, the vector (4 polarisations) gives $\frac{Bs}{\sinh Bs}(2+2\cosh2Bs)$ and the ghosts give $-2\frac{Bs}{\sinh Bs}$. Inserting $\frac{x}{\sinh x}=1-\frac{x^2}6+O(x^4)$ and $\cosh2x=1+2x^2+O(x^4)$ for $x=Bs$,

and the $+4$ in $\tfrac{11}3=-\tfrac13+4$ comes from the spin term $\cosh2Bs$ Proved here (one-loop level; the expansion is a two-line computation). $\tfrac{11}3$ is the coefficient of asymptotic freedom. The table and the details of the derivation are in 12-07; here we use only the conclusion.

• In an isometric comparison (a map isometric in the fluctuations: the identity map, translation in the group, gauge transformation, Migdal–Kadanoff-type decimation), the estimate (7) is the first-order trace $\tfrac12\operatorname{tr}((M_B-M_0)M_0^{-1})$, and the contribution of the spin term is exactly 0. The estimate is worse than the orbital value alone, so what it can give is at best the $-\tfrac13$ side, that is, $\beta'\ge\beta+0.023$ (per doubling): the direction against asymptotic freedom, the same direction as $U(1)$ Proved here (one-loop level).
• In a comparison that reshapes the fluctuations (changing the shape of the fluctuations by an optimal linear map of determinant 1), the exact $\tfrac12\ln\det$ is reproduced with error $\sim\Delta F^2/n$ ($n$ is the number of degrees of freedom), giving $\beta'=\beta-0.258$ (per doubling) Proved here (one-loop level).

Where the numbers come from: with $\beta=4/g^2$ and $b_0=22/(48\pi^2)$, the one-loop movement of the coupling per doubling is $\beta-\beta'=4\cdot2b_0\ln2=0.2575$. For the orbital part alone, replacing $\tfrac{11}3$ by $-\tfrac13$ gives $0.2575\times(-\tfrac1{11})=-0.0234$ Proved here (arithmetic).

This explains as a number, at the one-loop level, why the method of Tomboulis (decimation = isometric) could not distinguish groups. The mechanism that sees asymptotic freedom lies in the reshaping of the fluctuations. What remains is to control the error in the estimate of the reshaped transport (the local KL near the vortex sheet and the higher loops) at intermediate coupling and in large fields, and this is open.

Always read the statements at the two ends with their conditions. One cannot say "(RG-cmp) holds at both ends, so only the width of the band remains". The strong-coupling side is a line, part of whose bookkeeping is sketched, in the range $u\le u_0\approx10^{-80}$, and the weak-coupling side is a computation at the one-loop level. A non-perturbative weak-coupling comparison has not been shown at any $\beta$.

Figure 1 sets out, over the $\beta$ of the Wilson action, the status of the comparison (RG-cmp) and of confinement interval by interval. The boundaries of the intervals are rough guides, and none of them is a threshold of a theorem.

 β:  0 ─────── 4×10⁻⁸⁰ ──────── 0.3–1 ───── 1.9 (roughening) ─────────────── 2.7 ─────────────── ∞
     │ strong end  │ lower end     │ middle                  │ upper end      │ one loop
     │ ASM         │ open          │ open                    │ ASM (paper 3)  │ PRV (one-loop level)
     │ Prop. S     │ finite-size   │ one non-abelian         │ semiclassical  │ reshaping comparison
     │ (6)         │ criterion,    │ inequality;             │ at fixed N     │ β−β′=0.258
     │             │ computer-     │ the four faces          │                │ isometric: β′≥β+0.023
     │             │ assisted count│ (§07)                   │                │
                           ↑
          numerics [RDG]: SU(2) β′ < β over the whole band (§05); U(1) reverses sign at β_c ≈ 1.01

• The strong-coupling end Assumption: Proposition S. With the constant $u_0\approx10^{-80}$ as it is, the proof cannot be set out in the book, so it is "Assumption".
• The lower end ($\beta\approx0.3$–$1$): open. The candidates are a Dobrushin–Shlosman-type finite-size criterion using the influence $\delta(L_0,\beta)$ on the inside when the centre magnetic flux on the boundary of the box is changed, and a road that shows the growth constant of the admissible sheets by computer-assisted enumeration and raises $u_0$ to $0.04$–$0.12$ (the range of the prospect is a physical estimate). Neither is expected to reach above $\beta\approx1$.
• The middle ($\beta\approx1$–$2.7$): open. One inequality that uses non-commutativity is needed, and there are four candidate local quantities to carry it (§07).
• The upper end (large $\beta$ at fixed $N$): for the direction of confinement there is a paper proof in Theorem A of paper 3, which came out of the study of this book ($-\ln\theta\ge c\beta^{-a}e^{-\beta(S_*+\varepsilon)}M^2$ at fixed $N$). However, it cites the general chessboard theorem and a Morse–Bott lemma, and the proof is not set out in this book, so we label it "Assumption" Assumption. The reverse estimate uniform in $M$, and whether the rate $S_*$ equals the minimal action $S_{\min}$, are open. Increasing $N$ is the same problem as the middle.
• One loop Proved here: (8) of §03 and the two comparisons. This is the perturbative level.

How the width of the band is counted depends on where the lower end is put. The strong-coupling expansion is considered numerically effective up to about $\beta\approx0.3$–$0.5$, and the record writes this band as "roughly 0.5 to 2.7 in the $\beta$ of the Wilson action" Reading (physical estimate). On paper the band is much wider, $[4\times10^{-80},\,2.7]$, because $u_0$ is so small.

We test (RG-cmp) directly with numerics. The procedure is to measure the vortex free energy on the fine lattice and read off, from the $\Delta F$ curve of the coarse lattice, the $\beta'$ of the coarse lattice that gives the same value. Every measurement makes samples that cross sectors by replica exchange and obtains the free-energy difference by Bennett's acceptance ratio (BAR). The errors are statistical; at some points the difference between two runs with different seeds (initial values of the random numbers) is 1–2 times the displayed error, so the errors may come out on the small side. Every number is a reading, not a proof Reading.

The curves on the coarse side are, for $SU(2)$, $\Delta F(4,8,\beta')$ = $0.05,0.00,0.31,0.63,1.43,2.97$ at $\beta'=2.2,2.3,2.4,2.5,2.6,2.7$, and for $U(1)$, $0.42,0.55,0.83,0.92,1.46$ at $\beta'=1.3$–$1.7$. For $SU(2)$, doubling lowers the vortex free energy ($2.97\to0.92$ at $\beta=2.7$), and the matching $\beta'$ is smaller than $\beta$. For $U(1)$ it rises ($0.83\to2.32$, about 3σ), and $\beta'\gt\beta$. The direction of the comparison separated for the two groups. The first $U(1)$ run (32 replicas) had a minimum exchange acceptance rate of 0.03 and mixed badly, and is not used.

The errors are 0.01–0.08 on $2^4$ and 0.03–0.26 on $4^4$. For $SU(2)$, $\beta-\beta'\gt0$ at every $\beta$ that can be read (1.75–3.0), that is, the step function of one step has no fixed point. The closer to strong coupling, the larger the movement in one step, in a form that connects to the strong-coupling values of Table 1. The decrease to 0.07 on the weak-coupling side is likely a lattice artefact, because $2^4$ is too small for the one-loop regime. For $U(1)$, the sign reverses between $\beta=1.0$ and $1.1$. This is at the position of the known phase transition $\beta_c\approx1.01$: in the confining phase $\beta'\lt\beta$, in the Coulomb phase $\beta'\gt\beta$.

For the $4^4$ curve we also used the points $\beta=2.6$ ($3.96\pm0.13$, 1 seed), $2.75$ ($5.18\pm0.10$) and $2.8$ ($5.60\pm0.27$). For $4^4\to8^4$ too, $\beta-\beta'\gt0$ at every $\beta$ measured, with a mean of about $+0.2$. This is close to the one-loop asymptotic value 0.258 and to the value of about 0.23 expected from the known values of $a\sqrt\sigma$ Reading.

How to read this. The test of the sign of the step function distinguishes whether there is a phase transition. $U(1)$ reverses at the position of the transition, and $SU(2)$ does not reverse in the range measured. This is the numerical face of criterion (m) of §08. However, the lattices are small, up to $8^4$, the $\beta$ points are few, and there are only 1–3 seeds. "For $SU(2)$ the sign does not change over the whole band" is a reading on these lattices and points.

The band is a bounded closed interval $I=[\beta_a,\beta_b]$ in $\beta$. A comparison that crosses infinitely many scales shrinks to finitely many steps on a bounded interval. We write this in the form of the following lemma.

Assumption (A) (the two-sided area law at fixed $\beta$ on the band $I$). There are constants $N_1,M_0\ge2$, $c_-,c_+\gt0$, $p\ge0$ and a function $\sigma_N(\beta)\gt0$ such that

• (A1) for all $\beta\in I$, $N\ge N_1$, $M\ge M_0$, $\;c_-N^2e^{-\sigma_N(\beta)M^2-pM}\ \le\ \Delta F(N,M,\beta)\ \le\ c_+N^2e^{-\sigma_N(\beta)M^2+pM}$.
• (A2) With $\sigma_*:=\inf_{N\ge N_1,\ \beta\in I}\sigma_N(\beta)\gt0$, for some $\delta\gt0$: if $\beta,\beta'\in I$ and $|\beta-\beta'|\le\delta$, then for all $N\ge N_1$, $\sigma_N(\beta')\le\sigma_{2N}(\beta)+\sigma_*$.

The upper side of (A1) is the area law (with an error $pM$ of the order of the perimeter), and the lower side is that "vortices are not too cheap", that is, an upper bound on the string tension. (A2) says that the rate is uniformly positive on the band and does not change much when $\beta$ is moved a little.

Lemma 1 (reduction of the comparison inside the band). Under Assumption (A), for $\beta\in I$, $\beta'\in[\beta-\delta,\beta)\cap I$, $N\ge N_1$, $M\ge M_0$,

holds. $K$ does not depend on $N$, $M$ or $\beta$.

Proved here Proof. Since $2N\ge N_1$ and $2M\ge M_0$, the upper side of (A1) gives $\Delta F(2N,2M,\beta)\le c_+(2N)^2e^{-\sigma_{2N}(\beta)(2M)^2+p(2M)}=4c_+N^2e^{-4\sigma_{2N}(\beta)M^2+2pM}$. The lower side gives $\Delta F(N,M,\beta')\ge c_-N^2e^{-\sigma_N(\beta')M^2-pM}$. Setting $\eta:=4\sigma_{2N}(\beta)-\sigma_N(\beta')$, (A2) gives $\eta\ge3\sigma_{2N}(\beta)-\sigma_*\ge2\sigma_*$ (using $\sigma_{2N}(\beta)\ge\sigma_*$). The ratio is

The second inequality is the maximum $9p^2/(4\eta)$ of the quadratic $-\eta M^2+3pM$, and the third is $\eta\ge2\sigma_*$. ∎

Corollary. The band can be crossed in at most $\lceil(\beta_b-\beta_a)/\delta\rceil$ steps, and the loss over them is $K$ to the power of the number of steps, a constant independent of $N$ and $M$ Proved here. The same proof goes through if $\beta'$ is lowered further, as long as it satisfies $\sigma_N(\beta')\le4\sigma_{2N}(\beta)-2\sigma_*$.

Calibration. Proposition S at the strong-coupling end ((6)) gives (A1) with $p=0$ and $c_\pm=2(1\pm C'u)$. Then (9) holds with $K=4(1+C'u)/(1-C'u)$, which agrees with $4(1+O(u))$ of the corollary in §03 Proved here (Proposition S is an assumption, so the agreement is an agreement of form).

What Lemma 1 says. (RG-cmp) inside the band reduces to "confinement at fixed $\beta$ (two-sided asymptotics with an error of the order of the perimeter)" at each $\beta$ of the band. The comparison across scales itself is not needed. The band is entered when the physical box is large, so (A1) is needed only in the range where $N,M$ is sufficiently larger than the correlation length $\xi(\beta)$.

What Lemma 1 does not say. Assumption (A) is unproved at every $\beta$ of the band. The upper side is itself the unsolved problem of confinement at intermediate coupling on the four-dimensional $SU(2)$ lattice. As far as we searched, there is no proof of confinement on the four-dimensional $SU(2)$ lattice at $\beta\approx1$. The lower side (an upper bound on the string tension to the accuracy of the order of the perimeter) has not been written out either. The lemma does not depend on the group; all the non-commutativity is in Assumption (A). For $U(1)$ ($\beta\in(1.01,2.7)$), Assumption (A) is false, so no conclusion follows. This lemma is a reduction, not a tool, and we do not say it has made the band easier.

Where the difficulty gathers Reading (physical estimate). After Lemma 1, what remains inside the band are problems at fixed $\beta$ (estimates uniform in the volume, non-convex large fields, the absence of a phase transition). The difficulty of crossing scales does not vanish from inside the band; it gathers at the upper end of the band and at the seam with ultraviolet control. At the upper end of the band, $\xi(\beta)$ exceeds 10 in lattice units, and the road of checking (A1) by computations in finite boxes stops on size. And beyond the upper end of the band ($\beta\to\infty$) is the range taken on by ultraviolet control (6-07), and the two must be joined with the same constant. In terms of the three roots of 11-07 §05, inside the band it is (R3) (the non-convexity of large fields) and excluding a phase transition, and the seam at the upper end is (R2) (there is no expansion that joins the ultraviolet and the infrared).

Example. On the strong-coupling side ($u\le u_0$), a statement of exactly the form (A) exists as Proposition S (at the level of a line of proof). The work of pushing the lower end of the band up to $\beta\approx0.3$–$1$ (the lower end of §04) is, in the words of Lemma 1, "widening the interval on which (A1) holds", and needs no comparison across scales.

It has been narrowed down to this: crossing the band in the middle needs one inequality that uses non-commutativity. On which local quantity to write it in, there are four candidates. We call them "faces" in the sense that it is enough to show any one of them uniformly in the scale, in a form that fails for $U(1)$. Table 5 lists only the names and the expected behaviour; the detailed examination is in 13-10. Every row is an estimate Reading (physical estimate).

A numerical example. Measuring the column face $f(\ell)$ by the twist bias $\mathbb E|\theta_B|$ on $T^2_4\times\ell\times\ell$ with the outer frame held fixed, $SU(2)$ ($\beta=2.3$) gives $0.23,0.07,0.09$ at $\ell=4,5,6$, and the Coulomb phase of $U(1)$ ($\beta=1.3$) gives $0.61,0.49,0.41$ ($0.25$ at $\ell=8$); for $\ell\ge5$, $SU(2)$ is 1/5 to 1/7 of $U(1)$ Reading. However, the $SU(2)$ values are close to the statistical floor, and at this size $e^{-\sigma\ell^2}$ cannot be told apart from a power.

For each of the four faces, the comparison across scales (for example $f(2\ell,2N,\beta)\le Kf(\ell,N,\beta')$) is a local version with the same difficulty as the global (RG-cmp), and has not become easier. What changing the face gains is that the object of the proof becomes "a statement about a finite region with its boundary held fixed".

Criterion (m). $U(1)$ has, inside this band ($\beta\approx1.01$), a phase transition to a non-confining phase. An argument that closes the band must show that $SU(2)$ has no phase transition, distinguishing it from $U(1)$. This is of the same weight as the unsolved problem that "four-dimensional $SU(2)$ has no transition to a non-confining phase at zero temperature", and proposals that do not satisfy this condition are discarded from the start.

The sign of the step function in §05 was the numerical face of this criterion. Put another way, a proof that closes the band is a proof that the direction $\beta'\lt\beta$ of the comparison does not change over the whole band for $SU(2)$. Purely structural arguments (reflection positivity, duality, the transfer matrix) hold in the same form for $\mathbb Z_2$ and $U(1)$, so they alone do not satisfy the criterion (13-04). Renormalisation that does not distinguish groups (the isometric comparison, the Migdal–Kadanoff flow) does not satisfy it either (§03).

Even without closing the whole band at once, there are intermediate goals that satisfy criterion (m).

1. $\Delta F(L,L)\to0$ (at any speed). In the Coulomb phase of $U(1)$, $\Delta F(L,L)$ tends to the classical half-flux value $\pi^2\beta_R/2$ (with the renormalised $\beta_R$), and in the non-confining phase of $\mathbb Z_2$ it increases, so showing this would exclude the known non-confining phases Assumption (the behaviour of the phases is an estimate). It does not give the mass gap. Numerically, for $U(1)$ ($\beta=1.2$), $2^4\to4^4$ is flat at $3.10\to3.68$, and for $SU(2)$ ($\beta=2.75$) it is $5.95\to5.10$ Reading.
2. $\ln(1/\theta(L,L))\ge cL^2/\ln L$ (nearly the area law). Through the Tomboulis–Yaffe inequality Cited, the Wilson loop decreases faster than the perimeter law (of the type $\langle W\rangle\le2\exp(-cA/\ln A)$).
3. The band itself: $\ln(1/\theta(L,L))\ge cL^2$ (13-01).

(a) We show neither (RG-cmp) nor (Mon) at any $\beta$ of the band.

(b) The strong-coupling end is a line, part of whose bookkeeping is sketched, in the range $u\le u_0\approx10^{-80}$. The weak-coupling end is at the one-loop level, not a non-perturbative claim. Do not read "it holds at both ends" with these two conditions removed.

(c) Lemma 1 is a reduction, and Assumption (A) contains an unsolved problem. We do not say that Lemma 1 made the band easier.

(d) The numbers of §05 are readings on small lattices up to $8^4$, with few $\beta$ and 1–3 seeds, and are not evidence that there is no phase transition.

(e) The four faces of Table 5 are candidates, and none of them has an estimate uniform in the scale.

(f) The relation between (Mon) and the monotonicity of the string tension includes a step that is a reading (§02). Nor do we say that (Mon) is false.

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

[01] PRV DER    ∂_β ΔF = ⟨ΔS⟩
[02] CIT GIBBS  Gibbs variational principle (7)
[03] PRV MON    (Mon) ⇔ E₁ ≥ E₀ ⇔ θ increases ⟸ [01]                    Y₁ > 0
[04] ASM HARD   (Mon) weighs as much as β-monotonicity of the tension ⟸ [03]
[05] ASM PART   partial results on (Mon)
[06] ASM STRG   Proposition S (strong-coupling end)                     u ≤ u₀ ≈ 10⁻⁸⁰
[07] ASM RSTR   comparison at the strong-coupling end, K ≈ 4 ⟸ [06]
[08] PRV HK     11/3 = −1/3 + 4                                         one loop
[09] PRV ISO    isometric comparison gives β′ ≥ β + 0.023 ⟸ [02][08]    one loop
[10] PRV SHP    reshaping comparison gives β′ = β − 0.258 ⟸ [02][08]    one loop
[11] ASM THA    Theorem A of paper 3                                    fixed N
[12] RDG DBL    direction of (4,8)→(8,16) separates by group
[13] RDG STEP   sign of the step function: SU(2) unchanged, U(1) reverses
[14] PRV RED    Lemma 1 (under Assumptions (A1)(A2))
[15] PRV FIN    the band takes finitely many steps ⟸ [14]
[16] PRV CAL    calibration ⟸ [06][14]                                  agreement of form
[17] RDG LOC    difficulty at the upper end and the UV seam ⟸ [14]
[18] RDG FACE   the four faces ⟸ [09][10][12][13]
[19] RDG CRIT   criterion (m) ⟸ [13]
[20] ASM MID    ΔF(L,L) → 0 excludes non-confining phases ⟸ [19]

The check is to see by eye that every number to the right of a "⟸" is smaller than the number on the left. There are no cycles. Assumptions (A1)(A2) are written as conditions in row [14] and are not made nodes, because the assumption rows do not contain the quantity of the conclusion (the comparison that crosses the band). Among the nodes of this chapter there is no node for the comparison at any $\beta$ of the band, none for confinement, and none for the mass gap.

What this chapter can say

Sources and reproduction

Terms used in this chapter — the crossover band, β-monotonicity (Mon), the renormalisation comparison (RG-cmp), the isometric comparison, the comparison that reshapes the fluctuations, the step function, the four faces, criterion (m), the reduction lemma. Their meanings are in A-02 Glossary.

Next chapter: 13-07 The band seen from the side of time — the statement corresponding to the band on the road of Part 10, and why the first-step wall "15/4 against 4" is a miniature of the band and not the band.

← 13-05 The chessboard and simultaneous flipsContents13-07 The band seen from the side of time →