2026-10-01 · chapter Yang–Mills and the mass gapPart 8 A dictionary of two vocabulariesscale and limits

Dictionary C: scale and limits — words of flow translate into maps, words of limits into questions of existence

The words fall into three groups. Words of flow (renormalisation group, β function, $\Lambda$) translate into properties of the maps of the steps. Example: "the β function is negative" becomes "a step that doubles the lattice spacing lowers $\beta$ by an almost constant amount on the weak-coupling side" (§02, §03).

Words of limits (continuum limit, $a(\beta)$, universality) translate into the question "does the limit exist, and is it unique?". Example: "universality" is the uniqueness claim "changing the details of the action leads to the same limiting measure" (§05, §06).

Words of geometric scale ($N$, an extra direction, flow time, topology) look at the theory by adding another scale. Classically, many of them agree by theorems. Example: an "instanton" is exactly an anti-self-dual connection (§07, §08).

Notation: $\beta=2N/g^2$ is the coefficient of the lattice coupling (not the inverse temperature of statistical mechanics), $b$ is the factor of one step (multiplying the lattice spacing $a$ by $b$), and $\kappa=4Nb_0$ (0.3715 for SU(2), A-01). Writing asymptotic freedom (4-06) in the lattice coupling gives the running per step. For SU(2), doubling $a$ lowers $\beta$ by about 0.2575, and halving $a$ raises it by the same amount.

(c) is column (c) of 8-01 §02 (the direction reachable by finite measurement). The rows about limits and uniqueness (C7, C9–C11) are out of reach, and for the $\lambda_1$ of C18 a single test function gives an upper bound (1-10). "None" in (b) means, as in 8-01 §04, "not yet made". C4, C9 and C10 have lattice numbers (at the level of physics), treated in §03 and §05.

The renormalisation group is the operation of integrating out fluctuations at short scales and moving to an effective action at long scales. Wilson (1971) rewrote Kadanoff's theory of scaling in the form of differential equations (abstract: 'The Kadanoff theory of scaling … is cast in differential form') Known. The existence of the map and convergence to a fixed point are premises in physics. In mathematics this becomes a sequence of steps that estimates the compositions of the one-step map $R_b$ uniformly (6-05).

For four-dimensional pure gauge theory, Bałaban (1987, 1989) built a sequence of effective actions and showed UV stability in finite volume. Chatterjee (2018) §6 calls the existence of convergent subsequences UV stability Known. Convergence of the orbits and limits of observables are unproved (6-07). Agreement of the sign of the β function (C2) with an exact flow is a theorem for the two-dimensional Gross–Neveu model (Gawędzki–Kupiainen 1985). In four dimensions it goes as far as Bałaban's recursion in the small-field approximation.

Polchinski's equation (C3)

Polchinski (1984) wrote, for four-dimensional $\lambda\phi^4$, the change of the effective action as the momentum cutoff is moved continuously as an exact renormalisation group equation. On that basis he showed perturbative renormalisability Known. The same equation exists on the side of mathematics. Splitting the covariance of a Gaussian measure into a sum over scales and integrating one layer at a time from the fine scales, the remaining effective action $V_t$ obeys this equation. Rewritten for $e^{-V_t}$, it is convolution with a Gaussian kernel, that is, the flow of the heat equation Known. The pair is equation to equation, so the (a) of C3 is Settled.

There are also criteria standing on this equation. The log-Sobolev inequality is an inequality of the form $\operatorname{Ent}_\mu(f^2)\le2C\int|\nabla f|^2d\mu$. It is a stronger version of the Poincaré inequality $\operatorname{Var}_\mu(f)\le C\int|\nabla f|^2d\mu$ (1-10), with the variance replaced by the entropy $\operatorname{Ent}_\mu(g)=\int g\ln g\,d\mu-\int g\,d\mu\,\ln\!\int g\,d\mu$, and it guarantees how fast a stochastic dynamics returns to equilibrium. Bauerschmidt–Bodineau (2021) extended the Bakry–Émery criterion (1-10) to multiple scales by controlling Polchinski's equation, and showed an asymptotically optimal log-Sobolev inequality for the massive continuum sine-Gordon model (a two-dimensional model, with coupling $\beta_{\rm sG}\lt6\pi$) Known. The criterion is a tool standing on the pair, so it goes on the side of the conditions. There is no application to four dimensions, so (b) is "None".

MCRG (Swendsen 1979) fixes $\Delta\beta$ so that observables blocked with $b=2$ match a lattice with coupling $\beta-\Delta\beta$. The first results for SU(3) (Gupta et al. 1984) reported no asymptotic scaling for fundamental-representation coupling $K_F\lt6.1$ Known. $\Delta\beta$ is the amount by which $\beta$ is lowered when $a$ is doubled, the same quantity, measured by another method, as "the step that halves $a$" in Table 4 of 4-06. The mathematical partner is the one-step map $\beta\mapsto\beta-\Delta(b)$ ($\Delta(b)$ is the decrease per step, not the $\Delta$ of the mass gap), and with the same projection (the way of matching to the Wilson form) chosen, they are the same quantity (C4).

Consider one step within the same way of fixing things (scheme). If at leading order at weak coupling ($\beta\to\infty$) $\Delta$ does not depend on $\beta$, then two steps in succession and a single multiplication by $b_1b_2$ are the same, so

(Cauchy's functional equation under monotonicity). The constant $C$ is not fixed by additivity alone; comparing with (1) (one loop) gives $C=\kappa$. From two loops on, $\Delta$ keeps a $\beta$-dependence of order $O(1/\beta)$, and (3) is the leading-order formula that ignores it.

Integrating (1), the one-loop formula is $\beta=-\kappa\ln(a\Lambda)$ (the one-loop part of equation (8) of 4-06). A change of the way of fixing things, $\tilde\beta=\beta-c+O(1/\beta)$, only moves the map by conjugation and does not change the leading term of $\Delta$. What changes is the reading of $\Lambda$: from $\tilde\beta=-\kappa\ln(a\Lambda e^{c/\kappa})$, $\tilde\Lambda=\Lambda e^{c/\kappa}$, that is, $\Lambda$ is multiplied once only by $e^{c/\kappa}$ (C5). Dashen–Gross (1981) found this ratio to be $\Lambda_{\overline{\rm MS}}/\Lambda_{\rm L}=38.852704\,e^{-3\pi^2/(11N^2)}$ (Wilson action) Known.

For SU(2), $c_2=1.1096$ is about 4.3 steps of the running $\kappa\ln2=0.2575$ of one step with $b=2$. Moving from the Wilson action to $\overline{\rm MS}$ is the same as lowering $\beta$ once only by about 4.3 steps, that is, taking once only about 4.3 extra steps that double $a$. A map whose constant shifts at every step, $\Delta(b)=\kappa(\ln b-\delta(b))$, breaks (3) and moves the reading of $\Lambda$ step by step. The record kept on this machine calls $\delta(b)$ in this form "the shortfall" (10-02, 8-05).

Kadanoff (1976) rederived Migdal's (1975) recursion formula as an approximation to decimation, and proposed the scheme of potential (bond) moving (abstract) Known. Here we write one step as a pair of two operations. Bond moving gathers $b^{d-2}$ plaquettes in one place and turns the weight $f$ into $f^{\,b^{d-2}}$. Decimation combines $b\times b$ plaquettes of a plane into one. Decimation is exact in two dimensions (6-02).

We set down the notation a line at a time. Let the eigenvalues of an element $U$ of SU(2) be $e^{\pm i\theta}$, and write the weight of one plaquette of the Wilson action as $e^{\beta\cos\theta}$ ($\beta=4/g^2$). $\chi_n$ is the character of the $n$-dimensional irreducible representation (spin $(n-1)/2$), $\chi_n(U)=\sin n\theta/\sin\theta$, with dimension $d_n=n$. $I_n$ is the modified Bessel function $I_n(x)=\frac1\pi\int_0^\pi e^{x\cos\theta}\cos n\theta\,d\theta$. Expanding the weight in characters as $f=a_1\sum_nd_nc_n\chi_n$, we call the coefficients $c_n$ ($c_1=1$) the normalised coefficients, and decimation is $c_n\mapsto c_n^{\,b^2}$. 6-02 writes the weight as $e^{\beta\,\mathrm{Re}\,\mathrm{tr}\,U}$, so the $\beta$ of 6-02 is the $\beta/2$ of this chapter.

The approximation lies only in bond moving. On the lattice made by repeating the two operations in a nested way (a hierarchical lattice), (4) is exact by construction. For spin systems there are the hierarchical lattices of Griffiths–Kaufman (1982). For gauge theories Ito (1985) showed permanent confinement for groups such as SU(N) in a four-dimensional hierarchical lattice gauge theory on which the MK recursion holds exactly (abstract: 'hierarchical lattice gauge theories in which the Migdal-Kadanoff approximate renormalization recursion formulas hold exactly') Known. The "approximation" of physics is, in mathematics, "an exact map on another lattice" (C6; (b) is at the level of this hierarchical lattice).

Example: at strong coupling $c_2\approx\beta/4$, so one step with $d=4$, $b=2$ is $\beta'\approx4\beta^4$, and with decimation alone $\beta^4/64$ (at $\beta=0.1$, $3.895\times10^{-4}$ and $1.560\times10^{-6}$) Computed. The latter is the strong-coupling formula of one step of bookkeeping in the record (10-02), and the missing factor $b^{d-2}$ is known in the record too.

Where the flow goes, and the decrease per step

ComputedThe numbers in this figure were computed on this machine (the weight function iterated with 8000 points and 400 characters; $\beta$ is read from the $c_2$ of (4), and for U(1) from $I_1/I_0$). Without JavaScript it can be read from the next two paragraphs.

For $d\lt4$ the flow goes towards strong coupling, for $d\gt4$ towards weak coupling, and $d=4$ is the border. For SU(2) at $d=4$, all the starting points examined ($\beta_0\le20$) go towards strong coupling ($\beta_0=8$ reaches $\beta\lt0.05$ in 31 steps, and 20 in 79 steps) Computed. For U(1), starting points with $\beta_0\ge1$ stop at values of about $\beta\approx1.4$ or more and lie on a line of fixed points (5-06). The first step goes up, since it moves from the Wilson form to the settled form of the recursion; this too is a one-off shift.

The decrease per step after the form has settled at weak coupling (SU(2) at $d=4$, the 10th step from $\beta_0=60$) is 0.2503, 0.2965 and 0.3126 for $b=2,3,4$ Computed. The step given by one loop, $\kappa\ln b$, is 0.2575, 0.4081 and 0.5150. The MK values match the formula $(1-b^{-2})/3$ of the detour below (0.2500, 0.2963, 0.3125) Computed. For $b=2$ it is 97% of the one-loop value, but two steps with $b=2$ (0.5006) and one step with $b=4$ (0.3126) differ, so (3) is not satisfied. The $a(\beta)$ of MK depends on $b$.

Detour: why the decrease per step is (1−b⁻²)/3

On the continuum limit (C7), Chatterjee (2018) §5 summarises the usage of physics: if the correlation length diverges at a critical point $\beta_c$ one says 'the model has a continuum limit' Known. The scaling limit is the operation of shrinking the lattice to $\varepsilon\mathbb Z^d$, normalising the field and letting it converge weakly as a distribution (2-08); once what is held fixed and the normalisation are specified, it is the same operation. The non-triviality that the physical word includes is not in the definition, and the limits of the four-dimensional Ising type and of $\phi^4$ are Gaussian (Aizenman–Duminil-Copin 2021, 6-08).

There are two ways of choosing $a(\beta)$ (C8). The explanation in §5 fixes the correlation length, and Problem 5.2 quantifies existentially with Wilson loops (the same formula as equation (4) of 8-03).

The $d$ here is the coefficient of the area law (the number corresponding to the continuum string tension), a different symbol from the dimension $d$ in (8). Problem 5.2 does not presuppose 5.1, and the existence of either $\varepsilon(\beta)$ is unproved. Physics fixes the scale by the string tension (Creutz 1980) or $r_0$ (Sommer 1994) Physics.

Scaling (C9) is that dimensionless ratios stop depending on $a$, and asymptotic scaling (C10) is that $a(\beta)$ follows the two-loop formula.

In the lattice numbers the two behave separately. The deviation on the right shows up in the report that MCRG for SU(3) sees no asymptotic scaling for $K_F\lt6.1$ (Gupta et al. 1984). In Table 4 of 4-06 too, the data for the step that halves $a$ (0.35–0.52) do not reach the two-loop value (about 0.61) Physics. The left is supported by dimensionless ratios lining up at one value in extrapolations that assume $a^2$ corrections (the $m/\sqrt\sigma$ of Lucini–Teper 2001, the $\sqrt{8t_0}/r_0$ of Lüscher 2010) Physics. Fits for SU($N$) with $N\gt4$ do not pin down $a(\beta)$ in units of $\Lambda_{\overline{\rm MS}}$ (Allton–Teper–Trivini 2008) Physics. The mathematical partners are, for the left, the limit of the pair in Problems 5.1 and 5.2 (B9 of 8-03), and for the right, the asymptotic form of $\varepsilon(\beta)$; neither has a theorem (5-05, 5-09).

Universality (C11) is that the same limit comes out independently of the details of the action. Griffiths (1970) and others stated it as a hypothesis, and Wilson (1971) explained it by irrelevant variables. In mathematics it is uniqueness of the limit, and the theorems are in two dimensions: Ising on isoradial graphs (Chelkak–Smirnov), and the Wilson, Villain and Manton actions going to the same measure for the YM measure of the trivial bundle on the two-dimensional torus (Chevyrev–Shen) Known. In four dimensions, that the Wilson action and improved actions give the same theory is a premise.

Symanzik's (1983) improvement (C12) is the idea of reducing the errors at finite $a$ systematically by adding irrelevant terms. Under a single action, the error of a dimensionless ratio is expanded in $a$.

For Yang–Mills the logarithmic corrections to $a^2$ are also said to be weak (Husung–Marquard–Sommer 2020) Known. The record's $\delta(b)=c_2/b^2+c_3/b^3+O(b^{-4}\ln b)$ is an expansion in the factor $b$, a different variable, so the pairing with Symanzik remains an analogy (Unsettled). Example: over 20 steps, the left is $4^{-20}s\approx9\times10^{-13}s$, and the right, with $b$ fixed, is $20\kappa\delta(2)$. In an example where the factor increases by one at each step, $b_j=j+2$, the leading terms alone give $2.39\kappa\delta(2)$ (the limit with the steps continued without end is $4(\pi^2/6-1)\approx2.58$ times) Computed.

The price of improvement (C13)

Lüscher–Weisz (1984) wrote that adding dimension-6 operators to the Wilson action means 'physical positivity is lost in general', and showed that the transfer matrix can still be defined. It has two properties. Complex eigenvalues can appear ('may occur'). The spectral decomposition of two-point functions always has contributions with negative weight ('there are always contributions … that come with a negative weight') Known. In the language of reflection positivity (5-03), the mathematical word of C13 is "loss of reflection positivity (contributions with negative weight)", grounded on the 'always'. (a) is Settled for actions with dimension-6 operators added (the judge is a theorem). The kernel that sets $\delta(2)=0$ in the record also has negative weights; whether it is of the same type is treated in 8-05.

Marginality (C14)

Shrinking as $A_\lambda(x)=\lambda A(\lambda x)$,

so at $d=4$ the action is scale invariant (the border in Figure 1). In mathematics this is the critical Sobolev exponent. The $L^p$ norm $\|F\|_{L^p}=(\int|F|^p\,d^dx)^{1/p}$ gives stronger control the larger $p$ is, and under shrinking $\|F_\lambda\|_{L^p}^p=\lambda^{2p-d}\|F\|_{L^p}^p$. The exponent that does not change under a change of scale, $p=d/2$, is the critical exponent (1-11, 1-12). Uhlenbeck's (1982) weak compactness (the theorem extracting a convergent subsequence from a sequence of connections) needs $p\gt d/2$ for boundedness of the $L^p$ norm of the curvature. The four-dimensional action has $p=2=d/2$, exactly at the threshold Known. The problem statement writes the outlook that asymptotic freedom rescues the case where the Sobolev inequality is at the borderline (0-03). The agreement goes only as far as classical dimension counting (6-09, the four-dimensions article).

The four words of this section look at the theory by adding one scale outside four-dimensional spacetime: the size $N$ of the group, the depth from the boundary into the interior, and the size of rolled-up extra dimensions.

Large N and planar diagrams (C15)

't Hooft (1974) showed that with $\lambda=g^2N$ fixed and $N\to\infty$, planar diagrams dominate (perturbation theory, 7-04). Two things agree as theorems: planar maps of matrix models (Guionnet–Maurel-Segala), and Wilson loops of the strongly coupled SO($N$) lattice written as sums over trajectories of strings (Chatterjee, Basu–Ganguly) Known.

The master field (C16)

The idea that as $N\to\infty$ fluctuations disappear and gauge invariants are fixed by a single configuration. In mathematics it corresponds to a deterministic limit of non-commutative distributions. A non-commutative distribution regards the totality of expectations of traces of products, $\frac1N\mathbb E\,\mathrm{tr}(X_{i_1}\cdots X_{i_k})$, for a set $X_1,X_2,\ldots$ of random $N\times N$ matrices, as the "distribution" of the matrices (the language of free probability). For planar two-dimensional Yang–Mills, Lévy showed that Wilson loops approach a deterministic limit in this sense (7-04) Known.

The radial direction of holography (C17)

Since the infrared of the interior is mirrored in the ultraviolet of the boundary (Susskind–Witten 1998), the radial direction is read as the scale of the boundary (7-08). What is certain in mathematics is only the expansion of the metric near the boundary (Fefferman–Graham). It expands the metric of the interior in powers of the depth $z$ from the boundary, in the form $ds^2=(dz^2+g_z)/z^2$, $g_z=g_{(0)}+z^2g_{(2)}+\cdots$ ($g_{(0)}$ is the metric of the boundary) Known. The side of physics too says the correspondence with the cutoff is 'left unanswered' (Heemskerk–Polchinski 2010) Physics.

The KK tower (C18)

Rolling up extra dimensions into an internal space $K$ makes a sequence of masses appear. For a massless scalar on an unwarped product,

(7-06) Known. $\lambda_1(\Delta_K)$ is the smallest non-zero eigenvalue of the Laplacian on $K$. No construction is known that separates the KK tower from $\Lambda$ while keeping geometric calculation (7-10). Pure Yang–Mills has no extra dimensions, so (b) is Not applicable.

The four words of this section agree with mathematical words by theorems for classical fields. Where they are carried into quantum fields, they become conditional or at the level of physics.

Instantons (C20)

The conventions of this section are the same as A-01: $g$ is put in front of the action and not included in $F$: $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu-i[A_\mu,A_\nu]$ ($A_\mu=A^a_\mu T^a$). The Hermitian generators are normalised as $\mathrm{tr}\,T^aT^b=\tfrac12\delta^{ab}$, so the action is $S=\frac{1}{2g^2}\int\mathrm{tr}\,F_{\mu\nu}F_{\mu\nu}=\frac{1}{4g^2}\int F^a_{\mu\nu}F^a_{\mu\nu}$. Setting the dual to $\tilde F_{\mu\nu}=\tfrac12\varepsilon_{\mu\nu\rho\sigma}F_{\rho\sigma}$ ($\varepsilon$ totally antisymmetric with $\varepsilon_{1234}=1$), $\int(F^a\mp\tilde F^a)^2\ge0$ bounds the action below by the topological charge.

Equality holds only for (anti-)self-dual fields (3-03). (12) of 3-03 is written in the canonical normalisation $D_\mu=\partial_\mu-igA_\mu$, and rewriting with $A=gA_{\text{3-03}}$ (and so $F=gF_{\text{3-03}}$) agrees with (10). With the solution of BPST (1975), the construction of all solutions by ADHM (1978), and Uhlenbeck's (1982) "finite-action fields on $\mathbb R^4$ come from $S^4$", they are Settled as classical objects Known. Approximating the path integral by sums over instantons is a separate, quantum claim.

Topological charge (C19)

The topological charge is the second Chern number. The second Chern number is an integer determined by $\frac{1}{8\pi^2}\int\mathrm{tr}\,F\wedge F$ for a connection on a closed four-dimensional manifold (for example $S^4$) (in the convention with $A$ anti-Hermitian, A-01), unchanged when the connection is varied continuously (the four-dimensional version of the first Chern number of 1-07). It differs from the $Q$ of (10) only by a sign convention (the detour in 3-05). On the lattice, from fields with each plaquette $U_p$ close enough to the identity (action density below a threshold: admissible) a bundle can be built, and it is an integer (Lüscher 1982, 3-05) Known. Otherwise the sectors are 'a dynamical property of the theory' (Lüscher 2010, SU(3)) Physics.

The Wilson flow (C21)

The Wilson flow is the equation $\dot B_\mu=D_\nu G_{\nu\mu}$, $B_\mu|_{t=0}=A_\mu$, which smooths the field by adding a flow time $t$. $G_{\mu\nu}$ is the field strength of $B$, and $D_\nu$ the covariant derivative by $B$. It is steepest descent of the action, and the equation is exactly the Yang–Mills heat flow. The radius $\sqrt{8t}$ is the value at $d=4$ of the spread $\langle|x|^2\rangle=2dt$ of the heat kernel $e^{t\Delta}$. Its action on quantum fields goes as far as the three-dimensional case under a GFF assumption (Cao–Chatterjee) and two-dimensional YM and three-dimensional YMH (a model including a Higgs field, CCHS) through stochastic quantisation Known. Four dimensions is at the stage of Lüscher's 'suggest' Physics.

The Gribov region (C22)

Gribov (1978), faced with the ambiguity of the Landau gauge $\partial_\mu A_\mu=0$, restricted the integral to fields where the FP determinant is positive (3-04, 4-08). The FP operator $M(A)=-\partial_\mu D_\mu(A)$ is the operator expressing the change in the gauge condition when the field is moved a little by a gauge transformation, and its determinant is the Faddeev–Popov determinant that multiplies the path integral. $D_\mu(A)X=\partial_\mu X-i[A_\mu,X]$ is linear in $A$, so

and $\Omega$ is convex ($M\ge0$ in the sense of positive semi-definite as an operator). Zwanziger (1982) showed 'bounded in every direction', and Dell'Antonio–Zwanziger (1991) showed that every orbit enters this region Known. The picture that the measure on it produces confinement is at the level of physics (7-02).

In classical four-dimensional Yang–Mills, the scale invariance of (8) makes the trace $T^\mu{}_\mu$ of the energy-momentum tensor vanish (3-03). The trace is a quantity that measures locally the change of the action when the field is shrunk by $x\mapsto\lambda x$, and it is 0 if the action does not change under a change of scale. In the quantum theory, integrating the fluctuations needs a cutoff (a lattice spacing, say), and a length enters there. The coupling $g(\mu)$ runs with respect to that length, so scale invariance breaks. Writing the way it breaks locally is the trace anomaly.

Here we write $F$ in the canonical normalisation ($D_\mu=\partial_\mu-igA_\mu$, the convention of 3-03). It is related to the field of the convention of (10) by $F_{(10)}=gF$, and writing (12) in the convention of (10), the coefficient becomes $\frac{\mu\,dg/d\mu}{2g^3}$.

This is a formula cited in standard form (the normalisation differs between references; Collins–Duncan–Joglekar 1977) Physics, and the coefficient is the β function itself. Under asymptotic freedom ($\mu\,dg/d\mu\lt0$) the trace does not become 0. This is the local mechanism of dimensional transmutation (4-07). There is no counterpart on the side of mathematics, because there is no four-dimensional continuum theory and no place to put this operator (as with the Λ of 8-03). The 7 rows of Table 0 whose (a) is Settled are agreements of equations, definitions or classical objects, and no row has (b) "exists" in four-dimensional continuum quantum theory.

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