2026-10-01 · chapter Yang–Mills and the mass gapPart 6 Constructive field theoryrenormalisation group

The sequence of renormalisation group steps — integrating out one scale at a time, splitting into small and large fields as we go

What we wanted to show in stage 3 was that expectation values are bounded independently of the lattice spacing $a$ (formula (8) of 6-01). In four-dimensional Yang–Mills, by asymptotic freedom (4-06), the coupling $g$ is small at fine scales and becomes $O(1)$ at coarse scales. Every scale has a small number, but there is no small number common to all scales. So the scales are integrated one step at a time.

The transformation with factor $L$ is repeated $K$ times, and at each step only the coupling of that scale is used as the small number ($L$ is unrelated to the size of the box in 6-01, and $a_0$ is fixed). Writing $\rho_k$ for the weight (effective density) after step $k$, $Z=\int\rho_0=\int\rho_1=\cdots=\int\rho_K$. The problem statement explains this transformation as "integrating out the large-momentum degrees of freedom and restoring the scale" (JW §6.5), and the introduction of Bałaban 1985 says it follows the form proposed by K. Wilson. Known

Example: with $L=2$, $a_0=1$ and $a=2^{-20}$, $K=20$. Each halving of $a$ adds one step, so what must be shown are estimates independent of $K$.

In one step, one fixes the variables of the coarse lattice and integrates the fine variables so as to keep those values. Written for a scalar field, the average over a block $B(y)$ of side $L$ points is (2), and Dimock, who expounds Bałaban's method, fixes the average loosely with a Gaussian weight ($\alpha\gt0$ is arbitrary, $N$ a constant that preserves the integral).

Then the lattice is shrunk to unit spacing and the field rescaled, giving $\rho_{k+1}$ (Dimock I (26)(27)). Known

Example: look at two-dimensional $U(1)$. An element of $U(1)$ can be written as $e^{i\theta}$ with the angle $\theta$. A unit square $p$ of the lattice is called a plaquette; the product of the variables on its four sides taken in order along the orientation is written $U(\partial p)=e^{i\theta_p}$, and $\theta_p$ is called the angle of the plaquette (5-01 §03). Putting the weight $e^{\beta\cos\theta_p}$ on each plaquette is the Wilson-type action ($\beta$ is the inverse lattice coupling; the larger it is, the weaker the coupling. A different quantity from the β function of §06). In two dimensions with free boundary the plaquette angles are independent (6-02 §02), and the angle of a $2\times2$ block is the sum of the four angles. For a sum of independent angles, the Fourier coefficients of the distribution are the products of the individual coefficients. Using the coefficient of one angle

the coefficient of the angle of the block becomes $c_n(\beta)^4$ in one step. Setting $I_n(\beta)=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{in\theta}e^{\beta\cos\theta}\,\mathrm d\theta$, $c_n=I_n(\beta)/I_0(\beta)$, and $I_n$ is a special function called the modified Bessel function. If the action were still of Wilson type after one step, the $\beta'$ fixed by $c_1$ should also match $c_2$.

ComputedTable 1 One step ($L=2$) of two-dimensional $U(1)$ with Wilson type. $\beta'$ is the value fixed by $c_1(\beta')=c_1(\beta)^4$, and the mismatch is $c_2(\beta')/c_2(\beta)^4-1$.

It does not match. The form of the action changes in one step, and new terms are born. Computed If the weight is chosen from the start so that its coefficients are $c_n=e^{-n^2/2\beta}$, then $c_n^4=e^{-n^2/(2\cdot\beta/4)}$, and the form is kept exactly. This weight is the heat kernel on the circle (also called Villain type): the distribution on the circle obtained by adding up copies of the density of the normal distribution with variance $1/\beta$ shifted by $2\pi$, that is, the normal distribution wrapped round the circle (6-02 §04). In general one follows the main local terms and gathers the rest into corrections $E_k(X)$ for each lump $X$ (Dimock I (105)(108); the form of the polymers of 6-04). Known

If the weight is Gaussian (a free field), (3) can be written in closed form for any number of steps. The density after $k$ steps is determined by the configuration $\phi_k$ that minimises the action of the fine field given the coarse values $\Phi$ (the minimal configuration, or background field) (Dimock I (45)–(52)).

$\Delta$ is the lattice Laplacian (not the $\Delta$ of the mass gap), $Q_k$ the average over $k$ steps, $Q_k^{\mathsf T}Q_k$ the projection onto functions constant on blocks, $\bar\mu_k$ the mass squared, and $\alpha_k=\alpha(1-L^{-2})/(1-L^{-2k})$ (Dimock I (32)) a constant gathering the weights of $k$ steps. Fluctuations about the minimal configuration are Gaussian variables, and integrating them leaves a constant $Z_k$. The point is the locality of $G_k$. Even if $\bar\mu_k$ is close to 0, the term acting on the averages plays the role of a mass, and when $x$ is in a cube $\Delta_y$ and the support of $f$ in a cube $\Delta_{y'}$ ($\Delta_y$ is a large cube of side $M=L^m$, and $d(y,y')$ the distance measured in units of these cubes)

holds (Dimock I (85), where $\gamma=\tfrac12\gamma_0$. The literature is Bałaban, CMP 89 (1983) and others). Known The massless $(-\Delta)^{-1}$ changes in two dimensions only with the logarithm of the distance.

Example: on a two-dimensional periodic lattice (32 blocks on a side), we lined up $(-\Delta+\ell^{-2}P)^{-1}$ ($P$ is the projection onto the averages over blocks of side $\ell$) by the distance $r$ in block units.

ComputedTable 2 $G(r)/G(0)$. The bottom row is $\gamma$ from fitting $e^{-\gamma r}$ to the values for $r=4$ to $12$ (the decay rate per block). The rightmost column puts the mass $\ell^{-2}$ everywhere.

The decay rate is about 1.1 per block and hardly depends on $\ell$. Although the constraint acts only on the block averages, it is about the same as putting a mass everywhere (1.055–1.064 for $\ell=2,4,8$ by the same fit), and if anything slightly faster. This locality, independent of the step, is why the cluster expansion (6-04) can be used at each step. Computed

Analysis in perturbative form can only be used when the field is small. The distinctive feature of Bałaban's method is that at each step the lattice is split into the small-field region $\Omega$ and its outside, and the sum is taken over all ways of splitting (Dimock II §1).

$\chi$ is the indicator function that the small-field condition holds on $\Omega$, and $\zeta$ that the condition fails somewhere in each cube of $\Omega^{c}$. In the words of the introduction of Dimock I (§1.1), 'The large field region is treated crudely but makes a small contribution due to the fundamental stability of the interaction'. The next step splits within the small-field region of the previous step, so the regions shrink, $\Omega_1\supset\Omega_2\supset\cdots$. Known

For gauge fields the condition is put on plaquettes. At the first step (bare coupling $g_0$), Bałaban 1985 imposes on each plaquette $p$

($c$ is a sufficiently large constant), and takes as the small-field region the union of blocks sufficiently far from the plaquettes where it fails (ibid. (4)(7) on pp. 256–257; notation adapted to this book). Known

We estimate the sizes as a rough guide. The weight is roughly $e^{-|U(\partial p)-1|^2/g^2}$, so the typical size is about $g$, and the threshold is $p(g)$ times that. Large fields then carry, as a rough guide, a factor of order $e^{-O(1)\,p(g)^2}$. This is smaller than any power of $g$. The exposition for scalar fields says that a factor $e^{-O(1)\,p_0^2}$ appears for each large-field cube (Dimock II §1; the $p_0=(-\log\lambda_0)^{p}$ there is the size of the threshold, a different symbol from the exponent $p_0$ in (7)). Known Example: with $c=1$, $p_0=3$ and $g=0.1$, $p\approx36.0$. Taking $O(1)$ to be 1, $e^{-p^2}\approx e^{-1298}$, which is in a different league from $g^{10}=e^{-23}$. Computed

ComputedThe numbers in this figure were computed on this machine (a 32×32 sample of standard normals embedded at about 0.1 resolution; the angles $\theta_p=z_p/\sqrt\beta$ and the block sums are computed on the spot).

ComputedTable 3 The sample of Figure 1 ($\beta=64$, $\varepsilon=0.5$; step 0 is 0/1024). "Exact" is the probability under the wrapped normal distribution with variance $4^k/\beta$.

The model of the figure is the heat-kernel type of §02, and the effective coupling at step $k$ is exactly $\beta_k=\beta/4^k$. In two dimensions the coupling strengthens at each step, and with the threshold fixed, large fields fill everything within a few steps. The place to stop before the coupling becomes $O(1)$ is the $a_0$ of (1). In four dimensions the strengthening is much slower (§06).

Gauge fields add three things. Known (i) Group elements cannot be rescaled. Bałaban 1985 writes 'we use improvement of regularity properties of typical gauge fields instead of non-existing scalings'. (ii) Group elements cannot be added and divided. The variables on coarse edges are made from those on fine edges, but the average of group elements leaves the group. How to make the average is itself the subject of a whole paper (Bałaban, CMP 98 (1985) 17–51, bibliographic data only). For charged fields, one averages gauge-covariantly by multiplying by phases along paths (formula (20) of Dimock's paper on QED). (iii) The gauge is chosen locally, scale by scale. To write $U=e^{igA}$ and expand, the gauge must be fixed, but no globally continuous choice exists (Singer 1978, 3-04, 4-08). On the lattice the group is compact, so the definition of $Z$ needs no gauge fixing, and only local choices are needed. According to the problem statement, the gauge is chosen for each place in space-time and each scale ('phase cells'), and is rechosen with the steps from an ultraviolet gauge to an infrared gauge (JW §6.5; describing the work in three dimensions).

The reason it is local is elementary. Define the operator norm of a matrix $A$ as the factor by which it stretches vectors the most, $\|A\|=\max_{|v|=1}|Av|$. A unitary matrix does not change lengths, so $\|U\|=1$. For unitary matrices $U,V,W$

(the former from $UV-1=U(V-1)+(U-1)$ and $\|U(V-1)\|\le\|U\|\,\|V-1\|=\|V-1\|$, the latter because neither $W$ nor $W^{-1}$ changes lengths). Paper Under a gauge transformation the plaquette variable changes to the form $WU(\partial p)W^{-1}$, so by the latter, condition (7) measured in the operator norm does not change under gauge transformations. On a two-dimensional block of side $\ell$, setting the vertical edges and the bottom horizontal edges to 1 (complete axial gauge), a horizontal edge at height $y$ is the product of the inverses of the variables of the $y$ plaquettes directly below it, so repeated use of (8) gives $\|U_b-1\|\lt\ell\,\varepsilon_1$. Paper Example: for $\varepsilon_1=0.05$, $\ell=8$ gives within 0.4. $\ell=64$ gives 3.2, which says nothing since always $\|U-1\|\le2$. The gauge is effective over a range of about a block.

The main thing that changes from step to step is the effective coupling $g_k$. In four dimensions, cutting into steps the one-loop running ($1/g^2(\mu)=1/g^2(\mu_0)+2b_0\ln(\mu/\mu_0)$, obtained by integrating the β function $\mu\,\mathrm dg/\mathrm d\mu=-b_0g^3+O(g^5)$, $b_0=11N/48\pi^2$ of 4-06 §01; the same formula from the physics side is formula (2) of 7-01) gives

and it strengthens by a fixed amount at each coarsening. In three dimensions $g^2$ has the dimension of mass, and the dimensionless coupling at step $k$, $g_k^2=g^2L^ka$, is geometrically small on the fine side. Known

From here on this is a rough-guide argument using only formula (9); it is not Bałaban's estimate itself, and not a claim at the level of a theorem. The difference shows up in the sum. If each step produces a correction of order $g_k^2$, then in three dimensions $\sum_kg_k^2$ is finite, but in four dimensions $g_k^2\approx1/(2b_0\ln L\cdot(K-k))$ and the sum keeps growing like the logarithm of the number of steps, so it cannot be piled up as an error. What this rough guide shows is that corrections of that size must be renormalised into the running of the coupling itself and followed. If the remainder is of order $g_k^3$, the sum stays bounded. The title of the paper extending to four dimensions (Bałaban, CMP 109 (1987); from the reference list of JW) contains 'a coupling constant renormalization in 4D'. Known

ComputedThe numbers in this figure were computed on this machine (partial sums of formula (9); they agree with Table 4).

ComputedTable 4 $SU(3)$, $L=2$, $g_K^2=1$ at the right end. $g_0^2$ is the value at the finest step.

Example: to go from $g^2=1$ at the right end to $g^2=0.1$ at the finest step takes about 93 steps for $SU(3)$ ($a\approx9\times10^{-29}$) and about 140 steps for $SU(2)$. $\sum g_k^2$ grows by about 7.2 each time $K$ doubles, while $\sum g_k^3$ settles at about 21.2 and the three-dimensional $\sum g_k^2$ at 2. Computed

The proof is an induction on the step $k$. The hypothesis is that "$\rho_k$ satisfies a fixed form and estimates". The form consists of the main action on the small-field region, exponentially small corrections $E_k(X)$ localised on lumps $X$, and the sum over large-field regions (Dimock II §1). One shows that the same form and estimates hold after one step (3), and the key is that the constants do not depend on $k$ or $a$. The estimate of the main theorem of the three-dimensional paper has the following form (Bałaban 1985 (5); notation adapted to this book).

$A$ is the Wilson action, $U_k(U)$ the minimal configuration, $\chi$ the indicator function of the small-field condition, $|T_k|$ the number of points of the lattice at step $k$, and $C$ depends neither on $a$ nor on $k$. Integrating gives upper and lower bounds on the partition function uniform in $a$ (ibid. (6)). In four dimensions, the 1989 paper states ultraviolet stability in four dimensions as its main theorem (Theorem 1) under the hypothesis 'If the sequence of the effective coupling constants is contained in an interval ]0, γ] with a sufficiently small positive γ'. Known

Example: for three-dimensional $\phi^4$, Dimock's Theorem 1 is of the same type: with the counterterms chosen, $e^{-c\,\mathrm{Vol}}\le Z/Z_{\mathrm{free}}\le e^{c\,\mathrm{Vol}}$ holds independently of the ultraviolet cutoff and the volume (6-03 §06).

Magnen–Rivasseau–Sénéor (MRS 1993) treated four-dimensional $SU(2)$ with a continuum momentum cutoff (with an infrared cutoff, in the trivial topological sector). They use axial gauge for large fields and a background-field-dependent gauge for small fields, and restore with counterterms the gauge invariance broken by the cutoff. The introduction states explicitly that the paper does not give a detailed proof ('we do not provide a detailed proof'). We saw the introduction's comment on Bałaban's results in the Seam of §07. Known The advantage of starting from the lattice is that gauge invariance and reflection positivity are kept (JW §6.5, 5-03). In exchange, the task remains of restoring the broken rotation symmetry in the limit (stage 5 of 6-01).

Where it stops is the right end of (1): at scales longer than $a_0$ the expansion has no foothold. MRS give as their reason for not removing the infrared cutoff 'this would lead to large values of the coupling constant, and presumably to so-called non-perturbative effects corresponding to confinement', and the problem statement says that a gap uniform in the volume needs new ideas (JW §6.5). Example: increasing $K$ in Figure 2 does not move $g_K^2=1$ at the right end. A way of adding steps beyond the right end is not among the tools of this chapter. Continued in 6-10 and 6-11.

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Next chapter to read: 6-06 Ultraviolet stability of three-dimensional Yang–Mills (the place where this method was first used: what was shown, and what is not yet constructed).

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