2026-10-01 · chapter Yang–Mills and the mass gapPart 6 Constructive field theoryφ⁴₂ and φ⁴₃

Constructing scalar fields — until φ⁴ satisfies the axioms in two and three dimensions

On the map of 6-01 (the six stages), the examples of interacting theories in which all six stages closed were $\phi^4$ in two and three dimensions. The field carries only one real number at each point, with no gauge symmetry and no constraints, and the difficulty is narrowed to products at a point (the ultraviolet) and volume (the infrared). The Euclidean action is the action of the free field of mass $m_0$ plus the polynomial of formula (3) of the problem statement.

$\lambda\gt0$ is the coupling constant, and $\sigma=\alpha\lambda+\beta\lambda^2$ and $c=\gamma\lambda+\delta\lambda^2+\varepsilon\lambda^3$ are counterterms made to depend on the cutoff (JW §6.2). The $d$ in $\phi^4_d$ is the dimension of space-time, and in two dimensions $P(\phi)_2$, extended to a polynomial $P$ bounded below, was handled by the same methods. The target is one sentence of the problem statement: 'For 0 < λ ≪ 1, one can choose σ(λ), c(λ) so the field theory described by (3) exists, is unique, and has a mass equal to m such that |m − m₀| is small' (JW §6.2). In four dimensions the lattice limit collapses to a Gaussian (6-08). Known

In Table 1 of 6-01 (where it breaks) we saw that in two or more dimensions the variance at one lattice point, $c_a=G_a(0)$, becomes infinite as $a\to0$. A Wick product is a product with lower powers subtracted so that its expectation is 0, and it is fixed step by step by the following conditions (Summers 2016 §3.1).

For a single variable with variance $c$, ${:}x^2{:}=x^2-c$ and ${:}x^4{:}=x^4-6cx^2+3c^2$: Hermite polynomials with the scale adjusted. The point is that for a Gaussian field with mean 0 and covariance $C$

holds. The variance of the smeared ${:}\phi^n{:}(f)$ is $n!\iint f(x)f(y)C(x-y)^n$, and at short distance $C$ behaves like $\log(1/|x|)$ if $d=2$ and like $|x|^{-(d-2)}$ if $d\ge3$, so

In two dimensions every $n$ is finite; in three dimensions only $n\le2$ is, $n=3$ is the logarithmic borderline, and $n=4$ diverges like $1/a$. Known

ComputedTable 1 The lattice $\int C^n\approx a^d\sum_xG_a(x)^n$ (m = 1. The side of the periodic box is 8 in two dimensions and 4 in three). The bottom row for two dimensions is the continuum, infinite-volume $\int_0^\infty2\pi r\,(K_0(r)/2\pi)^n\,\mathrm dr$.

Every two-dimensional column approaches the continuum value. In three dimensions $\int C^2$ converges, the increment of $\int C^3$ each time $a$ is halved approaches about 0.0044 (the continuum prediction is $\log2/(16\pi^2)=0.00439$), and $\int C^4$ roughly doubles each time. The two-dimensional ${:}\phi^4{:}(f)$ is a random variable, but the three-dimensional one is not. These two columns reappear in §06 as diagrams with two vertices.

Even if $V=\int_\Lambda{:}\phi^4{:}\,\mathrm dx$ ($\Lambda$ is a finite box) is a random variable, to make the measure $e^{-\lambda V}\mathrm d\mu_C/Z$ we need $e^{-\lambda V}$ to be integrable. The classical $\phi^4$ is non-negative, but the Wick product is not. Since ${:}x^4{:}=(x^2-3c)^2-6c^2$,

and in two dimensions $c_a=\frac1{2\pi}\log\frac1a+O(1)$, so the lower bound runs away to $-\infty$ at the rate $(\log\frac1a)^2$. Jaffe (2000) says that early progress in two dimensions owed much to Nelson's 1965 paper, which showed the stability (being bounded below) of the Hamiltonian of $\phi^4$ with space taken to be a circle. Known

The argument that it is integrable even though the lower bound runs away has three steps. (i) The cut-off $V_a$ is bounded below by formula (5). (ii) The variance of $V-V_a$ becomes small like a power of $a$, as in the convergence of Table 1. (iii) For a polynomial of degree 4 in a Gaussian field with standard deviation $s$, the probability of exceeding $t$ is of order $\exp(-\mathrm{const}\cdot(t/s)^{1/2})$. $V\lt-M$ needs $V-V_a$ to be large and negative, and choosing $a$ to match $M$ makes that probability decrease faster than $e^{-pM}$. The conclusion is (Summers 2016 §3.1) Known

Here $X\in L^p(\mathrm d\mu_C)$ means $E[|X|^p]\lt\infty$, that is, the expectation of the $p$-th power is finite ($E$ is the expectation in the free-field measure $\mu_C$). For $p=1$ the normalising constant $Z=E[e^{-\lambda V}]$ is finite, and the measure $e^{-\lambda V}\mathrm d\mu_C/Z$ can be made.

ComputedThe numbers in this figure were computed on this machine. $V_a=a^d\sum_x{:}\phi_x^4{:}$ was evaluated on 20000 samples of the free lattice field (6000 for a = 1/64 in three dimensions) in a periodic box of side 1 with m = 2. The horizontal axis is $V_a$, the vertical axis the density.

ComputedTable 2 The distributions of Figure 1 (excerpt). The standard deviation is exact from formula (3); the 0.1% point and $\log E[e^{-V}]$ (λ = 1) are from the samples (the error of the latter is ±0.002 in two dimensions and ±0.005–0.17 in three). In brackets, the Gaussian approximation $\mathrm{Var}(V)/2$.

In two dimensions, even as the lower bound drops from −1.5 to −6.7, the 0.1% point stays at −0.9 to −1.0 and $\log E[e^{-V}]$ settles at about 0.05. The lower bound is far away, but the probability of getting there is extremely small: this is the content of formula (6). In three dimensions, each halving multiplies the standard deviation by $\sqrt2$ and roughly doubles $\log E[e^{-V}]$, approaching the Gaussian approximation. A constant growing like $1/a$ has to be subtracted at order $\lambda^2$, and this corresponds to the $\delta$ of the problem statement (§06). Computed

The first construction (the Glimm–Jaffe series of four papers, 1968–1972) took the form of solving the field equations as operators in Minkowski space-time. The equation in two dimensions is

($\phi_\kappa$ is the field with Fourier components cut at $|k|\le\kappa$, $c_\kappa\sim\log\kappa$. Jaffe 2000 §3). One shows that $H(g)=H_0+\lambda\int{:}\phi(0,x)^4{:}\,g\,\mathrm dx$, cut off in space by a function $g$, is essentially self-adjoint on Fock space (a unique self-adjoint operator is fixed by the cut-off expression), bounded below, and has a simple lowest eigenvalue, and then uses the fact that signals do not exceed the speed of light to obtain a Poincaré-invariant vacuum as $g\to1$ (stage 2 → stage 4. Summers 2016 §2). The representation is not equivalent to the Fock representation (Haag's theorem of 4-09), but it is equivalent when restricted to a bounded region. Known

In the early 1970s, Nelson's Markov fields (only named here) and the Osterwalder–Schrader axioms (4-10) opened the road of making the Euclidean measure

(given meaning by formula (6)). On a two-dimensional lattice, the difference term, for pairs of neighbouring points $\langle xy\rangle$, becomes

and the last term, inside $e^{-S}$, appears as $e^{+\phi_x\phi_y}$ and makes neighbours prefer the same sign. It is a ferromagnet whose spins range over all real numbers (the title of Guerra–Rosen–Simon 1975 says 'as classical statistical mechanics'). There correlation inequalities (of GKS type: inequalities saying that correlations are non-negative in systems where neighbours prefer the same sign) apply, and if $P$ has the form of an even polynomial minus $h\phi$ ($h\ge0$), the Schwinger functions are non-negative and, with suitably chosen boundary conditions, increase with the box. Together with a uniform upper bound, the infinite-volume limit is obtained by monotone convergence (stage 4), with no restriction on the size of the coupling (Summers 2016 §3.1), though the problem statement says that correlation inequalities tend to be limited to scalar bosons and abelian gauge theories (JW §6.1). Reflection positivity is inherited by the limit (stage 5, 6-01). Known

Glimm–Jaffe–Spencer (1974) carried it through to stage 6. When $\lambda/m_0^2$ is small enough, they showed the infinite-volume limit and the OS axioms by the cluster expansion, and obtained exponential decay of correlations (formula (9) of 6-01). For the mass operator $M=\sqrt{H^2-P^2}$ ($P$ is the momentum)

has been shown (Summers 2016 §3.1). $0$ is the vacuum, the isolated $m$ is the one-particle state, and above $m_1$ are two or more particles. With an isolated one particle, Haag–Ruelle scattering theory (the general theory that assembles scattering states in the far past and far future from states of isolated particles; only named here) can be used, and Osterwalder–Sénéor (1976) showed that the scattering matrix is non-trivial and that the perturbation series is its asymptotic expansion (Summers 2016 §3.1). On the other hand, the cluster expansion used in the proofs of existence and of the gap is not a power series in $\lambda$ but a convergent expansion whose $n$-th term has the form $\lambda^nG_n(\lambda)$ ($G_n$ bounded as $\lambda\to0$) (Jaffe 2000 §4). The perturbation series diverges (4-12) and is only an asymptotic expansion of the exact answer. The mechanism is treated in 6-04. Known

Example: if $\lambda=0$, the two-point function is $C(x)$, which decays at large distance like $e^{-m_0|x|}$, and the gap is $m_0$. The weak-coupling result says that this gap is not closed by a small interaction. The source of the gap is the $m_0$ written into the action (§09).

The counting of diagrams in perturbation theory gives an idea of what increases (4-03, 4-04). A connected diagram with $n$ vertices and $E$ external lines has $I=2n-E/2$ internal lines and $L=n-E/2+1$ loops. A loop integral carries momentum to the power $d$ and a propagator to the power $-2$, so the superficial degree of divergence is

and the diagrams with $\omega\ge0$ are the ones that can diverge in the ultraviolet ($\omega=0$ is logarithmic). A diagram with one vertex consists only of lines joined to itself, and vanishes with Wick products.

ComputedThe values in this figure are formula (11) computed on the spot. Table 3 agrees with the computation on this machine.

ComputedTable 3 The types (n, E) with ω ≥ 0, and ω.

Two of the types remaining in three dimensions are exactly the columns of Table 1. (2, 0) is the vacuum bubble joining two vertices by four lines, $\int C^4$ ($1/a$), and (2, 2) is the self-energy joining them by three lines, $\int C^3$ (logarithmic). Moving $d$ in Figure 2 from 2.66 to 2.67 brings (2, 0) into the types that can diverge. The borderline is $d=8/3$, which agrees with the borderline $n=4$ of formula (4). Computed

The problem statement says that in two dimensions $\alpha,\gamma$, and in three dimensions all five, are made to depend on the cutoff (JW §6.2). KnownThe problem statement does not say which constant corresponds to which diagram, but reading the order in $\lambda$ as the number of vertices, $\alpha,\gamma$ correspond to $n=1$, $\beta,\delta$ to $n=2$ and $\varepsilon$ to $n=3$, which matches the number of types in Table 3. ComputedJaffe (2000 §6) also says that in three dimensions, besides Wick products, one needs a logarithmically divergent mass counterterm and two vacuum-energy counterterms, divergent linearly and logarithmically. $\lambda$ itself is not renormalised. Known

Moreover, in three dimensions the Wick-ordered Hamiltonian is not a densely defined operator (an operator defined on a subspace containing vectors arbitrarily close to any vector), the state vectors have to be rescaled by a constant going to 0 with the cutoff (what the problem statement calls wave-function renormalisation), and the representation is not equivalent to the Fock representation (Jaffe 2000 §6). Jaffe calls this entirely different from two dimensions (§04), where it is equivalent to the Fock representation on bounded regions. The tool of control is the localisation of Glimm–Jaffe (1973) into phase-space cells (boxes divided by scale in both position and momentum), and the induction from estimates at one scale to estimates at twice the scale leads to the idea of the renormalisation group (ibid.; 6-05). On that basis Feldman–Osterwalder (1976) and Magnen–Sénéor (1976) independently showed the OS axioms in the limit with the cutoff and volume removed, when $\lambda/m_0^2$ is small (Summers 2016 §3.2; the former puts the mass gap in its title). The restriction on $\lambda/m_0^2$ was later removed by other methods by Seiler–Simon (1976) and Park (1977), and Burnap (1977) showed the existence of one-particle states so that Haag–Ruelle scattering theory can be used (Summers 2016 §3.2; original papers from bibliographic data only). Known

$\phi^4_2$ and $\phi^4_3$ then became examples for checking whether new methods can reproduce known results. Each method includes different stages in its output. Brydges–Fröhlich–Sokal (1983) used the lattice, the Schwinger–Dyson equations and correlation inequalities to show existence and non-triviality at weak coupling as subsequential limits of lattice theories. Stochastic quantisation (6-09) makes the measure $e^{-S}$ as the invariant measure (a measure whose distribution does not change under time evolution by this equation) of

Hairer (2014), with regularity structures, constructed solutions of the stochastic quantisation equation (12) corresponding to $\Phi^4_3$ in finite volume (as local solutions up to a time at which they might blow up). The abstract describes this as solving the problem of constructing a natural Markov process symmetric with respect to that measure, but the text does not show that the solutions are global or that the measure is invariant; it says these are expected. Gubinelli–Hofmanová (2021) showed on $\mathbb R^3$ that every limit point is non-Gaussian, reflection positive and translation invariant, and the abstract says they verified the OS axioms except for rotation invariance and clustering. In finite volume (the three-dimensional torus) there is also a method writing the measure variationally by stochastic optimal control (Barashkov–Gubinelli 2020). Note that the problem statement, for the existence part of the Yang–Mills problem, does not accept weak existence by compactness alone as a solution, unless the properties of the limit (mass gap, axioms) are shown separately (JW footnote 2). Known

KnownTable 4 Four theories (the mass dimension of the coupling is formula (7) of 6-01, the types of divergence are Table 3).

Four things can be carried over. (1) Reading the form of the counterterms from perturbation theory and making them depend on the cutoff. (2) Piling up estimates scale by scale (phase-space cells became the sequence of renormalisation group steps. 6-05). (3) Properties given by inequalities, such as reflection positivity, are inherited by the limit. (4) If exponential decay of correlations is shown by a weak-coupling expansion, it becomes the gap.

At the centre of what cannot be carried over is the source of the gap. For $\phi^4_2$ and $\phi^4_3$, $m_0\gt0$ was written into the action and it was shown that $|m-m_0|$ is small. The Yang–Mills action has no mass term, and the expected scale of the gap depends on the coupling in the form $e^{-\mathrm{const}/g^2}$, which cannot be expanded in powers of $g$ (4-07). The mass that would be the foothold of a weak-coupling expansion is absent from the start. In addition, in four dimensions divergent diagrams appear at every order, and the field must preserve gauge invariance as a connection (5-01). KnownPhysics

On the other hand, the problem statement cites the fact that four-dimensional $\phi^4$ is not asymptotically free as evidence of triviality, and names Yang–Mills, whose coupling becomes weak at short distance, as the natural candidate for constructive field theory in four dimensions (JW §6.2). Where it stops is seen in 6-10.

What this chapter can say

Sources and reproduction

Next chapter to read: 6-04 The cluster expansion (the mechanism of the tool used in §05).

← 6-02 Solvability in two dimensions and the Yang–Mills measureContents6-04 The cluster expansion →