2026-10-01 · chapter Yang–Mills and the mass gapPart 6 Constructive field theoryfour dimensions, ultraviolet stability

Four dimensions: Bałaban and Magnen–Rivasseau–Sénéor — the ultraviolet side has reached "the level of control" in finite volume, but the construction is not complete

In three dimensions, refining the lattice weakened the coupling by itself (6-06 §02). In four dimensions the coupling $g$ has no dimension of length, and only asymptotic freedom weakens the coupling at fine scales (4-06). The coupling $g(\ell)$ at length scale $\ell$ obeys, for pure gauge $SU(N)$,

(the PDG formula (9.3) converted to the conventions of this book). Known Keeping only one loop,

and each halving of the scale increases $1/g^2$ by $2b_0\ln2$. Example: halving the lattice spacing from $g^2=0.2$, for $SU(3)$ $1/g^2$ goes from 5 to 5.0966 and $g^2$ decreases by 1.9%; for $SU(2)$ it decreases by 1.3%. In three dimensions $g_0^2=g^2a$ would halve. Computed

Because of this slowness, the sum of corrections of order $g_k^2$ at each step keeps growing like the logarithm of the number of steps, and cannot be piled up as an error (6-05 §06). The corrections must be renormalised into the coupling itself, and the title of the first paper of the four-dimensional series contains 'a coupling constant renormalization in four dimensions'. Known

Bałaban's work is a series that piles four-dimensional papers (1987–89) on foundations shared with three dimensions (1983–85), and the survey calls it 'a long series of papers spanning six years' (Chatterjee 2018 §6). Known

KnownTable 1 The order of the series (bibliographic data cross-checked with Chatterjee 2018, Dimock 2011 and Crossref).

The problem statement calls this extension 'a qualitative breakthrough', and summarises it as 'an analysis of asymptotic freedom to control the renormalization group flow as well as obtaining quantitative estimates on effects arising from large values of the gauge field' (JW §6.5). Going by the titles, the 1987 paper handles small fields and the running of the coupling, and the two 1989 papers handle large fields (the $\Omega^c$ of 6-05 §04). Known

This is the theorem in the introduction of the last paper.

'Theorem 1. If the sequence of the effective coupling constants is contained in an interval ]0, γ] with a sufficiently small positive γ, then the effective densities ρ_k have the form, and satisfy all the conditions and bounds, described in Sect. 2 [III].'

The sequence of effective couplings is the couplings $g_k$ for each step $k=0,\dots,K$ (from the fine steps to the unit lattice). "Form, conditions and estimates" refers to the form of the induction in 6-05 §07. Extracting only the logical form,

The author continues that the hypothesis can be removed because it follows from basic inequalities that come from Theorem 2 of an earlier paper, but 'The proof of Theorem 2, which is based on second order perturbative calculations, is very awkward and long ... and has not been published yet, so we have Theorem 1 with the assumption'. As of 1989, ultraviolet stability in four dimensions is a theorem with a hypothesis. Within what we searched (the bibliographic data of the author's papers from 1989 on), no paper containing the proof of Theorem 2 was found. Known

From the theorem follow 'the ultraviolet stability bounds of the same type as for superrenormalizable models in [16]', with constants $E_\pm$ 'independent of k, T_η, U_k'. The analysis of expectation values of loop variables and the like is left as future work. Known

This means estimates of the same type as for super-renormalisable models (reference [16] of the original paper). Three-dimensional Yang–Mills is super-renormalisable (formula (7) of 6-01), so the form can be read as being of the same type as formula (10) of 6-05. $T_\eta$ is the paper's lattice and $|T_\eta|$ its number of points, corresponding to the $|T_k|$ of formula (10) of 6-05. $U_k$ is the field configuration at step $k$. The point is that the constants depend neither on the step, nor on the size of the lattice, nor on the field (this paragraph is this chapter's reading).

Example: the effective couplings $g_k$ of the paper are defined step by step within the paper, and are not the running (2) itself. Reading them with the one-loop running (2), $g_k$ is larger on the coarser side, and the maximum of the sequence is $g_K$ at the unit lattice, so hypothesis (3) shrinks to the single condition $g_K\le\gamma$ (this chapter's reading).

If the scale of the unit lattice where the sequence of steps stops is $a_0$ (formula (1) of 6-05), the hypothesis is that the coupling is small at the scale of the unit lattice, written here as $g^2(a_0)\le\gamma$. Extending formula (1) to the coarse side, write $\ell_*$ for the length at which $g^2=1$. $\ell_*$ is a rough-guide length, extended to where perturbation theory cannot be trusted, and an indication of the scale at which physics thinks confinement and masses appear. Physics Integrating formula (1) with $u=g^2$,

The right-hand one is the one-loop approximation. The statement of the theorem only says that $\gamma$ is 'sufficiently small', so the $\gamma$ below are values for illustration. Here $\gamma$ is used as an upper bound on $g^2$ (it differs from the $g_k\le\gamma$ of formula (3) only by a relabelling of $\gamma$).

ComputedThe numbers in this figure were computed on this machine (formula (4); solid line two loops, dashed line one loop; agrees with Table 2).

ComputedTable 2 $\log_{10}(\ell_*/a_0)$ (one loop / two loops), and the number of points of the unit lattice on a four-dimensional torus of side $\ell_*$ (two loops).

In the range of Table 2 ($\gamma=0.1$ to $0.5$), $a_0$ lies 3 to 42 orders of magnitude finer than $\ell_*$, and the four-dimensional torus of side $\ell_*$ has $10^{12}$ to $10^{167}$ points of the unit lattice. Computed

How to read it (this chapter's reading). The sequence of steps controls only scales finer than $a_0$. Beyond that is a finite-dimensional integral on the unit lattice, which is bounded since the group is compact, but the constant has the form $e^{E|T|}$. $|T|$ is the number of points of the unit lattice, corresponding to the last step of the $|T_k|$ of formula (10) of 6-05 (the $|T_\eta|$ of §03). For a torus of side $\ell_*$, $|T|$ is of the size of the number of points above. Ultraviolet stability is not information about the scale $\ell_*$.

The same words are used with different strengths. Known As estimates: 'The ultraviolet stability means that the actions ρ_K have bounds independent of the lattice spacing ε' (Bałaban 1985). As subsequences: 'The existence of a convergent subsequence is known as ultraviolet stability' (Chatterjee 2018 §6). As "a form": for MRS and Bałaban, 'a form of ultraviolet stability' was shown (Chevyrev 2022 §1.1).

The tool that goes from estimates to subsequences is compactness, and its simplest form is the Bolzano–Weierstrass theorem.

The limit need not be unique. Example: $x_K=\sin K$ has $|x_K|\le1$ but does not converge: at $K=33,77$ it is 0.99991, 0.99952, and at $K=11,55$ −0.99999, −0.99976, so one can take a subsequence approaching 1 and one approaching −1. Computed

The introduction of MRS comments on Bałaban's result as follows. 'From this result the existence of an ultraviolet limit for gauge invariant observables such as "smoothed Wilson loops" should follow, at least through a compactness argument using a subsequence of approximations; but the limit is not necessarily unique.' Footnote 2 of the problem statement does not accept weak existence by compactness as a solution, unless the properties of the limit (gap and axioms) are shown separately (JW §6.5; see also 6-06 §06). Known

Magnen–Rivasseau–Sénéor did not use a lattice, and started from a continuum momentum cutoff. The object is the regularised axial-gauge Schwinger functions of pure four-dimensional $SU(2)$, in the trivial topological sector, with a fixed infrared cutoff, and with the ultraviolet cutoff removed. Known We define the words in turn.

• Axial gauge: a choice of gauge that sets the component in one coordinate direction to 0 ($A_4=0$ and so on; gauge fixing is in 4-08).
• Trivial topological sector: treating only configurations with instanton number (3-05) 0.
• Schwinger functions: the expectation values of products of fields with one gauge fixed, $\langle A(x_1)\cdots A(x_n)\rangle$.

The centre is the estimate of the size of the field. The counterterms restoring the gauge invariance broken by the cutoff hold the field to about $\lambda^{-1}$ ($\lambda$ is the coupling). In axial gauge, the measure after gauge fixing remains a positive weight. This is the positivity of axial gauge, and MRS give 'lack of positivity' as the reason Feynman gauge and the like cannot be used as starting points. Together with this positivity, the field is about $\lambda^{-1/2}$ in the sense of probability. So one treats it by splitting into small-field and large-field regions. Written with the curvature of MRS,

Expanding the action $|F|^2$ gives quadratic, cubic and quartic terms. The two on the right are the ratios of the cubic and quartic terms to the quadratic term, rough guides to size without counting derivatives (this chapter's reading). At $|A|\sim\lambda^{-1}$ both ratios are 1, and at $|A|\sim\lambda^{-1/2}$ they are $\lambda^{1/2}$ and $\lambda$. At $\lambda^{-1}$ the non-linear terms are as large as the quadratic term, and MRS too write that information at $\lambda^{-1}$ alone is 'not enough to start perturbation theory'. Example: for $\lambda=0.01$, at $|A|\sim100$ the ratios are 1 and 1, and at $|A|\sim10$ 0.1 and 0.01. Computed

In the small-field region the gauge is replaced by a background-field-dependent one suited to perturbative calculation. The conclusion is 'The ultraviolet limit as ρ → ∞ of the Schwinger functions ... exists; furthermore these functions in the ultraviolet limit satisfy the Slavnov identities'. The Slavnov identities are relations that invariance under small gauge transformations leaves among gauge-fixed correlation functions (4-08). Known

The authors themselves also write the reservations. 'we do not provide a detailed proof of this statement', and a detailed proof 'would presumably be several hundred pages long'. It is restricted to axial gauge, does not treat instantons, and does not remove the infrared cutoff. Removing it would make the coupling large, leading 'presumably to so-called non-perturbative effects corresponding to confinement'. Known

MRS say the role of their cutoff is 'quite similar to the role played by compactness of integration over the gauge group in Balaban'. As formulas,

and both are first upper bounds on the field. Known The lattice side is written with group elements and the continuum side with Lie algebra elements, so we match the sizes. Writing $U_\ell=e^{igB_\ell}$ ($B_\ell$ a Lie algebra element) for $SU(2)$, the rotation angle can be taken from $0$ to $\pi$, so $|B_\ell|\le\pi/g$. That is, on the lattice side too the Lie-algebra field is held to about $1/g$, and both sides take the form "field ≲ 1/coupling" (this chapter's reading). Example: for $SU(2)$, $\|U-1\|=2$ happens at $U=-1$, rotation angle $\pi$. Computed Also in common is that upper bounds alone cannot start perturbation theory.

KnownTable 3 The two roads.

MRS point out that Bałaban's work does not treat gauge-fixed Schwinger functions, and write 'one can consider in fact that the ultraviolet problem for the Yang-Mills field theory is not yet understood until this point is clarified'. Conversely, MRS leave reflection positivity and the infrared for later. Known

We divide stage 3 (6-01) and what lies beyond into levels 0–7 (the rows of Table 4. 5 is that the limit is not a free (Gaussian) field, i.e. non-triviality, and the problem statement further requires the short-distance behaviour to agree with the predictions of asymptotic freedom. 7 is a mass gap uniform in the volume). Putting the words of the literature on arrows,

and only level 1 has a theorem (in four dimensions, with a hypothesis). Known

KnownTable 4 The levels of control ("prospect" is what the literature says "should follow" or "could probably be shown". What the problem statement asks for is the existence and properties of the theory, and level 1 is a step of method).

"Has reached the level of control" means the state in which level 1 goes through (in four dimensions, with a hypothesis) and parts of levels 2–4 are written as prospects or sketches. The problem statement writes of $T^4$ 'while the construction is not complete, there is ample indication that known methods could be extended to construct Yang–Mills theory on T4', and says that a detailed construction on a compact space alone would be 'a major breakthrough' (JW §6). A 2004 report also writes 'The state of the art remains the works of Balaban and of Magnen, Rivasseau and Sénéor' (Douglas 2004). Known

Example: as a yardstick, for weakly coupled three-dimensional $\phi^4$, everything corresponding to levels 1–7 is a theorem (JW §6.2, 6-03 §06). What corresponds to level 5 is that the limiting measure is non-Gaussian ('a remarkable non-Gaussian measure' of JW §6.2). That level 5 is a property that must be shown separately is treated in 6-08, and that the setters ask for new ideas at levels 6 and 7 in 6-10.

What this chapter can say

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Next chapter to read: 6-08 Triviality of φ⁴ in four dimensions (why the non-triviality of level 5 is a property that must be shown separately).

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