2026-10-01 · chapter Yang–Mills and the mass gapPart 6 Constructive field theory1/0 and exp(−1/g²)

Detour: "if 1/0 is given a symbol" — assigning a value erases the manner of approach, and the scale expected for the gap has the form $e^{-1/g^2}$

"If $1/0$ were built into mathematics as a symbol, wouldn't the situation of field theory, troubled by divergences, change?" The question can be read in three ways. (i) Assign a value: give $1/0$ or $0/0$ a single value ($\infty$, $0$, a new symbol $\bot$ and so on). (ii) Replace: exchange the singular point for the set of directions of approach, or a boundary. (iii) Change the axioms.

(iii) becomes a different problem, because the problem statement names the properties: 'Existence includes establishing axiomatic properties at least as strong as those cited in [45, 35]' (the Wightman and Osterwalder–Schrader axioms) (0-03). Known This chapter examines (i) and (ii), and arrives at the following.

Example: $\sin x/x$ has the form $0/0$ at $x=0$, yet the limit is fixed at 1. On a line there are two ways of approach, from the left and from the right, and both give the same value. In the plane it is not so (§04).

There are several systems that assign a value to $1/0$, all built without contradiction, but each gives something up (Table 1).

KnownTable 1 Systems that assign a value to $1/0$ (IEEE 754 computed; transreal from bibliographic data only).

A meadow adds to a commutative ring (a set in which addition, subtraction and multiplication obey the usual rules; division is not assumed) the operation of inverse $x\mapsto x^{-1}$, and imposes only

From this follows $0^{-1}=0$. A field (a commutative ring in which one can divide by elements other than 0; $\mathbb Q$ and so on) with the convention $0^{-1}=0$ is a meadow, and so is its direct product (componentwise operations). For example, for $x=(1,0)$ in $\mathbb Q\times\mathbb Q$, $x^{-1}=(1,0)$ and $x\cdot x^{-1}=(1,0)\ne(1,1)$. Computed

Checking IEEE 754 double precision on this machine: in NumPy 1.26, 1/0, 1/(-0), 0/0 give inf, -inf, nan; 0.0 == -0.0 is True and nan == nan is False; and Python's 1.0/0.0 raises the exception ZeroDivisionError. Computed The same numbers give different answers. "The value of $1/0$" is a convention of the system. The signed zero is a symbol that keeps one bit of "which side it came from", and in exchange the reciprocals differ although $0=-0$. The tools of Yang–Mills (Hilbert spaces, spectra; 1-10) rely on $0\cdot x=0$, subtraction and equality. Nearby are non-standard analysis, which has $1/\varepsilon$ but does not define $1/0$, and Colombeau's algebras, which allow products of distributions at the cost of weakening equality (1-11).

§2.1 of Saitoh's draft (2023) regards $b/a$ as the solution of the equation $ax=b$, and fixes it by Tikhonov regularisation. Known

If $a\ne0$ it is the ordinary $b/a$; if $a=0$, $x_\lambda=0$, so $b/0=0$. Replacing the penalty by $\lambda(x-c)^2$ gives

and the answer at $a=0$ is $c$ itself. Example: for $(a,b)=(0,1)$, always $x_\lambda=c$ from $\lambda=1$ down to $10^{-8}$. For $(a,b,c)=(2,1,5)$, $1.4,\ 0.511,\ 0.5001,\ 0.5$, approaching $b/a$. Computed

Formula (3) is also the trivial case of the solution of the equation $ax=b$ by the Moore–Penrose generalised inverse (draft §2.1), and is a consistent definition. What can be said is that the value of $b/0$ comes not from the equation but from the added penalty: the path of regularisation. What renormalisation in field theory tracks is this dependence on the path (§06).

In the plane the limit changes with direction.

On the half-line $(r\cos\theta,\,r\sin\theta)$ at angle $\theta$, $f=\sin2\theta$, independent of $r$. Fixing one value at the origin does not make $f$ continuous. Known

The blow-up is the operation of replacing the origin by the set of directions (the projective line $\mathbb{RP}^1$: the circle with $\theta$ and $\theta+\pi$ regarded as the same). A direction is written $[u:v]$. $[u:v]$ is $(u,v)\ne(0,0)$ viewed up to non-zero multiples, that is, the name of a line through the origin ($[1:2]=[-2:-4]$). Precisely, one makes the set of pairs of a point and a direction

Away from the origin the direction is fixed uniquely, so it is the same as the original plane; only over the origin does a circle sit, and the whole is an open Möbius band (1-05). On it, $f=2uv/(u^2+v^2)$ is continuous, including on the circle over the origin.

ComputedThe values in this figure compute formula (5) within the figure (shade is $|f|$, the two colours the sign).

The picture on the right represents the point $(r\cos\theta,\,r\sin\theta)$ by $\theta\in[0,\pi)$ and signed $r$. Points with $r<0$ lie on the side of direction $\theta+\pi$, but $\sin2\theta$ does not change under $\theta\to\theta+\pi$, so the values form vertical stripes and do not break when crossing the central horizontal line $r=0$. $\theta=\pi$ is the same direction as $\theta=0$, and there the sign of $r$ is exchanged, so $(\pi,r)$ on the right edge is glued to $(0,-r)$ on the left edge. Since it is glued upside down, the band becomes a Möbius band. A symbol that assigns one value to $0/0$ has no place to carry these stripes.

The examples where replacing the singular point has worked all keep the manner of approach. Known

• Adding one point at infinity. By the removable singularity theorem of Uhlenbeck (1982), solutions of the Yang–Mills equations with finite action on $\mathbb R^4$ come, up to gauge transformations, from smooth solutions on $S^4=\mathbb R^4\cup\{\infty\}$ (3-05). Intuitively, the finite action constrains the manner of approach to infinity.
• Replacing collisions by directions. Fulton–MacPherson (1994) replace the diagonals (where two or more points coincide) of configuration space (all tuples of positions of $n$ points) by a boundary that records the direction and scale of collision. Part II of Axelrod–Singer (1994) used a compactification of this type to show topological invariance of the perturbation theory of Chern–Simons theory, one of the gauge theories in three dimensions. Bergbauer–Brunetti–Kreimer (2009) showed that extension to wonderful models (De Concini–Procesi; a generalisation of the Fulton–MacPherson type of compactification) agrees with Epstein–Glaser renormalisation.

The same holds within Yang–Mills. Singer (1978) noted that, because there is no continuous global gauge fixing, 'an infinity in the determinant of the Jacobian' arises in the Faddeev–Popov method, and named the road of gluing local gauge fixings together with a partition of unity (3-04, 4-08). The lattice $\Delta=m_{\mathrm{lat}}/a$ has the form $0/0$ as $a\to0$, and what has meaning is the limit of the dimensionless ratio $m_{\mathrm{lat}}/\sqrt{\sigma_{\mathrm{lat}}}$ (5-09). What has worked is on the side of geometry and perturbation theory, and it has not yet reached non-Gaussian measures and the mass gap.

Ultraviolet divergences come as families of regularised quantities. A small model:

The left diverges like $\ln(1/\eta)$ by cutting the lower end, the right like $1/\delta$ by shifting the power. Written as "$1/0$" they are the same, but their rates and forms differ, and still their difference goes to the same value.

ComputedTable 2 Two regularisations (SciPy on this machine).

Choosing a scale $\mu$ and subtracting as $I_R(p;\mu)=\lim_{\eta\to0}[I(p;\eta)+\ln(\eta\mu)]=-\gamma-\ln(p/\mu)$ ($\gamma$ is Euler's constant), the finite part depends on $\mu$ but the difference does not. Computed This freedom, and how the finite part moves when $\mu$ is moved, are the seed of the renormalisation group (4-04).

In field theory, symbolisations that carry "how it diverges" have already been built rigorously (4-05): the BPHZ subtraction, the Epstein–Glaser method in which no infinity ever appears, and the Hopf algebra of Connes–Kreimer, which Birkhoff-decomposes the pole $1/\varepsilon$ of dimensional regularisation as a formal symbol. Perturbative renormalisability of Yang–Mills was established in 1971–72 by 't Hooft, 't Hooft–Veltman, Lee–Zinn-Justin and others (followed by an organisation through BRS symmetry), and Costello gave another proof in the BV formalism. Known Symbolisation has been realised in a form that carries more than $1/0$, and the problem still remains. What remains is not a problem of assigning values.

is a calculus staple. For $x>0$, $h^{(n)}(x)=P_n(1/x)\,e^{-1/x}$ with $P_{n+1}(u)=u^2(P_n(u)-P_n'(u))$ (for example $P_2=u^4-2u^3$), and since $u^ke^{-u}\to0$ as $u\to\infty$, $h^{(n)}(0)=0$ for every $n$. The Taylor series is identically 0, yet $h$ is not 0 for $x>0$. Known

$e^{-1/x}$ goes to 0 faster than any power, but it starts winning late: the maximum of $e^{-1/x}/x^k$ is $k^ke^{-k}$ at $x=1/k$. Example: $e^{-1/x}/x^{10}$ is $4.5\times10^5$ at $x=0.1$, $2.1\times10^4$ at $x=0.05$, and only at $x=0.02$ does it reach $1.9\times10^{-5}$. Computed

In physics this form appears in the scale $\Lambda$. The one-loop running coupling is $\alpha_s(\mu^2)=(b_0\ln(\mu^2/\Lambda^2))^{-1}$, and the review of the Particle Data Group (PDG) explains $\Lambda$ as 'the scale where the perturbatively-defined coupling would diverge' (the "$1/0$" of the one-loop running coupling). Solving the other way,

(pure $SU(N)$, $\alpha_s=g^2/4\pi$; PDG's $b_0=(33-2n_f)/12\pi$ with $N=3$, $n_f=0$; 4-07). (10) has the form of (9) as a function of $g^2$, so quantities proportional to $\Lambda$ look like 0 at every order of a power series in $g$. The mass gap is expected to be proportional to $\Lambda$. Physics Example: for $SU(3)$, $g^2=1$ gives $\Lambda/\mu=7.6\times10^{-4}$ and $g^2=2$ gives $2.8\times10^{-2}$ ($g^2$ is hypothetical). Computed §6.6 of the problem statement also writes 'the mass gap is not classically visible' (6-10).

The series with coefficients $n!$

does not converge for any $x>0$, even though each term is finite. The terms decrease until $n$ reaches $1/x$, and increase beyond. By Stirling's formula the smallest term is of order $\sqrt{2\pi/x}\,e^{-1/x}$, and the accuracy of truncation levels off there (Figure 2). The Borel sum forms $B(t)=\sum t^n=1/(1-t)$ by dividing the coefficients by $n!$, and returns via $\int_0^\infty e^{-t}t^n\,dt=n!$.

On the path of integration, at $t=1/x$, there is a pole. The "$1/0$" of this chapter appears here: in the Borel plane. Avoiding the pole by a small semicircle $t=1/x+\rho e^{i\phi}$, going over the top moves $\phi$ from $\pi$ to $0$, and going underneath from $-\pi$ to $0$, so $\int dt/(t-1/x)=\int i\,d\phi$ is $-i\pi$ above and $+i\pi$ below. The integrand is $\frac{1}{1-xt}=-\frac1x\cdot\frac{1}{t-1/x}$, so the sign is reversed, and putting $e^{-t}\approx e^{-1/x}$ near the pole, the contribution of the semicircle is $+i\pi e^{-1/x}/x$ above and $-i\pi e^{-1/x}/x$ below. Writing $S_+$ for the sum going over the top and $S_-$ for the sum going underneath,

and they disagree. PV is the principal value, approaching symmetrically from both sides of the pole, and $\mathrm{Ei}(y)=\mathrm{PV}\!\int_{-\infty}^{y}e^{s}\,ds/s$ is the exponential integral. The two differ only in the sign of the imaginary part, whose size is $\pi e^{-1/x}/x$. The difference between above and below is $S_+-S_-=2\pi i\,e^{-1/x}/x$, twice that in size. Both the error of optimal truncation and the discrepancy are of the size of $e^{-1/x}$ times a power of $x$ (Table 3). Computed

ComputedThe values in this figure compute the terms $n!\,x^n$ and $\pi e^{-1/x}/x$ within the figure.

ComputedTable 3 The numbers in the table were computed on this machine (SciPy). $N^\ast$ is the index of the smallest term. The rightmost column is the size of the imaginary part of the Borel sum; the difference between above and below is twice that.

To keep the discrepancy, add $e^{-1/x}$ as an independent unit.

The choice of how to avoid the pole moves to the choice of the coefficient $\sigma$, and the information about the manner of approach is recorded not in the value of $1/0$ but in the coefficient of $e^{-1/x}$. Such formal sums are called transseries, and research in this direction resurgence (4-12). Known

In Yang–Mills the perturbation series is thought to diverge like $\alpha_s^n\,n!$ (renormalons). The $n!$ of (infrared) renormalons is thought to be a contribution from the region of the loop-momentum integral where the momentum is close to $\Lambda$, near the point where the one-loop running coupling diverges. There are said to be separate ultraviolet renormalons, coming from the side of large momenta. The PDG writes 'closely connected with non-perturbative contributions', while saying they are 'not usually visible' at the orders available. Physics Results of resurgence at the level of physics exist for the two-dimensional $CP(N-1)$ model on $\mathbb R\times S^1$ (twisted boundary conditions; Dunne–Ünsal 2012) and for four-dimensional gauge theories with adjoint fermions compactified on $\mathbb R^3\times S^1$ (Argyres–Ünsal 2012), but not for pure Yang–Mills on $\mathbb R^4$.

Systems that assign values hold together by weakening the rules of operations or fixing the path to one, and throw away the information about the manner of approach. What has worked are replacements that keep the manner of approach, and symbols that carry the rate of divergence (the Birkhoff decomposition of $1/\varepsilon$, the renormalisation group), both on the side of geometry and perturbation theory. Known The scale of the gap and the expected $\Lambda$ have the form $e^{-1/g^2}$, invisible at every order. Physics The remaining obstacles — estimates uniform in the volume, the infinite-volume limit, non-triviality, reflection positivity, the gauge orbit space — are not problems of assigning values (6-11).

Detour: from the side of logic

What this chapter can say

Sources and reproduction

Next chapter to read: 7-01 How physics sees it. Part 7 begins with how physics handles the scale $\Lambda$.

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