2026-10-01 · chapter Yang–Mills and the mass gapPart 4 Quantisationdivergent series, Borel sums, renormalons

Detour: the perturbation series diverges — the factorial in the coefficients is often the shadow of an $e^{-c/g^2}$ that the series does not show

The problem of this chapter is that the series of coefficients rendered finite by renormalisation does not itself converge (a different matter from the divergences of 4-04). Even so it has uses. If, fixing the order $N$ and letting $u\to0$, the truncation error vanishes as fast as $u^{N+1}$, we say that the series is asymptotic to $f$, and write $f(u)\sim\sum a_nu^n$.

The order of the limits is the reverse of convergence. The coefficients are determined uniquely by $f$ Known, but the converse does not hold. $e^{-1/u}$ approaches 0 faster than any $u^N$, so $f$ and $f+e^{-1/u}$ have the same series (6-12 §07). The theme of this chapter is that this "part not shown in the coefficients" is recorded in the way the coefficients grow.

Example. That all the coefficients of $e^{-1/u}$ are 0 can be checked in the form of (1) divided by $u^N$. Putting $v=1/u$, $e^{-1/u}/u^N=v^Ne^{-v}$, and as $u\to0$ ($v\to\infty$) the exponential beats the power and it approaches 0. This is so for every $N$, so $a_0=a_1=\cdots=0$. An example in the other direction is the model (2) of this chapter. (14) of 4-03 §08 shows $\bigl|Z(u)-\sum_{n\lt N}a_nu^n\bigr|\le|a_N|\,u^N$ for $u\ge0$, and $|a_N|$ can be taken as the $K_N$ of (1). $K_N$ grows factorially with $N$, but (1) needs only an estimate for each fixed $N$, so the series is asymptotic to $Z$.

The abstract of Dyson (1952) says it presents an argument leading 'tentatively' to the conclusion that the power series of quantum electrodynamics all diverge even after renormalisation Known. Its core is usually presented in the form: "for $e^2\lt0$, like charges attract and the vacuum collapses, so the series cannot continue smoothly to both sides of $e^2=0$". It is a heuristic argument. For a small model everything can be computed.

For $u=-s\lt0$ the integral does not exist (the "collapse of the vacuum" in the model). Even so, turning the variable as $x\to e^{-i\theta/4}x$ brings the quartic term of $u=s\,e^{i\theta}$ back to $s\,x^4$, so $Z$ extends to complex $u$. There are two reasons why the contour may be turned. One is that the integral of a function complex-differentiable on and inside a closed path, taken along that path, is 0 (Cauchy's theorem Known; we use it in the form of the integral formula in §03). The integrand is the exponential of a polynomial in $x$, so it is complex-differentiable everywhere. The other is that on the far arc of the sector between the original real axis and the turned line, the real part of the quartic term stays positive and acts at the rate $|x|^4$, so the contribution of the arc vanishes as $|x|\to\infty$. Approaching the negative axis from above,

takes a non-real value, and from below its complex conjugate. The values above and below the negative axis disagree: this is the source of the divergence. The imaginary part is fixed by the height of the wall. $S_{-s}(x)=\frac12x^2-\frac{s}{24}x^4$ takes its maximum $\frac{3}{2s}$ at $x^2=6/s$, and the rough probability of leaking from the valley over the wall is $e^{-3/(2s)}$. Strictly, there is no tunnelling in a zero-dimensional integral. What is meant is that the value $\frac{3}{2s}$ of the action at the saddle point $x^2=6/s$ of the integrand of (3) fixes the exponent of the imaginary part, and the leading term as $s\to0$ is $-e^{-3/(2s)}/\sqrt2$ (Table 1).

The tool we use is Cauchy's integral formula: a function complex-differentiable on and inside a closed path satisfies $f(u)=\frac{1}{2\pi i}\oint\frac{f(w)}{w-u}\,dw$ (anticlockwise) at points inside Known. $Z$ is complex-differentiable on the plane minus the negative axis and decreases as $|w|^{-1/4}$ at infinity, so taking the path to be a large circle and the upper and lower sides of the negative axis, only the difference $2i\operatorname{Im}Z$ between the sides remains.

This is the dispersion relation (numerically integrating the right-hand side with the imaginary part computed from (3), it agrees with the left-hand side to order $10^{-6}$ at $u=0.5,1,2$ Computed). Expanding $\frac1{s+u}=\sum_n(-u)^n/s^{n+1}$ and integrating term by term,

(since it is not complex-differentiable at $u=0$, what is obtained is an asymptotic series), and for large $n$ only the imaginary part $\approx-e^{-3/(2s)}/\sqrt2$ near $s\to0$ matters. From $\int_0^\infty e^{-c/s}s^{-n-1}ds=\Gamma(n)/c^n$,

The factorial comes from $\Gamma(n)=(n-1)!$, and the base $2/3$ is the inverse of the coefficient of the wall height $3/(2s)$; the alternation of signs comes from the singularity lying on the negative axis.

Bringing this line of reading large orders from the action at a saddle point into field theory is attributed to Lipatov (1977), and the review of Beneke (1999) credits Lipatov 1977 with factorial divergence due to instantons (classical solutions in imaginary time; defined in §06) Known.

The large orders of the ground-state energy $\frac12+\frac34g-\frac{21}8g^2+\cdots$ of $H=\frac12p^2+\frac12x^2+gx^4$ in quantum mechanics were studied by Bender–Wu (1969, 1973) (abstracts) Known. The ratio of the known form $E_n\approx(-1)^{n+1}\frac{\sqrt6}{\pi^{3/2}}3^n\Gamma(n+\tfrac12)$ (the text of the source not checked) to the coefficients computed with rationals up to order 150 is 0.832 at $n=10$ and 0.991 at 150 Computed. The base 3 is the inverse of the coefficient of the action $\frac{1}{3|g|}$ for passing through the wall of $V=\frac12x^2-|g|x^4$ in imaginary time.

The naive method is to stop where the terms are smallest. The terms $\Gamma(n)(2u/3)^n$ of (6) decrease up to $n\approx3/(2u)$ and grow beyond it. Putting Stirling's formula $\Gamma(n)\approx\sqrt{2\pi/n}\,(n/e)^n$ in there gives

The floor of the error is the size of an exponentially small quantity fixed by the height of the wall, and the series can say nothing finer than that. The "part not shown in the coefficients" of section 01 appears as a limit on precision (the ratio on the right moves because of $\sqrt u$).

Why the error is about the smallest term. In this model, by (14) of 4-03 §08, the error of stopping after $N$ terms does not exceed the first discarded term $|a_N|u^N$ Known. Stopping where the terms are smallest, the error is at most that smallest term. Since the signs of the coefficients alternate, the actual error settles at about half the smallest term (0.47 to 0.50 times in the 6 rows of Table 3 Computed). Thus "the best error ≈ the smallest term ≈ (7)". How to read Table 3. Each row fixes one $u$, compares the partial sums from order 0 to order $N$ ($N\lt60$) with direct numerical integration $Z(u)$, and shows the $N$ with the smallest error and that error. The best $N$ comes around $3/(2u)$. The rightmost column is the error divided by $e^{-3/(2u)}$, and stays within 0.07 to 0.26 as $u$ varies from 0.1 to 2. That is, the error becomes small at the same rate as $e^{-3/(2u)}$.

ComputedThe numbers in this figure were computed on this machine (Simpson's rule and the recurrence for the coefficients; agreeing with Table 3).

The Borel sum first divides by $n!$ to make the series converge, then multiplies back with $\int_0^\infty e^{-t}t^n\,dt=n!$.

$B$ is called the Borel transform, and the complex plane of $t$ the Borel plane. Term by term $\frac1u\int_0^\infty e^{-t/u}\frac{t^n}{n!}dt=u^n$, so the asymptotic series of $\tilde Z$ is the original series. For the zero-dimensional model, from $(4n-1)!!=16^n(\tfrac14)_n(\tfrac34)_n$ ($(a)_n=a(a+1)\cdots(a+n-1)$),

The hypergeometric function ${}_2F_1(a,b;c;z)$ extends to the plane minus a cut running from $z=1$ Known. The only singularity in the Borel plane is the single point $t=-3/2$, at the same position as the coefficient of the wall height. The path of integration $t\ge0$ does not touch the singularity, and (8) returns to $Z(u)$ as Table 3 shows Computed.

The theorem guaranteeing that the Borel sum agrees with the original function requires analyticity of the original function in a disc tangent at the origin, and a uniform estimate bounding the $K_N$ of (1) by $A\sigma^NN!$ ($A,\sigma$ constants independent of $N$). (1) alone is not enough. Beneke cites this as the Watson–Nevanlinna–Sokal theorem (Sokal 1980), and writes 'These requirements are too strong for renormalizable theories' Known.

For a series whose terms have the same sign and grow as $a_n\approx K\,a^n n!$ ($a\gt0$), the singularity comes at $t=1/a$ on the positive axis, on the path of (8). Going round above or below gives two sums whose imaginary parts disagree by an amount of size $e^{-1/(au)}$. We look at the simplest case, $a_n=n!\,a^n$. The Borel transform is $B(t)=\sum_n(at)^n=\frac{1}{1-at}$, with a pole at $t=1/a$. Subtracting the integral going round below from the one going round above gives an integral once clockwise round the pole, and from the residue

(the imaginary part of each is $\pm\pi e^{-1/(au)}/(au)$). This difference is of a size that shows up in no power of $u$. Tables and figures are in 6-12 §08. Beneke shows, in examples where the exact function is known, that the two ambiguities, of the Borel integral and of the exponentially small terms, cancel each other, and sets up the following chain Known.

He then warns 'it is not rigorous'. It is not a theorem Physics.

Example. The gap of the double well of 2-05 §07 has size $e^{-S_0}$ and appears at no order of perturbation theory. A classical solution that moves in imaginary time from one valley to the other is called an instanton, and its action is the $S_0$ of (9) in 2-05. A solution moving the other way is an anti-instanton (for the counterpart in Yang–Mills, see 3-05). The ambiguity of the Borel sum of the series around a valley has size $e^{-2S_0}$ (the weight of an instanton–anti-instanton pair), and is said to cancel against the ambiguity of the pair's contribution (Bogomolny 1980, Zinn-Justin 1981) Physics. The way the series diverges announces the existence of terms the series does not show.

Renormalisable theories have a source of $n!$ other than saddle points: renormalons (according to Beneke, found by Gross–Neveu 1974, Lautrup 1977 and 't Hooft 1977, with the name coming from 't Hooft's Erice lectures Known). The source is the bubble chain diagram: a propagator with $n$ one-loop bubbles (loops in which a pair of particles is created and annihilated) strung on it; summed over $n$, it becomes the one-loop running coupling $\alpha(k)$ inserted into a single line (Beneke §2.2) Known. It is a different thing from the vacuum bubbles of 4-03 §03.

We use a model that shrinks this to one variable. Suppose there is a quantity measured by injecting a momentum $Q$ from outside, and the integral over the loop momentum $k$ inside it carries the running coupling $\alpha(k)=\alpha/(1-b\alpha\ln(Q^2/k^2))$ ($\alpha=\alpha(Q)$, $b=\beta_0/4\pi$, $\beta_0=11N/3$ for pure $SU(N)$; for conversion between the four ways of writing the coefficient, Table 1 of 4-06; 4-04 §05, 4-06). Expanding $\alpha(k)$ in $\alpha$, the order $n+1$ contains the $n$-th power of $\ln(Q^2/k^2)$. About the rest of the integrand we assume only that, measured with $d\ln k^2$, it vanishes in proportion to $k^4$ at small $k$ ($x^2$) and decreases as $1/k^2$ at large $k$ ($x^{-1}$). With $x=k^2/Q^2$,

each term is finite, yet the coefficients grow as $n!$ (the closed form agrees with numerical integration to 6 digits Computed). The reason is the position of the peak. The infrared integrand (measured in $\ln x$, $x^2(\ln\frac1x)^n$) peaks at $x=e^{-n/2}$, and as the order rises the peak runs away to smaller momenta. The ultraviolet peak is at $x=e^n$. (2.24) and (2.25) of Beneke give the same positions for the bubble chain diagram Known. On the infrared side the signs are the same (positive axis, with ambiguity), and on the ultraviolet side they alternate. Changing the power at small $k$ to $k^a$, the infrared renormalon sits at $u=a/2$ on the scale of §08, and the size of the ambiguity is $(\Lambda/Q)^a$ (Beneke §2.2) Known. This model has $a=4$.

ComputedThe numbers in this figure were computed on this machine (the formula of (11) evaluated inside the figure; agreeing with Table 4).

Where does the peak stop? At one loop $\Lambda^2/Q^2=e^{-1/(b\alpha)}$ (4-07). The term $r_n^{\mathrm{IR}}\alpha^{n+1}$ is smallest at $n^\ast\approx2/(b\alpha)$, and then the peak is at $e^{-n^\ast/2}\approx\Lambda^2/Q^2$. The order at which the series stops being useful is the order at which the loop momentum reaches $\Lambda$, and the smallest term has the size of $(\Lambda/Q)^4$ times a factor of the order of a power of $\alpha$.

Beneke notes that the positions of renormalons do not depend on the higher coefficients of the β function, and are in general logically separate from the Landau pole Known.

Transforming the series in $\alpha$ in the manner of (8) and measuring positions by $u=\beta_0t/4\pi$, $e^{-t/\alpha}=(\Lambda^2/Q^2)^u$, so a singularity at position $u$ corresponds to an ambiguity of size $(\Lambda/Q)^{2u}$. We prepare two terms. The operator product expansion is a physical prescription that expands a quantity measured at short distance $1/Q$ as a sum of "coefficient × $(\Lambda/Q)^{d}$ × the vacuum expectation value of a combination of fields of dimension $d$" Physics. In pure Yang–Mills, the gauge-invariant combination of fields of lowest dimension is $F^a_{\mu\nu}F^a_{\mu\nu}$, of dimension 4, and its expectation value $\langle F^2\rangle$ is called the gluon condensate. There is no gauge-invariant combination of dimension 2 ($A_\mu A_\mu$ has dimension 2 but changes under gauge transformations).

For the two-point correlation functions of QCD that Beneke summarises (quantities measured by injecting a momentum $Q$ from outside; an example is the correlation of quark currents) Physics, the infrared renormalons are at $u=2,3,\dots$ (there is none at $u=1$ because the operator product expansion has no term of dimension 2, Parisi 1979; $u=2$ corresponds to the $(\Lambda/Q)^4$ of the gluon condensate, and the weight $k^4$ of the model of §07 is this $u=2$), the ultraviolet renormalons at $u=-1,-2,\dots$, and the instanton–anti-instanton pairs at $t=4\pi n$ (Bogomolny–Fateyev 1977). At which $u$ they are present depends on the observable. Converting to $t=4\pi u/\beta_0$ with the $\beta_0$ of pure $SU(N)$, the renormalons move towards the origin in proportion to $1/N$, and the pairs stay at $4\pi$. For $N=3$, $u=2$ is at $t=2.285$ and the pair 5.5 times farther; for $N=10$, 18 times Computed.

The weight of a pair $e^{-4\pi/\alpha}=e^{-16\pi^2/g^2}$ is $(\Lambda/Q)^{22N/3}$, a far smaller quantity than the $(\Lambda/Q)^4$ of $u=2$ (for a single one, (14) of 4-07 §08). Beneke also expects that in QCD the instanton singularities are far away and do not determine the large orders Physics.

Observation on the lattice. Bali–Bauer–Pineda (2014) computed numerically the perturbative coefficients of the average plaquette of pure SU(3) up to order 35 in $\alpha$, and write 'The on-set of the dominance of the dimension four renormalon associated to the gluon condensate is clearly observed' (abstract) Physics. The PDG review also says that the renormalon divergence is 'not usually visible' at the orders available, but is 'closely connected' with non-perturbative contributions.

On the side of theorems. (4) to (6) and (9) for the zero-dimensional model are checkable mathematics Known. There are papers with Borel summability in their titles, for the anharmonic oscillator (Graffi–Grecchi–Simon 1970) and for $P(\phi)_2$ and $\phi^4_3$ (Eckmann–Magnen–Sénéor 1975, Magnen–Sénéor 1977; 6-03) (bibliographic data only). According to their titles, they show Borel summability for the eigenvalues of the oscillator and for the Schwinger functions (correlation functions) of $P(\phi)_2$ and $\phi^4_3$ respectively. In those cases, by the Watson–Nevanlinna–Sokal theorem of §05, the series determines the value uniquely.

On the side of expectations. For four-dimensional Yang–Mills the coefficients are well defined and finite at each order (4-05), but the premises for asking about the relation between the series and the theory are missing. Beneke writes 'Since there is no rigorous non-perturbative definition of R in theories such as QED and QCD, we cannot even ask whether series expansions are asymptotic' Known. The problem statement (Jaffe–Witten §6.1) also writes that in most known examples perturbation series are 'divergent expansions' and 'even Borel and other resummation methods have limited applicability' Known.

To sum up: (i) the $n!$ in the coefficients is the shadow of quantities of size $e^{-c/g^2}$ that the series does not show Physics. Precisely, in the model, in the sense of (4) to (6), the exponent $c$ of the imaginary part at negative coupling fixes the base $1/c$ of the coefficients. For Yang–Mills it is an expectation at the level of the chain (10), and ultraviolet renormalons (with alternating signs) carry no ambiguity of exponentially small terms. (ii) If the gap is proportional to $\Lambda\propto e^{-8\pi^2/(\beta_0g^2)}$, it appears at no order of the series (4-07 §05), and the only clue left is on the side of how the series diverges Physics. (iii) The formalism that assembles the unseen terms from the way the series diverges (transseries) is in 6-12 §09, and for pure Yang–Mills on $\mathbb R^4$ it is not established even at the level of physics.

What this chapter can say

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Next chapter to read: 5-01 Wilson's lattice. The detour continues in 6-12.

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