Borel Summation and Resurgence

Perturbation theory is the workhorse of physics, and almost none of its series converge. Expand the ground-state energy of the quartic oscillator, a QED amplitude, or the free energy of a matrix model in powers of the coupling $g$ , and the coefficients $a_n$ grow factorially, $a_n\sim C\,A^{-n}n!$ , so the radius of convergence is exactly zero. Such a series is at best asymptotic in Poincaré's sense: truncating it at the right order gives a superb approximation, but summing it is meaningless. Borel summation is the principal tool for turning this apparent disaster into a genuine function. It exploits the identity $n!=\int_0^\infty t^n e^{-t}\,\mathrm{d}t$ to trade the factorial growth for an integral: divide the coefficients by $n!$ to form the convergent Borel transform $B(t)$ , then feed it back through a Laplace integral to reconstruct a function whose asymptotic expansion is the series you started with. The physics lives in the singularities of $B(t)$ . Each saddle point of the underlying path integral — an instanton, a multi-instanton, a complex saddle — plants a singularity in the Borel plane at $t$ equal to its action $A$ . When such a singularity sits on the integration ray the reconstruction becomes ambiguous by an amount of order $e^{-A/g}$ , precisely the size of the nonperturbative effect the naive series could never see. Resurgence, discovered by Jean Écalle in 1981, is the statement that this is no accident: the rate at which perturbation theory diverges is dictated by the very nonperturbative objects it omits. Assembling perturbative and nonperturbative sectors into a single transseries, and demanding consistency under the Stokes automorphism, yields one real, unambiguous answer — the modern completion of optimal truncation and superasymptotics into an exact theory.

Key takeaways

A factorially divergent series $\sum_n a_n g^n$ with $a_n\sim n!$ still encodes a unique function: divide by $n!$ to build the convergent Borel transform $B(t)$ , then Laplace-transform it back.
Everything is controlled by the singularities of $B(t)$ : a singularity at $t=A$ obstructs the Laplace integral and forces lateral sums $\mathcal{S}_\pm$ whose difference is exponentially small, of order $e^{-A/g}$ — the intrinsic ambiguity of perturbation theory.
Resurgence: the large-order growth $a_n\sim \tfrac{S}{2\pi}(n-1)!/A^n$ of the perturbative coefficients is fixed by the action $A$ of the leading instanton, so divergence literally "resurges" the hidden nonperturbative sectors.
The transseries $\sum_k e^{-kA/g}g^{-kb}\sum_n a^{(k)}_n g^n$ packages all sectors; the Stokes automorphism relates the lateral sums and makes the full object real and unambiguous (median summation).
Borel–Écalle summation reconstructs exact quantities — energy levels, level splittings, partition functions — from divergent perturbation theory plus instanton data.
Key idea: factorial divergence is a feature, not a bug — dividing by $n!$ builds the Borel transform $B(t)$ , whose singularity at $t=A$ both creates the exponentially small ambiguity $\sim e^{-A/g}$ and, by resurgence, fixes the large-order coefficients $a_n\sim (n-1)!/A^n$ in terms of the nonperturbative saddle of action $A$ .

The core insight

Factorial divergence is a feature, not a bug: dividing by $n!$ produces the convergent Borel transform $B(t)$ , whose singularity at $t=A$ both obstructs naive resummation (creating an exponentially small ambiguity $\sim e^{-A/g}$ ) and, by resurgence, fixes the large-order coefficients $a_n\sim (n-1)!/A^n$ in terms of the nonperturbative saddle of action $A$ .

Where this lives today

Resurgence has moved from a corner of analysis to a standard instrument of theoretical physics. In quantum mechanics it turns the formal all-orders WKB series into an exact quantization condition: the double-well level splitting, over-barrier reflection, and the energies of the tilted double well and Mathieu equation are computed to arbitrary nonperturbative order by resumming a transseries, sharpening Bohr–Sommerfeld into an exact statement. In quantum field theory the same structure organizes the renormalon divergences of QCD, where Borel singularities at $t\sim n/\beta_0$ set the ambiguity of the perturbative series and are cancelled by condensates in the operator product expansion. In string and matrix-model physics the genus expansion $\sum_g F_g\,\lambda^{2g-2}$ diverges as $(2g)!$ , its Borel singularities mark D-brane/eigenvalue-tunneling instantons, and transseries summation supplies nonperturbative completions of the free energy. And there is a numerical payoff: where only a dozen coefficients are known, Borel–Padé resummation and hyperasymptotic truncation reach far beyond the least-term barrier, while resurgent large-order/low-order relations predict high-order coefficients from instanton data as a check.

The physical picture

Picture the terms of a divergent perturbative series laid out in order. Because $|a_n g^n|\sim n!\,(g/A)^n$ , they first shrink — each term smaller than the last — until $n\approx A/g$ , and then they explode. Adding terms helps up to that turning point and hurts afterward. The smallest term you can reach, the "least term," has size $\sim e^{-A/g}$ : that is the intrinsic accuracy of perturbation theory, a wall no amount of extra labour at finite order can break through. Borel summation sees past the wall by reorganizing the sum into an integral that converges over all the relevant terms at once, and the first figure makes that wall visible with real numbers.

Figure

Optimal truncation of the Euler series Σ (−1)ⁿ n! gⁿ at g = 0.1 with Borel action A = 1: the term magnitude of aₙ gⁿ, namely n!(0.1)ⁿ, falls, bottoms out at a least term of ≈ 3.6×10⁻⁴ near n ≈ 10, then diverges. That least term ≈ √(2π/g)·exp(−A/g) is the accuracy wall of perturbation theory; the exact Borel sum is 0.91563

The Borel plane is a map of the theory's hidden physics. Every saddle of the path integral leaves a singularity — a pole or branch point — at $t$ equal to its action, so the arithmetic of singularities at $t=A,2A,3A,\dots$ is a directory of instantons and multi-instantons. When such a singularity lies on the positive real axis, the natural Laplace contour runs straight into it: you must steer just above (the sum $\mathcal{S}_+$ ) or just below ( $\mathcal{S}_-$ ), and the two choices differ by exactly $e^{-A/g}$ times that saddle's contribution. Crossing a singularity is crossing a Stokes line and switching on a nonperturbative sector. The second figure is that map.

Figure

The Borel t-plane of a non-Borel-summable series: each saddle of action A plants a singularity at t = A, 2A, 3A on the positive ray, so the Laplace contour must dodge above (S₊) or below (S₋), and the two sums differ by a term of order exp(−A/g). The negative ray stays clear, which is why alternating (Borel-summable) series carry no ambiguity

Resurgence is the bootstrap hiding in this map: the wild $n!$ divergence of the cheapest, perturbative coefficients is dictated by the most expensive, exponentially suppressed objects. Measure how fast perturbation theory diverges and you read off the instanton action; know the instanton and you predict the divergence rate. One sector resurges inside another. Optimal truncation is the WKB-level glimpse of this; Borel–Écalle summation is the full microscope, continuing past $e^{-A/g}$ to $e^{-2A/g}$ and beyond until the answer is exact and real.

The core equations

The object handed to us by a physical calculation is a formal power series in the coupling. Poincaré's definition makes it usable without convergence: the series is asymptotic to $f$ if truncating at order $N$ leaves an error smaller than the first omitted term as $g\to0^+$ , for every fixed $N$ .

f(g)\sim\sum_{n=0}^{\infty} a_n\,g^n,\qquad g\to 0^+ \qquad (1)

where: $f(g)$ — the physical quantity (energy, amplitude, free energy) being expanded; $a_n$ — perturbative coefficients, typically growing as $a_n\sim A^{-n}n!$ ; $g$ — small coupling or expansion parameter. Read it as a recipe good only up to its least term: the " $\sim$ " is not an equality, and the sum over all $n$ does not exist. The first move of Borel summation is to divide out the factorial. Because $a_n/n!\sim A^{-n}$ grows only geometrically, the resulting power series has a finite radius of convergence and defines an honest analytic function near the origin.

B(t)=\sum_{n=0}^{\infty}\frac{a_n}{n!}\,t^n \qquad (2)

where: $B(t)$ — the Borel transform, convergent in the disk $|t|<|A|$ set by the nearest singularity; $t$ — the Borel-plane variable, dual to $g$ . Read it as the analytic seed: $B(t)$ and its continuation — singularities and all — carry every bit of information the divergent series encodes. The second move undoes the division with Euler's integral $n!=\int_0^\infty t^n e^{-t}\,\mathrm{d}t$ , packaged as a Laplace transform of $B$ . This is the Borel (Borel–Laplace) sum, a genuine function of $g$ .

\mathcal{S}f(g)=\frac{1}{g}\int_0^{\infty} B(t)\,e^{-t/g}\,\mathrm{d}t \qquad (3)

where: $\mathcal{S}f(g)$ — the Borel sum of the series; $e^{-t/g}$ — the Laplace kernel, which suppresses large $t$ whenever $g>0$ . Read it as the inverse of equation (2): integrating $B$ term by term regenerates the original series (Watson's lemma), so $\mathcal{S}f$ is the function the series was asymptotic to. Everything nontrivial happens when $B(t)$ has a singularity on the integration ray $t>0$ . The contour cannot pass through it, so one deforms infinitesimally above or below, defining two lateral sums.

\mathcal{S}_\pm f(g)=\frac{1}{g}\int_0^{\infty e^{\pm i0}} B(t)\,e^{-t/g}\,\mathrm{d}t \qquad (4)

where: $\mathcal{S}_\pm$ — sums along rays rotated by an infinitesimal angle $\pm\varepsilon$ above/below the positive axis; $e^{\pm i0}$ — the prescription that steers around real-axis singularities. Read it as the origin of ambiguity: a Borel-summable series has no singularity on the ray and $\mathcal{S}_+=\mathcal{S}_-$ , but a non-Borel-summable one splits into two answers. The difference between the lateral sums is fixed by the discontinuity of $B$ across its cut, and it is exponentially small in the coupling — the fingerprint of a nonperturbative effect that no finite order of $g$ can reproduce.

\big(\mathcal{S}_+-\mathcal{S}_-\big)f(g)=2\pi i\,\frac{S}{g}\,e^{-A/g}\big(1+\mathcal{O}(g)\big) \qquad (5)

where: $A$ — the location of the leading Borel singularity, equal to the action of the leading instanton or complex saddle; $S$ — the Stokes constant weighting that nonperturbative sector. Read it as the size of the ambiguity: either lateral sum alone leaves an imaginary part of order $e^{-A/g}$ that genuine physics must cancel. The same singularity that creates the ambiguity dictates the growth of the coefficients. A dispersion relation ties a singularity of $B$ at $t=A$ to the high- $n$ Taylor coefficients — this is the central resurgence relation.

a_n\;\xrightarrow{\,n\to\infty\,}\;\frac{S}{2\pi}\,\frac{(n-1)!}{A^{\,n}}\left(1+\frac{c_1 A}{n-1}+\cdots\right) \qquad (6)

where: $a_n$ — perturbative coefficient of the zero-instanton (vacuum) sector; $c_1$ — the first subleading coefficient of the one-instanton sector. Read it both ways: measure the growth of $a_n$ and you read off $A$ and $S$ (the instanton action and Stokes constant); know the instanton and you predict every large-order coefficient, including the subleading $1/(n-1)$ tower. To make the answer unambiguous you must sum more than the perturbative sector. A transseries organizes the full solution as a double expansion — a series in the exponential $e^{-A/g}$ (instanton number) and, within each sector, a power series in $g$ .

\tilde f(g)=\sum_{k=0}^{\infty}e^{-kA/g}\,g^{-k b}\sum_{n=0}^{\infty}a^{(k)}_n\,g^{n} \qquad (7)

where: $k$ — instanton number (sector index); $a^{(k)}_n$ — perturbative coefficients around the $k$ -instanton saddle; $b$ — the characteristic exponent from the one-loop fluctuation determinant. Read it as the true expansion of the observable: the $k=0$ piece is ordinary perturbation theory, and each higher $k$ is a nonperturbative correction the naive series omits. Finally, the lateral sums and the transseries sectors are locked together by the Stokes automorphism, and averaging them yields the real, unique answer — the median (Borel–Écalle) sum.

\mathfrak{S}=\mathcal{S}_-^{-1}\circ\mathcal{S}_+,\qquad \mathcal{S}_{\mathrm{med}}=\mathcal{S}_+\circ\mathfrak{S}^{-1/2} \qquad (8)

where: $\mathfrak{S}$ — the Stokes automorphism relating the two lateral sums across a Stokes line; $\mathcal{S}_{\mathrm{med}}$ — the median summation. Read it as the resolution: $\mathfrak{S}$ encodes how much of the nonperturbative sector switches on, and the "half-jump" $\mathfrak{S}^{-1/2}$ symmetrizes the two lateral choices into a manifestly real function.

Derivation ADivergence of the Euler series and its Borel sum

The goal is to see the whole machine run once on a case simple enough to solve in closed form. We take the prototypical Euler series with coefficients $a_n=(-1)^n n!$ — the asymptotic expansion of a real integral — form its Borel transform, locate the single singularity, do the Laplace integral exactly, and confirm the reconstructed function has the original series as its asymptotic expansion. Because the singularity lands on the negative axis, off the Laplace ray, the series is Borel summable and the answer is unique — which is exactly why alternating-sign series are the friendly case.

Step 1. Write the coefficients: $a_n=(-1)^n n!$ . Their magnitude grows like $n!$ , so the radius of convergence is zero and $\sum a_n g^n$ is purely asymptotic — no value of $g\neq0$ makes it converge. Why: this is the generic behaviour of a physical perturbative series, reduced to its barest solvable form.

Step 2. Form the Borel transform by dividing each coefficient by $n!$ : $B(t)=\sum_{n=0}^\infty \frac{(-1)^n n!}{n!}t^n=\sum_{n=0}^\infty(-t)^n=\frac{1}{1+t}$ . The factorial cancels exactly, leaving a geometric series that converges for $|t|<1$ and continues analytically to a simple pole at $t=-1$ . Why: dividing by $n!$ is the entire trick — it converts factorial growth into geometric growth and hands us an ordinary analytic function.

Step 3. Locate the singularity: the only singularity of $B(t)$ is the pole at $t=-1$ , so $A=-1$ . It sits on the negative real axis, away from the Laplace contour $t>0$ . Why: the sign of the coefficients has pushed the Borel singularity off the integration ray, so the reconstruction will be unambiguous — this is the Borel-summable case.

Step 4. Do the Laplace integral. Substituting $B(t)=1/(1+t)$ into equation (3) and using the exponential integral $E_1(z)=\int_0^\infty e^{-zs}/(1+s)\,\mathrm{d}s\cdot e^{z}$ gives $\mathcal{S}f(g)=\frac{1}{g}\int_0^\infty\frac{e^{-t/g}}{1+t}\,\mathrm{d}t=\frac{1}{g}\,e^{1/g}\,E_1\!\left(\tfrac{1}{g}\right)$ , a finite smooth function for every $g>0$ . Why: the divergent sum has become a convergent integral with a closed-form value — the whole point of the construction.

Step 5. Check consistency. The known asymptotics $E_1(z)\sim e^{-z}\sum_n(-1)^n n!/z^{n+1}$ as $z\to\infty$ , with $z=1/g$ , regenerate exactly $\sum_n(-1)^n n!\,g^n$ . Numerically at $g=0.1$ the integral gives $\mathcal{S}f=0.915633$ , and the optimally truncated partial sums straddle it: $S_9=0.915456$ , $S_{10}=0.915819$ . Why: this is the defining requirement — the Borel sum is the unique function whose asymptotic expansion is the series we started with. *The results $B(t)=1/(1+t)$ and $\mathcal{S}f(g)=e^{1/g}E_1(1/g)/g$ are confirmed numerically: at $g=0.1$ the exact Borel sum is $0.915633$ and the least term is $3.63\times10^{-4}\approx\sqrt{2\pi/g}\,e^{-1/g}$ .*

Go deeper · Derivation B · 2 minLarge-order coefficients from a Borel singularity

Derivation BLarge-order coefficients from a Borel singularity

The goal is to prove the resurgence relation $a_n\sim (n-1)!/A^n$ by direct computation, showing that a single singularity of $B(t)$ at $t=A>0$ controls the entire tail of the perturbative series. We model the leading Borel singularity as a logarithmic branch point — the shape produced by a one-instanton sector with its fluctuation series — expand it, read the Borel coefficients, and multiply back by $n!$ to recover the factorial divergence with its rate set precisely by the singularity position. The same cut then produces the Stokes discontinuity of equation (5), closing the loop between large orders and the exponentially small ambiguity.

Step 1. Model the singularity: take $B(t)=-\frac{S}{2\pi}\log\!\left(1-\frac{t}{A}\right)$ , and expand the logarithm as $-\log(1-t/A)=\sum_{n=1}^{\infty}\frac{1}{n}\left(\frac{t}{A}\right)^n$ , convergent for $|t|<A$ . Why: a log branch point is the generic one-instanton singularity, and its expansion is elementary, so we can read coefficients directly.

Step 2. Read the Borel coefficients term by term: $[t^n]B(t)=\frac{S}{2\pi}\,\frac{1}{n\,A^n}$ . These are the coefficients of the analytic seed $B$ , still perfectly tame. Why: the singularity position $A$ has entered as the geometric factor $A^{-n}$ , and the log has supplied the extra $1/n$ .

Step 3. Undo the Borel transform. Since $a_n=n!\,[t^n]B(t)$ by definition of equation (2), we get $a_n=n!\,\frac{S}{2\pi}\,\frac{1}{n\,A^n}=\frac{S}{2\pi}\,\frac{(n-1)!}{A^{\,n}}$ . The factorial divergence has reappeared, and its rate is fixed by $A$ . Why: this is the resurgence relation itself — the growth of perturbation theory is the singularity of $B$ read backwards.

Step 4. Interpret physically: $A$ is the instanton action and $S$ the Stokes constant, so the divergence of the vacuum perturbation series is completely determined by the one-instanton sector. A pure simple pole $B\sim\frac{S}{2\pi}\frac{1}{A-t}$ would instead give $a_n=n!\frac{S}{2\pi}A^{-(n+1)}$ ; the log softens $n!$ to $(n-1)!$ . Why: it says the "cheap" perturbative coefficients literally count the "expensive" nonperturbative objects.

Step 5. Get the ambiguity from the same cut. The discontinuity of $B$ across $t>A$ is $\mathrm{disc}\,B=i\,S/1$ (a constant from the log), so $\mathrm{disc}\,\mathcal{S}f=\frac{1}{g}\int_{\mathrm{cut}}\mathrm{disc}\,B\,e^{-t/g}\,\mathrm{d}t\sim -\frac{S}{g}\,e^{-A/g}(1+\cdots)$ , matching equation (5). Why: the one object $B(t)$ delivers both the large-order growth (Step 3) and the exponentially small Stokes jump — this is why they are the same physics. *Cauchy's coefficient formula and the dispersion relation independently yield the same $(n-1)!/A^n$ law. For comparison, a simple pole $B\sim\frac{S}{2\pi}\frac{1}{A-t}$ gives instead $a_n=n!\,\frac{S}{2\pi}A^{-(n+1)}$ , consistent with the geometric expansion of $1/(A-t)$ .*

Go deeper · Derivation C · 2 minReality from the transseries and the median sum

Derivation CReality from the transseries and the median sum

The goal is to show why the ambiguity of a single lateral Borel sum is not a defect but a signpost, and how the transseries cancels it to leave a real answer. On the positive ray both the perturbative and the one-instanton sectors are present, and each lateral sum $\mathcal{S}_\pm$ carries an unphysical imaginary part of order $e^{-A/g}$ . We compute that imaginary part, write the one-instanton contribution with a free transseries parameter, and fix the parameter by demanding a real total — the reality condition that turns a two-valued recipe into a single physical observable and defines the median summation.

Step 1. Extract the perturbative imaginary part. From the Stokes discontinuity, $\mathcal{S}_+f^{(0)}-\mathcal{S}_-f^{(0)}=2\pi i\,\frac{S}{g}e^{-A/g}+\cdots$ , so each lateral sum carries $\mathrm{Im}\,\mathcal{S}_\pm f^{(0)}=\mp\frac{\pi S}{g}e^{-A/g}$ . Why: this is the ambiguity of equation (5) split symmetrically between the two contours — a manifestly unphysical imaginary energy or amplitude.

Step 2. Write the one-instanton sector: $\tilde f^{(1)}=\sigma_1\,e^{-A/g}\sum_n a^{(1)}_n g^n$ , where $\sigma_1$ is the (complex) one-instanton amplitude — the transseries parameter left undetermined by perturbation theory alone. Why: perturbation theory around the vacuum cannot know $\sigma_1$ ; it is genuinely new nonperturbative data, but its imaginary part is constrained.

Step 3. Impose reality. The total $f=\mathcal{S}_\pm f^{(0)}+\tilde f^{(1)}+\cdots$ must be real, which forces $\mathrm{Im}\,\sigma_1=\pm\tfrac12$ (in units of $S$ ), so that the $e^{-A/g}$ imaginary part of the one-instanton sector cancels the imaginary part of the lateral perturbative sum. Why: the ambiguity was pointing at exactly the object that removes it — the instanton is not optional, it is required for a real answer.

Step 4. Assemble the median sum: $\mathcal{S}_{\mathrm{med}}=\tfrac12\big(\mathcal{S}_++\mathcal{S}_-\big)f^{(0)}+\big(\text{real part of instanton sector}\big)$ , which is real and unambiguous to this order and equals $\mathcal{S}_+\circ\mathfrak{S}^{-1/2}$ of equation (8). Why: averaging the two lateral choices deletes the imaginary parts, and the matched instanton term supplies the true nonperturbative contribution.

Step 5. Read the lesson: the divergence of perturbation theory, its ambiguity, and the existence of instantons are three faces of one structure. Resurgence supplies the cancelling sector automatically, so the "disease" of divergence is the mechanism of its own cure. Why: this is the conceptual payoff — nonperturbative physics is encoded in, and demanded by, the perturbative series. *The $\mathrm{Im}\,\sigma_1=\pm\tfrac12$ structure and the median summation are realized explicitly in double-well and periodic-potential quantum mechanics and in matrix models.*

Limiting cases

The character of Borel summation is set entirely by the growth rate of $a_n$ and the location of the Borel singularities relative to the positive ray.

Case	Result	Meaning
Borel summable	$a_n\sim C\,(-A)^{-n}\,n!,\ A>0$	Singularity at $t=-A<0$ , off the positive Laplace ray. The integral converges to a unique, real answer — alternating-sign series (the Euler series, the quartic oscillator) live here.
Non-Borel-summable	$a_n\sim C\,A^{-n}\,n!,\ A>0$	Singularity sits on the ray $t>0$ . Lateral sums $\mathcal{S}_\pm$ differ by $\sim e^{-A/g}$ ; reality is restored only by the full transseries and median sum.
Faster than factorial	$a_n\sim (n!)^k,\ k>1$	The ordinary Borel transform still diverges; use the Gevrey- $k$ (Borel–Leroy) transform $B_k(t)=\sum a_n t^n/\Gamma(kn+1)$ before Laplace integration.
Convergent series	$a_n\sim r^{-n},\ \|g\|<r$	The Borel transform is entire and the Laplace integral reproduces the ordinary analytic sum; Borel summation reduces to plain summation — a consistency limit.

Common trapBorel summation is a single push-button that always gives one clean number

It is not, on two counts. First, the ordinary Borel transform only tames factorial growth: if the coefficients grow like $(n!)^2$ or faster, $\sum a_n t^n/n!$ still diverges and you must climb to a Gevrey- $k$ / Borel–Leroy transform $\sum a_n t^n/\Gamma(kn+1)$ before integrating. Second, and more subtly, when a Borel singularity lies on the integration ray — the generic "non-Borel-summable" case that includes the double well, Yang–Mills, and most interesting theories — the lateral sums $\mathcal{S}_+$ and $\mathcal{S}_-$ genuinely differ, and quoting either one alone yields an answer with an unphysical imaginary part of order $e^{-A/g}$ . The cure is never a single integral: it is the transseries plus the Stokes automorphism, which supplies the nonperturbative sector that cancels the imaginary part and produces the real, median-summed observable.

Worked examples

1 · Optimal truncation and the exact sum of the Euler series (fully worked)

The series $\sum_{n=0}^\infty(-1)^n n!\,g^n$ has Borel action $A=1$ . At $g=0.1$ , find the optimal truncation order, the resulting accuracy, and the exact Borel sum, and show the two agree.

Step 1. Locate the least term. The term magnitude is $|a_n g^n|=n!\,(0.1)^n$ , and successive terms shrink while the ratio $|a_{n+1}g^{n+1}/a_n g^n|=(n+1)g$ stays below one, i.e. for $n+1<1/g=10$ . The turning point is $n^*\approx A/g=10$ . Evaluating, $|a_9 g^9|=|a_{10}g^{10}|=362880\times10^{-9}=\mathbf{3.63\times10^{-4}}$ , the least term. This is the accuracy wall $\approx\sqrt{2\pi/g}\,e^{-A/g}=\sqrt{2\pi/0.1}\,e^{-10}=3.60\times10^{-4}$ .

Step 2. Truncate optimally. Summing to $n^*$ gives partial sums that bracket the true value: $S_9=0.915456$ and $S_{10}=0.915819$ . The best a truncated asymptotic series can do is pin the answer to within about one least term, here $\sim2\times10^{-4}$ — superasymptotic accuracy, but no better without new ideas.

Step 3. Compute the exact Borel sum. Because $A=1>0$ but the coefficients alternate, the Borel singularity is the pole at $t=-1$ , off the ray: the series is Borel summable. From Derivation 1, $\mathcal{S}f(0.1)=\frac{1}{0.1}e^{10}E_1(10)$ . With $E_1(10)=4.15697\times10^{-6}$ and $e^{10}=22026.466$ , this is $10\times(22026.466)(4.15697\times10^{-6})=\mathbf{0.915633}$ .

Step 4. Compare. The exact value $0.915633$ sits squarely between $S_9=0.915456$ and $S_{10}=0.915819$ , and the truncation error $\approx1.8\times10^{-4}$ is indeed about half the least term. Optimal truncation reaches $\sim e^{-A/g}$ accuracy; Borel summation reaches the exact value $0.915633$ with no ambiguity.

2 · Reading the instanton action off the perturbative coefficients

A divergent series has coefficients $a_1=0.5,\ a_2=0.25,\ a_3=0.25,\ a_4=0.375,\ a_5=0.75,\ a_6=1.875$ . Using resurgence, find the location $A$ of the leading Borel singularity and hence the size of the leading nonperturbative effect.

Step 1. Use the resurgence law. Equation (6) says $a_n\sim\frac{S}{2\pi}\frac{(n-1)!}{A^n}$ , so the ratio of consecutive coefficients is $\frac{a_{n+1}}{a_n}\to\frac{n!/(n-1)!}{A}=\frac{n}{A}$ . Therefore $A\approx n\,a_n/a_{n+1}$ , a quantity that should stabilize as $n$ grows. The factorial ratio isolates $A$ cleanly.

Step 2. Tabulate the estimator. Computing $n\,a_n/a_{n+1}$ : $1\times0.5/0.25=2.0$ ; $2\times0.25/0.25=2.0$ ; $3\times0.25/0.375=2.0$ ; $4\times0.375/0.75=2.0$ ; $5\times0.75/1.875=2.0$ . The estimator equals $\mathbf{A=2}$ at every order — here the coefficients were built from $a_n=(n-1)!/2^n$ exactly, so convergence is immediate.

Step 3. Read off the physics. The leading Borel singularity sits at $t=A=2$ on the positive real axis, so this series is non-Borel-summable, with lateral sums differing by a term of order $e^{-A/g}=\mathbf{e^{-2/g}}$ . That exponential is the magnitude of the leading nonperturbative (one-instanton) effect. The divergence rate of perturbation theory alone predicts an instanton of action $A=2$ and a nonperturbative correction $\sim e^{-2/g}$ — resurgence in one line.

Physics of the portrait

Borel summation is where three threads of nineteenth- and twentieth-century mathematics meet: Borel's rescue of divergent series, Stokes' discovery that exponentials switch on across lines in the complex plane, and Écalle's realization that the two are the same phenomenon, organized by the arithmetic of a single transform.

Origins · 1 minJean Écalle and how this was discovered

JÉ

Jean Écalle

b. 1950 · resurgence theory and alien calculus

Jean Écalle is the French mathematician who, in the three-volume Les fonctions résurgentes (1981–1985), created the theory of resurgence and its computational engine, alien calculus. The problem was old: divergent series had been made respectable by Borel around 1899 and by Hardy's Divergent Series (1949), and Stokes had noticed in 1847 that asymptotic expansions gain and lose exponentially small terms across special lines — but there was no systematic account of how the singularities of a Borel transform in different sectors relate. Écalle's answer was that "resurgent functions" — those whose Borel transforms have only isolated, controllable singularities — form an algebra closed under a new derivation, the alien derivative $\Delta_A$ , which measures the discontinuity across the singularity at $t=A$ and moves you between sectors; the Stokes automorphism is built from exponentials of these alien derivatives. Physicists arrived independently from the large-order side: Bender and Wu showed in 1971–1973 that the anharmonic-oscillator coefficients grow factorially at a rate set by instantons, and Zinn-Justin and Jentschura welded this to the transseries for the double well. Since the late 2000s, work by Mariño, Aniceto, Dunne, Ünsal and others has made resurgence a working tool across quantum mechanics, matrix models, gauge theory, and topological strings — realizing in physics the exact structure Écalle built in pure analysis.

History

1826

— Abel warns that divergent series are "an invention of the devil"

1899

— Émile Borel introduces his summation method for divergent series

1952

— Dyson argues the QED perturbation series must diverge

1973

— Bender and Wu link anharmonic-oscillator large orders to instantons

1981

— Écalle founds resurgence theory in Les fonctions résurgentes

Going further

References. Stone & Goldbart, Mathematics for Physics, chapter on asymptotic analysis · Arfken & Weber, Mathematical Methods for Physicists, asymptotic series · Riley, Hobson & Bence, Mathematical Methods for Physics and Engineering · C. M. Bender & S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers · G. H. Hardy, Divergent Series · J. Écalle, Les fonctions résurgentes (1981) · M. Mariño, Instantons and Large N (2015) · E. Delabaere & F. Pham, Resurgent methods in semi-classical analysis · C. M. Bender & T. T. Wu, Phys. Rev. D 7, 1620 (1973). Key numerical anchors: the Euler–Borel example gives $B(t)=1/(1+t)$ , $\mathcal{S}f(g)=e^{1/g}E_1(1/g)/g$ , exact value $0.915633$ at $g=0.1$ , and least term $3.63\times10^{-4}$ ; the resurgence law $a_n=\frac{S}{2\pi}(n-1)!/A^n$ follows from $B(t)=-\frac{S}{2\pi}\log(1-t/A)$ via term-by-term expansion and Cauchy coefficients; the coefficient-ratio estimator gives $n\,a_n/a_{n+1}\to A=2$ for $a_n=(n-1)!/2^n$ ; the anharmonic-oscillator coefficients satisfy $a_n\sim(-1)^{n+1}\frac{\sqrt6}{\pi^{3/2}}3^n\Gamma(n+\tfrac12)$ , with the alternating sign placing the Borel singularity at $t=-1/3$ .

Prerequisites. Asymptotic Series and Poincaré Expansions · Watson's Lemma · Laplace's Method for Integrals · Method of Steepest Descent and Saddle Points · Stokes Phenomenon and Anti-Stokes Lines. Leads to. Optimal Truncation and Superasymptotics · WKB Approximation · Bohr–Sommerfeld Quantization.

Ask the tutor: Want to watch resurgence run end to end — from the factorial growth of a real perturbative series to the instanton action it predicts and the median sum that makes it exact?

Is a divergent series really useless, or can it still give numbers?

It is far from useless — a divergent asymptotic series is often the best numerical tool available. The rule is optimal truncation: sum it only up to its least term, at order $n^*\approx A/g$ , and stop. The error is then about one least term, of size $e^{-A/g}$ , which for a small coupling can be astonishingly good — the Euler series at $g=0.1$ pins its value to four digits. What you must not do is add terms past the minimum, where the factorial growth takes over and the partial sums fly apart. Borel summation goes one better, reaching the exact value, but even without it a divergent series is a perfectly quantitative object.

What is the difference between a Borel-summable and a non-Borel-summable series?

It comes down to where the Borel transform $B(t)$ has its nearest singularity. If that singularity lies off the positive real axis — typically because the coefficients alternate in sign, pushing it to negative $t$ — the Laplace integral in equation (3) runs along a clear ray and returns a single, unambiguous, real answer: the series is Borel summable. If instead a singularity sits on the positive axis (fixed-sign coefficients, the usual case for interesting theories), the contour must dodge above or below it, giving two lateral sums that differ by $\sim e^{-A/g}$ . That series is non-Borel-summable in the naive sense, and you need the transseries and median summation to get a real answer.

What is resurgence, in one honest sentence?

Resurgence is the statement that the coefficients of a perturbative series and the properties of the nonperturbative sectors it omits are two encodings of the same information: the large-order growth $a_n\sim (n-1)!/A^n$ reads off the instanton action $A$ , and conversely the instanton data predict every large-order coefficient. Formally, the singularities of the Borel transform in one sector "resurge" as the Borel transforms of the other sectors, related by Écalle's alien derivatives. Practically, it means you can bootstrap between cheap perturbative data and expensive nonperturbative physics in either direction.

Why must the perturbative ambiguity be exactly the size of an instanton?

Because the same object — the Borel transform $B(t)$ — produces both. A singularity of $B$ at $t=A$ makes the two lateral sums differ by an amount proportional to $e^{-A/g}$ (equation 5), and that same singularity, read as a Taylor-coefficient generator, fixes the large-order growth of the series (equation 6). The exponential $e^{-A/g}$ is precisely the semiclassical weight of a saddle of action $A$ , i.e. an instanton. So the ambiguity is not a coincidence that happens to match the instanton; it is the instanton, seen from the perturbative side, and reality forces the genuine instanton sector to cancel it.

What exactly is a transseries?

A transseries is the expansion an observable actually has when perturbation theory alone is not enough. Instead of a single power series in $g$ , it is a sum over sectors labelled by an instanton number $k$ , each weighted by $e^{-kA/g}$ and by a power $g^{-kb}$ from the fluctuation determinant, and each carrying its own power series in $g$ (equation 7). The $k=0$ sector is ordinary perturbation theory; $k=1$ is the one-instanton correction, and so on. Crucially the transseries also contains free parameters (like $\sigma_1$ ) that encode the Stokes data; fixing them by reality and consistency is what makes the whole object a single well-defined function.

How does this connect to the Stokes phenomenon and WKB?

They are the same physics in different languages. The Stokes phenomenon is the abrupt switching-on of an exponentially small term $e^{-A/g}$ as you cross a Stokes line in parameter space; in the Borel picture, that switching is exactly what happens when the Laplace contour is forced from one side of a singularity to the other, and the Stokes constant $S$ is the strength of the jump. WKB is the leading approximation of this structure: the all-orders WKB series is itself Borel-summed across Stokes lines using the connection formulae, and resurgence promotes the formal Bohr–Sommerfeld condition into an exact quantization condition whose nonperturbative terms are the tunnelling and reflection amplitudes.