Instantons and Large-Order Relations

When an observable admits analytic continuation between saddle sectors, a neighboring saddle can determine the late coefficients of perturbation theory around the reference saddle. The leading form is a factorial divided by a power of the action difference, with a phase fixed by that complex action. This relation is exact in suitable finite-dimensional and quantum-mechanical problems and a controlled semiclassical inference in some field theories; it is not a universal proof that every QFT Borel singularity is an instanton.

Required background. Large-order growth and the Borel transform fixes the transform convention and coefficient–singularity relation. Quantum-mechanical instantons and tunnel splitting fixes the double-well action and determinant normalization.

Helpful background. Instanton measures, zero modes, and determinants explains which fluctuation and collective-coordinate factors determine the power multiplying an exponential sector.

From a neighboring sector to late coefficients

Let the perturbative sector be

$$\Phi_0(g)\sim\sum_{n=0}^{\infty}a_n g^n.$$

Assume that its convention-matched discontinuity has the small-$g$ form

$$\operatorname{Disc}\Phi_0(g) \sim 2\pi i K\,e^{-A/g}g^{-\beta} \sum_{k=0}^{\infty}c_k g^k,$$

where $A$ is the action difference between the relevant sectors and the constant $K$ includes the chosen orientation and Stokes normalization. Deforming the coefficient contour to the discontinuity gives integrals of the form

$$\int_0^\infty \mathrm dg\,g^{-n-1-\beta+k}e^{-A/g} =A^{-n-\beta+k}\Gamma(n+\beta-k).$$

Therefore

$$a_n \sim K\sum_{k=0}^{\infty} c_k\,\frac{\Gamma(n+\beta-k)}{A^{n+\beta-k}}.$$

In particular,

$$a_n \sim Kc_0\frac{\Gamma(n+\beta)}{A^{n+\beta}} \left[ 1+\frac{c_1}{c_0} \frac{A}{n+\beta-1} +O(n^{-2}) \right].$$

This is the factorial-over-action relation. The leading saddle fluctuation fixes $Kc_0$ and $\beta$; its higher-loop coefficients determine the $1/n$ corrections. Mariño derives the same Hankel-contour relation in the chapter convention used here in Mariño 2015, §§3.3 and 3.5, pp. 89–104.

If $A=|A|e^{i\varphi}$ and a real observable receives equal contributions from a conjugate pair, the leading terms combine into

$$a_n\sim 2|Kc_0|\frac{\Gamma(n+\beta)}{|A|^{n+\beta}} \cos\!\left(\arg(Kc_0)-(n+\beta)\varphi\right).$$

Thus an off-axis pair generates oscillatory signs. If several singularities have the same $|A|$, all must be retained. If a nearer renormalon exists, the instanton does not control the leading growth.

The normalized symmetric double well

Keep exactly the normalization used in the tunneling chapter,

$$S_E[x]=\frac1g\int\mathrm d\tau \left[ \frac12\dot x^2+\frac12(x^2-1)^2 \right], \qquad g\ll1.$$

The minima are $x=\pm1$, the small-oscillation frequency is $2$, and

$$x_I(\tau)=\tanh(\tau-\tau_0), \qquad S_I=\frac{4}{3g}.$$

Use the dimensionless spectral quantity $\mathcal E_{\mathrm{pert}}$ that appears in the Chapter 4 Euclidean kernel as $e^{-\mathcal E_{\mathrm{pert}}T}$. With the overall semiclassical factor $1/g$ kept in the displayed action, this convention is equivalently $\mathcal E_{\mathrm{pert}}=E_{\mathrm{can}}/g$ for the canonical energy normalization in which $E_{\mathrm{can}}=O(g)$. Its series about either well begins

$$\mathcal E_{\mathrm{pert}}(g) \sim1-\frac14g-\frac9{64}g^2-\frac{89}{1024}g^3-\cdots.$$

A single instanton changes wells and controls the parity splitting, but it does not return the reference vacuum to itself. The first sector with the same endpoints is an instanton–anti-instanton pair, so the leading positive Borel action for the common perturbative series is

$$A=2\mathcal S_I=\frac83,$$

where $\mathcal S_I=4/3$ is the dimensionless one-instanton action. This endpoint selection rule is essential: inserting $A=4/3$ would predict the wrong coefficient scale.

The conventional double-well Hamiltonian in Mariño 2015, §§1.8–1.9, pp. 38–54 uses coupling $g_M=g/8$ and energies $E=2E_M$. Translating its large-order result gives

$$a_n \sim -\frac6\pi \frac{\Gamma(n+1)}{(8/3)^n} \left[1+O(n^{-1})\right].$$

The negative, nonalternating tail agrees with a singularity on the positive Borel axis. The pair sector contains an attractive quasi-zero-mode integral and a prescription-dependent imaginary part. On the next pages that ambiguity cancels the perturbative lateral ambiguity; the cancellation also fixes the normalization entering the large-order formula. Zinn-Justin’s original multi-instanton calculation makes the endpoint sectors and logarithms explicit in Zinn-Justin 1981, pp. 125–140.

A practical coefficient test

Given coefficients through order $N$, a useful normalized sequence is

$$R_n \equiv \frac{a_n A^{n+\beta}} {\Gamma(n+\beta)}.$$

If one singularity dominates, $R_n\to Kc_0$ with an expansion in $1/n$. The ratio

$$\frac{a_n}{a_{n-1}} \sim\frac{n+\beta-1}{A}$$

tests $A$ and $\beta$, while Richardson transforms or explicit $1/n$ fits can expose subleading fluctuations. A responsible test varies the fit window, includes plausible competing singularities, and checks the inferred $A$ against an independently computed saddle action. Agreement of one ratio at modest order is not an identification.

The earliest quantitative oscillator analyses demonstrated this strategy for anharmonic potentials; see Bender and Wu 1969, pp. 1231–1260. Lipatov extended saddle estimates of large perturbative order to specified renormalized scalar-field quantities in Lipatov 1977, §§1–4, pp. 216–223, PDF. The hypotheses and renormalization details of that calculation must accompany any field-theory use.

Shared method and status checks

The complete passage from a gauge instanton through zero modes, stabilizers, determinants, and endpoint control is shown in the one-instanton measure chain. The Borel and transseries map locates the coefficient test without identifying its singularity prematurely. The exact and rigorous status comparison distinguishes a benchmark relation from a universal QFT claim.

Where the inference fails

The derivation needs analyticity sufficient to deform a contour, a controlled continuation between sectors, and the correct observable-specific boundary conditions. It can fail or become incomplete when:

• a renormalon or other singularity is nearer than the proposed instanton;
• an integration-cycle coefficient vanishes, so an existing saddle does not contribute;
• zero modes, negative modes, or collective-coordinate endpoints are mishandled;
• actions accumulate or have equal modulus; or
• renormalization and infinite-volume limits are not uniform in perturbative order.

These are scientific stop conditions, not small corrections. Exact WKB gives stronger results for particular one-dimensional spectral problems, but its full development lies beyond this page.

Common pitfalls

Using the one-instanton action for a vacuum-to-vacuum series. The admissible neighboring sector is fixed by endpoints and quantum numbers. In the symmetric double well the common perturbative energy first communicates with the pair sector at $2S_I$.

Dropping the phase of $A$. Complex actions determine oscillatory coefficient patterns. Replacing $A$ by $|A|$ destroys this diagnostic.

Calling a fitted singularity a saddle. A fit determines analytic scales. The saddle equations, contour coefficient, and fluctuation normalization must be checked independently.

Exercises

1. Starting from the first two terms of the general large-order formula, derive the $1/n$ correction to $R_n$.

Solution

Dividing by $\Gamma(n+\beta)/A^{n+\beta}$ gives

$$R_n =K\left[c_0+c_1A\frac{\Gamma(n+\beta-1)} {\Gamma(n+\beta)}+\cdots\right] =Kc_0\left[1+\frac{c_1A}{c_0(n+\beta-1)}+\cdots\right].$$

2. Translate Mariño’s double-well asymptotic $a_n^{(M)}\sim-3^{n+1}\Gamma(n+1)/\pi$ using $g_M=g/8$ and $E=2E_M$.

Solution

The coefficient of $g^n$ is $a_n=2a_n^{(M)}/8^n$. Hence

$$a_n\sim -\frac{2}{8^n}\frac{3^{n+1}}\pi\Gamma(n+1) =-\frac6\pi\frac{\Gamma(n+1)}{(8/3)^n}.$$

The inferred Borel action is $8/3=2(4/3)$, the pair action in the fixed normalization.

References

• Bender, Carl M., and Tai Tsun Wu. “Anharmonic Oscillator.” Physical Review 184 (1969): 1231–1260. doi:10.1103/PhysRev.184.1231.
• Lipatov, L. N. “Divergence of the Perturbation-Theory Series and the Quasi-Classical Theory.” Soviet Physics JETP 45 (1977): 216–223. Journal PDF.
• Mariño, Marcos. Instantons and Large N: An Introduction to Non-Perturbative Methods in Quantum Field Theory. Cambridge University Press, 2015. doi:10.1017/CBO9781107705968.
• Zinn-Justin, Jean. “Multi-Instanton Contributions in Quantum Mechanics.” Nuclear Physics B 192 (1981): 125–140. doi:10.1016/0550-3213(81)90197-8.