Perturbation theory in one integral

Source

Concept: Z(λ) = (1/√2π)∫e^{−x²/2−λx⁴}dx — the whole Feynman expansion with spacetime shrunk to a point. Coefficients a_n = (−1)ⁿ(4n−1)!!/n! from the exact recurrence a_n+1 = −a_n(4n+3)(4n+1)/(n+1); exact Z by Simpson quadrature on [−12, 12], 12000 panels.
Serves: Perturbation theory is asymptotic, not convergent: the partial sums close in on Z, come nearest at N* ≈ 1/(16λ) with error of order e^−1/16λ, then run away. Flipping the coupling shows why no convergent series could ever have existed.
Inquiry: Predict the fate of the zig-zag past N = 4; step to N = 25 and meet the closest-approach reveal; then predict the value of the wrong-sign integral before flipping the sign.
Misconception: "If the sum runs away, take more terms — or blame the computer." The coefficients are exact integer ratios, the arithmetic is exact, and more terms make it strictly worse past N*.

Pedagogy principles applied

Principle	How this sim honors it
Predict before reveal	Two gated predictions: the fate of the partial sums, and the value of the wrong-sign integral. Both are committed before the corresponding panel behaviour is on screen.
No lookup leak	The term-size panel, the error readout and the closest-approach reveal are staged: they appear only after the prediction they would have answered.
Misconception encoded in choices	"It converges", "rounding freezes it", "the tails cancel" are offered as options; the rebuttal appears only if selected.
Invariant made visible	\|a_nλⁿ\| is sign-blind: the bars do not move when the coupling flips, while the exact value vanishes. Same series, nothing to converge to.
Honest axes	The integrand window is fixed at x ∈ [−8, 8]; the partial-sum axis is clamped to Z ± 0.2 with off-scale arrows; the term axis is logarithmic and computed from the full N = 0…25 range, so it never rescales mid-run.
Operational numbers	Every displayed number comes from the same series and quadrature the panels draw, and each one is read or compared by a card.
Toggleable formalism	The recurrence, the 16ⁿn! growth law, N* ≈ 1/(16λ) and the wrong-sign turnaround live behind the ⚛︎ The Physics revision sheet.

Teach me

Integrand exp(−x²/2 − λx⁴) · fixed window x ∈ [−8, 8], never rescaled

exact integrand (shaded area ÷ √2π = Z) N-term truncation (its area ÷ √2π = S_N)

Partial sums S_N against N = 0…25 · y-window clamped to Z ± 0.2

Guided Inquiry

1 · One integral, all of perturbation theory

Let’s start small. Spacetime shrunk to a point: this one integral is the whole theory, and expanding it in λ term by term is the diagram expansion. At λ = 0.01, press +1 term four times. What do you think the first dots will do?

2 · Predict: does it keep improving?

Now push that idea further. Those first terms straddle the exact line and tighten. Suppose you kept pressing +1 term all the way to N = 25 — what do you predict the zig-zag will do?

3 · Step on to N = 25

Let’s test that. Press +1 term on to N = 25 (or drag N), and keep an eye on |S_N − Z| in the sidebar, the dots, and the term-size bars. On the way, what did the distance to the line do?

4 · Why it turns

So why does it turn around? The bars fall to a valley, then climb without limit: |a_n| grows like 16ⁿn! while λⁿ only shrinks geometrically. The Closest-approach panel names the best N, with 1/(16λ) right beside it. Try dragging λ upward and watch that pair move together.

5 · Predict: flip the coupling

Now for a twist. Back at λ = 0.01, suppose the interaction were −λx⁴, with λ as small as you like. What do you predict for the value of (1/√2π)∫e^{−x²/2+λx⁴}dx?

6 · Flip it

Time to try it. Press ⚠ −λx⁴ and watch the tails swing up through the marker at x*, the shading flood outward, and the readouts. What does Exact Z(−λ) read now?

7 · Same series, nothing to converge to

Here’s the punchline. The bars didn’t move — |a_nλⁿ| is sign-blind — yet the exact value is gone. The marker’s height e^−1/16λ is bitwise the same number as the floor your closest error rode: the closest error is a fixed fraction of that floor (you’ll find the ratio in Closest approach). A series convergent at λ would have to work at −λ too.

8 · Asymptotic, not convergent

So here’s what we’ve found. Radius of convergence zero: best truncation N* ≈ 1/(16λ), best error ≈ e^−1/16λ. Think of it as a ladder whose top rung is set by the coupling.

Listen

Terms kept N 0

0 · bare Gaussian25

Coupling λ · log slider 0.0100

0.0050.200

Sign of the interaction

+λx⁴ in the action ⇒ integrand e^(−x²/2−λx⁴); both drawn exactly, same fixed window

Compare these two

Series sum S_N (N+1 terms)1

Exact Z(λ) — the line0.9740486

Distance from the exact line

|S_N − Z|—

Closest approach

—

The whole theory: propagator = Gaussian, interaction = quartic. Exact value by Simpson quadrature on [−12, 12] with 12000 panels (the integrand is below 1e−31 beyond |x| = 12).

Expand e^−λx⁴ and integrate term by term. (4n−1)!! is the number of pairings of 4n fields — the Feynman diagrams at order n.

The exact recurrence used everywhere in this sim: a₀ = 1. It gives a₁ = −3, a₂ = 52.5, a₃ = −1732.5, all as exact integer ratios.

Factorial coefficient growth against geometric λⁿ: radius of convergence zero. Terms fall, reach a valley, then climb without limit.

Optimal truncation and the non-perturbative floor: the closest the series ever gets is an O(1) multiple of e^−1/16λ, a number no polynomial in λ can produce.

Wrong sign: the integrand stops falling at x* = 1/(2√λ), where its height is that same exponential, then grows forever — the area is infinite, so Z(−λ) has no value.

The Physics

Asymptotic perturbation theory · what this sim shows, the physics behind it, and every control

At a glance

This sim expands one integral, a field theory shrunk to a single point, as a power series in its coupling λ, and compares every partial sum with the exact value. Many particle-physics predictions are partial sums of a series in a small coupling, so this is the situation they are in.

What you'll learn

Where the terms of the series come from: pairing up fields, which is counting Feynman diagrams.
How the size of each term depends on its order n and on λ.
What limits the accuracy of the best partial sum, and how that limit depends on λ.
What the sign of the interaction tells us about the series.

The integral Z is the area under the solid curve. The dashed curve is the integrand with e^−λx⁴ cut off after its first correction.

What's on screen

Top panel, solid: the integrand e^{−x²/2−λx⁴}. Its area ÷ √2π is the exact Z.
Top panel, dashed: the integrand of the N-term truncation. Its area ÷ √2π is the series sum S_N.
Partial-sums panel: one dot per N against the exact value Z (amber line).
Term-size panel (log axis) and readouts (sidebar): |a_nλⁿ| for each term, and S_N, Z and their distance. The panels appear as the inquiry reaches them; Free Exploration opens them all.

See itStart at λ = 0.01 with N = 0, the bare Gaussian. Watch for: S_N reading 1 while Z reads 0.9740486.

The controls

Press ⌖ on a row to see where that control sits. A panel that is not on screen yet is reported as "not showing yet".

Control	What it changes	Range
Terms kept N	How many orders of the series are summed: S_N = a₀ + a₁λ + … + a_Nλᴺ, which is N + 1 terms.	0–25 opens at 0, the bare Gaussian
−1 / +1 term	Steps N down or up by one.	buttons
Coupling λ	The strength of the x⁴ interaction. The slider is logarithmic, so equal steps multiply λ.	0.005–0.200 opens at 0.0100
Sign of the interaction	✓ +λx⁴ is the physical sign. ⚠ −λx⁴ flips it, so the integrand becomes e^{−x²/2+λx⁴}. The control appears later in the inquiry.	two buttons opens on +λx⁴
Compare these two	The series sum S_N next to the exact Z. With the sign flipped, Z reads "no value".	readout
Distance from the exact line	\|S_N − Z\|, the gap between the two.	readout
Closest approach	A readout panel that opens later in a run. Section 05 explains what it reports.	readout
↻ Reset	Puts λ back to 0.0100 and N back to 0, and restarts the mode you are in.	top bar

Fixed: every quantity is dimensionless. Z is computed by numerical integration over x from −12 to 12, and the plotted window is x from −8 to 8. In Free Exploration, N opens at 7. There is no Play or Speed: nothing moves until you change a control.

One integral holds the whole series

The sim's theory is one integral. The x²/2 part is the free theory (the Gaussian, or propagator); λx⁴ is the interaction. Z = 1 at λ = 0.

The exact quantity the sim calls Z(λ).

Expand e^−λx⁴ in powers of λ and average each term over the Gaussian. The average of x⁴ⁿ is (4n−1)!!, the number of ways to pair up 4n fields.

The perturbation series. The coefficients alternate in sign.

The three pairings of four fields at order n = 1. They are the Feynman diagrams of order λ, so a₁ = −3.

So S₁ = 1 − 3λ, which is 0.97 at λ = 0.01. Each partial sum adds one more order of diagrams.

See itKeep one term beyond the bare Gaussian, at λ = 0.01. Watch for: S_N reading 0.97, which is 1 + (−3)(0.01).

Why the terms start growing

Each coefficient follows from the one before it. The sim uses this recurrence for every a_n.

Exact integer ratios, so no rounding is involved.

The ratio of successive terms is exactly λ(16n + 3/(n+1)). For large n it is about 16λn. That grows with n, so for any λ above zero the terms stop shrinking at some order and then grow without limit.

The coefficients grow like 16ⁿ·n!, which beats any geometric λⁿ.

At λ = 0.01 the terms shrink up to n = 7 and grow from n = 8 on, so the smallest term is n = 7. The radius of convergence of the series is zero.

See itKeep all 25 terms at λ = 0.01. Watch for: the term bars falling to a valley at n = 7 and then climbing, with the Distance reading over 1000.

The best the series can do

While the terms shrink, each one brings S_N closer to Z. Once they grow, each one pushes it away. The best stopping point is therefore near the valley, at N* ≈ 1/(16λ).

The best partial sum is accurate to about e^−1/16λ.

λ N

These sliders move the sim too.

Halving λ doubles N* and squares the scale e^−1/16λ.
The sim's Closest approach panel calls e^−1/16λ a floor. The closest distance is a fraction of it, from 0.078 to 0.28 across the λ slider, so it sets the size of the best error and is not a bound.
e^−1/16λ has every derivative zero at λ = 0, so no power series in λ can reproduce it.

See itStep one term past the best N at your λ. Watch for: the Closest approach panel naming N* and the closest distance, and its ratio to the scale (the panel's second line).

See itRun λ = 0.01 and then λ = 0.005, each at its best N. Watch for: N* doubling from 6 to 12, and the closest distance falling by a factor of about 700.

Why it can never converge

Flip the sign of the interaction and the integrand becomes e^{−x²/2+λx⁴}. It falls until x* = 1/(2√λ), where its height is e^−1/16λ, and then it turns and grows without limit.

The area is infinite for every λ above zero, so Z has no value at −λ.

The wrong-sign integrand at λ = 0.01. The dot marks x* = 5.00, where its height is the same number as the scale in section 05.

This is Dyson's argument from 1952, in miniature. A series with a non-zero radius of convergence would also define a function for small negative λ. No such function exists, so the radius is zero. The term sizes |a_nλⁿ| do not change with the sign of λ, so the bars stay put.

See itFlip the sign at your current λ. Watch for: Exact Z reading "no value" while the term bars stay where they were.

Where it matters

Quantum electrodynamics. Predictions such as the electron's anomalous magnetic moment are series in the small coupling α, calculated through five loops. The series is expected to be asymptotic, as in Dyson's argument, but its smallest term is expected only at an order far beyond any that has been computed.
QCD at low energy. The coupling is large there, so a series in it does not describe quantities such as hadron masses. They are computed non-perturbatively, for example on a lattice.
Effects invisible to every order. A contribution of the form e^{−const/coupling}, like the scale in this sim, is missed by each term of a perturbative series. This sim does not compute one; it only shows that the scale exists.
Reporting a result. A number from a truncated series should be quoted with its truncation error.

Summary

In one line. The perturbation series has terms that shrink and then grow without limit, so its best partial sum, at N* ≈ 1/(16λ), is accurate only to about e^−1/16λ.

Key points

The coefficients a_n count Feynman diagrams and grow like 16ⁿ·n!.
The term ratio is about 16λn, so the radius of convergence is zero.
Smaller λ allows more useful terms and a much smaller best error, but never convergence.
With the sign flipped there is no Z at all, which is why the series cannot converge.

Common mix-ups

"The sums run away because of rounding error." The coefficients are exact ratios. The growth is in the series itself.
"A small coupling means the series converges." A small λ gives more useful terms, about 1/(16λ) of them, but the terms still grow eventually.
"An asymptotic series is useless." Stopped near N*, it is accurate to 2×10⁻⁴ at λ = 0.01 and 3×10⁻⁷ at λ = 0.005.

Things to try on your own

Set λ = 0.2 and find the best N. Compare it with 1/(16λ).
Pick three values of λ, step N to the closest approach for each, and compare the distance with the scale e^−1/16λ.
Flip the sign, then drag λ across its range, and note what stays on screen and what does not.