The most interesting parts of physics usually come in through the hole outside perturbation theory
Almost all calculation in physics is perturbation theory ── expand in something small and add up first order, second order, third order. Quantum electrodynamics has been carried this way to twelve-digit agreement, the most precise theory humanity has built. But behind that success hides an odd fact. Perturbation expansions do not converge. Add terms and they approach up to a point, then start to diverge. And the size of the error at the closest approach is ── exactly \(e^{-1/g}\), the same size as tunneling and the BCS gap. That is no accident. Perturbation theory tells you, through the way it breaks, the size of what it is missing. This episode looks at that mechanism, and then collects the debt from Episode 5: how superconductivity and tunneling connect at "grade 2."
The electron's anomalous magnetic moment (\(g-2\)) has been expanded in the fine-structure constant \(\alpha=1/137\) to fifth order and agrees with experiment to twelve digits. In another field one would hesitate even to call that "agreement." There is no practical reason to doubt perturbation theory.
And yet in 1952, in barely two pages, Dyson showed something alarming.
If the power series in \(\alpha\) converged anywhere at \(\alpha>0\), it would converge on a disc around the origin in the complex plane. Then it would have to converge for \(\alpha<0\) too.
But \(\alpha<0\) is a world in which like charges attract ── the vacuum would produce electron–positron pairs without limit and collapse; there is no ground state at all. It cannot possibly converge.
Therefore the radius of convergence is zero. Perturbation series are non-convergent series.
Then why the twelve digits? Because they are asymptotic series ── they do not converge, but up to a point they keep getting closer. There is a place where they stop getting closer, and that becomes "the best precision that theory can reach."
When the perturbative coefficients grow as \(a_n\sim n!/S^n\), the size of the \(n\)-th term is \(|a_ng^n|\sim n!\,(g/S)^n\). This is minimised at \(n^*\approx S/g\), and (via Stirling) the size there is
$$|a_{n^*}g^{n^*}|\ \sim\ e^{-S/g}$$In other words ── "the error left at the optimal truncation" and "the term invisible to perturbation theory" are exactly the same size. At the very limit of its reach, perturbation theory leaks the identity of what it is missing.
For quantum electrodynamics \(g=\alpha=1/137\), so the optimal truncation is around order 137 and the error there is \(e^{-137}\approx10^{-60}\). You cannot get closer than that, in principle (though with only five orders computed so far, that day is a long way off).
Words alone are hard to believe, so let us actually compute. A very simple integral will do:
$$Z(g)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\!\!e^{-x^2/2-gx^4/4}\,dx$$This is the simplest toy in field theory, zero-dimensional \(\phi^4\) theory. Expanding in \(g\) gives coefficients \(a_n=(-1/4)^n(4n-1)!!/n!\), which grow factorially. The exact value comes from numerical integration.
In the figure below, add the partial sums one term at a time. At first they close in on the exact value (green), reach a closest approach at some order, and then start thrashing. Check that the error at closest approach sits at the same height as the \(e^{-1/(4g)}\) marker.
Here is the beautiful part. The reason the coefficients grow as \(n!/S^n\) is that a non-perturbative solution with action \(S\) exists. The causality looks backwards, but that is how it is.
| Phenomenon | Exponent | What rides on it |
|---|---|---|
| tunneling (Episode 1) | \(e^{-2S_E/\hbar}\) | Euclidean action ÷ \(\hbar\) |
| the BCS gap (Episode 5) | \(e^{-1/N(0)V}\) | inverse coupling |
| Yang–Mills instantons | \(e^{-8\pi^2/g^2}\) | inverse squared coupling. Fixes the vacuum structure of the strong interaction |
| the Schwinger effect | \(e^{-\pi m^2c^3/e E\hbar}\) | a strong field pulling electron pairs out of the vacuum. Critical field \(1.3\times10^{18}\) V/m ── not yet reached |
| the QCD scale \(\Lambda\) | \(\mu\,e^{-1/2b_0g^2}\) | a dimensionless coupling turning into a unit of energy (dimensional transmutation) |
| vacuum decay (Episode 7) | \(e^{-S_E/\hbar}\) | the bounce action. The lifetime of the universe |
Set out like this, you can see that a great many of physics' "why is it this size?" questions live in this column. The proton's mass, the superconducting transition temperature, the lifetime of the vacuum ── all decided on the exponent of \(e^{-1/g}\). The scales of the world are written in the part perturbation theory cannot see.
Finally, the promised "grade-2 connection."
Narrow the energy window around the Fermi surface from \(\Lambda\) downward (i.e. coarse-grain), and the attraction \(g\) between electrons grows on its own. The one-loop flow is
$$\frac{dg}{d\ln(1/\Lambda)}=+g^2\quad\Longrightarrow\quad g(\Lambda)=\frac{g_0}{1-g_0\ln(\Lambda_0/\Lambda)}$$The coupling diverges where the denominator vanishes, i.e. at
$$\Lambda_{\rm BCS}=\Lambda_0\,e^{-1/g_0}$$── which is exactly Episode 5's gap \(\Delta\approx2\hbar\omega_De^{-1/N(0)V}\) (with \(\Lambda_0=\hbar\omega_D\) and \(g_0=N(0)V\)).
In the classification from Episode 4 of "Renormalization That Clicks," the coupling in the Cooper channel is marginal at tree level (dimension exactly zero) and turns marginally relevant at one loop. So however small it starts, continued coarse-graining always reaches strong coupling ── that is what Cooper's "any attraction however weak binds" really is.
This structure has a name: dimensional transmutation. The input is a single dimensionless coupling \(g_0\); the output is \(\Lambda_0e^{-1/g_0}\), a scale of energy.
| System | Input | Scale that emerges |
|---|---|---|
| superconductivity | \(N(0)V\) | \(\Delta\sim\hbar\omega_De^{-1/N(0)V}\) |
| the strong interaction | \(g^2\) | \(\Lambda_{\rm QCD}\sim\mu\,e^{-1/2b_0g^2}\) (where the proton's mass comes from) |
Most of your body weight comes from this exponent. The bulk of the proton's mass is not the quarks' mass but the scale \(\Lambda_{\rm QCD}\) at which QCD becomes strongly coupled (see the sister series "Mass That Clicks"). A non-perturbative exponent sets how much matter weighs.
Established: that \(e^{-1/g}\) is non-analytic at \(g=0\) with vanishing Taylor coefficients at all orders; that perturbation series are generally divergent asymptotic series (Dyson's 1952 argument); that the optimal truncation is at \(n^*\approx S/g\) with error of order \(e^{-S/g}\); that the large-order behaviour \(a_n\sim n!/S^n\) gives the action \(S\) of the non-perturbative solution (Bender–Wu 1969–73, Lipatov 1977); the Yang–Mills instanton action \(8\pi^2/g^2\); the Schwinger effect and its critical field \(1.3\times10^{18}\) V/m; dimensional transmutation in QCD and \(\Lambda_{\rm QCD}\); and that the Cooper-channel coupling is marginally relevant and diverges at \(\Lambda_0e^{-1/g_0}\) (the renormalization-group formulation of Shankar 1994 and Polchinski 1992). All established results.
Points to note: (1) The zero-dimensional \(\phi^4\) integral in the figure is a toy model, not field theory. It is nonetheless a valuable example where the properties of asymptotic series can be shown exactly. (2) That integral is Borel summable, but real field theory is harder. QCD has "renormalons," a source of divergence distinct from instantons, which obstructs naive Borel summation. (3) Resurgence and trans-series are an active research field (Écalle's mathematics, the physics of Dunne, Ünsal and others). "Perturbative and non-perturbative can be organised into a single expansion" is demonstrated in simple systems but incomplete for general four-dimensional gauge theories. (4) The one-loop flow \(dg/d\ln(1/\Lambda)=g^2\) is schematic with the coefficient normalised to 1; the real coefficient depends on the density of states and so on. (5) "Invisible to perturbation theory" means invisible to the expansion in that coupling; other expansions (large \(N\), lattice computations) can see it. Lattice QCD is indeed a non-perturbative method.
Perturbation series do not converge (Dyson 1952). Quantum electrodynamics still agrees to twelve digits because they are asymptotic ── closing in up to order \(n^*\approx S/g\), then diverging. And the error at closest approach is exactly \(e^{-S/g}\). The term perturbation theory cannot see and the limiting precision of perturbation theory are the same size.
The coefficients grow as \(n!/S^n\) because a non-perturbative solution with that \(S\) exists (Bender–Wu, Lipatov). Perturbation theory reports the size of its own blind spot ── the attempt to unify both into one expansion is resurgence theory, still under active research.
Through this hole come tunneling, the BCS gap, instantons, the Schwinger effect, \(\Lambda_{\rm QCD}\) and vacuum decay. A single dimensionless coupling in, a scale of energy out (dimensional transmutation) ── most of your body weight is set by this exponent. Episode 5's debt is collected too: superconductivity and tunneling were connected at grade 2 (different solutions of one framework). Siblings, not parent and child.
Print / make a PDF: ⌘+P (Ctrl+P on Windows). On screen, move the term-count slider one step at a time to watch the partial sums approach the exact value and then thrash. "Go to the optimal truncation" jumps to the turning point. "See the answer" opens each solution.