"1/137 becoming 1/128" was the sound of conformal symmetry breaking
For seven episodes we have treated the conformal transformation as "swapping the ruler". In Episode 7 we saw light pass straight through the swap. But — quantum mechanics will not let you swap the ruler. Merely defining a field theory smuggles in a standard of resolution. As a result a theory that was perfectly conformally invariant classically loses the symmetry the moment it is quantised. This is called the trace anomaly (or conformal anomaly). And the most important line of this episode is: that breaking is exactly the running of \(\alpha\) we watched over and over in the previous series.
Consider a massless theory. In Episode 7's language, a theory carrying no standard of length. Yet the moment you try to compute anything in it, you run into trouble — summing all the way down to infinitely fine detail makes the answer diverge.
So in field theory you first decide "how finely will I look?". That standard is written \(\mu\) (the renormalisation scale). In the language of Episode 6 of the previous series, it is the "resolution knob".
Something fatal has just happened.
A massless theory had no standard of length.
But to define the theory, a standard \(\mu\) must be imported.
→ Conformal symmetry is lost at the moment of quantisation.
We handed a ruler to a theory that had none. In Episode 7 we wrote that light passes through a conformal transformation because it has no ruler; quantum theory forces a ruler on light as well.
Once \(\mu\) has been imported, coupling constants depend on the \(\mu\) at which they are measured. The strength of that dependence is the β function.
$$\beta(g)\equiv\frac{dg}{d\ln\mu}$$And the breaking of conformal symmetry is expressed by exactly that \(\beta\). The quantity \(T^\mu{}_\mu\), used in Episodes 5 and 7 as the indicator of whether conformal symmetry is broken, was zero classically — but quantum mechanically:
If the coupling runs (\(\beta\ne0\)), conformal symmetry is broken. If it does not run (\(\beta=0\)), it is not.
This equation says that two things which look entirely unrelated are one and the same.
| How a particle physicist says it | How a conformal transformation says it |
|---|---|
| The coupling runs (\(\beta\ne0\)) | Conformal symmetry is broken quantum mechanically |
| The coupling does not run (\(\beta=0\)) | Conformal symmetry is restored (a conformal field theory) |
| \(\alpha\) goes from 1/137 to 1/128 | The size of the breaking is being measured in the laboratory |
The title of that episode's first half was "1/137 was a number that moves". Look more finely and \(\alpha\) grows, reaching about 1/128 at energies of order the Z boson mass — an established experimental fact.
In one-loop QED the running can be written like this.
① One-loop running (for \(N\) species of charged particle)
$$\frac{d\alpha}{d\ln\mu}=\frac{2N}{3\pi}\alpha^2 \qquad\Longleftrightarrow\qquad \frac{d(1/\alpha)}{d\ln\mu}=-\frac{2N}{3\pi}$$Written for the reciprocal it is simply a straight line, of slope \(-2N/3\pi=-0.212\,N\).
② From the electron mass to the Z boson mass
$$\ln\frac{M_Z}{m_e}=\ln\frac{91.19\ \mathrm{GeV}}{0.511\ \mathrm{MeV}}=12.09$$③ With the electron alone (\(N=1\))
$$\frac{1}{\alpha(M_Z)}=137.036-0.212\times12.09=134.5$$The measured value is 127.95, so this is not enough — the other charged particles (muon, tau, quarks…) each join the running once their own mass is passed. Effectively \(N\simeq3.5\). The slope of the running is counting what charged particles the universe contains.
Set the knob to 0 and the line goes perfectly flat. That is what a conformally invariant world looks like — \(\alpha\) is the same at every resolution, so the marks on the ruler have no influence on physics. Our universe is not like that. The line is tilted. That tilt is the size of the breaking of conformal symmetry.
"A symmetry was broken" sounds like a loss, but this breaking does an extraordinary amount of work.
A massless theory has no standard of length. But once a coupling runs, a standard is born in the form of "the resolution at which the coupling takes some particular value". A scale wells up out of nothing — this is called dimensional transmutation. For the strong interaction that standard is an energy of a few hundred MeV, written \(\Lambda_{\rm QCD}\).
Now the point. Let us estimate the proton's mass from the masses of the quarks inside it.
A proton is three quarks: u, u, d
$$2m_u+m_d=2(2.16)+4.67=8.99\ \mathrm{MeV}$$Compared with the proton mass
$$\frac{8.99\ \mathrm{MeV}}{938.27\ \mathrm{MeV}}=0.96\%$$The quark masses (which come from the Higgs) are only 1% of the proton's weight. The remaining 99% comes from the scale \(\Lambda_{\rm QCD}\) — a scale created by the quantum breaking of conformal symmetry.
About 99% of your body weight comes not from the Higgs
but from conformal symmetry breaking quantum mechanically.
A massless theory manufactured mass. This is not a by-product of symmetry breaking — it is the very reason there is such a thing as size in our world. Episode 7 said that mass imports a scale; the thing that first made that scale was broken conformal symmetry.
If \(\beta(g)=0\), conformal invariance is restored at the quantum level. Such a point is called a fixed point, and the theory realised there is a conformal field theory (CFT).
Our world (the four-dimensional Standard Model) sits, unfortunately, at no fixed point. So \(\alpha\) goes on running.
There is worse news. Switch off every interaction, keep only free massless fields, and conformal symmetry still breaks if spacetime is curved.
$$T^\mu{}_\mu=\frac{1}{16\pi^2}\Big(c\,C_{\mu\nu\alpha\beta}C^{\mu\nu\alpha\beta}-a\,E_4\Big)$$The right-hand side is built from curvature squared. \(a\) and \(c\) are pure numbers fixed by the kinds of field present (scalar, fermion, vector). The curvature of spacetime counts how many fields there are. This matters in practice — in calculations of Hawking radiation and of fluctuations during inflation.
So the breaking of conformal symmetry cannot be avoided. Cut the interactions and it still leaks, as long as curvature remains.
We must now annotate the earlier episodes.
In Episode 4 we counted "one field added, one symmetry added, net zero". In Episode 5 we wrote "this is a rewriting, not physics". Both were classical statements.
Quantum mechanically, a conformal transformation is no longer a mere rewriting. Swapping frames generates anomaly terms and the calculations in the two frames stop agreeing exactly. The question "which frame is physical?", meaningless classically, can acquire meaning quantum mechanically — and that dispute is still unsettled, as we shall see in the finale (Episode 10).
The straight line in the figure is a one-loop estimate that ignores thresholds. The actual value \(1/\alpha(M_Z)=127.952\) is obtained by summing the effect of each particle joining the running once its own mass is passed (thresholds), together with the hadronic contribution from the strong interaction, and cannot be represented by a single \(N\). The \(N\) in the figure is an effective value — "the slope that would pass through the measured point" — not the actual number of particles.
Also, the value of \(\Lambda_{\rm QCD}\) depends on scheme and order and is not a unique number beyond "a few hundred MeV". The conclusion that 99% of the proton's mass comes from something other than quark masses is established by lattice QCD, but saying it "came out of a single number \(\Lambda_{\rm QCD}\)" is a simplification.
Quantising requires importing a standard of resolution \(\mu\), and conformal symmetry dies at that instant. The size of the breaking is measured exactly by the β function: \(T^\mu{}_\mu=(\beta/2g)F_{\mu\nu}F^{\mu\nu}\). So "the coupling runs" and "conformal symmetry is broken" are two ways of saying one fact. The \(1/137\to1/128\) of Episode 6 of the previous series was, all along, a measurement of how badly it is broken.
And the breaking is no loss. The running creates a scale out of nothing (dimensional transmutation), and that \(\Lambda_{\rm QCD}\) supplies 99% of the proton's mass — most of your body weight exists thanks to conformal symmetry being broken. The breaking stops only at a fixed point \(\beta=0\) (a conformal field theory), and the Standard Model is not at one. On top of that, cut the interactions and it still leaks in the form of curvature squared whenever spacetime is curved. So Episodes 4 and 5's "a rewriting, hence safe" was, all along, a statement about classical physics.
Print / PDF: ⌘+P (Ctrl+P on Windows). On screen the slider changes the slope of the running and shows the line going flat — conformally invariant — at β = 0. "Show answer" opens the solutions.