← quantum · contents

§ Phase x · The Particle Zoo · Chapter 03

Three generations

By 1930 physics seemed to have a tidy inventory of matter: protons, neutrons, electrons, and a guessed-at neutrino. Then in 1936 a cosmic-ray photograph turned up a track that looked exactly like an electron, except two hundred times heavier. Forty years later a third copy turned up. Nature had written out the recipe for ordinary matter and then handed in two more drafts. Why three? Why not four? The answer is, embarrassingly, that we measured it. We did not derive it.

In the closing months of 1935, a young physicist at Caltech named Carl Anderson was developing photographs in a dim Pasadena darkroom. He had a cloud chamber on the roof of the Norman Bridge Laboratory, a glass vessel of supersaturated alcohol vapor, dropped into a strong magnetic field, that revealed the path of any charged particle as a feathery white trail. Above Pasadena, every second of every day, cosmic rays from distant supernovae were raining down through the building’s roof. Most of them were electrons, which curved one way, or positrons (which Anderson himself had discovered in 1932), which curved the other. He knew those signatures the way a typist knows the alphabet.

But on a few plates he kept finding something that did not fit. A track that ionized the gas like an electron, curved through the magnetic field like an electron, but with the wrong radius. The curvature was too gentle. Either it was a very fast electron (but the ionization density said it was slow) or it was a slow heavy particle. The arithmetic kept coming back to the same number: a mass about two hundred times the electron, and a charge of plus or minus one. Anderson and his student Seth Neddermeyer wrote the result up in 1937. They had found a new particle. They did not know what to call it.

The physics community had a guess, and the guess turned out to be wrong. Hideki Yukawa had recently predicted that the strong nuclear force should be carried by a particle a few hundred times the electron mass, neither as light as a photon nor as heavy as a proton. Anderson’s new particle weighed about right, so for a decade it was called the “mu meson” and assumed to be Yukawa’s force-carrier. But the muon would not cooperate. It refused to feel the strong force the way Yukawa’s particle should. By 1947 a separate, much shorter-lived particle (the pion) was found, and that one really was Yukawa’s. The muon, it turned out, was something nobody had asked for: a heavy copy of the electron. Heavier by a factor of 207, identical in every other respect, unstable, and inexplicable.

The muon was a problem because the world appeared to be running fine without it. Stable matter (the matter you and I are made of, the matter in stones and stars) needs only four ingredients: the up quark, the down quark, the electron, and the electron’s neutrino. Two quarks build the proton (uud) and the neutron (udd). The electron orbits the nucleus. The neutrino skitters away from beta decays carrying off the missing energy. Together these four particles account for every atom in the periodic table. The hydrogen in your blood, the carbon in your bones, the iron in your hemoglobin, the calcium in your teeth: all of it is just clever arrangements of u, d, e, and ν_e. Add to this the photon and the gluons and the W and Z bosons that hold things together, and you have a complete recipe for the world as we know it.

The muon adds nothing to that recipe. It is heavier than the electron by a factor of 207, lives for an average of 2.2 microseconds, and decays into an electron, a neutrino, and an antineutrino. It plays no role in chemistry. It plays no role in nuclear physics. If you erased every muon in the universe tomorrow morning, your breakfast would be unchanged. So why is it there?

Worse: the muon does not come alone. In 1962, a Brookhaven experiment by Leon Lederman, Melvin Schwartz, and Jack Steinberger fired a beam of pions into a wall of steel sixty feet thick. The pions decayed in flight; the steel absorbed nearly every product of those decays. Nearly every product. The neutrinos slipped through. On the far side, in a spark chamber, the experimenters watched what those neutrinos did when they finally interacted. The answer was clean: they made muons, not electrons. The neutrino that comes out of a pion decay is not the same neutrino that comes out of a beta decay. There are two flavors of neutrino, one paired with the electron and one paired with the muon. The “second generation” was now a full set: a heavy charged lepton (μ), its own neutrino (ν_μ), and (it would soon emerge) a pair of heavier quarks called charm (c) and strange (s) to keep the books balanced.

Then, in 1975, Martin Perl at SLAC announced a third charged lepton. He had been looking for it on the suspicion that the pattern would continue. It did. The tau is so heavy (3,477 times the electron mass, nearly twice the proton mass) that it can decay into hadrons as well as into lighter leptons. It lives for less than a trillionth of a second. Its associated neutrino (ν_τ) was not directly observed until 2000, when the DONUT experiment at Fermilab caught four interactions, beautifully, in a target of emulsion. The third generation also brought two more quarks: the bottom (b), found at Fermilab in 1977, and the top (t), found at Fermilab in 1995 with a mass so absurd, 173 GeV (roughly as heavy as an entire gold atom), that no experiment before the Tevatron had enough energy to produce it.

So by the late 1990s the experimental picture was complete. Three generations of quarks: (u, d), (c, s), (t, b). Three generations of leptons: (e, ν_e), (μ, ν_μ), (τ, ν_τ). Each generation a faithful copy of the first in every quantum number that matters: electric charge, weak isospin, color, spin. The only thing that changes from one generation to the next is the mass. And the mass is wild. From the lightest neutrino (well under an electron-volt) to the top quark (173 GeV), the particles span fourteen orders of magnitude on a single ladder. That is a bigger range than the size of an atom compared with the size of the Sun.

It helps to put the same data in a different layout. Forget for a moment that the masses are wildly different and look at the quantum numbers, the spins and charges that determine what each particle does. The pattern is uncanny.

The visual point of that diagram is worth pausing on. Every quantum number a particle physicist cares about (electric charge, color, weak isospin, weak hypercharge, spin) is identical from one generation to the next. The charm quark has the same charge as the up quark, the same color states, the same weak couplings. It carries no new label that the up quark does not also carry. The only thing that distinguishes one generation from the next is mass, and through mass, lifetime. The heavy ones decay into the light ones; the light ones are stable.

That is why ordinary matter sees only the first generation. The muon decays into an electron in two microseconds. The tau decays in about three hundred femtoseconds. The charm and strange quarks last for picoseconds, bound up in mesons that fly a few millimeters from the production point before falling apart. The top quark lives for about 5 × 10⁻²⁵ seconds, so short that it never has time to form a bound state. Within a femtosecond of the Big Bang the universe was hot enough to keep these heavy particles in thermal equilibrium; ever since, they have been a kind of cosmic ornament, produced briefly in collisions or cosmic rays, gone again before anyone notices.

So now we have a brick wall to lean against. Three generations is not a theoretical prediction. It is an experimental measurement, accurate to about half a percent, taken from the way the Z resonance broadens by absorbing every available neutrino channel. The Standard Model accepts the number three as input, the way Euclidean geometry accepts the parallel postulate as input. Nothing in the model derives it. You could write down a perfectly consistent Standard Model with two generations, or four, or seventeen; the equations would look the same, just with more columns in the mass matrix. Nature picked three, and we know it picked three because we counted the neutrinos. We do not know why.

Γ(Z → ν ν̄) = G_F · M_Z³ / (12 π √2) ≈ 167 MeV per species

Γ_inv = Γ_total − Γ_vis ≈ 0.499 GeV

N_ν = Γ_inv / Γ(Z → ν ν̄) = 499 / 167 ≈ 2.99

There is a temptation, when you see a pattern repeated three times, to look for a deeper structure that generates it. Physicists have spent forty years trying. One class of attempts goes by the name “horizontal symmetry”: maybe the three generations are three copies of a single thing under some new gauge group, and the mass differences are spontaneous-symmetry-breaking effects, like the mass differences inside the weak doublets. Grand Unified Theories, supersymmetry, string theory landscapes, extra dimensions: most attempts to go “beyond the Standard Model” produce models in which three generations is a derived feature rather than an axiom. None of these models has yet been confirmed. None has even produced a prediction that experiment has decisively tested.

We are left, then, with a curious aesthetic problem. The Standard Model is the most accurately tested theory ever written; the electron’s anomalous magnetic moment is verified to one part in 10¹², a precision matched by nothing else in human science. But the same Standard Model accepts three generations and twenty-some free parameters as raw inputs and offers no explanation for any of them. Rabi’s question, “who ordered that?”, has never really been answered. The muon is still on the table, and the tau is on the table beside it, and across the room sit the strange, the charm, the bottom, the top, the muon neutrino, and the tau neutrino. If physics were a restaurant, somebody walked into the kitchen seventy years ago and ordered the same thing three times, but only the cook knows why, and the cook is not talking.