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§ Phase xii · Quarks & Hadrons · Chapter 02

Mesons and baryons

Half a century of cosmic-ray photographs and bubble-chamber tracks turned into a zoo, and the zoo turned into a periodic table. Mesons came in pairs, baryons came in triples, and a Swiss-army formula written by a Caltech eccentric in 1962 sorted them like cards on a felt table. Then a single missing card predicted a particle nobody had ever seen.

In the autumn of 1934, the young Japanese physicist Hideki Yukawa, working in Osaka with no European tradition behind him and no senior collaborator to argue with, sat down to solve a problem that had been bothering everyone for two years. James Chadwick had discovered the neutron in 1932. The nucleus, it was now clear, was a crowd of positively charged protons jostling with neutral neutrons in a volume only a few femtometres across. The electrical repulsion among the protons was enormous, and yet nuclei held together. Something else had to be pulling them in. That something had to be stronger than electromagnetism at short distances and yet completely invisible at atomic distances, otherwise it would have wrecked every spectroscopic measurement of the previous thirty years.

Yukawa borrowed an idea from quantum field theory. If the electromagnetic force was carried by the photon, a particle of zero mass that could reach across infinity, then a short-range force should be carried by a massive particle, one heavy enough that quantum mechanics forbade it from travelling more than a femtometre or so. From the known range of the nuclear force, about 1.4 femtometres, Yukawa calculated the mass of the carrier: roughly two hundred times the mass of the electron. He called the particle the meson, from the Greek mesos meaning “middle”, because its predicted weight sat between the light electron and the heavy proton. The paper appeared in the Proceedings of the Physico-Mathematical Society of Japan in 1935. Outside Japan, almost nobody read it.

What changed that was cosmic radiation. By the late 1930s, balloons and mountaintop detectors had revealed a steady drizzle of charged particles raining down on the upper atmosphere, products of cosmic-ray collisions with nitrogen and oxygen. Inside the shower were odd tracks that did not match any known particle: heavier than electrons, lighter than protons, often decaying after a microsecond into smaller things. In 1936 Carl Anderson, working at Caltech, identified what looked like a perfect Yukawa meson with the right mass. The world celebrated. The nuclear force had found its mediator.

The celebration was premature. Through the 1940s, a string of careful experiments showed that Anderson’s particle, eventually renamed the muon, did not interact with nuclei. It happily ploughed through dense matter like a fat electron. A Yukawa meson should have been gobbled up by the first nucleus it hit. Something was wrong. The real Yukawa particle was hiding behind the muon, and it took the patience of Cecil Powell, working with photographic emulsions exposed at high altitude in the Andes mountains, to find it. In 1947 Powell’s group at Bristol caught the decay sequence: a heavier particle, the pion (or pi meson), stopping in the emulsion and producing a muon, which in turn decayed into an electron. The pion was the real Yukawa meson. It interacted strongly. It had been the protagonist all along. Powell got the 1950 Nobel, Yukawa had already received his in 1949, and the bestiary of hadrons opened its doors.

The pion was a meson by every measure: a particle of integer spin (in fact spin zero), much lighter than the proton, and unstable on a microsecond timescale. But as accelerators replaced cosmic rays in the 1950s, mesons multiplied like rabbits. The Bevatron at Berkeley, the Cosmotron at Brookhaven, the proton synchrotrons at CERN, all of them began spitting out new short-lived particles in numbers that nobody had asked for. The pion came in three charges: pi-plus, pi-zero, and pi-minus. Then came the K mesons, or kaons, heavier than the pion and behaving in ways that made nuclear physicists nervous. A new generation of resonances appeared with names like rho, omega, eta, and phi. Every one of them was a meson by the bookkeeping rule that defined the family: integer spin, produced in pairs, decaying eventually into pions or muons or electrons.

Meanwhile a parallel zoo was sprouting on the baryon side. The proton and neutron, the two members of what Werner Heisenberg had called the nucleon doublet, were joined by heavier siblings: the lambda, three sigmas, two cascade particles (also called xi), and four delta resonances. All of them were fermions of half-integer spin, all of them carried what came to be called baryon number equal to one, and most of them lived for less time than it took a sound wave to cross a hydrogen atom. Enrico Fermi, watching the proceedings from Chicago, was supposed to have said that if he had known the universe was going to be this complicated he would have studied botany. The botanists, at least, had Linnaeus. Particle physics did not, not yet.

The first hint of an underlying classification came from a quantum number Murray Gell-Mann invented in 1953 and called strangeness. Some of the new particles, notably the kaons and the lambda, had a curious property. They were produced copiously in collisions involving the strong nuclear force, in pairs, on a timescale of about 10⁻²³ seconds. Yet once produced they decayed slowly, on a timescale of about 10⁻¹⁰ seconds. That ratio, thirteen orders of magnitude, was the fingerprint of a conserved quantum number being broken: the particles could be made together via the strong force, but each one had to leak away via the much weaker weak interaction because it carried a property that strong-force interactions preserved. Gell-Mann named the property strangeness. Each particle of the new class carried a strangeness number S of plus or minus one or two. The kaon plus had S equal to plus one. The lambda had S equal to minus one. The cascade had S equal to minus two. The proton, neutron, and pion had S equal to zero.

According to the quark model, the properties of hadrons are primarily determined by their so-called valence quarks. For example, a proton is composed of two up quarks (each with electric charge 3 e, for a total of +3 e together) and one down quark (with electric charge 3 e). Adding these together yields the proton charge of +1 e. Although quarks also carry color charge, hadrons must have zero total color charge because of a phenomenon called color confinement. That is,…

The deeper question was whether strangeness was just a bookkeeping trick or a sign that something inside the new particles was changing. Gell-Mann answered it in 1962, in a paper that sorted every known hadron of the same spin and parity onto a flat plot whose axes were isospin (a relic of the old proton-neutron similarity) and hypercharge (a stand-in for strangeness plus baryon number). The picture that came out had a geometry no nineteenth-century chemist would have recognised but every group theorist did. The eight lightest baryons, the proton, neutron, lambda, three sigmas, and two cascades, formed a perfect hexagonal pattern: two doublets at top and bottom, a triplet on the middle line, and a singlet sitting at the centre. The lightest mesons did the same. The whole arrangement looked like a printer’s tray for a typesetter who only knew how to set hexagons.

Gell-Mann called the scheme the Eightfold Way, a knowing nod to the Buddhist path. It was independently discovered by the Israeli physicist Yuval Ne’eman, working at the same time in Imperial College London. The underlying mathematics turned out to be the representation theory of SU(3), the group of three-by-three unitary matrices with determinant one. Three was the magic number: not the three colours of QCD, which would come a decade later, but three flavours of fundamental constituent. The Eightfold Way said that hadrons came in family portraits whose size matched the patterns SU(3) could draw on paper, and those patterns were always built from groups of three.

The killer prediction came almost immediately. The lightest spin-three-halves baryons formed a ten-member pattern, a decuplet, that looked like a triangular arrangement on the same isospin-hypercharge axes. By 1962 nine of the ten slots had been filled in by particles known from cosmic rays and accelerator data. The bottom slot, with strangeness equal to minus three, was empty. Gell-Mann stood up at the 1962 Geneva conference and, with the certainty of someone reading a clear sentence, predicted a particle: charge minus one, strangeness minus three, mass around 1675 MeV, decaying slowly via the weak interaction because no lighter hadron could absorb its three units of strangeness in a strong-force transition. He even named it. He called it the omega-minus.

In February 1964 a team led by Nicholas Samios at Brookhaven, scanning bubble-chamber photographs from the new 33 GeV alternating gradient synchrotron, found a single beautiful track exactly where Gell-Mann said it would be. The omega-minus weighed 1672 MeV. It decayed as predicted into a cascade and a pion. The photograph went around the world in days. The Eightfold Way had passed the same test Mendeleev’s periodic table had passed almost a century before: it had pointed at an empty box and said “fill this in” and nature had obliged.

The success of the Eightfold Way left one question hanging in the air. Why three? Why did nature insist on building hadrons out of triples on the baryon side and pairs on the meson side, with patterns that matched SU(3) so neatly? Gell-Mann gave a tentative answer in a 1964 paper that was so short it ran to just two pages: maybe the patterns came from real underlying constituents, three of them, with fractional electric charge. He named them quarks, after a nonsense word from James Joyce’s Finnegans Wake, partly to signal that he was not yet sure they existed and partly because he loved a literary joke. George Zweig at CERN had the same idea independently and called them aces. Gell-Mann’s name stuck.

A quark would carry charge plus two-thirds or minus one-third, baryon number one-third, and one of three flavours: up, down, or strange. A baryon would be three quarks. The proton would be uud, charge plus one. The neutron would be udd, charge zero. The lambda would be uds, with one unit of strangeness from the s quark. The omega-minus would be sss, three strange quarks, exactly the corner of the decuplet that nothing else could reach. A meson would be one quark and one antiquark. The pi-plus would be u-dbar. The K-zero would be d-sbar. The eta would be a quantum-mechanical mixture of u-ubar, d-dbar, and s-sbar. Everything that had taken thirty years to catalogue fell out of three constituents and one simple combination rule.

↘ Derive: from the Gell-Mann–Nishijima formula to quark charges

Long before quarks, spectroscopists had noticed an arithmetic rule that worked across every hadron known. The charge Q of a hadron in units of the elementary charge e was always:

Q = I₃ + (B + S) / 2

where I₃ is the third component of isospin, B is the baryon number (one for baryons, zero for mesons), and S is the strangeness. The combination Y = B + S is the hypercharge. This is the Gell-Mann–Nishijima formula, named for Gell-Mann and the Japanese physicist Kazuhiko Nishijima, who arrived at it independently in the mid-1950s.

Now plug in the quark model. Assign every up quark I₃ = +½, B = ⅓, S = 0. Plug in:

Q(u) = +½ + (⅓ + 0)/2 = +½ + ⅙ = ⅔

So the up quark must carry electric charge plus two-thirds. For the down quark, I₃ = −½, B = ⅓, S = 0:

Q(d) = −½ + ⅙ = −⅓

For the strange quark, I₃ = 0, B = ⅓, S = −1 (by convention, the strange quark itself carries minus one strangeness):

Q(s) = 0 + (⅓ − 1)/2 = −⅓

Three fractional charges fall out of an old empirical formula the moment you grant that hadrons are made of three-fold constituents. That coincidence is what convinced Gell-Mann the quarks were probably real, even though no isolated fractional charge had ever been measured. The bookkeeping rule of the 1950s was secretly a statement about what was inside the proton.

The neat picture had one more chapter, and it came not from theory but from a basement at SLAC and a tunnel at Brookhaven on the same November weekend in 1974. At Stanford a team led by Burton Richter was scanning electron-positron collisions looking for resonances. At Brookhaven Samuel Ting’s group was bouncing protons off a beryllium target and looking for narrow peaks. Both groups, completely independently, saw a sharp spike at exactly 3.1 GeV: a meson far heavier than anything Yukawa or Gell-Mann had ever sketched, lasting a hundred times longer than the strong-force timescale predicted, and standing out like a clean musical note against the noisy background of the early seventies. Richter called it psi. Ting called it J. The double name stuck. The particle is the J/psi to this day.

The J/psi was not just one more particle. Its narrow width meant a long lifetime, and a long lifetime for a heavy hadron meant it could not decay through the strong force the usual way. The simplest explanation was that the J/psi contained a new flavour of quark whose conservation was being respected by the strong interaction. That flavour, predicted on theoretical grounds by Sheldon Glashow and others a decade earlier as a way to suppress certain rare kaon decays, was called charm. The J/psi was the bound state of a charm quark with its antiquark, charmonium, the cc-bar ground state. The accidental simultaneity of the Stanford and Brookhaven discoveries earned the joint Nobel for Richter and Ting in 1976 and gave the field a name it still uses: the November Revolution.

After 1974 the rest of the heavy-flavour story unfolded fast. The bottom quark turned up in 1977, again as a meson, this time at Fermilab: the upsilon, a b-bbar bound state weighing about 9.5 GeV. The top quark was harder. It is so heavy, about 173 GeV, that it decays before it has time to bind into a hadron at all. There is no top meson and no top baryon, only the bare quark, observed for the first time at Fermilab’s Tevatron in 1995 by detecting its decay products. The full six-flavour family of quarks was now complete, paired into three generations of two: up-down, charm-strange, top-bottom. Every meson is a member of one of these flavour pairs combined with an antiquark of another, and every baryon is a triple drawn from the six flavours. The combinatorics is enormous; the particle data tables now run to hundreds of pages.

What is striking, in hindsight, is how cleanly the pattern Gell-Mann sketched in 1962 survived the explosion. Add a fourth flavour, charm, and the SU(3) plot turns into an SU(4) prism whose lower slice is the old Eightfold Way. Add a fifth and a sixth, and the prism grows new floors. The geometry of hadrons is the geometry of how many quark flavours nature decided to grow, and the categories Yukawa, Powell, and Gell-Mann gave us, mesons and baryons, hold even at the top of that tower. The proton is still uud. The pi-plus is still u-dbar. The omega-minus is still sss. A photograph from the Andes in 1947 and a bubble-chamber print from Brookhaven in 1964 sit in the same family album as the LHC traces from the 2010s. The particles are heavier; the family rules are the same.

§

We have spent a chapter learning the family rules. Next we open the lid on the most familiar baryon of all, the proton, and look inside. The story turns out to be far stranger than uud. The three valence quarks are a faint signal in a roiling sea of gluons and short-lived quark-antiquark pairs, and the proton’s mass, almost all of it, comes from the binding energy of that sea, not from the quarks themselves.

next chapter → The proton, up close