The label three identical quarks forced

No two in the same state introduced the exclusion principle as the reason matter has volume. The pressure that is not a temperature found it holding up the electrons in a metal, the force with no force in it traced it to the symmetry of many-particle states under exchange, and the collisions the exclusion principle forbids found it silencing collisions in a Fermi sea. In each case it was a constraint on particles already known, telling them where they could and could not be.

Its most consequential use ran the other way. It was not a constraint on known particles but a demand that particles have a property nobody had seen. In the early 1960s the quark model explained the pattern of the strongly interacting particles by building them out of three kinds of quark, and it worked remarkably well except for one family of particles whose existence it could not accommodate. Taking the exclusion principle seriously for quarks required them to carry a hidden three-valued label. The label is colour, the charge of the strong force, and the chain of reasoning that led to it began with a spin and a symmetry.

A particle that should not exist #

The quark model of Murray Gell-Mann and George Zweig, in 1964, built every baryon — protons, neutrons and their heavier relatives — from three quarks, of three flavours, up, down and strange. The lightest baryons have the quarks in the lowest orbital state, with no orbital motion, and their spins combined either to one-half or to three-halves. The spin-three-halves family includes a particle called the $\Delta^{++}$, with a mass of 1,232 MeV and a charge of plus two. Its charge requires three up quarks, each of charge $+2/3$. Its spin of three-halves requires all three spins to point the same way.

That is three identical fermions in the same spatial state with the same spin. The force with no force in it found the rule behind the exclusion principle: a state of identical fermions must change sign when any two are exchanged. The $\Delta^{++}$ as described is unchanged by any exchange. Its spatial part is symmetric, because all three quarks are in the same orbital; its spin part is symmetric, because all three spins point the same way; its flavour part is symmetric, because all three are up quarks. There is nothing left to be antisymmetric in, and the state cannot exist. Yet the $\Delta^{++}$ had been known since the 1950s as a sharp resonance in the scattering of pions from protons, one of the best-measured particles there was.

Three values, antisymmetrised #

In 1964 Oscar Greenberg proposed that quarks obey a modified statistics, and the next year Moo-Young Han and Yoichiro Nambu proposed what became the accepted form: each quark carries an extra label with three possible values, and every observed baryon is in the one combination of those values that changes sign under the exchange of any two quarks. The whole state — space, spin, flavour and the new label — is then antisymmetric, as the exclusion principle requires.

The number three is forced. The antisymmetric combination of $n$ labels among three particles exists only if there are at least three values, because a totally antisymmetric state of three objects must give each a different value; with two values it vanishes. With exactly three there is exactly one such combination, which explains why each baryon comes in only one version rather than several. The label came to be called colour, with the three values named red, green and blue, and the rule that observed particles are in the antisymmetric combination became the statement that observed particles are colourless — the three colours of a baryon cancel as the three primary colours of light combine to white. None of this has anything to do with visible colour; the names were a mnemonic.

Ten particles that need it #

The $\Delta^{++}$ is not an isolated case. It belongs to a family of ten, all with spin three-halves, and every one has the same problem.

The figure plots the family’s masses against the number of strange quarks. They rise in almost equal steps as down or up quarks are replaced by the heavier strange one, and Gell-Mann used that regularity in 1962 to predict that the tenth member — three strange quarks, the $\Omega^-$ — should exist near 1,685 MeV and should be stable against strong decay. It was found at Brookhaven in 1964 at 1,672 MeV, a triumph for the scheme that also sharpened its problem. The $\Omega^-$ is three identical strange quarks, all in the ground state with spins aligned. The same is true of the $\Delta^-$, three down quarks. Every member of the family has a spin-and-flavour arrangement that is symmetric under exchanges, so every one needs colour, and the prediction of the $\Omega^-$ was a prediction of a particle made of three identical fermions in the same state.

The spin-one-half family, which includes the proton and neutron, escapes the problem without colour — its spin-and-flavour arrangement is mixed in symmetry — but in the colour picture it too is antisymmetric in colour, and the counting of which spin-and-flavour states can combine with an antisymmetric colour state explains why there are eight spin-one-half baryons and ten spin-three-halves ones, and no others among the lowest states. Two states where the counting says three found the photon’s two polarisations by counting states a symmetry allows; the baryon spectrum is the same kind of counting with colour included.

Where the sign under exchange comes from #

The exclusion principle is itself a consequence of something deeper, and it helps to see why the rule for quarks could not simply be relaxed. The turn that has to be made twice found that a particle of half-integer spin changes sign under a rotation of 360 degrees and returns to itself only after 720. The spin–statistics theorem, proved by Wolfgang Pauli in 1940 from the requirements of relativity and of energies bounded below, ties that sign to exchange: exchanging two identical particles is, in a precise sense, rotating one about the other, and particles that change sign under a full turn must change sign under an exchange. Every particle of half-integer spin is therefore a fermion, obeying the exclusion principle, with no exceptions in any consistent relativistic quantum theory.

That is why Greenberg’s first proposal, a modified statistics for quarks alone, was uncomfortable, and why colour was the better answer. A quark has spin one-half, so it must be a fermion; the only freedom is in what else it carries. Four states, and one of them is odd counted the states of two spins and found the one that is antisymmetric under exchange; for three quarks with three colours, there is exactly one colour state antisymmetric under every exchange, and it is the one nature uses. The exclusion principle was not bent. It was satisfied by a variable nobody had yet counted.

A baryon is an atom that cannot fill its shells #

The comparison with atoms is exact enough to be useful. In an atom the exclusion principle is what stops every electron falling into the lowest orbital: two electrons fill it, with opposite spins, and the third must go to the next shell. Lithium’s third electron is outside the first two, which is why lithium is a reactive metal and helium, whose two electrons fill the lowest shell, is inert — the pattern the order the shells fill followed through the table. A baryon is three quarks in the lowest orbital with room for them all, because the three colours give each a distinct state even when their spins align. Without colour, the $\Delta^{++}$ would be like a lithium atom with all three electrons forced into 1s with parallel spins: not a slightly different particle, but an impossible one.

The count of three also caps the pattern. A fourth identical quark with the same spin would find all three colours taken, and baryons have three quarks, not four. Mesons, a quark and an antiquark, are colourless in a different way — colour and anticolour cancelling — and they are the only other simple combination the rule allows.

Counting colours with electrons #

A label introduced to rescue a model is suspect until it does something else. Colour did three other things, each a direct count.

When an electron and a positron annihilate, they make a virtual photon, and the photon turns into any pair of charged particles light enough: a muon and an antimuon, or a quark and an antiquark, which then dress themselves as hadrons. The rate for each pair is proportional to the square of its charge. The ratio of the rate for hadrons to the rate for muons is therefore the sum of the squares of the quark charges — and each quark flavour comes in as many varieties as there are colours, each produced separately. The figure draws the prediction. Below the energy needed to make charm quarks, up, down and strange give $4/9 + 1/9 + 1/9 = 2/3$ per colour: two for three colours, two-thirds for one. The early measurements, in the first years of the 1970s, were puzzling because they straddled the charm threshold nobody yet knew was there; once the charm quark was found in 1974 the picture settled into the figure’s steps — about two below the threshold and about three and a third above it. One colour would have been a factor of three too low at every energy.

A decay rate that goes as the square #

The second count is sharper because the number enters squared.

The neutral pion decays almost always into two photons, in about $10^{-16}$ seconds. The rate can be computed exactly from the structure of the theory — it is fixed by what is called the axial anomaly — and the calculation runs through a loop of quarks to which each colour contributes an equal amplitude. The amplitude is therefore proportional to the number of colours and the rate to its square. The figure compares the prediction for one to four colours with the measured width of 7.80 electronvolts. One colour would give 0.86, nine times too small; four would give 13.8. Three gives 7.78, within one per cent. Jack Steinberger had computed this decay in 1949 treating the proton as the particle in the loop, and got the right answer for a reason that was obscure until colour made the quark loop give the same factor.

The Z boson, counted #

The third count came from the large electron–positron collider at CERN, which from 1989 produced millions of Z bosons.

The Z decays into every pair of fermions lighter than half its mass, at rates fixed by its known couplings. Each quark flavour, in each colour, is a separate decay channel. The figure computes the hadronic width for one to four colours, including the small correction for the strong force, and compares it with the measured 1.744 GeV. Three colours give 1.737, within half a per cent. The same measurements counted the number of neutrino types — three, from how much of the Z’s width goes into nothing visible — and together they gave the particle content of the standard model its final shape.

From label to force #

The step that turned colour from a count into a physical theory was to make it the charge of a force. In 1973 David Gross, Frank Wilczek and David Politzer showed that a theory in which colour is the source of a field — carried by eight gluons that themselves carry colour — has the property that the force weakens at short distances and strengthens at long ones. At high energies quarks inside a proton behave almost as free particles, which is what the deep scattering experiments of the late 1960s had seen; at the size of a proton the force becomes so strong that no quark can be pulled out alone. Only colourless combinations exist freely, which is exactly the rule that the antisymmetric colour state had imposed on the baryons. The theory, quantum chromodynamics, is now tested to a few per cent over a huge range of energies, and it accounts for the fact that nearly all the mass of ordinary matter is the energy of the colour field inside protons and neutrons rather than the masses of the quarks.

It is worth tracing what the exclusion principle did in this story. It did not predict a force. It demanded that three apparently identical particles be different in some way, and it fixed the number of ways at three. Everything else followed from taking that label seriously as a charge.

There is one further place the number three appears, and it is the strangest. The standard model is consistent only if certain quantum effects called gauge anomalies cancel between its particles, and for one generation of matter the simplest of those conditions is that the electric charges of all its fermions add to zero. An electron contributes $-1$ and a neutrino nothing. The up and down quarks contribute $+2/3$ and $-1/3$, a total of $+1/3$ — in each colour. With three colours the quarks contribute $+1$, and the generation’s charges sum to exactly zero. With any other number they would not, and the theory would be inconsistent. The same number of colours that the $\Delta^{++}$ demanded, that electron collisions counted and that the pion’s decay rate squares, is the number that makes the charges of quarks and leptons fit together — a coincidence for which a unified theory of the forces would be one explanation, and which no measurement has yet explained.

What the pictures cannot show #

The quark model of the figures treats baryons as three quarks in simple orbitals, and that is a caricature of what quantum chromodynamics describes: a proton is three valence quarks in a seething field of gluons and quark–antiquark pairs, and its spin, for example, is carried only partly by its quarks’ spins. The model’s success for the ground-state baryons is still not fully explained from first principles, though calculations on a lattice of spacetime points now reproduce the masses of the decuplet to a few per cent. The R-ratio figure leaves out the resonances near each threshold — the J/ψ and its relatives at the charm threshold, the Υ at bottom — which dominate the data just above each step, and uses a single simple correction for the strong interaction where the real correction depends on energy. The anomaly calculation of the pion’s width assumes the pion’s decay constant, measured separately.

Still open: why the quark model works as well as it does #

That a baryon is three quarks bound in a colourless state is established. What is not understood in detail is why a picture of three nearly independent quarks with effective masses of about 300 MeV each describes the lowest baryons so well, when the quarks in the underlying theory have masses of a few MeV and are bound by a field that dominates the proton’s energy. Lattice calculations reproduce the numbers without explaining the simplicity. Related questions — whether bound states of four or five quarks, or of gluons alone, exist as distinct particles, several of which have now been reported at colliders — test how far the colourless rule extends beyond the simplest combinations, and they are being settled one resonance at a time.

The habit worth carrying away is to treat a symmetry requirement as a demand for information. Three quarks that are identical in space, spin and flavour cannot share a state, so the exclusion principle required them to differ in a property nobody had seen, and required that property to take exactly three values — which electron–positron collisions, the $\pi^0$’s decay and the Z boson’s width all then counted, each coming out at three. A rule about where electrons may sit turned out to be a statement about the charge of the strong force.