In the summer of 1970, I spent some time at the Aspen Center of Physics, where I met Gell-Mann and we started working together. In the autumn we studied the results from SLAC on the deep-inelastic scattering of electrons and atomic nuclei. The cross-sections depend on the mass of the virtual photon and the energy transfer. However, the experiments at SLAC found that the cross-sections at large energies depend only on the ratio of the photon mass and the energy transfer – they showed a scaling behaviour, which had been predicted by James Bjorken.
In the SLAC experiments, the nucleon matrix-element of the commutator of two electromagnetic currents is measured at nearly light-like distances. Gell-Mann and I assumed that this commutator can be abstracted from the free-quark model and we formulated the light-cone algebra of the currents (Fritzsch and Gell-Mann 1971). Using this algebra, we could understand the scaling behaviour. We obtained the same results as Richard Feynman in his parton model, if the partons are identified with the quarks. It later turned out that the results of the light-cone current algebra are nearly correct in the theory of QCD, owing to the asymptotic freedom of the theory.
The Ω– is a bound state of three strange quarks. Since this is the ground state, the space wave-function should be symmetrical. The three spins of the quarks are aligned to give the spin of the omega minus. Thus the wave function of the Ω– does not change if two quarks are interchanged. However, the wave function must be antisymmetric according to the Pauli principle. This was a great problem for the quark model.
In 1964, Oscar Greenberg discussed the possibility that the quarks do not obey the Pauli statistics but rather a “parastatistics of rank three”. In this case, there is no problem with the Pauli statistics but it was unclear whether parastatistics makes any sense in a field theory of the quarks.
Two years later, Moo-Young Han and Yoichiro Nambu considered nine quarks instead of three. The electric charges of these quarks were integral. In this model there were three u quarks: two of them had electric charge of 1, while the third one had charge 0 – so on average the charge was 2/3. The symmetry group was SU(3) × SU(3), which was assumed to be strongly broken. The associated gauge bosons would be massive and would have integral electric charges.
In 1971, Gell-Mann and I found a different solution of the statistics problem (Fritzsch and Gell-Mann 1971). We considered nine quarks, as Han and Nambu had done, but we assumed that the three quarks of the same type had a new conserved quantum number, which we called “colour”. The colour symmetry SU(3) was an exact symmetry. The wave functions of the hadrons were assumed to be singlets of the colour group. The baryon wave-functions are antisymmetric in the colour indices, denoted by red (r), green (g) and blue (b):
Thus the wave function of a baryon changes sign if two quarks are exchanged, as required by the Pauli principle. Likewise, the wave functions of the mesons are colour singlets:
The cross-section for electron–positron annihilation into hadrons at high energies depends on the squares of the electric charges of the quarks and on the number of colours. For three colours this leads to:
Without colours this ratio would be 2/3. The experimental data, however, were in agreement with a ratio of 2.
In 1971–1972, Gell-Mann and I worked at CERN. Together with William Bardeen we investigated the electromagnetic decay of the neutral pion into two photons. It was known that in the quark model the decay rate is about a factor nine less than the measured decay rate – another problem for the quark model.
The decay amplitude is given by a triangle diagram, in which a quark–antiquark pair is created virtually and subsequently annihilates into two photons. We found that after the introduction of colour, the decay amplitude increases by a factor three – each colour contributes to the amplitude with the same strength. For three colours, the result agrees with the experimental value.
In the spring of 1972, we started to interpret the colour group as a gauge group. The resulting gauge theory is similar to quantum electrodynamics (QED). The interaction of the quarks is generated by an octet of massless colour gauge bosons, which we called gluons (Fritzsch and Gell-Mann 1972). We later introduced the name “quantum chromodynamics”, or QCD. We published details of this theory one year later together with Heinrich Leutwyler (Fritzsch et al. 1972).
In QCD, the gluons interact not only with the quarks but also with themselves. This direct gluon–gluon interaction is important – it leads to the reduction of the coupling constant at increasing energy, i.e. the theory is asymptotically free, as discovered in 1972 by Gerard ’t Hooft (unpublished) and in 1973 by David Gross, David Politzer and Frank Wilczek. Thus at high energies the quarks and gluons behave almost as free particles. This leads to the approximate “scaling behaviour” of the cross-sections in the deep-inelastic lepton–hadron scattering. The quarks behave almost as free particles at high energies.
The logarithmic decrease of the coupling constant depends on the QCD energy-scale parameter, Λ, which is a free parameter and has to be measured in the experiments. The current experimental value is:
Experiments at SLAC, DESY, CERN’s Large Electron–Positron (LEP) collider and Fermilab’s Tevatron have measured the decrease of the QCD coupling-constant (figure 2). With LEP, it was also possible to determine the QCD coupling-constant at the mass of the Z boson rather precisely:
It is useful to consider the theory of QCD with just one heavy quark Q. The ground-state meson in this hypothetical case would be a quark–antiquark bound state. The effective potential between the quark and its antiquark at small distances would be a Coulomb potential proportional to 1/r, where r is the distance between the quark and the antiquark. However, at large distances the self-interaction of the gluons becomes important. The gluonic field lines at large distances do not spread out as in electrodynamics. Instead, they attract each other. Thus the quark and the antiquark are connected by a string of gluonic field lines (figure 3). The force between the quark and the antiquark is constant, i.e. it does not decrease as in electrodynamics. The quarks are confined. It is still an open question whether this applies also to the light quarks.
In electron–positron annihilation, the virtual photon creates a quark and an antiquark, which move away from each other with high speed. Because of the confinement property, mesons – mostly pions – are created, moving roughly in the same direction. The quark and the antiquark “fragment” to produce two jets of particles. The sum of the energies and momenta of the particles in each jet should be equal to the energy of the original quark, which is equal to the energy of each colliding lepton. These quark jets were observed for the first time in 1978 at DESY (figure 4). They had already been predicted in 1975 by Feynman.
If a quark pair is produced in electron–positron annihilation, then QCD predicts that sometimes a high-energy gluon should be emitted from one of the quarks. The gluon would also fragment and produce a jet. So, sometimes three jets should be produced. Such events were observed at DESY in 1979 (figure 4).
The basic quanta of QCD are the quarks and the gluons. Two colour-octet gluons can form a colour singlet. Such a state would be a neutral gluonium meson. The ground state of the gluonium mesons has a mass of about 1.4 GeV. In QCD with only heavy quarks, this state would be stable but in the real world it would mix with neutral quark–antiquark mesons and would decay quickly into pions. Thus far, gluonium mesons have not been identified clearly in experiments.
The simplest colour-singlet hadrons in QCD are the baryons – consisting of three quarks – and the mesons, made of a quark and an antiquark. However, there are other ways to form a colour singlet. Two quarks can be in an antitriplet – they can form a colour singlet together with two antiquarks. The result would be a meson consisting of two quarks and two antiquarks. Such a meson is called a tetraquark. Three quarks can be in a colour octet, as well as a quark and an antiquark. They can form a colour-singlet hadron, consisting of four quarks and an antiquark. Such a baryon is called a pentaquark. So far, tetraquark mesons and pentaquark baryons have not been clearly observed in experiments.
The three quark flavours were introduced to describe the symmetry given by the flavour group SU(3). However, we now know that in reality there are six quarks: the three light quarks u, d, s and the three heavy quarks c (charm), b (bottom) and t (top). These six quarks form three doublets of the electroweak symmetry group SU(2):