When the era of particle accelerators started in the 1930s, physicists knew apart from the photon five more fundamental particles. Besides the muon, the electron and the neutrino these were the two nucleons proton and neutron, which were believed to have no substructure. In the following years many new particles were discovered in accelerator experiments and they could not be explained as composite objects of the known fundamental particles, but rather had to be added to the list of these “elementary” particles. This steadily growing list was rather unsatisfactory in the eyes of theorists. It was Gell-Mann who introduced a systematic approach to the particle zoo by proposing a new set of truly elementary particles called quarks in 1964 [1]. All hadrons that had been discovered so far were proposed to be composite objects of these quarks that should have spin 1/2 and come in different flavors.
Today the quark model is well established and provides the basis for the standard model of particle physics. Quarks are realized in six different flavors, up, down, strange, charm, top and bottom. These can be grouped in three doublets that are also referred to as the three generations of quarks:
u d
, c
s
, t
b
. (1.1)
They carry fractional electric charge, the quarks in the upper components +2/3e, those in the lower components−1/3e. The characterization as generations alludes to the fact that the quark masses increase several orders of magnitude from the left to the right, which made also their experimental discovery more and more challenging. The original quark model contained only the up, the down and the strange quark. These are therefore also known as the three light quarks, and only later the three remaining flavors were added to the theory. So it were Kobayashi and Maskawa who postulated the third generation in 1973 in order to accommodate for the observed CP violation in the decay of neutral kaons [2]. Hadrons containing the charm quark were discovered in 1974, the heavy top quark finally in 1995.
Although the quarks are subject to all four fundamental interactions, i.e., gravitational, electromagnetic, weak and strong, only the strong interaction is responsible for the fact that quarks cannot exist isolated but combine to hadrons. In the theory this is explained by
7
Figure 1.1: The spins= 1/2 baryon octet.
assigning a color charge red, green or blue to each quark and the opposite color (anti-red, anti-green or anti-blue) to the antiquark. Requiring all isolated states to be color neutral, one observes hadrons that are build out of three quarks, one of each color, or out of one quark and one antiquark with opposite colors. Hadrons containing three quarks are called baryons, systems with one quark and one antiquark mesons. Assigning a baryon number B = 1/3 to the quarks and B =−1/3 to the antiquarks, baryons haveB = 1 and mesonsB = 0.
All baryons that can be constructed from Nf different flavors are obtained by reducing direct products of the quark flavors that are identified with the fundamental representation of SU(Nf). In the case of the three light flavors up, down and strange we have:
3⊗3⊗3=10⊕8⊕8⊕1. (1.2)
The resulting direct sum of irreducible representations contains one decuplet of baryons, two octets and a singlet. Since the three quarks within these baryons are antisymmetric in color, and noting that the overall wavefunction is also antisymmetric due to the Pauli exclusion principle, one concludes that the spin-flavor part must be symmetric. The decuplet is sym-metric in flavor, whereas both octets have mixed symmetry and the singlet is antisymsym-metric.
For non-excited spin s = 1/2 states one thus ends up with only the baryon decuplet and one octet, because both octets are physically equivalent and the needed antisymmetric spin wavefuntion for the singlet does not exist.
A graphical representation of the baryon octet is given in Figure 1.1. The two axes represent the electric charge Q and the strangeness S of the baryon. The latter quantum number is defined by the number of strange antiquarks minus strange quarks. Note that the electric charge of all baryons is an integer multiple of the elementary chargee, although the quarks themselves carry fractional charges. In the top row of the graph we see the neutronn and protonp both with strangeness 0, as they are build from two u and one d, respectively from one u and two d quarks. Below them we find the sigma and lambda baryons, which contain one strange quark each. The last row finally summarizes particles with two strange quarks, the negatively charged and the neutral xi that were first observed in 1964.
In Figure 1.2 we summarize the spin-3/2 constituents of the baryon decuplet. The deltas in the top row again have strangeness zero. Their flavor content is from left to right: ddd, udd, uud and uuu. The sigma and xi baryons below have the same quark content as in the
1.1. QUARKS, BARYONS AND MESONS 9
Figure 1.2: The spins= 3/2 baryon decuplet.
s= 1/2 octet, namely dds, uds,uus and dss,uss. In the last row we finally face the omega Ω−, which consists of three strange quarks and therefore has strangeness −3. This baryon was not known when the quark model was introduced. Its discovery was therefore a great success that lead to a fast acceptance of the quark model.
One can proceed in a similar manner for the mesons. They are constructed by a reduction of the direct product of a quark and antiquark representation in SU(Nf). For the three light quark flavors one finds:
3⊗¯3=8⊕1. (1.3)
The graphical representation of the octet and singlet for spin 0 is shown in Figure 1.3. At the top of the octet we find a neutral and a positively charged kaon with strangeness +1. The related quark content isd¯sand u¯s. For strangeness zero one finds three pions, two of them charged, and in the center theη and η0 mesons. The flavor content of the two charged pions isd¯uandud, whereas the neutral pion and the etas are composed of¯ u¯uanddd. Finally there¯ are the K− and ¯K0 kaons that are the antiparticles of theK+ and K0.
The success of the quark model in taming the particle zoo was also important for the devel-opment of quantum chromodynamics as the underlying theory. This local (color) SU(3) gauge theory describes the strong interaction, which is responsible for the formation of hadrons, in terms of the fundamental degrees of freedom. These are the quarks that belong to the fun-damental, and the gauge fields, called gluons, that belonging to the adjoint representation of SU(3). The quantum character of the theory leads to a slight correction of the above pre-sented picture of mesons and baryons. At any time the valence quarks of the hadrons, which we actually quoted above, are surrounded by a cloud of gluons and quark-antiquark pairs.
The latter are so-called sea quarks, and since they always come in quark-antiquark pairs they do not change the net flavor content of the hadrons.
Today quantum chromodynamics is the key to study the internal structure and the bind-ing mechanisms of hadrons. Hand in hand with perturbative continuum QCD especially the non-perturbative approach of lattice QCD has proven very successful in enhancing our understanding of the strong interaction.
Figure 1.3: Thes= 0 meson octet and singlet.