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Hadrons and Quarks, an Introduction to the Theory of Mesons
and Baryons

Walter Pfeifer
38 pages, 10 line figures


Since fifty years physicists are convinced that protons, neutrons and related particles (hadrons) consist of subparticles named quarks. The properties of the quarks determine the feature of the hadrons. Although the quarks cannot be detected as free particles their configuration in the hadrons is known in detail.
In this book we restrict ourselves to low-mass hadrons and low-mass quarks. Higher states can be dealt with using the same technique. Here, a good agreement between theoretical and measured masses and magnetic moments of hadrons is presented.
The detailed developments and the numerous references to preceding places make this book easy to follow. However, knowledge of elements of quantum mechanics and Lie algebra is a prerequisite.
As in our previous publication, Pfeifer, W., 2008, we utilize the following symbols: operators are written in bold letters and three-dimensional vectors are marked with an arrow.


Preface 2
1 Hadrons 4
1.1 Characteristics of elementary particles 4
1.2 Conservation laws 5
1.3 Compiling of the properties of low-mass hadrons 6
1.4 Diagrammatic arrangement of hadrons 10
2 Quarks 13
2.1 Low-mass quarks 13
2.2 Mesons and quarks 14
2.3 Baryons and quarks 16
2.4 The isospin of the quark configurations, mesons 18
2.5 Meson representation with spins 20
2.6 Quark representation of baryons allowing for isospin and spin 21
3 Masses and magnetic Moments of Hadrons and Quarks 26
3.1 Masses of mesons and effective masses of quarks 26
3.2 Masses of baryons and effective masses of quarks 28
3.3 Magnetic moments of baryons and quarks 31
4 Epilogue 36
References 37
Index 38


1 Hadrons


Up to the year 1932 the proton and the neutron were the only known heavy elementary particles . They are named nucleons . Scattering experiments show that they attract themselves extremely strongly when they are very near (nearly touching themselves) independently oft their charges. One says that they interact strongly.

In the following years, investigating cosmic rays and using accelerators, other particles with strong interaction were found. Particles of this type being lighter than nucleons were named mesons . Most of the heavy ones are called baryons. Mesons and baryons  constitute the collective of the hadrons.

In the next section we deal with characteristic quantum numbers and data of hadrons.


1.1 Characteristics of elementary particles

The quotient of the charge and the mass  of a free moving particle, , is measured using a combination of magnetic and electric fields. We will give the charge as a multiple  of the elementary charge    including the sign. Because for free particles  is an integer, experimentalists can determine it and calculate the mass .

Most elementary particles (including hadrons) are endowed with as spin. As in atomic or nuclear physics the eigenvalue of the square of the spin operator    is , where  is a half-integer or integer (including zero) value. The -component  meets . For  and  we will use the symbol  and for  we take .

The isospin  is a virtual spin. To the nucleons W. Heisenberg assigned the isospin . The proton is characterized by the component  and the neutron by . Both nucleons which have nearly the same mass, form an isospin doublet . Three hadrons with about the same mass are summarised to a triplet  with isospin . The member with positive charge has , the neutral member has  and negative charge means . If a hadron with charge zero has no partner with comparable mass, it is interpreted as a charge singlet with .

Isospins of interacting particles are couplet with the same formalism as spins using Clebsch-Gordan  coefficients.

For hadrons, Gell-Mann and Nishijima have introduced the strangeness    as a quantum number as follows


The quantum number  (baryon charge ) is 1 for baryons and 0 for mesons. The fact that several hadrons are generated by strong interaction and decay by weak interaction first was denoted as a strange property. For these particles  holds.

Starting from the strangeness, the hypercharge    is defined as follows


It’s true that the marking hypercharge isn’t accepted generally, but the quantity  is useful for the geometrical representation of hadrons (section 1.4).

Originally, parity  gives the sign change of the wave function in question under reflection with regard to the origin of the spatial coordinates. The parity of elementary particles mostly is determined starting from known parities of interacting particles. The conservation law for parity (section 1.2) is applied. Per definition, nucleons have positive parity and -mesons (in ground state) has negative parity.

An antiparticle  has the same mass, spin and isospin as the corresponding particle. The sign of its charge, of its baryon charge  and of its isospin component  is inverted. Therefore the hypercharge  has a different sign.

1.2 Conservation laws

The  total (relativistic) energy  of a moving particle with rest mass  and (classic) impulse  reads


For  we have . In a reaction or in a decay the sum of the total energies of all involved particles is constant.

The total charge  i.e. the sum of the charges (taking into account their signs) of the involved particles is conserved.

The spin    of a system of particles is the vectorial sum of the individual spins formed using the addition rules of quantum mechanics (cf. Pfeifer, 2008, p. 43). The magnitude and the z-component of the spin are conserved.

The isospin    of a system is determined with the same formalism as the spin. If the particles have strong interaction the isospin and its component  are conserved (which isn’t true for electromagnetic and weak interactions)

For strongly interacting particles (e.g. hadrons) the sum of the baryon charges    is also a conserved quantity.

Because the algebraic expression for the strangeness    (equation 1.1.1) is a linear combination of conserved quantities, the strangeness is also conserved.

The parity  of a system of particles is the product of the parities (+1 or -1) of the constituents. This parity is conserved in reactions or decays.

1.3 Compiling of the properties of low-mass hadrons

Table shows the measured properties of the low-mass mesons with spin zero and negative parity. They are named pseudoscalar mesons. Most of the lifetimes of the given particles are near s. For three mesons which decay in two  they are much shorter.

Table 1.3.1. Properties of the pseudoscalar mesons


From Greiner, W., Müller, B., 1994, p.233

The table contains the particle-antiparticle pairs . The hypercharge  is calculated with equation 1.1.2. Basing on equation 1.2.1 the mass is given in the unit .

The next table contains mesons with spin 1 and negative parity. They are named vector mesons.


Table 1.3.2. Properties of vector mesons


From Greiner, W., Müller, B., 1994, p. 242.

There are also other meson multiplets with higher spins and masses.

The next table  shows the properties low-mass baryons . As is known, the proton is stable, the neutron has a lifetime of 15 min and most of the given baryons live near s long. Only the baryon  which emits a -quant has a much shorter lifetime.


This is only an excerpt. The whole publication can be ordered from the category «order for free» in this site.