Structure of the Universe

4. WEAK, ELECTROMAGNETIC, AND STRONG INTERACTIONS: STRUCTURE OF THE MICROCOSMOS

At sufficiently low energies, there are four types of fundamental interactions whose existence is well established. Most studied are two of them: The gravitational and the electromagnetic interactions. The foundations of the classical (non-quantum) theory of the two interaction types were laid long time ago by Newton and Maxwell.

In particle physics, the Standard Model encompasses all the particles known today and three of the fundamental interactions. The basic building blocks are two sets of matter particles, the quarks and the leptons (Table 1). These particles interact with each other through the exchange of gauge bosons (see Section 4.3, Table 2). The three fundamental interactions of the Standard Model are the electromagnetic, the strong, and the weak interactions, respectively. Gravity remains outside the Standard Model, but this does not invalidate the model as gravitational effects on particles are far smaller than the effects of the other interactions. The Standard Model has two components. One is the theory of strong interaction, called quantum chromodynamics. The other component is the theory that gives a unified description of electromagnetic and weak interactions, called the electroweak theory. The physical concepts used in the Standard Model are generalizations of concepts familiar in quantum electrodynamics (Kolb and Turner, 1990; Kaku, 1993).

Table 1. Fundamental fermions divided into two groups: leptons and quarks (Barnett et al., 1996).

Leptons (spin = )

Electric charge	Flavor	Mass (GeV / c²)	Flavor	Mass (GeV / c²)	Flavor	Mass (GeV / c²)

0	v_e , electron neutrino	< 1 10^-8	, muon neutrino	1.7 10^-4	, tau neutrino	< 2.4 10^-3

-1	e, electron	5.1 10^-4	, muon	0.106	, tau	1.777

Quarks (spin = )

Electric charge	Flavor	Approx. mass (GeV / c²)	Flavor	Approx. mass (GeV / c²)	Flavor	Approx. mass (GeV / c²)

	u, up	(2 - 8) 10^-3	c, charm	1 - 1.6	t, top	180

-	d, down	(5 - 15) 10^-3	s, strange	0.1 - 0.3	b, bottom	4.1 - 4.5

4.1.1 Gravitational interaction

Gravitational interaction that governs the motion of celestial bodies is characterized by Newton's constant G = 6.7 10^-8 g^-1 cm³ s^-2. An excellent approximation that describes the gravitational interaction of two point masses m, a distance r apart, is Newton's formula,

. (1)

All fundamental particles are affected by gravity. The relativistic generalization of Newton's theory of gravity is Einstein's theory of general relativity.

4.1.2 Electromagnetic Interaction

Electromagnetic interaction determines the motion of charged bodies and acts only on charged particles. In the general case, their law of motion is described by Maxwell's equations. In the quasistatic approximation, an analogue to Newton's law, the Coulomb approximation,

, (2)

proves to work well. Here, e denotes the charge of each point mass. The quantum field theory of electromagnetism, quantum electrodynamics, is the best theory available for describing effects of the electromagnetic force. One of this theory's important features is its gauge symmetry, which means that when independent changes to local field values are made at different points in space, the equations of quantum electrodynamics are not changed. This symmetry is ensured only if the quantum description of a charged particle contains an electromagnetic field with its gauge boson; i.e., the gauge symmetry demands the existence of the electromagnetic force and the photon. The symmetry is also linked to the ability to renormalize quantum electrodynamics so that it yields sensible, finite results.

The magnitudes of Gm² and e² depend on the choice of the system of units. To facilitate comparison in the framework of quantum field theory, one combines these quantities with universal physical constants, the Planck constant and the velocity of light c, to obtain dimensionless constants. Thus, the nondimensional gravitational constant (see end of Section 4, Table 3),

, (3)

and the nondimensional electromagnetic fine structure constant (see end of Section 4, Table3),

, (4)

are obtained, e 10^-19 C being the electron (proton) charge. There is a difference in the definition of the two constants, _e being in a way more universal than _g. The definition of the number _e contains fundamental constants only, whereas the constant _g involves a mass m which is, generally speaking, arbitrary. To eliminate this arbitrariness, it is common to fix the value of m by setting it equal to the proton mass m_p. This choice is quite natural, for the proton is one of the two stable particles constituting the structure of the universe; the other one is the electron, with mass m_e. The choice between m_pand m_e is a matter of convention (m_p 1837 m_e).

4.1.3 Weak interaction

The weak interaction governs the decay of particles into lighter ones and acts upon all quarks and leptons, including those with no electric charge. Historically, the first decay discovered was the decay of a neutron within an atomic nucleus (the ß-decay), according to the reaction n p + e^- + (n, p, e^-, and denoting a neutron, a proton, an electron, and an antineutrino, respectively). Subsequently, the discovery of new particles was intensified by progress in the development of accelerators. It turned out that all newly discovered particles have a common property: Heavy particles decay into lighter ones. Numerous investigations led to the conclusion that many decays are controlled by a unique interaction, referred to as the weak interaction, which is characterized by the Fermi coupling constant g_F = 10^-49 erg cm³. The corresponding dimensionless coupling constant for the weak interaction is (see end of section 4, Table 3)

, (5)

The processes of collisions of neutrinos with matter are determined by the weak interaction as well.

There are many attempts to develop unified descriptions of all four interactions. In quantum theory every particle is associated with a particular field. How such a field transform under the Lorentz transformation depends on the spin of the particle described by the respective field. A zero-spin particle can be described by a scalar field (Higgs field), a spin-half particle by a spinor field, a spin-one particle by a vector field. The Lagrangian describing the field will carry information about the mass of the particle and its interactions. It is possible to construct a model that describes electromagnetic and weak interactions using a Lagrangian which possesses invariance under two transformation groups, U(1) and U(2).

The Higgs fields have a Lagrangian with a potential V() which has non-trivial minima. If such a system comes into contact with matter fields which are in thermal equilibrium at some temperature T, then the effective potential energy will acquire a temperature dependence. That is, V() will become V(, T). Such a temperature dependence can lead to several non-trivial effects - like phase transitions - in the early universe (Kolb and Turner, 1988). Even though the transformation group underlying the theory can be determined from some general principles, the detailed transformation properties of the fields representing specific particles cannot be derived from any fundamental considerations. These details are fixed using the known laboratory properties of these particles. For example, consider the fields describing the leptons. Given a spinor field one can construct its "right-handed" and "left-handed" components by the decomposition

, (6)

where is the 2 2 matrix

$gamma_sub_5 = {(0 1), (1 0)}$ . (7)

In the standard electroweak theory, the right-handed components behave as singlets (that is, they do not change) while the left-handed components transform as a doublet [that is, under an SU(2) transformation these fields are changed into linear components of themselves]. It is a consequence of this feature that in the simplest electroweak theory there is no necessity for the right-handed neutrino v_R and that the left-handed neutrino v_L is massless. Since the transformation properties of the fields are put in by hand into the theory, it is possible to generalize these models in many ways. In particular, it is possible - though not necessary - to have massive neutrinos in the theory. This arbitrariness is of great importance for the existence of dark matter in the universe and for the solar neutrino problem.

4.1.4 Strong interaction

The strong interaction was identified with the nuclear interaction which acts only on quarks and is ultimately responsible for binding protons and neutrons within the nucleus. The attempts to develop a consistent theory of nuclear interaction took a long time. A breakthrough was achieved with the progress of the dynamical theory of quark systems that led to the advent of quantum chromodynamics. In that scheme, the nuclear interaction was identified with the interaction in many-quark systems. It is instructive to trace briefly the evolution of the quark interpretation of nuclear interaction. To do so, we briefly outline the quark model proposed by Murray Gell-Mann (b. 1929) and George Zweig (b. 1937) in 1964. According to this model, each proton and neutron consists of three point-like particles which are referred to as quarks and possess a charge that is a fraction of the electron charge e: ±e or ±e. This theoretical conclusion was seemingly in contradiction to the experimental evidence that all the observable particles have an integer electric charge. Nevertheless, numerous experimental confirmations of the quark hypothesis (such as systematics of the elementary particles, the magnitude of the magnetic moments, the ratios of the interaction cross-sections, etc.) suggested that it deserves serious consideration. But then a profound question arose: How can the existence of quarks be reconciled with their nonobservability in experiments? At present, this problem is referred to as that of quark-confinement. A postulate is invoked which has rather a character of a magic: Quarks do exist, but in bound states. Even though no final solution of the confinement problem is available, one bases some expectations on the construction of a mathematical model that claims to provide a theory of the interaction between the quarks. It is this interaction that is identified with the strong interaction governing nuclear interaction. In 1954, Chen N. Yang (b. 1922) and Robert L. Mills (b.1927) proposed a theory which is basically different from electrodynamics, but accounts for the interaction caused by the transfer of zero-mass particles. The only such particle known at that time was the photon. The photon is the gauge particle in electrodynamics. That is why the Yang-Mills theory was considered just mathematical exercise. The picture changed, when a need emerged for a theory describing the dynamics of quarks. It seemed natural to consider the massless particles introduced by Yang and Mills to be responsible for the quark interaction. These particles were named gluons; by analogy with quantum electrodynamics, the quantum field theory of electromagnetism, one of the variants of the Yang-Mills theory is referred to as quantum chromodynamics. While gluons are analogues of photons, quarks are analogues of electrons. They carry not only color charges but also ordinary electric charge. E.g., a proton consists of three quarks (p = uud), a neutron consists of three quarks (n = udd), held together by continuing exchange of gluons. In the early 1970s the Yang-Mills equations were subjected to more scrutiny. As a result, the constant _s was found to exhibit quite remarkable behavior, as distinct from quantum electrodynamics. This constant determines the quark-quark interaction which is currently believed to be the true strong interaction. It should be remembered that from the viewpoint of contemporary field theory the interaction is mediated by gauge bosons, i.e., quanta of the corresponding field. Energy momentum and hence - according to the special theory of relativity - mass is transferred along with a quantum. Elaborate calculations have demonstrated that the strong interaction coupling constant _s essentially depends on the energy-momentum and the mass mtransferred. In a way, one had encountered a mass dependence of the constants before (e.g., _g _w), but quantum chromodynamics introduces a basic difference. In this theory, the dependence _s(m) is deduced from quantum field theory, and not postulated, as was done earlier for the constants, _g and _w on the basis of dimensional considerations. In addition, the variation of the constant _s with the mass m has a specific feature: _s decreases with increasing m. It should be remarked here that the terminology repeatedly used above might appear contradictory. On the one hand, one speaks of the constants ; on the other hand, one keeps stressing their dependence on m. In fact, the constants are only constant at a fixed m; they vary with changing m. That is why they are referred to as "running" constants. The final expression for the dependence of _s on mreads, in the asymptotic approximation when m m_p (see end of Section 4, Table 3),

$alpha_sub_s proportional to {a / (ln[m / m_sub_p])}$ . (8)

The quantity a depends on N_q, the number of the sorts of quarks. In a standard theory (N_q = 6), a 1. It follows from this formula that _s 0 as m . This is the phenomenon of asymptotic freedom. A similar dependence also follows from a more exact expression. Unfortunately, the latter has been also obtained by methods whose validity breaks down for m < m_p. A "true" expression for _s at small m is missing, owing to the fact that _s is large, thus rendering standard computation techniques inapplicable. One can only state that for a small characteristic mass m, corresponding to the proton (or neutron) size, r_N 10^-13 cm, the coupling constant is large. Furthermore, a rapid increase of the constant _s with r approaching r_N inhibits progress in solving another problem, namely, that of nuclear forces. Today, quantum chromodynamics is considered as a theory that describes the interactions among quarks and gluons, out of which atomic nuclei are made.

4.2 Classification of Fundamental Particles

Fundamental particles are classified with respect to various parameters. A particle classification appears to be provided by the value of the spin, s. The behavior of particles depends on whether their spin is characterized by an integer (0, 1, 2, ...) or a half-integer (, , , ...). Particles with a half-integer spin are referred to as fermions, while those with an integer spin as bosons. In the framework of quantum mechanics, the difference in the behavior of fermions and bosons is expressed by the kind of symmetry of the wave functions describing these particles. A system consisting of fermions obeys Pauli's exclusion principle, as distinct from a boson system on which no such exclusion principle is imposed. Pauli's principle reads as follows: No two fermions may be in exactly the same quantum state.

An excellent illustration of Pauli's principle is the atomic level structure underlying the periodic system of the chemical elements. It is known that the first period of this system is composed of two elements, hydrogen and helium. For the first period, the principal quantum number equals unity. The atomic states associated with the first period are therefore determined only by the value of the spin projection of orbital electrons. There are two such values. Thus, only two elements can occur in the first period. For the second period, the principal quantum number equals two, giving rise to eight possible different states and thus to eight elements, etc. The Pauli principle is one of the foundations of the very structure of the periodic system. If this principle did not work, all the atomic electrons would populate the ground energy level (i.e., the hydrogen level), and, consequently, the periodicity of the system as well as the valency of chemical bonding would vanish. It is the Pauli principle that prevents atomic electrons from occupying the energetically most favorable ground state.

Another basis for the classification of the fundamental particles is their interaction. All particles participating in the strong interaction are referred to as hadrons (from the Greek word "hadros", meaning "strong"). All fermions which do not participate in the strong interaction are called leptons. A special place in this classification is reserved for the bosons, particles which mediate the interactions. The hadrons, in turn, are subdivided into the baryons and the mesons. The baryons are fermions; the lightest baryon is the proton. The hadrons with integer spin are referred to as mesons; the lightest meson is the pion ( 140 MeV).

4.3 How Fundamental Particles Interact

Particles interact by exchange of gauge bosons; the exchange in the process of interaction involves not only energy, momentum, and mass, but also the internal quantum numbers: spin, isospin, charge, and color. The properties of the exchange particles in the context of quantum field theory determine the interaction to a great extent. All exchange particles are bosons. The properties of the exchange particles are summarized in Table 2.

4.3.1 Graviton

The graviton is a massless particle with zero charge and a spin s = ±2 and has not been detected owing to its extremely weak interaction. Although most physicists have no doubts about the existence of gravitons, some caution is advisable, since the quantum theory of gravitation itself is far from being complete. Because of the weakness of the gravitational field, there is no hope for rapid progress in detecting and investigating gravitons. Because the graviton is a massless particle, the gravitational interaction is long range.

Table 2. Properties of the force carriers (Barnett et al., 1996).

	Bosons (spin = 0, 1, 2,...)

	Electric Charge	Mass (GeV/c²)

Unified elektroweak (spin = 1)
, photon	0	0
W^-	-1	80
W⁺	+1	80
Z⁰	0	91
Stronger or color (spin = 1)
g, gluon	0	0

4.3.2 Photon

Photons have spins of s = ±1 and a rest mass of zero, and are their own antiparticles. The electromagnetic interaction has a long range because the photon is massless.

4.3.3 Intermediate bosons

The weak interaction is mediated by three intermediate gauge bosons W^± and Z⁰, which have masses of 80 and 91 GeV, respectively. Since the range of an interaction is inversely related to the mass of the gauge boson, the weak interaction has an extremely short range.

4.3.4 Gluons

The gluons, like the quarks, are not observable in the free state. However, in the late 1970s, considerable progress in the indirect verification of gluons was achieved by investigating the annihilation of high-energy positrons and electrons with hadron generation. It turned out that three hadronic jets occur in such processes; two jets are attributed to quarks and the third one, to gluons. The experimental data on the three-jet processes accompanying positron-electron annihilation are in good agreement with the predictions of quantum chromodynamics, which indirectly confirms the existence of gluons, one of the basic elements of that theory. The strong interaction is mediated by a set of gauge bosons, containing at least eigth gluons.

The data presented in Table 3 list the properties of the four interactions. A new entry here is the value of the interaction radius. For the gravitational, the weak, and the electromagnetic interaction, the magnitude of the radius r is determined from the uncertainty relation, /m_Bc, m_B denoting the mass of an exchange particle. In the case of the strong interaction, the interaction radius, r_N , may be regarded either as an empirical constant or as the distance at which the value of the coupling constant _s becomes unity.

Table 3. Properties of the four fundamental interactions.

	Coupling constants

Interaction	Analytic expression	Numerical value at m = m_p	Interaction radius (cm)

Gravitational	_g = Gm²/c	10^-38
Weak	_wg_Fm²c/³	10^-5	10^-17
Electromagnetic	_e = e²/c	1/137
Strong	_s = a/[ln(m/m_p)], m m_p	1	10^-13