
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 ßdecay),
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 cm^{3}. The corresponding
dimensionless coupling constant for the weak interaction is (see end of section 4, Table 3)
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 zerospin particle can be described by a scalar field (Higgs field), a spinhalf particle by a spinor field, a spinone 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
nontrivial 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 nontrivial 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 "righthanded" and "lefthanded" components by the decomposition
where is the 2 2 matrix
In the standard electroweak theory, the righthanded components behave as singlets (that is, they do not change) while the lefthanded 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 righthanded neutrino v_{R} and that the lefthanded 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 manyquark 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 GellMann (b. 1929) and George Zweig (b. 1937) in
1964. According to this model, each proton and neutron consists of three pointlike
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
crosssections, 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 quarkconfinement. 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 zeromass
particles. The only such particle known at that time was the photon. The photon is the gauge
particle in electrodynamics. That is why the YangMills 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 YangMills 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 YangMills 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 quarkquark 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 energymomentum 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),
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.
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).
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.


4.3.2 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.
The weak interaction is mediated by three intermediate gauge bosons
W^{±} and Z^{0}, 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.

