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NIKHEF/97-004
Gauge Theories of Matter and Forces
and the Unication of Particle Interactions
J.W. van Holten
NIKHEF, Amsterdam NL
Abstract
The multitude of levels of structure in the physical universe is related to the existence
of dierent fundamental interactions at various scales of distance. At vary small scales
they are described by quantum gauge theories. A non-mathematical exposition of the
role of quantum gauge theories in the Standard Model of particle physics is presented.
Some prospects for unication of these gauge interactions are explored. The use of
scalar elds in the breaking of symmetry between various gauge forces and in connecting
bosons and fermions through supersymmetry is discussed.
Matter and forces
In the natural world there are many levels of structure. The forms of matter
we are familiar with from everyday experience, like solids, uids and gases, are
aggregates of atoms and molecules, which have sizes of the order of Angstrms
(a hundred-millionth of a centimeter). These atomes and molecules themselves
are specic arrangements of one or more atomic nuclei with a cloud of electrons
orbiting around them. Indeed, the specic composition and arrangement of the
subatomic particles together determine the molecule's chemical properties, which
in turn explain to a large extent the properties of solids and gases.
As far as we know, electrons are elementary particles in the sense that they do
not have an observable substructure nuclei, however, are complicated objects,
arrangements of electrically charged protons and chargeless neutrons. Protons
and neutrons in turn have an internal structure described in terms of particles
called quarks. Later we have to say more about these objects. For the moment we
only observe that quarks and electrons |although they appear in very dierent
environments, the electrons forming the outer layers of the atomic and molecular
structures, whilst the quarks are buried two levels deep inside the nuclei| are akin
in many respects. Indeed, the standard model of particle physics classies them
in a single family of elementary particles, having no observed substructure on
scales as small as 10;16 m (one millionth of an Angstrm). Theoretical physicists
even argue convincingly that they probably are structureless at least down to the
inconceivably small distances of about 10;30 m.
The hierarchy of structures observed for smaller and smaller distances can also
be extended in the other direction: from solids, uids and gases to solar systems
of planets and stars, and on to galaxies and clusters of galaxies. Ultimately even
the universe as a whole seems suciently homogeneous and isotropic on a large
scale to be treated as a single system as far as its global properties and evolution
are concerned. The ordering of scales, from clusters of galaxies to subatomic
particles, is often visualized as a ladder, with the dierent rungs of the ladder
representing dierent levels of structure.
Of course, no matter how convenient, such a one-dimensional view of the
structure of matter and of the universe is not quite correct. It happens to be
useful in the regime of terrestrial temperatures and pressures, in which atoms
and molecules are stable objects. But it does not represent the physics of other
environments, for instance the properties of matter in stars. On the one hand, in
an ordinary star like the sun the temperatures are too high to allow stable atoms
to form: the electrons and nuclei have so much energy that they do not bind,
and matter exists in the form of a gas of charged particles immersed in a eld of
strong radiation, a plasma. On the other hand, in the extremely dense objects
known as neutron stars, with a radius of about 10 km and a mass one and a half
times that of the sun, the pressure of the gravitational force is so high that the
protons and electrons are not able to resist it and merge to form stable neutrons.
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As a result, a neutron star is like a giant atomic nucleus, with a mass larger than
that of the sun. Its structure is determined by the balance between gravitational
and nuclear forces, rather than the weaker electric and magnetic forces governing
the structure of molecules and solids.
If the pressure of the gravitational force is increased further, as in a star still
heavier and more compact, even the nuclear forces can no longer balance it. Such
a star can collapse under its own weight into an object known as a black hole, as
nothing can escape the strong pull of its gravity, not even light. In this state of
matter no substructure can be identied at all from the outside.
Thus we see that the structure of matter, and actually the whole hierarchy
of structures, can be dierent at dierent places in the universe, depending on
local conditions. In addition, it also varies as a function of time: we know that
the universe has evolved to its presently observed state by expansion from a hot
dense beginning, when no stars, no atoms or molecules, and not even nuclei were
present. Indeed, the early universe consisted mostly of light and other forms of
radiation.
The question now arises what determines the various forms of matter existing in
dierent regimes. The answer to this question is found in the variety of forces at
work in nature, the dierence in their strength and the corresponding dierence
in the characteristic scales on which they act. At least four dierent fundamental
forces are known to govern the dynamics of physical structures:
- The gravitational force, which keeps together the earth and moon, and the sun
and its planets. It clearly acts primarily on astronomical distance scales.
- The electro-magnetic force, responsible for the phenomena of electricity and
magnetism. This force shapes the structure of atoms and molecules. As such it
is the driving force behind almost all chemical and biological phenomena.
- The weak force, which drives radioactive transformation processes of unstable
particles and nuclear isotopes. It is the main force responsible for nuclear energy,
including the energy produced by stars like the sun.
- The strong color force, which binds the quarks in the proton and neutron. Together these two particles are refered to as nucleons because of their role as main
constituents of atomic nuclei.
The characteristic scales of these forces correspond to the dierent layers of structure discussed above, with gravity acting on the largest scales, and the color force
at the smallest scales that have been probed.
The typical size of a system in which a force is eective is an indication of
the strength of the force clearly, the stronger an attractive force acting between
two bodies, the tighter the binding between them is and the closer they will be.
In that sense gravity is the weakest force, whilst the color force is the strongest.
However, this is not the whole story. Under the right circumstances gravity can
balance the thermodynamic pressure of a hot gas (in a star like the sun), it can
be as strong as the electrostatic coulomb forces (in a white dwarf star) or as the
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nuclear forces (in a neutron star). And in extreme cases it can dominate other
forces completely: during gravitational collapse to a black hole, and during the
initial split second after the Big Bang which marks the beginning of our universe.
Similarly, in an atomic nucleus electrostatic repulsion between charged protons
balances nuclear forces keeping protons and neutrons together. The lesson to be
learned from this is, that the strength of a force depends on the distance over
which it acts, but also on the number and density of various types of particles in
the system, and on the intensity of various kinds of radiation present in short:
on the environment and the medium in which the force is at work.
What makes our universe interesting is that the interplay of these forces creates dierent environments on dierent length scales, and in dierent regimes of
pressure and temperature: the nuclear environment, the molecular environment,
the uid state, and so on. As a result the strength of a force also changes with
the length scale, and with the density or the energy of the physical system in
which it acts. This is true not only in macroscopic bodies, like dierent kinds of
stars, but also in the microscopic realm of atomic and subatomic particles. In
this context the eect that the medium regulates the strength of the interaction
between particles, and the way they respond to forces, is called a renormalization
of the interaction.
The vacuum
One might think naively that renormalization is not important when one discusses
simple laboratory situations in which the interaction between isolated particles
far away from other matter is studied. However, this intuitive idea turns out
to be unjustied: the vacuum, the empty physical space in which the particles
move, behaves itself like a very non-trivial medium. The specic properties of this
medium will be discussed in some detail later. What is important here, is that it
is not quite a material medium in the ordinary sense, but one that should more
properly be called a virtual medium. The term virtual is taken from quantum
theory, and refers to the eects of quantum mechanical uctuations which are
due to physical degrees of freedom not directly observable themselves.
As without further explanation this may seem a somewhat obscure statement,
a brief rst excursion into quantum physics is in order here. Quantum theory is
the basic mathematical framework for describing dynamical processes down to the
level of the smallest physical systems known. Strangely enough, the description it
oers is inherently statistical in nature: the expected result of the measurement
of a physical quantity |like the energy of a particle, or the intensity of a light
wave| is given in terms of a probability distribution, rather than by a precise
number xed by the physical boundary conditions. In a quantum system the
physically observable quantities can uctuate in time around an average value.
It must be stressed that in spite of its statistical nature, quantum theory gives
a complete description of the behaviour of physical systems. On the one hand
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it allows to compute the full probability distribution |the expected value and
the uctuations| for any physically observable quantity. On the other hand the
outcome of individual measurements are a priori unpredictable except for the
probability distribution: there is no completely deterministic scheme which can
reproduce the full set of statistical distributions of quantum theory.
The statistical nature of the description of physical systems also applies to the
vacuum. In particular the number of particles, although strictly zero on average,
uctuates: the physical state of a system containing no particles cannot be distinguished sharply from a state in which there are equal numbers of particles and
anti-particles, as long as these states have the same energy. Of course a system
containing observable physical particles (or anti-particles) does in principle have
a higher energy than the vacuum: the relativistic equivalence of mass and energy
implies that any particle carries at least the energy necessary to create its mass.
However, in quantum theory this is only true on the average for long times. In
short time intervals the energy of an individual particle can uctuate around this
value, the allowed uctuations being larger as the period becomes shorter.
The upshot of this story is, that particles and anti-particles can be present
in the vacuum for very short periods of time in which they contribute no energy
to the system. These particles cannot be individually observed, as this would
violate energy conservation therefore they are called virtual particles. Their collective eect however is to change the physical properties of the vacuum: virtual
particles contribute to a renormalization of electro-magnetic and other forces in
an observable way. Thus it is approporiate to refer to the quantum uctuations
in the vacuum as representing a virtual medium.
With the vacuum behaving like a material medium due to virtual particles, its
properties depend on the number and type of particles present in nature. If one
could change the properties of the particles, one could change the nature of the
vacuum. Moreover, a large change of the vacuum would be a phenomenon much
like that normally associated with a phase transition: a qualitative change in the
nature of a medium, like the change from liquid water to solid ice if one lowers
the temperature by extracting energy, or from liquid to vapour if the temperature
is increased by supplying energy in the form of heat. A kind of phase transition
might indeed happen in the virtual medium of the vacuum if the temperature
and/or density of matter and radiation are raised to a high enough level. Such
phase transitions are thought to have occurred in the early universe when it was
still very hot and very dense. Because of the dierent environment such a phasetransformed vacuum provides, the properties of particles and their interactions
also become qualitatively dierent in that case: massive particles may become
massless, and new kinds of charges may appear on which new forces act. Indeed
the properties of particles and of the vacuum are in direct correspondence, the
one determining as well as re
ecting the other.
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The hierarchy of scales and forces
The variety of physical structures we encounter in the universe on dierent scales
and in dierent regimes of density and temperature, from protons to stars, nd
their origin in the variety of forces of dierent strength and range. In most cases
these regimes are well-separated in the sense that the physics on one scale is
relatively independent of the precise substructure and dynamics on the smaller
scales. Practically this implies that it is possible to formulate theories of physics
on a given scale entirely in terms of a set of degrees of freedom relevant at that
scale, without having to take into account the detailed microscopic dynamics.
This is fortunate, as otherwise all of science would have to be formulated in the
language of particle physics, or maybe even the physics at a still deeper level. It
would make scientic descriptions of natural phenomena impossibly complicated.
It should be stressed, however, that the phenomenological separation of various length scales, each characterized by its elementary degrees of freedom and
dominant forces, is presently not a logical necessity, unless as some form of the
anthropic principle: we would not be here if it were not true. Imagine for example
a world in which the strength of the gravitational force is comparable to that of
the weak force which drives radio-active decay processes in nuclei1. Then stars
and planets, rocks or rain clouds would never form: all matter in the universe
would collapse and end up in black holes in a very short time. It is the very
hierarchy in strength between the gravitational and the other known forces, more
in particular the weak nuclear force, which allows for the diversity of distance
and time scales in our universe. If it were not for this hierarchy, the universe
could not be as large and as old as it is, there would be no luminous matter in
the form of stars, no chemical elements and ultimately no possibility for life on
earth.
At present the variety of physical phenomena and dierent characteristic processes at dierent distance and time scales in our world appear just a contingency,
not a necessity. However, this conclusion is only valid if it were indeed possible for
the constants of nature |like Newton's constant of gravity and Fermi's constant
of weak interactions| to have any other values than they actually have. But
this in itself is questionable maybe the consistency of the laws of physics requires
the hierarchy of their values as an unescapable necessity. Then the existence of a
universe like ours would not be an unlikely accident, but a quite natural outcome
of the laws of physics. This issue is at the center of debate in current theoretical
physics, and much eort is devoted to nding evidence for or against one or the
other possibility. It is commonly accepted that there may exist a unique physical theory able to provide a consistent microscopic description of all matter and
forces in the universe. If so, it is well conceivable that this theory allows for only
one free parameter, from which all other constants of nature can be computed, at
I.e., a world in which Newton's gravitational constant GN , measured in natural units, would
be as large as Fermi's weak interaction constant GF .
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least in principle, including the full hierarchy of their magnitudes. Such a theory,
if it exists, is then a unied theory of matter and forces.
To meaningfully discuss these ideas and speculations, it is necessary to have
a more detailed understanding of physics in the domain of subnuclear particles
and the forces at work there. Actually at this microscopic level the distinction
between these concepts becomes somewhat hazy and a new way of thinking about
matter, force, space and time is required. The most complete framework for the
description of matter and its interactions at these distance scales is relativistic
quantum eld theory. Some elements of this theoretical framework have already
been touched upon in the discussion of the role of the vacuum. Other features
will be high-lighted below. I begin with a brief sketch of the structure of matter
at the smallest distance scales we can currently probe in experiments.
The standard model of particle physics
At the sub-atomic level matter is composed out of quarks and leptons, two kinds of
particles known as fermions, and their anti-particles2. Fermions are characterized
by the property that they possess half a quantum unit of intrinsic rotation, or spin,
implying that their quantum-theoretical description is not invariant when they
are rotated through 360 , but only after rotation through 720 . Of the two types
of fermions mentioned the quarks are the more dicult to study experimentally,
because they are only found locked up inside nuclear particles: triplets of quarks
form baryons, the family of particles including the nucleons (proton and neutron)
and pairs of one quark and one anti-quark combine into mesons, like the -mesons,
which play a role in the strong interactions between the nucleons in atomic nuclei.
The forces responsible for binding quarks inside a baryon or a meson are not
of electric or gravitational origin, although in many respects they are similar to
the electro-magnetic interaction. In particular they act through some kind of
charges, of which there are however three types, instead of one as in the electromagnetic case. To distinguish them these charges are represented by colors: the
red, green and blue charge. Consequently, the interactions between these charges
are called color forces but obviously the reference to color is only symbolic.
The leptons, which derive their name from the observation that they are
generally much lighter than quarks, are often encountered as single free particles.
This is possible because they do not carry any color charge. Therefore they do
not participate in the strong color interactions and they are not locked up inside
mesons or baryons. For the same reason they are not found in atomic nuclei.
Leptons can carry electric charge, in amounts equal in magnitude to the proton
charge, but of opposite sign. Three such charged leptons are known: the electron,
the muon and the -lepton. The only intrinsic dierence between them seems to
be their mass, the electron being the lightest and the -lepton the heaviest. Each
Anti-particles have the same mass and spin as ordinary particles, but complementary electric charge and other properties.
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of these charged leptons has an electrically neutral partner: a neutrino, of which
there are three kinds as well. The neutrinos are known to be extremely light, so
light that for all we know they may be massless.
The lightest leptons, the electron and its corresponding neutrino, are found
abundantly in nature. In atoms the electrons form the cloud of negative charge
which neutralizes the positive nuclear charge and is responsible for almost all
chemical properties of atoms. neutrinos are emitted in great quantities by stars
like the sun, where they are generated by the nuclear processes in their interior.
It is also believed that there is a cold gas of low-energetic neutrinos lling the
universe, much like the cosmic microwave background radiation, as an afterglow
of the dense, hot early phase of the universe.
As neutrinos are electrically neutral and colorless (recall that none of the
leptons carry color charge) they do not participate in electro-magnetic or color
interactions. Instead, neutrinos are made in the decay of charged leptons or in
the transmutation of quarks in nuclear reactions. These processes involve another
kind of force: the weak interaction. The main weak interaction process for the
production of neutrinos is -decay, in which a lepton or quark changes its electric
charge by one unit. For example, a free muon at rest decays after about onemillionth of a second into a muon-neutrino, under simultaneous production of
an electron and its corresponding anti-neutrino. In a similar, but much faster
-decay process the -lepton changes into neutrinos and a muon. neutrinos and
anti-neutrinos can also be made in pairs in the annihilation of matter and antimatter, for example the annihilation of electrons and positrons. In that case no
charge is transferred from the initial to the nal particles.
Like the leptons, quarks can be grouped in pairs or doublets, the members
of a doublet diering by one unit of charge however, whereas a lepton doublet
consists of one charged and one neutral particle, the quarks in a doublet carry
2/3 or ;1=3 unit of electric charge. Furthermore for every known lepton doublet
there is a quark doublet, or rather a color triplet of doublets: a red, green and
blue doublet. Therefore the color-triplets of quarks, like the leptons, also appear
in three almost identical families, diering only in mass.
The lightest quark doublet consists of a pair bearing the unimaginitive names
up and down (u d). They are the main constituents of the proton, which is a
conguration of (uud), and the neutron containing (udd). The heavier quark
doublets are called charm and strange (c s), and top and bottom (t b). The
(u c t)-quarks all have 2/3 units of electric charge, whilst the (d s b)-quarks
carry ;1=3 unit of electric charge. Quarks can also transform into each other by
-decay processes for example, the conversion of a d-quark into a u-quark causes
the transmutation of a neutron into a proton, accompanied by the emission of an
electron and its anti-neutrino. This process accounts for the instability of the free
neutron, which has an average life-time of almost 15 minutes. The process and its
reverse (proton ! neutron + neutrino + positron) are important as production
processes of (anti-)neutrinos, especially in astrophysics | e.g. in stellar interiors.
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>From the above we can infer that there are two types of weak interactions:
in one type electric charge is transfered from one of the initial particles to one of
the particles produced the -decay processes are of this kind. In the other type
of weak interaction no charge is exchanged. Besides the annihilation of matter
and anti-matter, an example of the latter is the scattering of neutrinos o other
particles. Such a scattering process always proceeds via the weak force, but the
neutrino remains electrically neutral all along. In the literature these two kinds of
weak interaction are refered to as charged current and neutral current processes,
respectively.
Summarizing, according to the standard model of particle physics there are
three families of quarks and leptons, distinguished only by their mass. These
particles can interact via three kinds of forces: weak, electro-magnetic and color
forces. In addition we know that these particles fall in a gravitational eld, and
therefore are also expected to interact gravitationally among themselves. However, these gravitational forces are so extremely weak compared to the other
ones that they can not be detected. Therefore they are generally ignored for
all practical purposes, at least at the distance and energy-scales of present-day
experiments.
Particles and elds
The picture of the microscopic structure of matter I have sketched largely covers
the experimental facts, but is still imprecise. This is because it describes matter
and forces in the language of classical macroscopic physics: the notion of particle
is used as if we are dealing with solid, hard mass points, whilst the forces have
only been discussed in terms of their eects on such particles, without specifying
any detailed mechanism other than some passing references to various types of
charges and elds. I must now ll in some details to get a more accurate picture.
This means above all, that we have to leave behind our classical ideas of particles
and elds of force, and replace them by something else, covering both at the same
time: quantum elds.
The concept of a quantum eld is best explained by considering the illustrative example of the electro-magnetic eld. At the macroscopic level the electromagnetic eld shows up in two kinds of physical phenomena. In the rst place it
provides a force aecting the motion of charged particles for example, the electric
eld between the plates of a condensor and the magnetic eld of the earth both
accelerate electrically charged particles like protons or electrons. On the other
hand, the electro-magnetic eld also shows up in the form of radiation. Electromagnetic radiation is energy stored in and transported by the eld itself at the
speed of light, independent of the presence of any charged particles. The phenomenon of radiation shows, that the eld itself has physical degrees of freedom
|in fact innitely many: each mode of vibration is one| which contribute to
the energy, momentum and angular momentum of the physical system of which
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the eld is part (this may be the whole galaxy or even larger portions of the
universe including the intergalactic space). Thus the electro-magnetic eld is as
real a physical system as any particle, molecule or material body.
As a macroscopic physical system with innitely many degrees of freedom, like
a crystal or a membrane, a eld such as the electro-magnetic eld can be assigned
a temperature and other thermodynamic properties. The spectrum of vibrations
and their intensities at each frequency, for a eld in thermodynamic equilibrium
at a xed temperature, is known as black-body radiation. It is possible to derive
the thermodynamic equation of state, which relates the temperature, pressure and
energy of black body radiation, from the mechanical properties |the vibration
modes| of the eld using statistical arguments.
However, it turns out that the kind of statistics needed is rather unusual,
and requires the individual modes of the eld to behave in a very non-classical
way in particular, the energy E of any frequency mode is quantized in amounts
of an integer number of smallest parcels, the energy of one parcel, or quantum,
being proportional to its frequency : E = nh , where n is the number of quanta
(n = 0 1 2 :::). The constant of proportionality h is Planck's constant, and has
the dimensions of energy times time. Not only the energy, also the momentum
P stored in a vibrational mode of the eld turns out to be quantized in a similar
way, in xed amounts inversely proportional to the wavelength3 : P = nh=.
The correct thermodynamic equation of state for black body radiation can now
be obtained by treating the eld as a gas of indistinguishable and non-interacting
particles with energies xed by the frequencies of the various vibration modes of
the eld.
This particle picture of radiation is reinforced by the result of experiments in
which light is scattered with charged particles like electrons: one obtains perfect
agreement between theory and experiment if one describes the process as an
elastic collision between an incoming photon of xed energy and momentum, and
a point particle with the mass of an electron. The nal energy and momentum of
the photon and the electron after the collission then follow in a straightforward
way from the corresponding conservation laws, much as if one were colliding
billiard balls on a frictionless table.
The result is the more striking, because we also know that we can produce
diraction and interference patterns of light on a screen by letting it pass through
one or more narrow slits. This implies unequivocally that light is a wave phenomenon. Quantum eld theory asserts that both properties are part of the
complete description of the eld: depending on the circumstances it can behave
like a wave eld with characteristic modes of vibration, or a gas of indistinguishable particles with characteristic energies and momenta.
The quantum properties of the electro-magnetic eld, including its particle
interpretation in terms of massless photons, have their counterparts for all the
3
The wavelenght and frequency are related to the velocity of light by c = .
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force-elds we encountered before: the strong color eld is quantized in terms of
eight massless particles called gluons, and the weak force as mediated by a triplet
of massive particles called Z and W . Of these the Z is electrically neutral
and behaves much like a heavy photon it is responsible for neutral current weak
interactions. In contrast, the pair of oppositely charged W is responsible for
the charged-current weak interactions, including -decay.
The charges of the W quanta provide one dierence with the photon, which
does not carry any charge at all. Indeed, there is a general distinction between
abelian elds of force, whose quanta carry no charges, and hence are not themselves subject to the force they mediate and non-abelian elds of force, whose
quanta carry some kind of charges themselves and therefore interact with one another. For instance, the electro-magnetic eld is an abelian eld, as the photon
carries no electric charge, whilst the eld of the strong color force is non-abelian,
with gluons carrying specic combinations of color and anti-color charges. Also
the weak interactions have a non-abelian component, though the massive W and Z are not quanta of denite weak charges4.
Like the Compton scattering of electrons and photons, also the other forces
show their particle nature most clearly in scattering experiments, of the type
realized in high-energy accelerators. Indeed it is very common to describe such
scattering processes in terms of the exchange of particles, although interference
eects between alternative ways to realize a nal conguration of particles actually take place.
If elds of force can behave like material particles, it is only natural that
particles of matter should have an interpretation as excitations of a eld with
wave-like properties. This is indeed the case. Particles like electrons can show
interference and diraction phenomena, for example when traveling through the
ordered arrays of atoms in a crystal, or in an electron microscope. Wave properties
can also be ascribed to the other leptons and the quarks. There is only one
dierence with the quantum elds of force like the electro-magnetic eld: whilst
the electro-magnetic eld can have an arbitrary number of quanta (photons)
with the same energy, momentum and spin, the fermionic elds of matter can
possess at most one quantum of a certain type, with a certain energy, at a time.
This experimentally observed fact is known as the Pauli-principle. It is of great
importance in understanding the electronic properties of atoms and the structure
of the periodic table of the elements.
The restless quantum eld: physics of the vacuum
An important property of quantum elds is that they are never completely quiet.
As has already been explained, even at zero temperature they produce uncontrollable uctuations, which are revealed in the zero-point energy, the energy still
present in the eld in its lowest energy state. This state is by denition the vac4
A more detailed discussion is presented in a later section.
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uum state, as it is the state of the quantum eld in which there are no observable
particles |eld quanta like photons, gluons or electrons| present. Although the
zero-point energy itself cannot be measured, it can be changed by varying the
external parameters which control the vibration modes of the eld, for example
the volume of the system these changes are observable. A famous example of this
phenomenon is the Casimir eect, an attractive force observed to act between the
two plates of a condensor even in the absence of a voltage. It can be interpreted
as a negative pressure from the eld, because the zero-point energy of the eld
in the vacuum between the plates decreases when their distance, and hence the
volume, is varied.
In the particle language the vacuum uctuations of the eld are ascribed to
the presence of unlimited numbers of virtual particles and anti-particles, which
make themselves felt but can not be liberated to become directly observable. To
render these particles observable one must provide energy, at least an amount
equivalent to twice the rest mass of the particle, as this is the energy needed to
create a pair of one particle and one anti-particle by exciting the sea of virtual
particles while conserving electric or other charges. Pair creation is frequently
observed in the laboratory, for example when highly energetic charged particles,
like electrons or muons, travel through a strong external electric eld. When the
eld breaks the speed of the charged particle suciently strongly, it is forced to
give up some of its energy to the eld for conversion into electron-positron or
other particle/anti-particle pairs.
The fact that the vacuum state of a quantum eld still shows observable and
non-trivial behaviour as a result of particle- and charge-
uctuations is one more
testimony to the physical reality of such elds. It has far-reaching consequences
for the properties of particles and the physical processes involving them.
Consider a charged particle like an electron, moving through empty space.
By empty we mean, that no other observable localized particles are present.
However, there is still the sea of virtual particles and anti-particles that belong
to the vacuum conguration of the eld. These particles feel the charge of the
physical electron, exerting a repulsive force on the virtual particles with a charge
of the same sign, and an attractive one on the virtual particles with an opposite
charge. Thus the vacuum uctuations become polarized: the density of positive
vacuum charge is on the average higher near the electron, and tends to screen
the negative electron charge. As a result the net charge of the electron observed
from some distance away is less than the `bare' charge as seen from very short
distances. It is then no longer possible to speak of the charge of the electron, but
only of the charge as observed from innity, or from some nite distance5.
Eectively, the vacuum polarization diminishes the static force between two
This charge is probed by sending another charged particle to some distance close to the
electron, and observing how strongly it is deected by the electro-static force between them. As
the test-charge can come in closer the more energy it has, the observed charge of the electron
is often expressed as a function of the energy of the probe, instead of the distance.
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bare charges compared to the value predicted by Coulomb's law. However, it
is more useful and more common to represent the eect as a renormalization of
the charge by the vacuum uctuations. Earlier we have already introduced this
idea of renormalization by the presence of a (virtual) medium. Now we have a
picture of the actual mechanism by which the eect operates. From this picture
it is clear that the eectively observed charge changes as a function of distance
(or energy of the probe) it is often refered to as the `running charge', in contrast
to the bare charge, the value that would be measured in the absence of vacuum
polarization.
I have used electro-magnetism to illustrate the phenomena of vacuum polarization and running charge. This case is relatively simple, as the electro-magnetic
eld is abelian and the photon itself carries no charge. Things are more complicated for non-abelian interactions and charges. For instance, the gluons of the
strong interactions carry color charges, and therefore contribute to color-charge
polarization of the vacuum. However, the gluon charges are not simple like those
of quarks, but rather specic combinations of color and anti-color charge carried by one particle. As a result the interactions of gluons are quite dierent
from those of single charges like quarks. For example, a gluon can neutralize
its charges by attaching itself to a quark an an anti-quark with complementary
color charges. It can also neutralize its charge by attaching to other gluons with
complementary colors. In this way the color elds can build up narrow bundles
of color ux: string-like congurations of the color eld connecting separate color
charges. The color forces build up by these ux strings become very strong, and
tend to increase with distance, rather than decrease, like for electric forces.
The property that the eective color forces grow stronger with increasing distance, and weaker at short separations, is known as asymptotic freedom. In fact,
the force between two color charges increases so rapidly with the distance, that it
is impossible to separate color charges from each other: full separation requires
an amount of energy sucient to form a new quark/anti-quark pair long before
the color charges are at a macroscopic distance. These new quarks then attach
themselves to the original charges to neutralize the net macroscopic color charge.
As a result it is impossible to create free color charges at macroscopic distance
from each other, and color charges are permanently conned inside mesons and
baryons, the simplest color-less combinations of quarks and/or anti-quarks that
can be made.
The discussion presented here is very qualitative, but some details depend
on numerical factors. In particular, it is evident that the amount of vacuum
polarization depends on the number of charged particles that can participate in
the uctuations at a given level of energies, or equivalently in a certain range of
distance scales. Then also the running of the charges depends on this number. For
example, the simple color charges of quarks contribute to the vacuum polarization
dierently from the color charge combinations of gluons as a result the presence
of a large number of dierent types of quarks (more than sixteen) can destroy
14
asymptotic freedom. The running of the color charges has actually been measured
in the range of energies between ordinary nuclear energies (the MeV range) and
the Fermi scale of the weak interactions (about 100 GeV). They decrease in
agreement with asymptotic freedom, at a rate consistent with ve types of quarks,
or rather six, of which one is very heavy6 and therefore contributes very little in
this energy range.
A similar conclusion must hold for the non-abelian component of the weak
charges, at least on distance scales within reach of the weak force, about 10;16
cm. This is a subtle point, as the weak charges are actually mixed with the
electric ones, and the electric and weak charges rst have to be disentangled
before one can discuss their scale dependence. For the present purpose we can
however ignore this complication and be satised with the observation that the
weak charges run only very slowly. In the energy range presently accessible to
experiment it can not yet be measured.
Unication
The interactions of quarks and leptons take place through exchange of quanta of
the electro-magnetic, weak and strong force. The generic name for these quanta
is vector bosons or gauge bosons, and the eld theories describing their dynamics
are called gauge theories. One of these theories is an abelian gauge theory, with
neutral quanta, the other two are non-abelian gauge theories with charge-carrying
gauge bosons. The various kinds of (bare or renormalized) charges of the gauge
bosons, and of the quarks and leptons, always appear in certain multiples of some
xed unit, one for every kind of charge this eect is known as charge quantization.
Now consider a unit charge of each of the gauge theories describing the quantum force elds in nature. Recall that the magnitude of these charges eectively
changes with the length scale, and that at smaller distances they increase for
abelian forces, whilst they decrease for non-abelian ones like the color force. It
follows that if one could go to suciently small distances (which requires experiments with extremely energetic particles) at some point the strength of the forces
between two unit charges of each kind will become equal, at least if there are no
drastic changes in the way nature behaves at those very small distance scales. Of
course, the precise scale where the equality is obtained does depend on the exact
number of charge-carrying particles that can be observed at that scale, as I have
already explained above. But that does not change the basic observation that
dierent rates of running of the various charges imply that at some scale they
must become of equal magnitude.
One can now ask if it might be possible for all charges to become equal
at the same time, i.e. at some universal distance scale. Amazingly, with the
known particles of the standard model (quarks, leptons and gauge bosons) this
is almost true if one adds the eect of one other type of particle we have not
6
The top-quark has a mass of about 175 GeV/c2 , well above the Fermi scale.
15
encountered so far: scalar bosons, which are predicted to play a role in the weakinteraction sector of the standard model. I will discuss this important aspect
of weak interactions in more detail in the next section. It is also the basis for
a further extension of the standard model with more scalar and fermion elds,
involving a new feature refered to as supersymmetry. Supersymmetry imposes
some stringent relations between the properties and interactions of bosons and
fermions it is a speculative idea, but it turns out that it is exactly what is needed
to tune the running of the various kinds of charges so that they become equal at a
common scale. The universal scale at which the equality takes place is extremely
small, even compared to the scales of subatomic and subnuclear particle physics
we have considered so far: about 10;30 cm. However, it is food for thought that
it should be possible at all with such relatively simple ingredients as the standard
model and its supersymmetric extension provide.
Suppose now for the sake of argument, that all subatomic forces indeed become equally strong at this universal scale. What would be the implications?
The most interesting possibility is, that it signies that the three forces have a
common origin. This origin could be found in a quantum gauge eld with a larger
number of charges | at least ve kinds| all directly related and on an equal
level, like the three color charges of the strong interactions. At distances smaller
than the universal scale somewhere near 10;30 cm, this unied gauge eld is supposed to provide a correct decription of ultra-microscopic dynamical phenomena.
However, at larger distances the symmetry between the various charges must be
broken, so dierent subsets of gauge interactions can evolve with distance in a
dierent way, with the result that at the level of length scales presently accessible
to particle physics experiments, around 10;16 cm, the strength of the weak, the
electro-magnetic and color forces have become vastly dierent. How the breaking
of the symmetry between the various charges and interactions can be achieved is
explained below. What is worth stressing is, that the scenario developed here is
a realistic one, for which there is some circumstantial evidence. If true, there are
many interesting new physical phenomena to be discovered beyond the standard
model.
Broken symmetries and the origin of mass
Having sketched a scenario for unication of the three fundamental forces playing
a role in the physics of quarks and leptons, we must address two important
questions which require clarication. First, we need a mechanism to break the
symmetry between various charges of the unied force. And second, we need a
plausible argument why the origin of the observed forces and structure of matter
is found at a scale smaller by factor of about 10;14 compared to the weak and
color interaction scale. Particles with such small ranges have typical masses of
some 1016 protons. If this scale is fundamental, why do not all our particles,
quarks and leptons, have masses in this order of magnitude? Why are they so
16
extremely light compared to the mass and energy scale of our unication scenario?
One of the very interesting ideas of modern physics is, that there might exist
a link between these two questions. This link is provided by the physics of scalar
elds. In classical physics a scalar eld is characterized by its amplitude, a pure
number, at every point in space and time it does not have directionality, and as
a result scalar waves possess no spatial polarization. In this respect scalar waves
are unlike electro-magnetic waves, which are specied by both an amplitude and
a direction of the eld: the electro-magnetic eld has the characteristics of a
vector, rather than a pure number. As the electro-magnetic eld vector is always
at right angles to the direction of propagation, electro-magnetic waves have two
independent polarization states. They can be taken, for example, as waves with
right circular polarization, if the eld-vector rotates clock-wise around the axis of
motion or waves with left circular polarization if the eld vector rotates counter
clock-wise. In contrast, scalar elds have no polarization states.
In quantum eld theory the particle property corresponding to polarization
is the intrinsic rotation, or spin. Spin, like charge, is quantized in xed units.
Taking the spin of the photon as a unit, all vector particles like gluons, W and
Z have one unit of spin. Fermions like quarks and leptons carry half a unit of
spin. Scalar particles, the quanta of scalar elds, are distinguished by having no
spin at all.
Because of the absence of spatial polarization, or spin, scalar elds do not
have to vanish in the vacuum, even though this is the state of lowest energy.
Most importantly, a constant non-zero scalar eld does not spoil the rotation
and translation symmetry of empty space, as would say an electric or magnetic
eld. However, in general a scalar vacuum eld does produce observable eects.
In particular if the scalar eld carries a charge of some kind, its vacuum value generates equal, non-zero densities of charge and anti-charge which are constant in
space and time. Any other particle with a similar charge, or a gauge boson which
interacts with this kind of charge, is then aected in its motion. Of course, constant charge densities do not generate potential dierences, and charge-carrying
particles are not accelerated by the vacuum eld. However the eld can exert a
drag, aecting the ease with which particles respond to other forces, and thus it
contributes eectively to the mass of charged particles and gauge bosons. This
mechanism of generating an eective mass for particles, by their interactions with
a background charge density in the vacuum, is known after their originators as
the Brout-Englert-Higgs mechanism.
The size of the drag, and hence the mass, depends on the magnitude of the
charge densities. Moreover, the response of particles to other external forces
depends on the frequency or wavelength of the external force eld. If the potential
dierences, produced by the external eld over a volume in which there is on
average one vacuum charge/anti-charge, are much larger than the vacuum eld,
the reaction of a particle to the external eld is strong, and hardly in
uenced
by the vacuum eld. The force exerted by the vacuum charge density on the
17
particle compared to the external force is important only at longer wavelengths
of the external eld, when it varies relatively slowly. The upshot of this analysis is,
that the eective mass of the particle increases for small accelerations over longer
distances, whilst for large accelerations over short distances it is unaected by
the vacuum scalar eld.
This is precisely the kind of eect we are looking for. If a unied non-abelian
gauge eld theory of weak, electromagnetic and color forces is combined with the
idea of scalar elds with non-zero vacuum value, we can construct the following scenario. The unied eld quanta are gauge bosons carrying many dierent
combinations of unied charges, like the color combinations carried by gluons.
At distances shorter than the unication scale of about 10;30 cm all these gauge
bosons are massless. The lowest energy conguration of the scalar elds is one
in which some of them get a non-zero constant vacuum value, generating corresponding charge/anti-charge densities for those unied charges not felt by the
standard model gauge elds. Then the gauge bosons carrying the right charges
to interact with these scalar elds become massive by the Brout-Englert-Higgs
mechanism, when probed by external elds of long range compared to the unication scale. The typical range of these heavy gauge bosons will be of the order
of the unication scale, and their mass of the order of 1016 proton masses.
Other gauge bosons, which do not feel the in
uence of the vacuum scalar
elds, remain massless also over distances large compard to the unication scale.
The dierences in vacuum charge/anti-charge densities due to scalar elds, and
the resulting dierences in the mass of unied gauge bosons, then break the
symmetry between the unied charges and between the corresponding forces:
at the unication scale the interactions are all equally strong, the theory being
symmetric under the exchange of the unied charges at very short distance scales
but beyond the unication scale only the long-range gauge-elds survive. These
must ultimately correspond to gluons, photons and weak vector bosons, which
are still massless in this scenario.
Note that symmetry breaking, like renormalization, is the result of non-trivial
eects of elds in the vacuum. However, whereas renormalization only implies a
quantitative change in properties like the charge and mass of particles, symmetry
breaking implies a qualitative change: particles which were originally massless
now become heavy, whilst charges which were free, dynamical degrees of freedom
become unobservable at large scales. Thus symmetry breaking provides a mechanism for the phase transitions that were mentioned in our earlier discussion of
the role of the vacuum. Indeed, the energy density in the vacuum (zero-point
energy) is dierent in the phase with broken and non-broken, explicit symmetry.
It is clearly an important question how such a scenario for unication of the
standard model interactions could be empirically tested. The best test would
be to nd evidence for the existence of the heavy gauge bosons of the unied
gauge theory. These gauge bosons are too heavy to be produced directly in any
accelerator experiment, but they could be observed indirectly through their eects
18
on quarks and leptons. For example, in any unied theory there are heavy gauge
bosons which can transform a quark into a lepton. This eect makes the proton
an unstable particle, and one could try to detect its decay. Now the extreme
short range of the interaction makes the decay very slow the proton life-time is
much larger than the age of the universe. Nevertheless, if one takes a sample of
some 1029 protons (as are found in 100 tons of water) one might have a chance
of observing a few decays per year. Experiments to this eect are being carried
out at several places around the world.
Another more indirect test is to search for scalar elds which play a role in the
symmetry breaking. Some of these scalar elds are unobservable because they
are used to generate an eective mass for the gauge particles, or because their
quanta are very heavy as well. But some scalars, carrying weak interaction type
of charges, must remain light, almost massless on the scale of unication, like the
surviving long-range gauge bosons. Indeed this is an absolute requirement, as a
second stage of symmetry breaking is necessary at the much larger distance scales
of the standard model to generate an eective mass for the W - and Z -boson of
the weak interaction. Evidence for such relatively light scalar elds might show
up in accelerator experiments. This point is further elaborated below. First I
wish to mention two other aspects of the unied gauge eld theory.
One is, that for reasons also detailed below, all quarks and leptons need to
be massless at the scale of unication. Their mass should arise as a result of
the Brout-Englert-Higgs mechanism, as their charges interact with the constant
scalar background elds. Now as these masses are of the order of the Fermi
scale or less, the most economical way to generate quark and lepton masses is
by using the same scalar elds as generate the mass for the weak vector bosons.
Theoretically this is very well possible indeed.
Secondly the hierarchy of scales, represented by the Fermi scale of the standard
model as compared to the unication scale, is now the result of two successive
stages of symmetry breaking. The question how such a hierarchy of distance
scales comes out naturally from a unied theory thus becomes a question about
the ratio of the mass-scales generated by the vacuum values of the scalar elds:
those responsible for symmetry breaking between unied charges on the one hand,
and those for breaking the symmetry between the weak charges on the other
hand. As yet there is no convincing answer as to the why of this hierarchy. It
is however possible to give an explanation as to how such a hierarchy could be
dynamically stable, given that it is there. This involves the use of supersymmetry,
as mentioned earlier.
A supersymmetric theory is one in which there are as many bosons as fermions,
with exactly the same charges. In the ideal case the masses of corresponding
Bose- and Fermi-particles would be equal. However, this is completely unrealistic. Fortunately, like the symmetry between various charges is broken by masses
for certain gauge bosons, others remaining massless or very light, the symmetry
between fermions and bosons may be broken by dierent masses as well, without
19
spoiling the nice properties of supersymmetric theories. One such nice property
is, that the quantum uctuations of Fermi- and Bose-elds tend to oppose each
other. Therefore the eects of vacuum uctuations on running charges, dynamically generated masses, and so on, are considerably less in a supersymmetric
quantum eld theory than in the non-supersymmetric versions. In a supersymmetric version of the standard model the charges run slower and a large hierarchy
between the scales of the standard model and the unication scale is more easily established. A bonus of the supersymmetric unication scenario is also, that
it is much easier to construct theories in which true unication of the electromagnetic, weak and color forces is realized within the accuracies derived from
measurements over distance scales up to the Fermi-scale.
>From the experimental point of view it might be much easier to establish
supersymmetry then direct unication, because in the supersymmetric extension
of the standard model there are many new particles, fermions and bosons, which
should be detectable by experiments with high-energy accelerators in the foreseeable future.
The weak interactions revisited
The symmetry of our unied model of the gauge interactions is broken in two
steps. At a very small distance scale a very large mass is generated for gauge
bosons not corresponding to any of the known standard model forces. And at a
distance scale 1014 times larger a mass is generated for the W and Z -boson of
the weak interactions, which have masses of the order of the Fermi-scale, about
100 proton masses. This explains why at still larger scales, like the atomic scale
of 10;8 cm, this force is indeed very weak: the range of a massive particle like
the Z -boson is too short to be signicant over atomic distances.
Now the second step in the process is actually somewhat more complicated
than the rst one, when the unied gauge symmetry is broken. This is because
in the second step masses are generated for some gauge bosons in a model with
eectively three separate kinds of forces, rather than one. The Brout-EnglertHiggs mechanism itself need not be changed, but its implementation then is. The
crucial point is, that the charges arising in the unied model are not precisely
those we know from the standard model. In particular, electric charge is a linear
combination of two unied charges known as isospin and hypercharge. Therefore
the photon, as the quantum of the electro-magnetic eld, arises by a corresponding linear superposition of the gauge bosons of these two charges. The other
independent combination is the Z -boson, which becomes massive at standard
model scales due to a scalar eld carrying the particular combination of isospin
and hypercharge to which only the Z -bosonis sensitive. Note, that the photon
and the Z -boson are both electrically and color-neutral. Hence at short distance
scales where both are massless they can mix to give back the right combinations
of elds coupling to the unied charges.
20
Finally we turn to discuss the masses of quarks and leptons. It has been
mentioned that these particles are supposed to get their masses from the vacuum
value of the same scalar elds as the W - and Z -bosons. Indeed, there is a strong
theoretical prejudice, that at distance scales shorter than the weak interaction
scale these fermions must be massless. This prejudice is based on the chiral nature
of the charges of the weak interaction, meaning that particles with left-handed
spin have charges dierent from those with right-handed spin. Now the notion
of left- or right-handed spin can make sense only if one species an axis. The
axis can be taken as the line of motion of the particle | recall the discussion of
the circular polarisation of the electro-magnetic eld. But this axis has a welldened meaning, on which any arbitrary observer can agree, only if the particle
is massless. Indeed, a particle with nite non-zero mass can always be brought
to rest, in which case there is no axis whatsoever to refer to. Only massless
particles, which always move at the speed of light with respect to any observer,
have an intrinsic direction of motion and axis of spin to refer to. Therefore only
in the case of massless particles it is possible to assign dierent charges to leftand right-spinning ones in an unambiguous way.
To be somewhat more specic, in the standard model electric charge is actually
the sum of isospin and hypercharge. Now a left-handed electron is assigned isospin
;1=2 and hypercharge ;1=2, giving ;1 unit of electric charge. However, the
right-handed electron has no isospin, and hypercharge ;1. Therefore it has the
same electric charge as a left-handed electron. Then a massive electron, which is
sometimes left- and sometimes right-handed depending on the observer, can be
assigned an electric charge of ;1 unit, without specic isospin or hypercharge:
these latter charges are only well-dened individually at short ranges when all
quarks, leptons and weak vector bosons are massless.
For neutrinos it is even simpler. A left-handed neutrino is assigned isospin
1=2 and hypercharge ;1=2. Therefore it has no electric charge. A right-handed
neutrino, if it exists, should then have neither isospin nor hypercharge. As it also
has no color charge, such a particle does not interact with the rest of the world
through any standard model gauge eld. We do not know if such a neutrino
exists, but if it does the only particles with which it can interact are massive
scalars and the very heavy gauge bosons of the unied gauge force. To this one
may also add gravity, but in any case their interactions must be extremely weak,
to the point that they are unobservable in experiments today. For this reason
also we do not know if neutrinos have a mass. Experimental investigations of this
question are very subtle, though not completely hopeless. A discussion of this
topic is however beyond the scope of this exposition.
Beyond gauge unication
In my sketch of quantum gauge elds and their role in the physics of quarks
and leptons I have strongly emphasized a unied point of view, by discussing
21
the generation of mass as a clue to unication of the forces which we know to
act between them. This view is biased, because the masses of the fermions and
the weak vector bosons can be generated by the Brout-Englert-Higgs mechanism
really independent of whether there is unication at a much smaller scale of distances or not. It is nevertheless very suggestive that with the necessary scalar
bosons taken into account it becomes so easy to construct a consistent and quantitatively correct scheme for unication by studying the running of the various
gauge charges.
Yet, even if such a scheme would turn out to describe correctly the physics
of the fundamental particles of matter at these extremely small scales, this is
not the end of the story. We still need an account of the role of gravity in our
quantum eld theory models. This is a very intriguing and very hard problem,
aecting not only our understanding of matter and forces, but of space and time
as well. If supersymmetry would show up however, it would mean a giant step
towards solving this problem also, as it would point directly to a supersymmetric
theory of gravity known as supergravity. For mathematical reasons this theory is
an excellent starting point for a quantum theory of gravity. Even if it is not the
last word in this respect, and there may be more surprises like a quantum theory
of strings behind all these schemes, the discovery of supersymmetry is one of the
strongest pieces of evidence for unication of all interactions, including gravity,
one might ever hope to get.
This nally returns us to the question we posed in the beginning: is the
hierarchy of structures on dierent scales which we observe in the universe a
contingency or a necessity? The more we learn about quantum eld theory
and the way its pieces t together, the more it seems that it is very tightly
constrained and that there are really very few options for nature to be organised
dierently from the way it is. If unication works as I have sketched, there is
only one fundamental type of gauge force, with charges that appear in a highly
symmetric way. In addition, supersymmetry may relate bosons to fermions. Only
by some dynamical symmetry breaking do forces of dierent strength make their
appearance in physics at larger distance scales. Hierarchies of scales are generated
in the process. If nature actually follows this course, the Copernican view is to
prevail once more: our universe doesn't have to be special to be interesting.
22
References
An somewhat more mathematical but still elementary introduction to gauge
theories can be found in:
- J.C. Taylor, Gauge theories in particle physics in: The New Physics, ed.
P. Davies (Cambridge Univ. Press, 1989)
It also contains a contribution on unication of the gauge interactions:
- H. Georgi, Grand unied theories (id.)
A classic introduction to cosmology is:
- S. Weinberg, The rst three minutes (Deutsch, 1977)
An excellent introduction to the history of particle physics and quantum eld
theory is:
- A. Pais, Inward bound (Oxford Univ. Press, 1986).
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