THE STRUCTURE OF MATTER

 The Rise of Particle Physics      Yukawa's proposal    Quantum field theory      Interacting fields    QED      Strange Particles    Parity Violation       Fermi's theory of weak interactions    The Strong Interaction     QCD    The Electroweak Theory     The Higgs Boson    Conclusions       Glossary

The Rise of Particle Physics

The quest for the search of ultimate constituents of matter has fascinated the human mind since the recorded times of Democritus the Greek philosopher who suggested that it consists of indestructible particles called atoms.

Later studies on the nature of matter revealed that atoms consisted of protons and electrons. The planetary model of structure proposed by Ernest Rutherford in 1911 suggested that electrons moved about a central nucleus in much the same way as planets about the Sun. This though inadequate to account for the stability of the atom paved the way for a much refined model put forward by Neils Bohr based on Max Planck's quantum hypothesis.

Further refinements to Bohr's model were made as it was not agreeing with the atoms beyond hydrogen. The need for a successful theory for microscopic behaviour agreeing with other complicated atoms which consist of several particles led Erwin Schroedinger to propose wave mechanics. He took into account the Louis de

Broglie's hypothesis of wave-particle duality which states that matter exhibited wave-like characters and vice versa according to the equation = h/p where h is the Planck's constant and p is the momentum. This is the birth of a new mechanics built to suite the motion of microscopic particles which was termed quantum mechanics.

In quantum mechanics the Schroedinger equation H = E governs the motion of the microscopic particle where H is the Hamiltonian operator and and E are the wave function and the energy associated with the particle respectively, much the same way that the Newton's law F = ma governs the motion of objects in classical mechanics.

With the discovery of the neutron in 1932 by James Chadwick, the total number of particles rose to three together with the proton and the electron and the scientists thought for a while that they have found all the constituents of matter and that the study of physics be over in a few years. Needless to say, they were wrong, and a new and an exciting area of physics was awaiting to be born and named Particle Physics.

Modern particle physics can be thought of as starting with the advent of mesons. For these are not constituents of everyday matter, as are the protons and the electrons, but are those that first proposed to provide a description of nuclear forces. The subsequent discoveries of a bewildering variety of mesons heralded an unexpected richness in the structure of matter, which took scientists many decades to understand.

The core of particle physics, relativistic quantum field theory, is a synthesis of quantum mechanics and relativity. For this reason particle physicists find that a system of units in which h = c = 1 ( h = h/2 and c is the velocity of light.) is not only convenient but is a manifesto that quantum mechanics and relativity are the basic physical laws governing their area of physics. Taking the de Broglie relation: = h/p; Planck's formula: E = h; and Einstein's famous formula E=mc² ; it is seen that the above choice of units will make all energies, masses, and momenta have the same units (e.g., electron volts (eV)). These are the same as inverse lengths and times, larger energies and momenta inevitably corresponding to shorter times and lengths.

Yukawa's proposal

In attempting to describe the features of the strong nuclear force, which binds the protons and neutrons together thus giving stability to the nucleus, physicists in the 1930s had to satisfy two basic requirements. Firstly, as the force acts in the same way on both protons and neutrons, it must be independent of electric charges and, secondly, as the force is felt only within the atomic nucleus, it must be of short range.

In 1935 the physicist Hideki Yukawa suggested that the nuclear force between protons is mediated by a massive particle, now called the pi-meson or pion, denoted by . One proton feels the presence of another proton due to this messenger particle. This can be thought of as two kids throwing a ball at each other. Due to the exchange of the ball through throwing between them in turn, one feels the existence of the other person. Here, the ball is like the pion and the kids the protons. It is the mass of the mediating particle which ensures that the force it carries extends over only a finite range. This is indicated by Werner Heisenberg's uncertainty principle E t h/2. The Yukawa potential is given by e ^{-m r}/r where m is the mass of the pion and r is the distance.

From the -particle scattering experiments, we know that the effective range of the strong force is about 10^-15 m, which gives a pion, a mass about 300 times that of the electron. To account for all the possible interactions between nucleons, the pions must come in three charge states. For instance, the proton may transform into a neutron by the emission of a positively charged pion or, equivalently, by the absorption of a negatively charged pion. But the proton may also remain unchanged during a nuclear reaction, which can be explained only by the existence of an uncharged pion. The three pions are denoted as ^{+ 0 -} .

In 1937 Carl Anderson discovered a negative charge state with a mass some 200 times that of the electron. At first, the particle was thought to be Yukawa's pion but later studies revealed that it is not so and that this particle was fundamental. It was named the muon (denoted by ) and is now categorized under leptons. It is recorded that hearing the news of the muon physicist I.I.Rabi asked "who ordered that?". Yukawa's pion was finally found in 1947 by Cecil Powell, C. Lattes and G. Occhialini. The mass of the charged pions was determined to be 273 times the mass of the electron and the neutral pion to be 264 times that of the electron.

A few of the most often used terms in particle physics are:

nucleons: neutrons and protons;

hadrons: all particles affected by the strong nuclear force;

baryons: hadrons which are fermions (half integral spin particles) - e.g.: nucleons;

mesons: hadrons which are bosons (integral spin particles) - e.g.: pions;

leptons: all particles not affected by the strong nuclear force - e.g.: electron, muon.

Quantum field theory

In the most sophisticated form of quantum theory, all entities are described by fields. Just as the photon is most obviously a manifestation of the electromagnetic field, so is an electron taken to be a manifestation of an electron field and a proton of a proton field. Any one individual electron wave function may be thought of as a particular frequency excitation of the field and

A comparison of the four fundamental forces of nature

may be localised to a greater or lesser extent dependent on its interactions.

Interacting fields

The idea of quantum fields can be used to calculate the probabilities of the creation and destruction of their quanta in various reactions,

and to provide us with descriptions of the behaviour of the quanta between creation and destruction (the wave functions).

Hamilton's variational principle can be used to derive Newtonian mechanics, quantum mechanics and quantum field theory. The Lagrangian L for any system is the difference between its kinetic energy (KE) and its potential energy (PE).

L = KE - PE

Hamilton's principle states that the evolution of any system is such as to minimise L.

Thus if x and v are the position and the velocity of a ball respectively,

L(x,v) = 0 gives F= ma.

This entirely general principle of Hamilton can also be used in quantum mechanics. In the quantum version, however we do not deal with the total Lagrangian L directly, but with the Lagrangian density L. The total Lagrangian can then be found by integrating the Lagrangian density over all space.

QED

This is the abbreviation used for Quantum Electrodynamics. It is the relativistic quantum field theory describing the interactions of electrically charged particles via photons. The

discovery of the perturbation expansion revealed the existence of an infinite number of ever-

decreasing quantum corrections to any electromagnetic process. The renormalisabilitiy of QED found in 1949 by Richard Feynman, Julian Schwinger, Sini-itiro Tomonaga and Freeman Dyson, means that we can avoid apparently infinite contributions to the perturbation expansion by careful definition of the electron and photon. Therefore we can calculate the value of observable parameters of electromagnetic processes to any desired degree of accuracy, being limited only by the computational effort required to evaluate the many hundreds of Feynman diagrams which are generated within the first few orders of the perturbation expansion.

The theoretical predictions of QED agrees to within 1 part in billion with the experimental values (example: the g factor of the electron). This success makes QED the most precise picture we have of the physical world so far as the electromagnetic phenomena are concerned.

Strange Particles

In 1947 the physicists G.D. Rochester and C.C. Butler observed more new particles, about a thousand times more massive than the electron. As these particles are often associated with V-shaped tracks, they were first called V particles. Their origin and purpose were a total mystery. For the following six years, the V particles were observed in cosmic ray experiments and two kinds became apparent. There are those whose decay products always include a proton and are called hyperons, and there are those whose decay products consist only of mesons and are called K mesons or kaons.

The hyperons and kaons soon became known as the strange particles because of their anomalous behaviour. They were observed frequently enough to indicate production by the strong nuclear force, and so we would expect a decay time typical of a strong nuclear process (i.e. about 10^-23 s). But, from the length of their tracks it was possible to estimate their average lifetimes at about 10^-10 s, the time scale typical of weak interaction processes. This behaviour seemed to contradict the microscopic reversibility of reactions and required explanation.

Parity Violation

As mentioned in the last section the decay of the strange kaons led to a great deal of confusion in the early 1950s. Two decay modes in particular seemed so different that they were at one time thought to originate from two different parent particles, called the and mesons. The ⁺ decayed into three pion states while the ⁺ decayed into two pion states.

However, detailed study of the two and three pion final states indicated that the and were indeed both manifestations of the same charged kaon, K⁺. The decays were thought to be incompatible because the parities of the two final states are different. If they originate from the same initial particle, they imply that parity is not conserved by the force responsible for the decays. This means that the force behaves differently in left-handed and right-handed coordinate systems: it can distinguish left from right, or image from mirror image.

Such a revolutionary conclusion was not seriously entertained until 1956, when Tsun Dao Lee and Chen Ning Yang pointed out that, although evidence existed for the conservation of parity by the strong and electromagnetic forces, there was no evidence for its conservation by the weak force. Lee and Yang's prediction was confirmed by C.S. Wu and E. Ambler using the decay of ⁶⁰Co.

Fermi's theory of weak interactions

Prior to the early 1960s, just three different leptons were recognised: namely the electron, the muon and the neutrino (together with their anti-particles). The most common weak interactions available for study are the radioactive decay of nuclei and the decay of the pions and kaons and it was predominantly these reactions which formed the basis for the first description of the weak interactions formulated by Fermi in 1933.

The simplest manifestation of decay is the decay of a free neutron into a proton, an electron and an antineutrino.

n p + e +

Fermi took this to be the prototype for the weak interactions, which he described as four fermions (and now known as the Four Fermi Theory) reacting at a single point in space-time. He expressed this mathematically by saying that, at a single point in space-time, the quantum-mechanical wave function of the neutron is transformed into that of the proton and that the wave function of the incoming neutrino (equivalent to the outgoing antineutrino we actually see) is transformed into that of the electron. So a description of this reaction is provided by multiplying the wave functions by unknown factors which effect the transformation, and by another factor G_F called the Fermi coupling constant. This is the quantity which governs the intrinsic strength of the weak interactions, and so the rate of decay. Accordingly the amplitude for decay is given by

M = G_F (_{p n} )(_e )

The factors contain the essence of the weak interaction effects which give rise to the transformation of the particles. The challenge was to discover the nature of these quantities (whether they are scalars, vectors, tensors, etc.). By examining the angles of emission between the outgoing products of decay and their various energies, it is possible to narrow down the choice. This took many years: their nature was not confirmed until the parity-violating effect of the weak force was known.

In 1956 Richard Feynman and Murray Gell-Mann proposed that the interaction factors be a mixture of vector and axial-vector quantities, to account for the parity-violating effects of the interactions. Their theory is now known as the V-A theory which was also found by Robert Marshak and George Sudharshan independently.

A vector quantity has well-defined properties under a Lorentz transformation. For instance, it will change sign if rotated through 180 and will appear identical after rotation through 360. An axial-vector quantity will transform just like a vector under rotations, but will transform with the opposite sign to a vector under improper transformations such as parity. Thus, if the interaction comprises vector and axial-vector components, it will look different after a parity transformation (the components might add together instead of canceling), which is just what we need to describe the weak interactions. By inserting this form of interaction factor into the amplitude M for decay, it is possible to calculate the features of particle emission in free neutron decay.

The Strong Interaction

As mentioned earlier, experiments done during the 1950s revealed two very interesting facts about the strong force. First, the strong force does not distinguish between protons and neutrons. In more technical language this means that the proton and the neutron transform into each other under isospin rotations, and the Lagrangian of the strong interaction is invariant under these rotations. Second, the structure of protons and neutrons is as rich as that of nuclei. Furthermore, many new hadrons were discovered that were apparently just as "elementary" as protons and neutrons.

The table of "elementary particles" in the mid-1960s displayed much of the same complexity and symmetry as the periodic table of elements. In 1961 both Murray Gell-Mann and Yuval Ne'eman proposed that all hadrons could be classified in multiplets of the symmetry group called SU(3). The great triumph of this proposal was the prediction and subsequent discovery of a new hadron, the omega minus (^- ). This hadron was needed to fill a vacant space in one of the SU(3) multiplets.

In spite of the SU(3) classification scheme, the belief that all of these so-called elementary particles were truly elementary became more and more untenable.The most contradictory evidence was the finite size of hadrons (about 10^-13 cm), which drastically contrasted with the point-like nature of the leptons. The hadronic zoo was eventually tamed by postulating the existence of a small number of "truly elementary point-like particles" called quarks. In 1963 Gell-Mann and, independently George Zweig realised that all hadrons could be constructed from three spin 1/2 fermions, designated u, d, and s (up, down, and strange). The SU(3) symmetry that manifested itself in the table of "elementary particles" arose from an invariance of the Lagrangian of the strong interaction to rotations among these three objects. This global symmetry is exact only if the u, d, and s quarks have identical masses, which implies that the particle states populating a given SU(3) multiplet also have the same mass.

The origin of these quark masses is one of the great unanswered questions. It is established however, that SU(3) symmetry among the u, d, and s quarks is preserved by the strong interaction. Nowadays, one refers to this SU(3) as a flavour symmetry, with u, d, and s representing different quark flavours. This nomenclature is to distinguish it from another and quite different SU(3) symmetry possessed by quarks, a local symmetry that is associated directly with the strong force and has become known as the SU(3) of colour. The theory resulting from this symmetry is called quantum chromodynamics (QCD).

QCD

The fundamental structure of QCD mimics that of QED that it too is a gauge theory. The role of electric charge is played by three "colours" with which each quark is endowed - red, green, and blue. The three colour varieties of each quark form a triplet under the SU(3) local gauge symmetry. A local phase transformation of the quark field is now considerably extended since it can rotate the colour and thereby change a red quark into a blue one.

There are eight independent symmetry transformations that change the colour of a quark and these must be compensated for by the introduction of eight gauge fields, or spin 1 bosons. The eight gauge bosons of QCD are referred to as "gluon", since they represent the glue that holds the physical hadrons, such as the proton, together. The gluons themselves carry colour and can interact with themselves. This, in effect, weaken the force of the colour charge at short distances. The opposite effect occurs in QED.

The weakening of colour charge at short distances goes by the name of asymptotic freedom. Asymptotic freedom was first observed in deep inelastic scattering experiments. Deep inelastic scattering is based on bombarding particles on protons and neutrons so as to investigate their inner structure. This is like firing a canon into a room which you cannot enter. But by the destruction of the room and seeing what comes out of it after the explosion, you can have an idea of what existed inside it.

Asymptotic freedom explains why hadrons at high energies behave as if they were made of almost free quarks even though one knows that quarks must be tightly bound together since they have never been experimentally observed in their free state. This is known as the quark confinement.

The self-interaction of the gluons also explains the apparently permanent confinement of quarks. At long distances it leads to such proliferation of virtual gluons that the colour charge effectively grows without limit, forbidding the propagation of all coloured particles. Only bleached, or colour-neutral states (e.g. baryons which have equal proportions of red, green and blue, or mesons which have equal proportions of red-antired, green-antigreen, and blue-antiblue) are immune from this confinement. Thus all observable hadrons are necessarily colourless, whereas quarks and gluons are permanently confined.

The Electroweak Theory

In 1967 and 1968 respectively, Steven Weinberg and Abdus Salam independently formulated a unified theory for the weak and electromagnetic interactions, based in part on the work developed previously by Sheldon Glashow. It incorporates the successful theory of QED and provides a description of the weak force in terms of the exchange of massive vector bosons W and Z. It has moreover introduced a weak neutral current in a natural fashion, and the discovery of the neutral-current reactions in 1973 was a great boost to the acceptability of the model. The model's cleverest feature, however is the way it ensures the masslesness of the photon, whilst giving mass to the weak interaction gauge bosons W and Z. This is achieved by the use of the Higgs mechanism and a suitable choice of Higgs fields. In addition, the theory predicts the existence of a spinless Higgs partcle , which has yet to be discovered. The production of W and Z bosons at CERN (European Center for Nuclear Research) in 1983 with precisely the predicted masses was a great triumph.

As was mentioned previously although the messenger particle of QED is massless (i.e. the photon) the mediators in the electroweak theory W and Z bosons are massive. The solution to this paradox lies in the curious way in which the SU(2) X U(1) symmetry is broken and is known as the spontaneous symmetry breaking. (In the above statement SU(2) and U(1) stands for symmetry groups associated with weak and electromagnetic fields respectively.).

Spontaneous symmetry breaking simply refers to the asymmetric solutions to symmetric equations. Consider , for example, an ordinary magnet. Its magnetic field clearly defines a preferred direction in space, but the equations governing the motions of the individual atoms in the magnet are entirely rotationally symmetric. How has this come about? The answer lies in the fact that the symmetric state is not the state of minimum energy, and that in the process of evolving towards the ground state, the intrinsic symmetry of the system has been broken.

The three families of quarks and leptons. The fundamental constituents of all matter as we understand it today consists of quarks and leptons. Combinations of quarks result in protons and neutrons. For example a proton is made up of two up quarks and a down quark. (All particles below are experimentally observed.).

u - up, d-down, c-charm, s-strange, t-top, b-bottom

The Higgs Boson

In the simplest version of the spontaneously broken electroweak model, the Higgs boson is a

complex SU(2) doublet consisting of four real fields. These four fields are needed to transform massless gauge fields into massive ones. A massless gauge boson such as the photon has only two orthogonal spin components, whereas a massive gauge boson has three. In the electroweak theory the W and Z absorb three of the four real Higgs fields and in so doing become massive. The remaining neutral Higgs field is not used up in this transformation and therefore should be observable as a particle in its own right. Unfortunately, its mass is not fixed by

the theory. However it can decay into quarks and leptons with a definite signature. It is certainly a

necessary component of the theory and is presently being looked for in high-energy experiments. Its absence is a crucial missing link in the confirmation of the standard model. (The standard model is the model of particles obtained by the unification of the strong, weak and the electromagnetic forces.).

Conclusions

Although many mysteries remain, the standard model represents an intriguing and compelling theoretical framework for our present-day knowledge of the elementary particles. Its great virtue is that all of the known forces can be described as local gauge theories in which the interactions are generated from the single unifying principle of local gauge invariance. The essence of the standard model is to put the physics of the apparently separate strong, weak, and electromagnetic interactions in the single language of local gauge field theories, much as James Clerk Maxwell put the apparently separate physics of Coulomb's, Ampere's, and Faraday's laws into the single language of classical field theory.

Just as the "undetermined parameters" ₀ and ₀ were related to the velocity of light through Maxwell's unification of electricity and magnetism, so the undetermined parameters of the standard model (eg. quark and lepton masses) might be fixed by embedding the standard model in some grand unified theory. One proposal to achieve this goal is the supersymmetry where each boson will be paired with its fermion partner.

Although hints of a solution have emerged, it is fair to say that we are still a long way from formulating an ultimate synthesis of all physical laws. One reason for this is that the role of gravitation still remains mysterious. This the weakest of all forces, whose effects are so dramatic in the macroscopic world, may well hold the key to a truly deep understanding of the physical world. Furthermore the role and importance of the plethora of particles observed and recorded in particle accelerators remain unanswered.

I have so far presented a brief sketch of some of the ideas of particle physics which is our door way to a fuller understanding of the structure of matter. The field of particle physics is becoming richer and even more complicated every day. Whatever the challenges the field of particle physics poses for the human mind, it is an enterprise worth the trouble, for then only, as Stephen Hawking says, we will be truly knowing the mind of God. n

Glossary

Wave function - The mathematical object in quantum theory which determines probability of different results of experiment. It is a complex quantity, so it has an amplitude (whose square of the modulus gives the probability) and a phase angle.

Yukawa potential - A potential used to explain the forces between nucleons. In the nucleus, stronger short-range forces are acting between the nucleons and Yukawa assumed that their potential energy varied according to e ^{-m r}/r rather than 1/r. The interaction is assumed to be caused by the virtual production of a pion that is exchanged between the nucleons.

Perturbation theory - Mathematical approximation in which a small disturbance added to an exactly soluble system is analysed by a series expansion in powers of the small disturbance.

Renormalization - Strictly, the rescaling of some parameter in a field theory. In practice nearly all renormalizations involve an infinite rescaling.

Feynman diagrams - Pictorial representations of mathematical expressions for the quantum field theoretical predictions of the scattering of elementary particles. Broadly speaking, the lines in a diagram resemble the path of a particle and the vertices correspond to particle interactions that are localized at a space-time point.

Hyperons - The collective name given to long lived baryons other than the nucleons.

Parity - The operation for studying a system or sequence of events reflected in a mirror.

Antiparticle - For every particle there exist an antiparticle. It has the same properties as that of the particle (e.g. same mass and charge) but differs in the sign of the charge. The positron is the antiparticle of the electron. It has the exact properties as that of the electron but has a positive electric charge the value being identical to that of the electron.

Tensors - An abstract mathematical entity, an operator, by which one vector can be converted into another vector by a linear modification of components. Tensor analysis is a generalization of vector analysis. A tensor of rank zero is a scalar and a tensor of rank one is a vector.

SU(N) - A mathematical structure known as a 'group' that describes operations on N objects. e.g. SU(2) applies to the two quarks or two leptons in a generation and SU(3) applies to the three colours of quarks.