Group Theory and Elementary Particles

Type of physical science: Elementary particle (high-energy) physics

Field of study: Systematics (particle physics)

A branch of mathematical theory developed in the nineteenth century, known as group theory, found application in the 1960's as an explanation of the symmetries of the fundamental particles composing atoms.

Overview

Group theory is a branch of mathematics which deals with matrices that have one or more components, called dimensions. These matrices are designated as "special unitary" or, in symbol form, as U, SU, or SU, depending on the number of dimensional elements present. A group is a set of operations or transformations that fulfill specified conditions. For example, the rotation of an object in three-dimensional space defines a group, and a combination of rotations may be performed in one operation. One rotation of a cube on an axis followed by a second rotation on a different axis may be considered the same as a third rotation.

If the operation defined by members of the group can be initiated in any order with an identical final result, then the group is called abelian after the Norwegian mathematician Niels Henrik Abel (1802-1829). Rotating a cube about a single axis is abelian because this operation produces the same result regardless of sequence. On the other hand, rotating the cube about three axes sequentially will produce a different result if the sequence is the vertical axis followed by the horizontal one. The final result clearly depends on which operation is performed first.

Therefore, the latter set of operations was nonabelian in nature.

A new model for classifying hadrons was introduced in 1961 by Murray Gell-Mann from the California Institute of Technology and Yuval Ne'eman of the Imperial College in London. The model was based on the assumption that hadrons were not the fundamental building blocks of matter but were, in turn, divided into families or subunits called supermultiplets. The model was named "the eightfold way" and was based on the mathematical properties of the SU group. With the development of new high-energy accelerators, physicists watched the emergence of many new particles. With existing theory, however, they were unable to explain how these particles were related.

With the eightfold model, hadrons are grouped into multiplets, with each particle in the multiplet represented by a point in a coordinate system of quantum numbers that are conserved in certain reactions. The eightfold way united various multiplets -- those of the same baryon number, spin, and parity (a property similar to but not identical to a reflection in a mirror)--into larger sets called supermultiplets. An eightfold-way graph consists of two axes, with the horizontal axis as a plot of the electrical charge and the vertical axis as a plot of the strangeness of the hadron. Strangeness is a property possessed by matter containing a strange quark, which has a fractional charge of -1/3 and forms a partnership with the charmed quark.

The technique that Gell-Mann and Ne'eman used is illustrated as a plot for mesons which includes pions and K particles. The K+ has a positive charge and a positive strangeness and is located at the top right of the diagram. The line representing the positive charge crosses the line for zero strangeness on the far right of the figure. This location for a particle with no strangeness and positive charge is the pion π+. At each point where the lines intersect, a particle is found, with the overall pattern being a hexagon with a pion π⁰ located in the center. The location of the particles on the hexagon correctly describes these two quantum numbers. When baryons are plotted in the same fashion but with one unit added to the strangeness axis, a similar hexagonal structure emerges. The difference is that two particles are located in the center instead of the one particle for mesons.

The pattern similarity in these two diagrams is quite remarkable, but to make them identical would require an eighth meson with no charge or strangeness located in the center spot along with the pion π⁰. This similar pattern of eights for both the baryons and the mesons implies an underlying cause. It was named the "eightfold way" by Gell-Mann after the "noble eightfold way of Buddhism," which refers to compliance with the eight commandments of the Buddhist religion.

Eightfold-way symmetry permitted physicists to define a new quantum number, unitary spin, which was conserved in strong interactions. The strong interaction is the strongest known physical force, and it involves the exchange between quarks of tiny massless particles called gluons that hold the nucleus together.

In 1964, Gell-Mann and George Zweig independently discovered that eightfold-way symmetry would occur naturally if all the elementary particles were composed of only three varieties of quarks. Two of these quarks, called "up" and "down" (u and d), would be adequate to build all the hadrons that possess zero strangeness. Strange hadrons are composed of a third variety of quark known as the strange quark (s). Quarks are quite odd, as they have electrical charges that are fractional amounts of the proton's charge. The up quark has a charge of +2/3, but the down quark has a charge of -1/3. Combining two up quarks with a down quark, one arrives at the same charge as the proton (2/3 + 2/3 - 1/3 = 1). The neutron, having no charge, would consist of two down quarks and an up quark: -1/3 + -1/3 + 2/3 = 0.

Strange quarks also possess a property termed "strangeness"; the more strange quarks present, the higher the degree of strangeness. The neutron and proton contain no strange quarks and have zero strangeness, but the strange quark has 150 million electronvolts more energy than either the up or down quarks. Therefore, clusters of three quarks have more energy if they have higher strangeness. For example, the ω minus particle, with three strange quarks, is 450 million electronvolts larger than the δ particle and has zero strangeness.

The up, down, and strange characteristics are collectively called the quark "flavors."

Every flavor of quark corresponds to an antiquark having the same mass and spin as its counterpart but having an opposite charge and strangeness. The strange antiquark has a strangeness of +1 and a charge of +1/3. The arrangement of these antiquarks in clusters gives rise to antibaryon particles; that is, the up, down, and strange antiquarks (u, d, s) have the opposite charge and strangeness in contrast with the up, down and strange quarks (u, d, s).

It is also known that quarks spin at the same rate as the electron or a half-integral value (1/2). As an odd number of halves always gives half-integers, a three-quark system will have a half-integer spin, exactly what is observed for baryons. Conversely, because an even number of halves adds to a full integer, quarks plus antiquarks equal the integer spins that are observed for mesons. For example, quarks and antiquarks with spins of +1/2 or -1/2 combine to either 0 or 1.

Nine mesons that have a spin of 0 are known, and nine spin 1 mesons also exist. If three quarks combine, then their total spin will be either 1/2 or 1-1/2, depending on the number of quarks of spin 1/2 that combine with quarks of spin -1/2. For example, protons each have a spin of 1/2, but other hadrons (such as the δ, σ plus, and ω minus particles) each have spin of 1-1/2.

By combining both quarks and antiquarks in pairs using the three quark flavors and three antiquark flavors, nine possible combinations result: uu, dd, ss, ud, us, du, ds, su, and sd.

The strangeness and electrical charge of each of these combinations is achieved by adding each contribution of the pair. The obtained results match the combinations of strangeness and charge that mesons have. For example, while a strangeness of -2 is found only in baryons, both baryons and mesons exist for a strangeness of 0 and a charge of +1.

A quark model may be obtained in a similar manner as for mesons and baryons by plotting charge on the horizontal axis and strangeness on the vertical axis. The hexagonal pattern for mesons results if antiquark triangles are drawn centered on the vertex of a quark triangle using six of the nine quark-antiquark possibilities.

Applications

When the eightfold-way model was first proposed, not all of the particles described by the SU diagrams for baryons and mesons were discovered. In fact, not one of the supermultiplets in either of the models was known completely. Particles that were yet to be discovered, however, neatly filled in the gaps predicted by the model and contributed to its success.

The eta particle had not been discovered, but the eightfold way not only predicted its existence but also that its energy should be 570 million electronvolts; its actual energy of 550 million electronvolts was very close to the prediction. An important test for the eightfold way was whether it had estimated the correct value of the spin of xi particles. The model indicated a spin of 1/2, similar to other baryons in the octet, and in 1963, the actual spin was measured and found to be 1/2.

Gell-Mann and Ne'eman extended their theoretical models to include a group of ten (decuplet) which they suspected might exist. A group of ten is obtained by extending the hexagon at the bottom corner, forming an inverted triangle. A particle which would fit into this lowermost position of ten members would have a strangeness of -3 and a charge of -1.

Gell-Mann named the particle the ω minus. The minus refers to its negative electrical charge, and ω, the last letter of the Greek alphabet, corresponds to the last particle in the group of ten.

Gell-Mann also correctly predicted the ω minus particle's mass. From a decuplet diagram, it is apparent that the δ particles have zero strangeness and an energy of 1.235 billion electronvolts. Sigma plus particles have a strangeness of -1 and an energy of 1.385 billion electronvolts. As one moves down the table, it is evident that the strangeness decreases by whole negative integers while the energy increases by 150 million electronvolts. On this basis, Gell-Mann predicted that the missing particle in the apex would have a strangeness of -3, a charge of -1, and an energy of 1.68 billion electronvolts.

The predicted ω plus particle was found in 1963 at the Brookhaven National Laboratory in New York and later at Centre Europeen de Recherche Nucleaire (CERN) in Geneva, Switzerland. Brookhaven had a new synchrotron in operation which was capable of accelerating protons to the energy necessary to create the heavy particle. Protons at 3 billion electronvolts smashed into a target and produced 5 billion-electronvolt kaons. The kaons were directed into a bubble chamber that was 3 meters in diameter (the world's largest at the time) and had been filled with liquid hydrogen. Photographs were taken of the tracks left by the collisions of the kaons with protons in the chamber. Of the one hundred thousand photographs taken, one clearly showed the presence of the new particle. Measurements showed that all the quantum numbers fit the theoretical model: Its strangeness was -3, its charge was -1, and its energy was 1.679 billion electronvolts.

In spite of its dramatic successes, the eightfold way was not a complete theory.

Although it was able to predict missing particles in a supermultiplet, it was not able to predict whole new supermultiplets. In addition, although it allowed for the prediction of hadron masses, it could not explain why hadrons had those masses.

Context

The models used in the description and prediction of elementary particles had their origin in a branch of mathematics known as group theory. This field dealing with matrices was developed in the nineteenth century by the Norwegian mathematician Sophus Lie (1842-1899).

The group theory that was used previously in describing the symmetry of crystal structures found important applications in the 1960's in elementary particle physics.

The first of the atomic forces to be described accurately by group theory was electromagnetism. Electromagnetism is described by a one-by-one matrix, known as U, which refers to a set of transformations that can be carried out on a single object. The U symmetry indicates that the electromagnetic force cannot change a particle's identity. A photon inside of the matrix can transform an electron only into another electron; it can interact with just one type of particle at a time.

The weak force and electromagnetism are described jointly by a theory with a symmetry represented by the product of the two groups: SU x U. SU refers to a special unitary with two rows and columns in the matrix element and represents all the possible transformations of two objects with the weak force. The objects are members of a "weak doublet" consisting of an electron and its corresponding neutrino of the same symmetry. The transformation elements of the matrix are three bosons: W⁺, with a weak charge and an electric charge of +1; W^-, with a weak charge and an electric charge of -1; and W⁰, which is neutral with respect to weak and electromagnetic charges. The boson particle mediates transformations between the members of each doublet. By emitting a W^- boson, the left-handed electron can convert into a left-handed neutrino. In the process, the electrical charge converts from -1 to 0, while the weak charge changes from -1/2 to 1/2.

The product SU x U only partially unifies the physical world because it includes only two forces. At nuclear distances that are greater than interatomic dimensions, the theory begins to break down.

The search for larger symmetries of elementary particles received a breakthrough in 1961 with the announcement of the eightfold-way model, which was in the form of a three-by-three element matrix of type SU. This model was successful in predicting the existence of several particles, including the ω minus particle. It was based on the existence of the up, down, and strange quarks that later helped to establish the underlying structure of the model. Yet, five quarks are now known to exist, with a sixth quark (the top quark) being a likely candidate for discovery. Incorporating these additional quarks into a symmetry model requires the extension of additional matrix elements.

In order to integrate the known elementary particles of leptons and quarks, higher-order symmetries, including SU, have been attempted. SU symmetry would encompass all the possible transitions between an integrated, five-component family. It also would have the advantage of incorporating the symmetries of the SU and SU matrices within it. An interesting prediction of the SU theory which has not yet been observed is proton decay.

Principal terms

BARYONS: particles that are composed of three quarks, such as neutrons and protons

BOSONS: particles that have integral spins (such as 0, 1, 2, 3), including kaons (k-mesons), pions, and photons

HADRONS: particles that are made of quarks, including baryons and mesons, that combine through the strong interaction

LEPTONS: particles that are not affected by the strong interaction, such as the electron, muon, tau, and their neutrinos

NEUTRINO: a tiny particle of the lepton family that has no detectable mass

QUANTUM NUMBER: a set of such particular aspects of a particle as electric charge, spin, and orbital angular momentum

QUARKS: elementary particles that have charges in increments of one-third (such as 1/3 or 2/3) and are the building blocks of hadrons

Bibliography

Close, Frank. THE COSMIC ONION: QUARKS AND THE NATURE OF THE UNIVERSE. New York: American Institute of Physics, 1986. Stimulating and exciting reading for both the lay reader and the scientist. Clear explanations include cartoons, diagrams, and photographs. Eightfold-way diagrams are developed sequentially, aiding comprehension.

Close, Frank, Michael Marten, and Christine Sutton. THE PARTICLE EXPLOSION. New York: Oxford University Press, 1987. A spectacular story of the accomplishments of the leading particle physicists and how their work has changed human understanding of the world. Three hundred photographs of the personalities, machines, and image tracks of the particles are provided, including the dramatic photograph of the discovery of the ω minus particle predicted by Murray Gell-Mann.

Eisberg, Robert, and Robert Resnick. QUANTUM PHYSICS OF ATOMS, MOLECULES, SOLIDS, NUCLEI, AND PARTICLES. New York: John Wiley & Sons, 1985. A more in-depth discussion of the symmetry theories for the advanced physics or chemistry student is found in chapter 18. A section on the color interactions of quarks (quantum chromodynamics) follows. Contains excellent appendices and includes a chapter on crystallography.

Georgi, Howard. "A Unified Theory of Elementary Particles and Forces." SCIENTIFIC AMERICAN 244 (April, 1981): 48-63. Georgi discusses the search for larger group symmetries that may account for all the attributes of the known or postulated elementary particles.

Halliday, David, and Robert Resnick. FUNDAMENTALS OF PHYSICS. Rev. ed. New York: John Wiley & Sons, 1988. The chapter entitled "Quarks, Leptons, and the Big Bang" provides short but informative discussions of the eightfold-way method and the quark model. Tables and illustrations of both models are included.

Ne'eman, Yuval, and Yoram Kirsh. THE PARTICLE HUNTERS. New York: Cambridge University Press, 1986. A well-known particle physicist and a physicist who engages in science writing combine their efforts to produce a nontechnical summary of the search for the fundamental particles of matter. Interesting historical anecdotes are highlighted on the margins of each page, including Ne'eman's contribution to the eightfold-way theory.

Pais, Abraham. INWARD BOUND. New York: Oxford University Press, 1986. A history of the physics of matter and forces written for the student by a noted physicist. The chronology covers more than one hundred years and gives the reader insights into the thoughts of the scientists involved and the events. The appendix contains a synopsis chronology.

Quigg, Chris. "Elementary Particles and Forces." SCIENTIFIC AMERICAN 252 (April, 1985): 84-95. A summary of the knowledge of high-energy physics, including quarks and their color attributes. Gluons and bosons (including the postulated Higgs boson) are also discussed. Speculation is given on the unification of the various theories and how these basic questions will be studied with the superconducting supercollider accelerator.

Rebbi, Claudio. "The Lattice Theory of Quark Confinement." SCIENTIFIC AMERICAN 248 (February, 1983): 54-65. The confinement of the quark in particles such as the proton is presented as a lattice on the structure of space and time, showing why free quarks are not observed. Excellent diagrams illustrate difficult concepts.

Trefil, James S. FROM ATOMS TO QUARKS. New York: Charles Scribner's Sons, 1980. A description of particle physics for the general reader by a noted physics professor. The groups of eight (octets) are diagrammed in a simplified way. A catalog of both stable and unstable particles and a glossary of terms are included in the appendix.

Eightfold-way meson symmetry chart

Quark-antiquark symmetry triangles

Grand Unification Theories and Supersymmetry

Leptons and the Weak Interaction

Quarks and the Strong Interaction

The Unification of the Weak and Electromagnetic Interactions