Invitation to a Contemporary Physics (2004)

8.7 Further Reading

Books

•Charles Misner, Kip Thorne and John Wheeler, Gravitation (W. H. Freeman, San Francisco, 1973).

•Stuart Shapiro and Saul Teukolsky, Black Holes, White Dwarfs and Neutron Stars (John Wiley and Sons, New York, 1983).

•Edwin Taylor and John Wheeler, Exploring Black Holes: Introduction to General Relativity (Addison Wesley Longman, San Francisco, 2000).

•Stephen Hawking, A Brief History of Time (Bantam Books, New York, 1988).

•K.S. Thorne, Black Holes and Time Warps: Einstein’s Outrageous Legacy

(Norton, New York, 1994).

The sun and other normal stars

•John Bahcall, M.H. Pinsonneault and S. Basu, Solar Models: Current Epoch and Time Dependences, Neutrinos and Helioseismological Properties, Astrophys. J. 555 (2001) 990–1012.

•John Bahcall, Frank Calaprice, Arthur MacDonald and Yoji Totsuka, Solar Neutrino Experiments: The Next Generation, Physics Today (July 1996), pp. 30–36.

•Hans Bethe, Supernovae, Physics Today (September 1990), pp. 24–27.

•Virginia Tremble, 1987A: The Greatest Supernova since Kepler, Rev. Mod. Phys. 60 (1988) 859–871.

White Dwarfs, Neutron Stars and Black Holes

•J.L. Provencal, H. Shipman, E. Hog and P. Thejll, Testing the White Dwarf Mass–Radius Relation with Hipparcos, Astrophys. J. 494 (1998) 759–762.

•Lars Bilden and Tod Strohmayer, New View of Neutron Stars, Physics Today (February 1999), pp. 40–46.

•Roger Blandford and Neil Gehrels, Revisiting the Black Holes, Physics Today (June 1999), 40–45.

•F. Melia and H. Falcke, The Supermassive Black Hole at the Galactic Center, Ann. Rev. Astron. Astrophys. 39 (2001) 309–352.

Web-sites

•http://www.stsci.edu/resources/ (Space Telescope Science Inst.).

•http://imagine.gsfc.nasa.gov/ (Imagine the Universe).

•http://heasarc.gsfc.nasa.gov/ (Astrophysics Archive Resource Center).

•http://math.ucr.edu/home/baez/relativity.html (Links to selected websites on relativity).

is the energy gain?

8.18(a) Calculate the thermal entropy of the sun, assuming it consists entirely of ionized hydrogen. Suggestion: use S = E/T , where E is the total thermal energy. (b) Calculate the quantum-mechanical entropy of a Schwarzschild black hole of one solar mass.

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A chemical element may have several isotopes. They have identical chemical properties, the same number of electrons in each atom, but they di er from one another in the mass of their nuclei. This simply suggests that it is the electrons of an atom, and not the nuclei, that are responsible for its chemical properties. But why should there be di erent nuclei for the same element? The mystery was finally solved when James Chadwick discovered the neutron in 1932. This is an electrically neutral particle that resides in the nucleus. The other particle in the nucleus is the proton which carries one unit of positive charge and is nothing but the nucleus of the hydrogen atom. All nuclei are made up of protons and neutrons, and these two particles are collectively referred to as nucleons.

The charge of the electron is equal and opposite to that of the proton, so it carries one unit of negative charge. Now, a given chemical element has a definite number of electrons. Since atoms are electrically neutral, there must be an equal number of protons as electrons in each atom. However, this argument does not limit the number of neutrons in the nucleus because neutrons are electrically neutral. Atoms which di er only by the number of neutrons in their nuclei are isotopes of one another. This is why a chemical element may have several isotopes.

From this reasoning one might conclude that one could obtain an infinite number of isotopes just by adding more and more neutrons. This turns out to be false because a nucleus with too many neutrons becomes unstable by undergoing beta decay. We will come back later to discuss what that is.

Now the laurel of elementarity is passed on to the electrons, protons, and neutrons.

Since the 1930’s many other new particles have been discovered in cosmic rays, or produced and detected in experiments using high energy accelerators. (A complete listing of them can be found on the Particle Data Group website http://pdg.lbl.gov/.) Most of these are hadrons, a name reserved for particles which respond to nuclear forces. Protons and neutrons are definitely hadrons, for what holds them together in a nucleus is by definition the nuclear force. In contrast, electrons are not hadrons; what keeps them held to the nucleus in an atom are the electric and magnetic forces.

With the presence of tens of hadrons, the stage is set for another shakedown. There were many hadrons and they all seemed to be unrelated. It was just like the old days with many unrelated chemical elements before the discovery by Mendeleyev. For that reason the rush was on to find a regularity, or if you like, some sort of a periodic table for them. Murray Gell-Mann and Yuval Ne’eman finally succeeded in the early 1960’s, and the regularity goes under the name ‘Eightfold Way,’ or more technically, an SU(3) symmetry. Just like Mendeleyev’s periodic table, which opened the way to the discovery of electrons, protons, and neutrons, and the inner structure of an atom, the Eightfold Way led to the discovery of an inner structure of the hadrons. In 1964, Murray Gell-Mann and George Zweig proposed that protons and neutrons, and all the other hadrons, were themselves made out of

three fundamental particles called quarks. The electron, however, is not a hadron and is still believed to be elementary.

All of this, so to speak, is ancient history. Nevertheless, it serves to illustrate how discoveries are made through an intimate interplay between theory and experiment. Puzzles lead to new ideas, and regularities lead to new structures. This is the same in all fields. What distinguishes this field of elementary particles from others, is the small size of the elementary constituents. As the substructures become smaller and smaller, we need a larger and larger ‘microscope’ to ‘see’ them. The only di erence is that these large ‘microscopes’ are called accelerators, and they occupy miles and miles of land. An optical microscope yields to an electron microscope when higher resolution is called for, because the beam of electrons in the latter has a smaller wavelength than the beam of light in the former, and better resolution can only be achieved by an illuminating beam with a smaller wavelength.

It is a fundamental fact of the quantum nature of matter that all particles in the microscopic world behave like waves. Moreover, the wavelength of a particle gets shorter as its energy gets higher. This is where an accelerator comes in: to produce a higher energy beam with enough resolution to probe the tiny world of elementary particles.

One never sees a cell directly from a beam of light without the complicated array of lenses in a microscope. Similarly, one cannot see what is illuminated by the beam emerging from an accelerator without a complicated bank of detectors and computers. What is di erent between the two, however, is the destructive nature of the latter. One can illuminate a sample under an ordinary microscope without destroying it, but it is virtually impossible to have a very energetic beam impinging on a target particle without shattering it to pieces. This complication adds substantially to the di culty of reconstructing the original image before it broke apart, but the other side of the coin is that such pulverization and disassembly allows us to look into the guts of the target particle. With the help of theory, the reconstruction can be achieved.

In what follows, we will adopt a pedagogical approach. We will skip all these reconstructions, and hence most of the direct experimental data, to concentrate on the physics that finally emerges. We will make free use of the license of hindsight to discuss the beauty and the intricacies of the physics in as logical a manner as possible without mathematics. Let us be forewarned however that this approach also contains an inherent danger of misconception that physics is purely a logical and theoretical science. It is definitely not; experiments are needed to confirm a conjectural theory and to guide our thinking in future developments. The final judge as to whether something is correct or not is experiment.

Accordingly, we will now skip over the intervening history in the second half of the 20th century, and jump directly to a summary of the elementary constituents of matter as we know them today.

These constituents fall into two categories: quarks and leptons. They are distinct because quarks experience nuclear forces but leptons do not.

332 Elementary Particles and Forces

There are three generations (sometimes known as families) each of quarks and leptons. Constituents in di erent generations are almost clones of each other, except that those in higher generations have larger masses. Constituents of the first generation make up the matter we are familiar with. Constituents of the second and third generations are unstable and cannot be found naturally on earth, but they can be produced by high energy collisions of ordinary matter.

Recent experiments show that there are probably no more than three generations of matter. But why three? Why do we even have a second and a third generation anyway? This nobody really knows. When the muon, a second generation lepton, was discovered in 1936, it was so astonishing, and so uncalled for, that it prompted I. I. Rabi to utter his famous remark: ‘Who ordered that?’ Almost seventy years later, we still do not know the answer.

We will now proceed to give a more detailed description of these constituent particles.

There are two quarks and two leptons in each generation. The quarks of the first generation are called the up quark (u) and the down quark (d), and the leptons of the first generation are the electron (e = e−) and the electron-neutrino (νe). Symbols in the parentheses are used to designate the particles; a superscript, when it is present, indicates the electric charge carried by that particle.

The quarks of the second generation are the charmed quark (c) and the strange quark (s), and the leptons are the muon (µ−) and the muon-neutrino (νµ). Finally, the quarks of the third generation are the top quark (t) and the bottom quark (b) (sometimes also known, respectively, as the truth and the beauty quarks), and the leptons are the tau (τ−) and the tau-neutrino (ντ ). All these particles have been experimentally observed.

If the electric charge carried by an electron e is counted as −1, then the electric charges of µ and τ are also −1, but those for the neutrinos νe, νµ, and ντ are zero (hence the name neutrino, which means ‘little neutral one’ in Italian). The electric charges of u, c, and t are +32 , and the electric charges of d, s, and b are −31 .

The proton (p), which has a unit (+1) of electric charge, is made up of two up quarks and a down quark. The neutron (n), which is electrically neutral, is composed of two down quarks and an up quark: p = uud, n = udd. Other hadrons are made up of three quarks, called baryons, or a quark and an antiquark, called mesons (see the following for a discussion of antiquarks).

There is an asymmetry between quarks and leptons in that leptons have integral units of electric charges while the electric charges of quarks are multiples of 13 . This factor of 3 is actually accompanied and ‘compensated’ by another factor of 3: each quark comes in three colors. We will explain more fully what ‘color’ is in Secs. 9.4.3 and 9.5.4, but whatever it is, it has nothing to do with the color we see with our own eyes. It is just a name invented to label the di erent varieties that exist for each of the quarks described above.

To this list of constituents of matter particles we must add the anti-matter particles, or simply antiparticles, for each of them. Antiparticles will be denoted

by a bar on top of the symbol for the particle. For example, the antiparticle for the u quark is called the anti-u quark and is denoted by u¯. The only exception to this notation is the antiparticle of the electron, called the positron, which is usually denoted as e+. Similarly, the antiparticles of µ− and τ− are denoted by µ+ and τ+, respectively. Antiparticles have the same mass but opposite electric charge as the particle. They also have all the opposite additive quantum numbers (see Sec. 9.4). When they encounter particles of the same kind, they annihilate each other. Since the earth is made out of particles (by definition), no antiparticles can live for long in our vicinity, so all the antiparticles observed to date are produced either by cosmic ray bombardments or by high energy accelerators.

The existence of antiparticles for matter was first predicted theoretically in 1929 by Paul Dirac long before any of them were found experimentally. This prediction was based on the theory of special relativity and quantum mechanics, and represents one of the greatest triumphs of theoretical physics in the twentieth century. Nowadays the existence of antiparticles is very firmly established experimentally.

Quarks and leptons are known as fermions, as they obey Fermi–Dirac statistics and the Pauli exclusion principle, forbidding two of them to occupy the same quantum state.

Beyond the quarks and leptons, present theory (called the ‘Standard Model,’ or SM for short) would also like to have — for technical reasons to be discussed in Sec. 9.5.3 — another electrically neutral particle, called the Higgs boson (H0). An intense search for the Higgs is underway, but as yet it has not been found. In fact, we are not completely sure that it has to exist as an elementary (rather than a composite) particle, and we do not know whether there should not be more than one of them. The form and shape of the future theory (which is more fundamental than the existing Standard Model) will depend quite critically on such details, which is why it is such an important matter to discover the Higgs particle or to rule it out experimentally.

The Higgs boson is certainly not a matter particle in the usual sense. Matter generally cannot disappear unless it is annihilated by its own antiparticle. In contrast, H0 can be created and absorbed singly.

The Standard Model also requires a set of gauge particles to transmit the forces (see Secs. 9.2 and 9.3). These particles are again not indestructible; in fact, forces are transmitted only because these particles can be freely absorbed and created (see Sec. 3). There are a number of gauge particles, each responsible for transmitting a di erent kind of fundamental force. As we shall see in Sec. 2, the fundamental forces are the strong or nuclear force, the electroweak (electromagnetic and weak) force, and the gravitational force. The corresponding gauge particles are the gluons (g) for the strong force, the photons (γ), W ± and Z0 particles for the electroweak force, and the graviton (G) for the gravitational force. With the exception of the graviton, all the other gauge particles have already been found.

Gauge and Higgs particles are bosons. They obey Bose–Einstein statistics and they always carry integer spins.

All in all, there are several kinds of elementary particles. Fermions are responsible for giving us the material universe (quarks and leptons), and bosons are used to transmit the forces (gauge particles) and provide symmetry breaking (the Higgs particle — see Sec. 9.5.3). These particles are summarized in Table 9.1.

9.1.1Summary

Matter is composed of two kinds of particles, quarks and leptons. Quarks experience the nuclear force but leptons do not. Each quark has three possible colors while leptons are colorless. Quarks and leptons come in three generations each. The first generation quarks are the (u, d), the second generation quarks are the (c, s), and the third generation quarks are the (t, b). The corresponding three generations of leptons are (e−, νe), (µ−, νµ), and (τ−, ντ ).

For each quark or lepton there exists an antiparticle with the same mass and opposite electric charge.

On top of these, there may also be one or several Higgs bosons.

Forces are mediated by the exchange of gauge particles. The gauge particles re-

sponsible for strong, electromagnetic, weak, and gravitational forces are respectively g, γ, (W ±, Z0), and G.

9.2 Fundamental Forces

Of equal importance to the identity of elementary particles is the nature of fundamental forces. What holds protons and neutrons together to form a nucleus? What binds electrons and a nucleus together into an atom, and atoms together into a molecule? For that matter, what holds molecules together in ourselves, and what holds us on the surface of this planet? Can we study the forces of Nature and classify them systematically?

At first glance this seems to be quite an impossible task. There are simply too many varieties of forces. After all, you can tie things together with a rope, hold them together with your hands, or bind them together with crazy glue. All these correspond to di erent forces. How can we possibly classify them all?