Invitation to a Contemporary Physics (2004)

The creation and absorption of the exchange particle are given by the vertex factors CA and CB. For the dominant forces transmitted by neutral gauge particles (γ, Z0, G), we can take CA = gNA, CB = gNB, where the parameter g determines the overall strength of the force and is known as the coupling constant. NA and NB are respectively the additive quantum numbers of A and B that couples to the gauge particle (see Secs. 9.4 and 9.5). A similar statement can be made for the exchange of W ±, but it is more complicated in that case because the charge of particles A and B after the emission/absorption of W ± is no longer the same as the charge before.

9.3.5Summary

Forces are transmitted by the exchange of particles. The mass m of the exchange particle determines the range R, and its spin s determines the nature of the resulting forces. For distances small compared to R, all forces obey the inverse-square law. For distance large compared to R, they taper o rather sharply, unless confinement is involved. In that case the force is independent of the distance.

9.4 Quantum Numbers and Symmetry

We have now met a large number of particles: the quarks, the leptons, and the various exchange particles carrying forces. What distinguishes them?

How do we tell two persons apart? By their individual characteristics, of course. John Doe might be 2 meters in height and 100 kg in weight, male, and have black hair and brown eyes. Jane Smith is, say, 1.8 meters tall, weighs 70 kg, female, and has blond hair and blue eyes. They are di erent and they look di erent. There is no problem in distinguishing them.

Similarly, we distinguish the di erent particles also by their special characteristics, except that these characteristics in physics are usually expressed as numbers. We shall refer to these numbers as quantum numbers. Other than the mass, they are usually discrete numbers, often integers, but not always. The spin s of the particle is such a quantum number, its electric charge is another.

Quantum numbers are either additive or vectorial. Unless otherwise stated, all quantum numbers are conserved. This means that the sum of these quantum number in an isolated system remains the same at all times. If they did change with time, they would not be permanent features of a particle, and we could not use them to label particles.

The two kinds of quantum numbers di er in how the sum is taken. For additive quantum numbers, it is just the arithmetic sum. Electric charge is an example of an additive quantum number. Its conservation allows the β-decay reaction n → p + e− + ν¯e to occur but not the reaction n → p + e+ + ν¯e, for example. Quark number and the lepton number are other additive quantum numbers. Quark number is defined to be the number of quarks minus the number of antiquarks, and the

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lepton number is defined to be the number of leptons minus the number of antileptons. We can check the conservation in the β-decay reaction n → p + e− + ν¯e. The initial state has 3 quarks and zero leptons, and the final state also has 3 quarks and 0 = 1 − 1 leptons.

One may regard quark-number and lepton-number conservations as the modern version of Dalton’s indestructibility of matter.

9.4.1 Vectorial Quantum Numbers

A vectorial quantum number is the quantum version of a conserved vector, such as angular momentum. Classically, if a vector has d components, then the addition of two vectors is equivalent to the addition of their respective components. If this were true also quantum mechanically, then a vectorial quantum number is no di erent from a collection of d additive quantum numbers.

Because of the uncertainty principle, a quantum mechanical vector can usually be specified only by r < d components. The other d − r components remain uncertain, just like the momentum of a particle with specified position is uncertain. r is called the rank of the vectorial quantum number.

If this were all, then again a vectorial quantum number is no di erent from a collection of r additive quantum numbers. However, the length and some other characteristics of the vector can also be specified. In general, an additional t discrete numbers (p1, p2, . . . , pt) are needed to determine the vector as completely as possible. In most of the vectorial quantum numbers we are going to deal with (the so-called SU(n) quantum numbers), the number t is equal to the rank r. These numbers pi do not add arithmetically, and this is what distinguishes a vectorial quantum number from a set of additive quantum numbers. By a vectorial quantum number, we now mean the collection of the r additive numbers and the t non-additive quantum numbers.

Spin is the simplest example of a vectorial quantum number. Let us recall from Sec. 9.3 and Fig. 9.2 what we know about it.

Classically, spin angular momentum is a vector Ns, specified by three components. Quantum mechanically, because of the uncertainty principle, only one of the three components (say the z-component) may be specified, although the length of the vector may be simultaneously determined as well. Both of these numbers are quantized; they are only allowed to take on certain discrete values. The length of the

vector must be of the form s(s + 1) , with s being an integer or a half-integer. For a given s, the z-component must be of the form ms , with ms being one of the 2s + 1 values −s, −s + 1, . . . , s − 1, s. In other words, r = t = 1, and p1 = 2s is a non-negative integer.

Now let us discuss how to find the sum of two spins, specified by (s1, m1) and (s2, m2). If their sum is specified by (s, ms), then clearly ms = m1 + m2. But what about s?

Classically, the length of Ns = Ns1 + Ns2 depends on the relative angle between the two vectors, Ns1 and Ns2. In particular, if the two vectors are parallel, then the length |Ns| of Ns is simply the sum |Ns1| + |Ns2|. If they are anti-parallel and if |Ns1| ≥ |Ns2|, then

|Ns| = |Ns1| − |Ns2|.

Quantum mechanically, the non-additive numbers s, s1, s2 are integers or half integers, which means that the equivalent angle between Ns1 and Ns2 is not arbitrary. The best s can do is to take on the discrete values between s1 + s2 and |s1 − s2|, in integer steps. This is precisely what happens. For example, the sum of s1 = s2 = 12 can yield s = 0 or s = 1. The sum of s1 = s2 = 1 can yield s = 2, 1, or 0.

We can verify these rules by counting the number of components. Take s1 = s2 = 12 . Each of Ns1 and Ns2 has two allowed orientations, so there should be 2×2 = 4 states altogether when they are vectorially added. Indeed, s = 0 has one state, and s = 1 has three states, making a total of four. Similarly, s1 = s2 = 1 each has three states, so altogether there should be nine states when they are vectorially added. Indeed, s = 2 has five states, s = 1 has three states, and s = 0 has one state, making a total of nine, as it should.

For more complicated vectorial quantum numbers, a branch of mathematics known as group theory is needed to tell us what the non-additive quantum numbers pi are, and how they are combined. See the next subsection for further discussions.

9.4.2Symmetry and Conservation Laws

Physics is the same today as a billion years ago. It is also the same here on earth as in other parts of the universe. The former, known as time-translational invariance, or time-translational symmetry, can be shown to lead to energy conservation. The latter, known as space-translational invariance, can be shown to lead to momentum conservation. We also get the same physics whether the laboratory is facing east or south. This rotational symmetry leads to angular momentum conservation. These examples illustrate the universal correspondence between symmetries and conservation laws. See Chapter 1 for more discussions on symmetry.

Symmetry is systematically studied in a branch of mathematics called group theory. A symmetry is specified by a group, which tells us the quantum number that is conserved, and how to add them. Additive quantum numbers typically correspond to the U(1) group. It is an abelian group, meaning that the order of symmetry operation is immaterial. Vectorial quantum numbers correspond to nonabelian groups, in which a di erent symmetry results if the order of two symmetry operations is interchanged. The spin angular momentum corresponds to the SU(2) group, which is the first of a series of groups known as SU(n), for n ≥ 2. It is useful to get acquainted with these groups, as they will come up later in several places.

Let us start from SU(2) which we already know. Its conserved quantum number is specified by a three-dimensional vector, which is the spin vector in the previous application. The length of this vector is discrete, given by a non-negative integer

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p1 = 2s. For a fixed p1, the allowed orientation of the vector is determined by an additive quantum number ms, which takes on discrete values between s and −s in integer steps. These p1 + 1 possible states are said to form a multiplet; the size (p1 + 1) of the multiplet is called its multiplicity, or dimension. Note that the sum of the additive quantum number ms over the multiplet is always zero. This actually is what the ‘S’ in SU(2) stands for.

The smallest non-singlet multiplet, given by s1 = 12 , is called a fundamental multiplet. The multiplet s1 = 1 has a multiplicity three equal to the dimension of the spin vector. It is called the adjoint multiplet. The adjoint multiplet can be obtained by vectorially adding two fundamental multiplets, because the vectorial sum of s1 = 12 and s2 = 12 yields both s = 1 and s = 0.

These properties can be generalized to SU(n). The SU(n) quantum number is specified by an (n2 − 1)-dimensional vector, whose allowed length and multiplicity are determined by n−1 non-negative integers (p1p2 · · · pn−1). Each SU(n) vectorial quantum number contains n − 1 additive quantum numbers, which are the generalization of ms. As pointed out before, the number of additive quantum numbers in a group is known as the rank of the group, so SU(n) has a rank of n − 1. The sum of each of these additive quantum numbers in a multiplet is always zero. We shall refer to this relation as the traceless constraint; this is actually what the ‘S’ in SU(n) stands for.

The multiplet (p1p2 · · · pn−2pn−1) and the multiplet (pn−1pn−2 · · ·p2p1) are said to be conjugate to each other. They have the same multiplicity. In physics, they often represent particles and antiparticles, respectively, especially for n > 2.

One often denotes a multiplet by its multiplicity. Conjugate multiplets are distinguished by putting an asterisk on one of them. For example, in SU(2), multiplets with s = 0, 12 , 1, 32 are denoted by 1, 2, 3, 4 respectively.

For n > 2, there are two fundamental multiplets of dimension n. They are n = (10 · · ·0) and n = (0 · · · 01).

The adjoint multiplet n2 − 1 = (10 · · ·01) has the same dimension as the conserved vector. It can be obtained by vectorially adding n and n , for such an addition yields a singlet 1 and an adjoint multiplet n2 − 1. The reason why the multiplicity of the adjoint is n2 − 1 but not n2 is intimately related to the traceless constraint of SU(n).

As mentioned before, these general properties of SU(n) are already present in the familiar case n = 2. Readers who do not follow these semi-quantitative details should have little problem understanding most of the remaining chapter.

The same symmetry may be applicable to di erent physical quantities. For example, charge conservation, quark number conservation, and lepton number conservation all correspond to a U(1) symmetry. Unlike energy, momentum, and angular momentum, these symmetries have nothing to do with ordinary spacetime. For that reason they will be referred to as internal symmetries. Di erent members of an internal multiplet correspond to di erent elementary particles.

We shall now proceed to discuss an internal SU(2) symmetry known as isospin.

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Classically, color is an 8-dimensional vector. Quantum mechanically, these vectors are discrete and are labeled by two non-negative integers, (p1p2), which are the counterparts of I in isospin. The size of the multiplet is determined by p1 and p2. For example, 1 = (00), 3 = (10), 3 = (01), and 8 = (11).

The particles within each multiplet are labeled by two additive quantum numbers, which are the counterpart of I3 in isospin.

Elementary particles carrying non-zero color have strong interactions (see the next section). Those with no color do not. Hence, only quarks and gluons carry a non-singlet color. Each of the six quarks u, d, c, s, t, b constitutes a color triplet 3 = (10), and each of the six antiquarks constitutes the conjugate color triplet 3 = (01). Gluons belong to the adjoint multiplet 8 = (11). All in all, there are eight gluons, three u-quarks, three u¯-quarks, etc.,

The color quantum number comes about in the following way. There is an excited state of the nucleon by the name of ∆++, made up of three u quarks (∆++ = uuu). Its spin is 32 , so the spins of the three quarks are all lined up along the same direction. It has no orbital angular momentum between the quarks. Its wave function is therefore symmetric under the interchange of any two quarks, because each of the orbital, spin, and isospin parts of the wave function is symmetric. That, however, is against the Pauli exclusion principle, which demands the wave function to be completely antisymmetric under an interchange. The simplest way out is to assume there are three varieties (three di erent ‘colors’) of u quarks, u = (u1, u2, u3), and that ∆++ is composed of u1u2u3. Then, there is no di culty in constructing an antisymmetric wave function in color so that the exclusion principle is satisfied.

This then is the origin of color, and the color triplet for each of the six quarks.

9.4.5Vacuum Condensates

At the beginning of the universe, temperature is high and energy is plentiful. Elementary particles too massive to be observed at the present may appear at that time. At high temperatures, particles might possess additional conserved quantum numbers which are no longer conserved at the present temperature of 2.7 K (see Sec. 10.7). This actually includes the weak isospin. In this subsection we will discuss how that happens.

The ground state of the universe is called the vacuum. It is the state with no particles, as the presence of each particle invariably brings along an additional amount of energy. As the temperature decreases from its initial value, the vacuum is believed to undergo a series of phase transitions. Let us see what they are and how it a ects the conservation of quantum numbers.

We are familiar with the phase transitions of water. For temperatures above 100◦C, it is in the vapor or gaseous phase. Between 0 and 100◦C, it is in the liquid phase, and below 0◦C, it is in the solid or ice phase.

Phase transitions are caused by the desire of a physical system to seek the lowest possible energy. Since thermal energy is part of the total energy budget, the ground

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happen, then the process viewed in a mirror can happen as well. This symmetry leads to a conserved quantum number known as parity. In 1957, Tsung Dao Lee and Chen Ning Yang discovered that while this is true for strong and electromagnetic interactions, it is not so for weak interactions. In other words, parity is not a conserved quantum number in the presence of weak interactions. This discovery of Lee and Yang has rather far-reaching consequences, so let us discuss it in some detail.

We need to introduce two concepts: helicity and chirality.

The spin component along the direction of motion is called helicity. In the case of quarks and leptons, whose spin is s = 12 , these components are either ms = 12 or −12 . The former is known as a right-handed helicity, because when you stand behind the particle, the angular motion of its spin turns like a right-handed screw. The latter is called a left-handed helicity. In relativistic quantum mechanics, helicity is a more important concept than spin. Spin-orbit coupling is present in a relativistic theory, so neither spin nor orbital angular momentum is conserved, though the total angular momentum still is. Since orbital angular momentum has no projection along the direction of motion, the projection of total angular momentum is equal to the projection of spin alone, so in spite of the spin-orbit coupling, helicity is still conserved.

Chirality is defined to be twice the helicity for a massless spin- 12 particle. If the particle is massive, the concept of chirality is mathematically well-defined, but it is hard to describe it in physical language.

A right-handed screw looks like a left-handed screw in a mirror. Thus parity transformation reverses helicity and chirality. Since the late 1950’s, the weak interaction has been known to be left-handed. This means that only quarks and leptons of left-handed chirality, and only antiquarks and antileptons of right-handed chirality, participate in the charged-current interactions mediated by the exchange of W ±. This preference of one handedness over another clearly violates parity (P) symmetry. It also violates charge conjugation symmetry (C) as we will explain, though it preserves the combined CP symmetry.

Charge conjugation symmetry means that if we change all particles to their antiparticles (γ, Z0, H0 are their own anti-particles, and W + and W − are anti-particles of each other), without altering any other quantum number, then the new process is still an allowed one occurring with the same probability. The weak interaction violates charge conjugation invariance because interactions occur for left-handed quarks and leptons but not for left-handed anti-quarks and anti-leptons. It is the right-handed anti-quarks and anti-leptons that participate in the interactions. So C symmetry is violated. Nevertheless, the weak interaction is still invariant under a combined CP operation, for this changes the left-handed quarks and leptons to right-handed anti-quarks and anti-leptons.

Strictly speaking, the CP symmetry is violated at a small level. Such violations are very important, for we probably owe our existence to it. We will come back to discuss this in a later section.

9.4.7Summary

Conserved quantum numbers may be additive, or vectorial. The former type add arithmetically, but the latter also contains non-additive quantum numbers which do ‘add,’ but in a peculiar way.

Examples of additive quantum numbers are electric charge, quark number, and lepton number. Examples of vectorial quantum numbers are isospin and color.

Strong and electromagnetic interactions are also invariant under charge conjugation (C) and parity transformation (P), but the weak interaction is not. CP is almost conserved in all weak interaction processes, but a small violation does exist.

9.5 Standard Model

The current theory of strong, electromagnetic, and weak interactions is known as the Standard Model, or SM for short. It has so far passed all the experimental tests thrown at it, so it is a very reliable theory. The only exception is neutrino oscillations, which will be discussed in the next section.

As for gravity, Einstein’s theory of general relativity works for macroscopic bodies, but owing to the weakness of its strength, quantum gravity has not yet been tested. In any case, we do not have a very good understanding of the theory of quantum gravity. We will discuss that problem further in the last section of this chapter.

The three fundamental forces in the SM are all described by gauge theories. The fourth, Einstein’s theory of gravity, is also a kind of gauge theory, though technically sometimes not regarded as such.

A gauge theory is a very special theory of force, probably the most elegant theory at present. The simplest and the oldest of these gauge theories is the Maxwell theory of electromagnetism. It will serve as a template for the other gauge theories, so let us study it in some detail.

9.5.1Electromagnetic Theory

Classical electricity and magnetism were unified and completed by James Clerk Maxwell in the late 19th century. Its quantum version was invented in the late 1920’s. This quantum theory is known as quantum electrodynamics, or QED for short. From then on it was just a matter of studying its properties and obtaining numbers from it to compare with experiments. Calculations of this sort actually ran into a snag which did not get resolved until the late 1940’s. The solution goes by the name of renormalization, a topic which we shall discuss at the end of this section.

In order to expose the elegance of the theory, and to motivate how similar considerations can be applied to other gauge theories, let us now pretend not to know the Maxwell theory, and see how far a few simple facts can lead us.

exchange particle must be massless. It must also be electrically neutral, for otherwise the charge of the electron will be altered after a photon is emitted. The fact that it transmits both an electric and a magnetic field shows that it must carry a non-zero spin. In fact, the photon spin is known to be 1.

We will now argue that this spin-1 nature is extremely important in keeping its mass zero, and hence the infinite range of the Coulomb force.

A particle with a mass m carries a minimum amount of energy equal to mc2. According to quantum mechanics, energy generally shifts with interactions, which leads to a shift in the mass m. For the photon, this shift must be positive, because there is no such thing as a negative mass. A positive mass will cause the Coulomb force to have a finite range, contrary to experimental observation.

There are two ways out. It is logically possible for the photon to start with a finite mass, and the forces of the universe are so finely tuned to make the final photon mass zero. Such an accidental zero is highly contrived and seems to be highly unlikely.

It is also conceivable to have some sort of a mechanism to protect the photon mass, so that no shift of it can occur. One such mechanism is the spin trick.

A particle with spin s has 2s + 1 possible spin orientations at rest. According to the principle of special relativity, this should continue to be the case when the particle is in motion.

Since a massless particle must move with the speed of light, and cannot be put at rest, this argument does not apply to them. In fact, the photon has spin 1 but it has only two helicities, corresponding to the transverse polarizations (right-handed and left-handed circular polarizations) of light. The longitudinal polarization is absent.

Conversely, if somehow we can keep the number of physically realizable helicities of a particle to be fewer than 2s + 1, then it must be massless. This is the spin trick.

A gauge theory is a theory in which the longitudinal polarization of the spin-1 boson (called the gauge particle) is decoupled from the rest. In other words, it is invariant under a gauge transformation in which the magnitude of the longitudinal polarization is changed. The Maxwell theory of electromagnetism is a gauge theory, which is why the photon has a zero mass even in the presence of interactions.

One can envisage a situation in which every point in space has one or more photons. Since none of them may have a longitudinal polarization, gauge invariance must be true at every point in space. Thus, the symmetry of gauge transformation is a local symmetry, which is a much more powerful symmetry than the global symmetries that lead to energy, momentum, angular momentum, charge, isospin, and color conservations.

In the presence of matter, it is possible to maintain this local symmetry only when the gauge particle couples universally to all matter particles. This means a U(1) symmetry must be present in the matter sector to produce an additive