Invitation to a Contemporary Physics (2004)

conserved quantum number, Q. For Maxwell theory, Q is just the electric charge. By universality, we mean that the coupling to matter is proportional to Q, whatever the other quantum numbers are. It is the democratic principle, if you like. Everybody with the same ability (Q) has the same rights, whatever his/her sex, religion, or ethnicity is.

This kind of gauge theory, coupled universally to an additive quantum number, is known as an abelian gauge theory, or a U(1) gauge theory.

9.5.2 Yang–Mills Theory

One might wonder whether there are gauge particles coupled universally to vectorial quantum numbers. Such a theory was constructed by Chen Ning Yang and Robert Mills in 1954, and is nowadays known as a non-abelian gauge theory, or a Yang–Mills theory.

As in the abelian gauge theory, the masslessness of the gauge particle is enforced by the spin trick, which in turn is realized by demanding a local gauge invariance. For that to be fulfilled, the gauge particle once again has to couple universally to the conserved vectorial quantum number.

The vectorial nature of the quantum number however brings about some complications.

The gauge particles always belong to the adjoint multiplet, so there are n2 − 1 of them in SU(n). Unlike the photon which carries no electric charge, and therefore cannot emit other photons, the non-abelian gauge particle carries the adjoint quantum number and can therefore emit other gauge particles, on accoant of universality. This a ects the nature of short-distance interactions, as we shall see in Sec. 9.5.5.

9.5.3Electroweak Theory

Weak interaction was discovered at the turn of the century in beta decay. The first theory to describe it was written down by Enrico Fermi in 1934 shortly after Pauli postulated the existence of the neutrino. Indeed it was Fermi who christened this elusive particle emerging from the beta decay n → p + e− + ν¯e to be the (anti-)neutrino.

Fermi’s theory was very successful in predicting the energy distribution of the electron, but when parity violation was discovered, it had to be modified to take into account parity and charge-conjugation violations. This was carried out successfully in the late 1950’s.

The resulting interaction resembles an isospin gauge theory. However, on the one hand, the mass of the would-be gauge particle has to be very, very large to explain the extremely short range of the weak forces; on the other hand, the mass of any gauge particle has to be zero to allow for gauge invariance. These two requirements do not seem to be compatible with each other.

This dilemma was solved using the Higgs mechanism, which we proceed to explain.

9.5.3.1 Higgs Mechanism

The Higgs mechanism enables a gauge particle to become massive.

To make it work we need to have a scalar (spin-0) particle present to form a vacuum condensate (see Sec. 9.4.5).

The gauge particle W , which couples universally to any isospin object, will interact with this condensate if the latter carries an isospin. As a result, W receives a drag as it travels through the vacuum, causing it to slow down from the speed of light. This is analogous to light traveling in a dielectric material. Its interaction with the molecules in the material causes its speed to be c/n < c, where n > 1 is the dielectric constant of the medium.

To travel slower than the speed of light, W must have a mass. In order to have a mass, each spin-1 W must acquire from somewhere a longitudinal polarization. In the Higgs mechanism, these extra degrees of freedom are supplied by three extra scalars, one for each of the three gauge particles W +, W 0, and W −.

With the Higgs mechanism to make W massive, the weak interaction may now be described by an (weak) isospin gauge theory. To that end we need to introduce four scalar particles: one for the vacuum condensate, and three to supply the longitudinal polarizations. In the SM, these four form an isodoublet (φ+, φ0),

densate also gives rise to a new particle, which is the Higgs particle H0 listed in Table 9.1.

Actually, a refinement is needed to describe Nature. In this refinement, weak and electromagnetic interactions are combined into a single electroweak theory. The W 0 boson mixes with another neutral gauge boson to form the Z0 boson and the photon γ of Table 9.1. We will now proceed to discuss this refinement.

9.5.3.2 The Electroweak Interaction

The electromagnetic and weak interactions in the SM are combined into a single electroweak theory by Sheldon Glashow, Abdus Salam, and Steven Weinberg. It is a combined abelian and non-abelian gauge theory. The non-abelian gauge boson W is coupled to the weak isospin, as described before. The abelian gauge boson B0 is coupled to an additive quantum number known as the (weak) hypercharge, defined to be Y = 2(Q − I3), where I3 is the third component of the weak isospin and Q is the electric charge of a particle. Table 9.2 below lists the elementary particles discussed in Table 9.1, together with their Q, I, I3, Y , and YZ quantum numbers.

Table 9.2: Electroweak quantum numbers of the Standard Model particles.

The YZ quantum number is defined to be I3 − (0.23)Q, whose significance will be discussed in the subsection ‘Mixing of Neutral Gauge Bosons.’ Only fermions in the first generation are listed. Those in the second and third generations have identical quantum numbers as the first. Anti-fermions have the same I and opposite Q and I3, and are not listed either. Gravitons (G) and gluons (g) are absent in the table because they do not participate in electroweak interactions.

It is important to note that particles with left-handed chirality (L) do not have the same isospin and hypercharge as particles with right-handed chirality (R), so parity is explicitly broken in this theory. However, it can be shown that parity is still conserved in the electromagnetic part of this theory.

Here are some items in the table worth noting:

•The 2I +1 members inside an isospin multiplet all have the same hypercharge

Y .

•The gauge boson B and the gauge bosons W are decoupled, because the

isospin of B is zero and the hypercharge of W is zero. Since B couples only to hypercharge Y and W couples only to isospin (I, I3), B is not subject to the isospin force and W is not subject to the hypercharge force.

•Only left-handed (L) components of a fermion are a ected by the chargedcurrent weak interaction mediated by W . The right-handed (R) components have no isospin so they are not a ected.

•The right-handed neutrino νR is assumed to be absent. If it were there, it should have I = I3 = 0 like all other right-handed fermions. It is electrically neutral, so Q = 0. As a result, Y = 0 as well. Like all leptons, it does not carry a color. Hence, except for gravity, it is not subject to any force at all. Such a particle is extremely di cult to detect, even if it were present. For that reason it is called a sterile neutrino.We will return to νR later when lepton mixing is discussed.

360 Elementary Particles and Forces

Fermion masses produced this way are known as Dirac masses. Neutrinos cannot have a Dirac mass even in the presence of a vacuum condensate because right-handed neutrinos νR are absent. However, there is a way to get a neutrino mass just with the left-handed neutrinos νL. We can do it by taking f in Fig. 9.3 to be νL, and f¯ also to be νL. This kind of a mass is known as a Majorana mass.

To do so, we identify νL with (f¯)L, so fermion number is no longer conserved. Another way of saying this is that the pair in Fig. 9.3 now has a fermionic number 2, and the vacuum has a fermionic number 0, so this process does not conserve the fermionic number.

All the other fermions carry an electric charge, so none of them may have a Majorana mass. Otherwise, electric charge is not conserved, contrary to observations.

9.5.4Strong Interactions

The nuclear force between nucleons was explained by H. Yukawa to be due to the exchange of pions (see Sec. 9.1). We now know that both nucleons and pions are made up of quarks and anti-quarks. How do the quarks/anti-quarks interact to bind them into the nucleons and pions?

If the interaction is given by a gauge theory, the only thing we have to know is what conserved quantum number it couples to; not isospin, nor electric charge, for those have been taken up by the electroweak theory already. The only thing left is color, and that turns out to be correct. The theory of strong interaction is therefore known as quantum chromodynamics, or QCD for short.

We shall now discuss what led to that conclusion.

9.5.4.1 Asymptotic Freedom

In the late 1960’s, an experiment performed by Jerome Friedman, Henry Kendall, Richard Taylor, and collaborators at the Stanford Linear Accelerator (SLAC), exposed an important property of quark interactions that was hitherto unknown.

In that experiment, high energy electrons thrown at a nucleon target were detected at large angles (a more precise technical description is large ‘momentum transfers’). As a result of the collision, energy is taken from the electron to create other particles. Friedman and his collaborators were interested in those electrons which su ered a great loss in energy. Collisions of this type are known as deep inelastic scattering.

The probability of detecting an electron depends on the energy loss of the electron, and the angle at which it is detected. James D. Bjorken had predicted earlier that in the deep inelastic scattering region, the probability depends only on one independent combination of these two variables. When the experiment was analyzed in the deep inelastic region, this Bjorken scaling was indeed obeyed. Richard Feynman then realized that this scaling phenomenon could be understood physically

if the nucleon was composed of many constituents, which he called partons, provided that the partons do not interact among themselves, nor radiate other particles when being knocked around in the deep inelastic scattering process. Since nucleons are made up of quarks bound together by gluons, these partons must be nothing but quarks, anti-quarks, and gluons.

There are two puzzling features in the parton picture. Firstly, a nucleon is supposed to contain three quarks, but the parton idea of Feynman requires that the nucleon contain very many quark partons. So how many quarks are there in a nucleon? Secondly, quarks and gluons interact strongly, but for Feynman’s idea to work, they must either have no interaction, or at best a very weak one. How can that be?

To understand the first puzzle, let us be reminded (Sec. 9.3) that every particle has a Yukawa cloud around it. The cloud surrounding a quark or a gluon is made up of quark-anti-quark pairs and other gluons. If we examine the particle with a large probe (one with a long wavelength), we will see the particle as a whole, endowed with its full Yukawa cloud. Such a quark is known as a constituent quark, and a nucleon is made up of three constituent quarks.

On the other hand, if the probe is small (short wavelength), it can only see a small fraction of the cloud. We may then pick up a quark (or an anti-quark) hidden in the cloud, without picking up the rest of the cloud. These are Feynman’s partons, since a high-energy electron scattered at large angles qualifies as a short wavelength probe. This argument shows that there can indeed be many quark partons.

The second puzzle is more di cult to understand. In fact, its resolution led to the discovery that the dynamical theory of the strong interaction is QCD.

The only way to reconcile the second dilemma is for a strongly interacting quark to become weak when bombarded in a deep inelastic scattering experiment. Such a property is known as asymptotic freedom. To find out whether there are theories with such a property, an intense theoretical e ort was undertaken. After many unsuccessful attempts, it was finally discovered by David Gross and Frank Wilczek, and by David Politzer, that this is possible if quarks interact through a non-abelian gauge theory. The reason for that will be explained in Sec. 9.5.5. As discussed in Sec. 9.4.4, we are forced by the Pauli exclusion principle to introduce this new quantum number ‘color’ for the quarks, and not for the leptons. Since the other quantum numbers ‘isospin’ and ‘electric charge’ have already been used by weak and electromagnetic interactions, it is reasonable to assume that ‘color’ is the quantum number that governs the new Yang–Mills gauge theory for strong interactions. This SU(3)C non-abelian gauge theory is now accepted as the correct theory of strong interaction between quarks. The subscript C is there to remind us that the SU(3) vectorial quantum number used is color.

Recall that the electroweak theory is an SU(2)L × U(1)Y gauge theory. Hence, the SM of strong, electromagnetic, and weak interactions is given by an SU(3)C × SU(2)L × U(1)Y gauge theory.

9.5.4.2Confinement

Quarks have never been seen in isolation, so QCD must carry with it the confinement property (see Sec. 9.3.2). Ordinary hadrons like nucleons and mesons are not subject to confinement because the quarks and anti-quarks within them conspire to form color singlets. As such they no longer interact directly with the gluon, which is why the dominant nuclear force between two nucleons is carried by the pion and not by the gluon.

The mechanism behind confinement is still not clear. This ignorance lies in our inability to calculate reliably strong interaction field theories, except when the momentum transfer is large and the e ective coupling small, as is the case in the deep inelastic scattering region. There are, however, approximate numerical calculations (known as lattice gauge theory calculations) which indicate that confinement indeed occurs in QCD.

We will now describe a qualitative model of confinement which has some intuitive appeal, but whose quantitative validity is not yet known. Nevertheless, it will serve to illustrate what kind of mechanisms might induce confinement.

According to this model, at temperatures below about 0.2 GeV, a colorless gluon condensate is formed in the vacuum. Like the Higgs condensate, this will happen only when the vacuum with the condensate is more stable than the one without. In order for color to remain conserved, the gluons in the condensate must pair up to form colorless objects. Ideally color electric and magnetic fields should be absent from this condensate, for otherwise they will break up the colorless condensate and increase its energy. However, if two quarks are present, a color field connecting them is inevitable. In that case, to minimize the energetic damage, this field should be concentrated in as small a volume as possible, and that is in a tube between the two quarks with a cross-sectional area like that of the quarks. As discussed in Sec. 9.3.2, the color force between the two quarks is then a constant, and confinement results.

If this scenario is correct, one might be able to achieve deconfinement by high energy heavy ion collisions, if a temperature of 0.2 GeV can be created and trapped by these ions to melt away the condensate. Experiments of this kind are underway in RHIC at the Brookhaven National Laboratory.

9.5.5 Renormalization

Has it ever occurred to you that it is a miracle we can discover any physical law at all?

Newton managed to discover his three famous laws of mechanics and many others without ever knowing any atomic or molecular physics, because he did not have to. Microscopic details are really not important to macroscopic physics. This is not to say that there is absolutely no connection between the microscopic and the macroscopic worlds. The mass of an object, its thermal conductivity, its tensile strength, etc., are controlled by the details of atomic and nuclear physics. Macroscopic

theories, however, treat these quantities as experimentally measured parameters. If we want to compute these parameters, then a detailed knowledge of atomic and/or nuclear physics will be needed.

This is the normal world. The theory in such a normal world is known as a renormalizable theory.

We could imagine a completely di erent world in which the details of atomic and nuclear physics are required to describe macroscopic physics. We can even write down theories with these characteristics. Such a world would be so complex that physical laws could never be discovered.

In our normal world, physics at di erent scales is described by di erent equations. In the macroscopic world we have Newtonian mechanics; in the microscopic world there is quantum mechanics. The motion of an object on a frictionless table can easily be described by Newtonian mechanics. In principle, it can also be described by the Schr¨odinger equations of its molecules, but this latter description is so horribly complicated that it is virtually useless. Similarly, The motion of water in the macroscopic scale is described by the ‘Navier–Stokes equation,’ but on the atomic and molecular scale, the motion of the water molecules would be given by the quantum-mechanical Schr¨odinger equation. It would be foolhardy to treat the waves in the ocean with more than 1026 coupled Schr¨odinger equations for the water molecules; we will never get anywhere with such complexity. The Navier–Stokes equation bears very little resemblance to the Schr¨odinger equation, though the former must be derivable from the latter. The Navier–Stokes equation contains a number of parameters, viscosity being one of them. In the classical regime, viscosity is measured, but it can also be calculated if the microscopic physics is known.

Renormalization is the procedure whereby we relate physics of one scale to physics of another scale.

In the water example above, the equations in the two scales are di erent: Navier– Stokes in one, and Schroedinger in the other. In other systems it is also possible for the two equations to be essentially the same, but the parameters in one scale are di erent from the parameters in another. In such examples the task of renormalization is simply to determine how these parameters change with scale. SM equations are of this type.

It is not di cult to understand why the SM parameters change with scale. Take for example, an electron in QED. Let r = c/µ be the size of the probe, measured with a parameter µ, which has the dimension of energy. Di erent µ see di erent amounts of the Yukawa cloud around the electron, hence a di erent mass. For that reason the e ective mass m(µ) depends on the probe scale µ. Since the cloud contains e+e− pairs, these pairs will shield the electric field of the bare electron at the center. This is like an electron placed in a dielectric material. The electric field it generates polarizes the medium which in turn shields the electric field to make it weaker. The larger µ is, the deeper we penetrate into the cloud, and the

less is the shielding. Hence, the e ective electric charge e(µ) is1 also a function of µ. We see from this example that the e ective parameters in quantum field theories generally depend on the scale µ of the probe. This is renormalization at work. Such parameters are known as running parameters (running mass, running coupling constants, etc.).

There is a fundamental di erence between the running coupling constant of an abelian gauge theory, and a non-abelian gauge theory. In an abelian gauge theory, as we saw above, e(µ) grows with increasing µ. In a non-abelian gauge theory, say QCD, the cloud is no longer color neutral. In fact, gluons can emit more gluons to spread the color throughout the cloud. When we increase µ to penetrate deeper into the Yukawa cloud surrounding a bare quark, on the one hand we are seeing less qq¯ shielding than in QED. This tends to increase the e ective color charge, or equivalently the e ective coupling constant gS. On the other hand, because of the spreading, we are seeing less net color charge in the cloud, and that tends to decrease the coupling strength gS. The final e ective coupling constant gS(µ) receives corrections from these two competing e ects. Calculation shows that the gluon spreading wins out, so that gS(µ) becomes a decreasing function of µ, opposite to the behavior in QED. For large energy or momentum scale, gS(µ) can become quite small. This is the origin of asymptotic freedom, observed in the deep inelastic scattering experiment in SLAC (see Sec. 9.5.4).

Similarly, the SU(2)L gauge theory is also asymptotically free.

Historically, renormalization was discovered in the late 1940’s, in QED. Quantum field theory with interaction is such a di cult theory that it can be solved only approximately, by making use of the smallness of the coupling constant e in QED to obtain an iterative solution. Upon the first iteration, every thing works well and the result agrees approximately with experimental results. If we iterate once more, instead of getting a small correction to the first iteration, one gets something which is infinite.

After the second World War, microwave equipments developed for radar became available for precise spectroscopic measurements. Using them Willis Lamb and Robert Retherford found an unexpected energy shift in an atomic hydrogen line, a shift that is very small but nevertheless cannot be accounted for using the first iterative solution. In response to the challenge to explain this Lamb shift, people were forced to tackle the infinities brought on by the second iteration.

The infinity comes from very short-range contributions, a range much shorter than the resolution of any existing probe. From the result of the first iteration, we know our QED theory to be a good description of Nature at the present energies, or probe resolution. The infinity occuring at the second iteration tells us that the QED theory is not a good description of reality at a much smaller distance scale, or else infinities should not occur. This is analogous to the Navier–Stokes equation,

1The actual electron mass with mc2 = 0.51 MeV is the parameter when µ = mc2. The coupling constant e with the measured value α = e2/ c = 1/137 is also the value at µ = mc2.