Invitation to a Contemporary Physics (2004)

which is true at the macroscopic scale, but not true on the quantum scale. Without knowing the physics at the quantum scale, parameters like viscosity cannot reliably be computed, so it must be measured. Similarly, in QED, since its description at the very small distance scale is wrong, we must take parameters like the electron mass and the coupling constant at their measured values, at least at one scale µ, with no attempt to compute them. The necessary machinery to accomplish this was developed by Richard Feynman, Julian Schwinger, and Sin-Itiro Tomonaga. The result agrees very well with the Lamb shift, as well as all other precision measurements of QED.

With renormalization, the SM becomes a precision theory for particle physics at the presently available energies.

9.5.6Summary

Gauge theories are theories whose forces are transmitted by massless spin-1 particles. To keep them massless, a gauge invariance is enforced to decouple the longitudinal polarization. Gauge invariance is a local invariance; it can be achieved only when the gauge boson couples universally and democratically to a conserved quantum number.

Abelian gauge theory is just the Maxwell theory. Non-abelian, or Yang–Mills gauge theories are more complicated. The former couples to an additive quantum number, and the latter couples to a vectorial quantum number. The gauge bosons in abelian theories do not couple directly among themselves, but those in a non-abelian gauge theory do.

The electroweak interaction is given by the gauge theory SU(2)L × U(1)Y . To make it work, a Higgs condensate is required.

The strong interaction is given by the gauge theory SU(3)C, also known as QCD. Asymptotic freedom exists in QCD; confinement may be present there as well.

The SM of the strong, electromagnetic, and weak interaction is given by a gauge theory based on the gauge group SU(3)C × SU(2)L × U(1)Y .

Parameters like masses and coupling constants run with the probing scale µ. In particular, the coupling e(µ) in QED increases with µ, and the coupling gS(µ) in QCD decreases with µ, giving rise to asymptotic freedom.

9.6Fermion Mixing

According to the electroweak theory, W couples universally to all isospin doublets. Among the left-handed fermions, there are three quark doublets, (u, d), (c, s), and (t, b), and three lepton doublets, (νe, e), (νµ, µ), and (ντ , τ). However, these fermions may not have a definite mass, so they may not be the fermions detected in experiments.

366 Elementary Particles and Forces

To understand why these fermions may not carry a definite mass, let us recall the mass generating mechanism discussed at the end of Sec. 9.5.3, and Fig. 9.3. Two virtual fermions (f1)L and (f2)R = (f2)L, or (f1)R and (f2)L = (f2)R, are created from the vacuum, with opposite momentum, opposite electric charge and color, and opposite quark and lepton numbers. In Sec. 9.5.3, we concentrated on the situation when f1 = f2, but there is nothing to prevent f1 =f2.

To conserve quantum numbers, if f1 is any one of (u, c, t), then f2 must be one of (u, c, t). If f1 is any one of (d, s, b), then f2 must be one of (d, s, b). If f1 is any one of (e, µ, τ), then f2 must be one of (e, µ, τ). The 3 × 3 combinations in each of these three categories can be arranged into three 3 × 3 matrices, called the mass matrices. Unless the matrix is diagonal, i.e., non-zero only when f1 = f2, the fermions f mix with one another and they will not have a definite mass. The states fm that have definite masses, known as mass eigenstates, are linear combinations of these fermions f, known as flavor states. This is similar to the mixing discussed in Sec. 9.5.3.3 to form the Z0 and the photon. The combinations can be computed mathematically once the mass matrix is known, but unfortunately we do not know what it should be. For that reason, we simply have to parameterize the combinations and determine the parameters by experiment.

We will discuss the quark sector below. The lepton sector is similar, and will be discussed in a subsequent subsection. The mixing parameters can be measured experimentally, and they may shed light on the origin of the interaction that causes the mixing.

9.6.1Quarks

There is a 3 × 3 mass matrix for (u, c, t), and another one for (d, s, b). In principle, neither of them is diagonal. In practice, as we shall explain, we can always assume the (u, c, t) matrix to be diagonal, so only linear combinations of the flavor states (d, s, b) in terms of the mass eigenstates (dm, sm, bm) have to be specified. The subscript m denotes quarks with definite masses.

The (u, c, t) mass matrix can be chosen to be diagonal because of the universality of charge-current electroweak interactions. W couples to each of the three left-handed quark isodoublets in exactly the same way. Hence it also couples the same way to any linear combination of (u, d), (c, s), and (t, b). We will choose the linear combination that makes the (u, c, t) mass matrix diagonal. With this new

combination, quarks in the flavor basis will change from (u, d), (c, s), (t, b) to something else, say (u , d ), (c , s ), (t , b ). By construction, (u , c , t ) = (um, cm, tm) have definite masses. As remarked above, W still couples universally to (u , d ), (c , s ), and (t , b ). We will now drop all the primes in the notation. In this way, we see that we may indeed assume the mass matrix of (u, c, t) to be already diagonal, as claimed.

violation than whatever can be provided by quark mixing alone to account for the present amount of matter (over antimatter) in the universe, so another stronger source of CP violation must occur somewhere, presumably at higher energies than we can reach right now.

9.6.2 Leptons

As in the quark sector, we may assume the charged-lepton mass matrix to be already diagonal, (e, µ, τ) = (em, µm, τm). The neutrino mass matrix and the neutrino mixings will now be discussed.

In what follows we shall denote the neutrino states with definite masses by ν1, ν2, ν3, rather than (νe)m, (νµ)m, (ντ )m. The latter notation is not only more cumbersome, it is also not justified if the mixing is large. In the case of quarks, the mixings are small, so we can uniquely associate the three mass eigenstates with the three flavor states. In the case of neutrinos, the mixings are large, so the former notation is better.

The corresponding neutrino masses will be denoted by m1, m2, m3.

In the SM, m1 = m2 = m3 = 0. It is so because there is no right-handed neutrino, so there can be no Dirac mass. Lepton number is conserved, so there can be no Majorana mass. A direct measurement of the ν¯e mass shows it to be small, being bounded by 3 eV/c2 or less.

The latest fit to available cosmological and astrophysical data by the WMAP collaboration suggests an upper bound of 0.23 eV/c2.

However, from the observation of solar and atmospheric neutrino deficits, to be discussed later, one deduces that at least two of the three masses must be non-zero. So m1 = m2 = m3 = 0 is ruled out by experiment.

In light of that, something must be missing in the SM. What then can we say about the neutrino mass matrix and neutrino mixing?

A neutrino mass may be a Majorana mass, or a Dirac mass (see Sec. 9.5.3). In the former case no right-handed neutrino is needed, but lepton number is violated because the vacuum state turns into a two-neutrino state (instead of one neutrino and one anti-neutrino). In the latter case the presence of a right-handed neutrino is required. If such a particle did exist, it is presumably not like the other right-handed fermions in the SM, because the neutrino mass is so much smaller than the other fermion masses.4 There are speculations of the existence of such a right-handed neutrino, either at high energy or in an extra dimension (see Sec. 9.7.3), or both. Since they are not directly visible right now, we can ‘integrate them out’ to get an e ective mass using only left-handed neutrinos. In doing so we are back to the Majorana mass again.

4The smallest charged fermion mass is that of the electron, which is 0.5 MeV/c2, some 200 000 times larger than the largest conceivable neutrino mass.

A Majorana-neutrino mass matrix is a complex symmetric matrix, because f and f¯ in Fig. 9.3 are both equal to νL. The mathematics of diagonalizing it is slightly di erent from that used to diagonalize the (d, s, b) mass matrix. Nevertheless, the flavor states (νe, νµ, ντ ) are still unitary linear combinations of the mass eigenstates (ν1, ν2, ν3) as in Eq. (9.1):

This time the matrix V is called the Pontecorvo–Maki–Sakata (PMNS) matrix. Since the mixings turn out to be large, it is not useful to parameterize it in the Wolfenstein manner. It is the Chau–Keung form2 of parameterization that is usually used.

In addition, since the mass matrix is complex, the resulting masses mi are complex as well. If we want to write them as positive numbers, we should replace mi by mieiδi . An overall phase does not matter, so we may set δ1 = 0, but we are left with δ2 and δ3, which have no counterpart in the mixing of quarks. These extra phase angles can be traced back to the Majorana nature of the neutrino masses.

We have assumed the neutrinos to be Majorana in order to get a mass. If so, there may be important consequences. Lepton number violation might turn into quark number violation, and possibly a mechanism to generate more protons than antiprotons, and more electrons than positrons in the universe, as required by observation. It is therefore important to ascertain directly whether the neutrino is Majorana or not. The smoking gun is the discovery of neutrinoless double β-decay. Two neutrons in a nucleus may decide to undergo the β-decay n → p + e− + ν¯e simultaneously. In such a decay, two ν¯e are produced. However, if the neutrino is Majorana, the two antineutrinos may combine into the vacuum, thereby producing a double β-decay without any neutrino. Measurements are being carried out to find such a reaction, but so far none have been observed.

9.6.3 Neutrino Oscillations

We will explain in this subsection the evidence for non-zero neutrino masses. We will also discuss the qualitative di erence between neutrino and quark parameters, and its possible implication.

The energy from the sun is primarily powered by the fusion reaction 4p → 4He + 2e+ + 2νe, in which two protons are turned into two neutrons through the inverse β-decay reaction p → n + e+ + νe. In addition to this nuclear reaction, there are several others which also produce neutrinos in the sun.

Since the early 1960’s, Raymond Davis and his group had been trying to detect these solar neutrinos. The experiment is a very di cult one because neutrinos interact very weakly, so a large chunk of matter is needed to capture some of them.

Moreover, cosmic rays overwhelm the solar neutrino events, so the detector must be heavily shielded, usually by putting it deep underground in a tunnel or a mine. Even so, natural radioactivities contained in the detector and the nearby rocks must be largely removed to give a chance for the solar neutrino events to show through. What Davis used as a target was a large 615 tons tank of dry cleaning fluid. Most of the neutrinos pass straight through, but an occasional one strikes a chlorine atom 37Cl in the fluid to turn it into a radioactive argon atom 37Ar. Periodically, the few argon atoms in the fluid are extracted and measured by counting its radioactivity. For over twenty years, Davis’ experiment was the only one set up to detect solar neutrinos. It was a di cult job and data accumulated very slowly.

What he found was quite astonishing. Only about 1/3 of the expected neutrinos were found. What happened to the other 2/3?

In the 1980’s, another large detector was built in Japan using water as the

ˇ

target, which the Cerenkov light emitted by the struck electrons as the signal for a neutrino. This group is led by Masatoshi Koshiba; the upgraded version of the detector is called the Super-Kamiokande. It uses 50 thousand tons of pure water as the target. Because of the large size of the tank, most solar neutrino data to date have come from this source. They found about 1/2 of the expected neutrinos.

By now there are also two gallium detectors in Europe, and a heavy water detector called SNO in Canada. They all find a deficit in solar neutrinos. These deficits are consistent with a similar deficit from reactor antineutrinos, measured after they travel some 180 km to reach a liquid scintillator detector known as KamLAND.

Di erent detectors are sensitive to di erent neutrinos and/or di erent energies. They are all needed to analyze the outcome.

The explanation for the deficit is neutrino oscillation. Unlike quarks which are detected in their mass states, neutrinos are produced by weak interaction, and also detected through weak interaction, so at both ends we are dealing with the flavor neutrino νe. In between the sun and the earth, it is the mass eigenstates νi that propagate with a definite frequency, when treated as a quantum mechanical wave. On account of relativistic kinematics, the frequency is proportional to m2i . To compute the deficit, we must use Eq. (9.2) to decompose the initial νe into the mass eigenstates νi, and at the detector use Eq. (9.2) again to convert each νi back to νe. If all the masses mi are the same, their propagating frequencies are identical, therefore all of them have the same phase at the detector. In that case, the mass eigenstates νi simply recombine into νe, and no deficit is present. If, however, the masses are di erent, the phases of the di erent νi’s will be di erent at the detector, so the νi’s will no longer recombine back into νe, for some of them will now combine into other kinds of neutrinos. In that case a deficit in νe is observed.

Conversely, the fact that deficits are observed tells us that we cannot have m1 = m2 = m3 = 0.

Super-Kamiokande also discovered a deficit in atmospheric neutrinos. Cosmic rays hitting the upper atmosphere generate many charged pions which eventually

Figure 9.4: (a) Quark mixing, and (b) neutrino mixing. Mass states appear at di erent columns, and flavor states appear at di erent rows.

decay, mostly via π+ → µ+ + νµ and π− → µ− + ν¯µ. The muons are themselves unstable; they decay via µ+ → e+ + νe + ν¯µ and µ− → e− + ν¯e + νµ. Combining these two decays, we expect to find two µ-type neutrinos and antineutrinos for every e-type neutrinos and antineutrinos. What is observed is more like one-to-one, not two-to-one.

The explanation for a deficit in the µ-like neutrinos is again neutrino oscillation. The solar neutrino deficit tells us about the oscillation of the e-type neutrino, and the atmospheric neutrino deficit tells us about the oscillation of the µ-type neutrino. There are other accelerator and reactor experiments which provide us with additional information. By analyzing these experiments carefully, one finds that neutrino mixings are large, and neutrino mass di erences are small. In fact, m22 −m21 (7.5 meV)2, |m23 −m22| (50 meV)2 (1 meV = 10−3 eV), tan2 θ12 0.4,

tan2 θ13 < 0.026, tan2 θ23 1.

Given the small mass di erences and the mass upper bound for ν¯e, it follows that all neutrino masses must be smaller than 3 eV or so. Even at this upper limit, the mass is a factor of 2 × 105 smaller than the electron mass, the lightest of the charged fermions. Neutrino mixings are also very di erent from quark mixings. For quarks, all three mixing angles are small. For neutrinos, there are two large angles and one small one. These two di erent mixing patterns are illustrated in Fig. 9.4.

Many experiments are being set up or planned to confirm the oscillation hypothesis, and to obtain better values for the neutrino parameters. Neutrino oscillation is the only confirmed phenomenon not explained by the SM, and as such it might provide a stepping stone to discover new physics beyond the SM. The fact that neutrinos behave so di erently from charged fermions also suggests the presence of new physics that only neutrinos can see. Neutrinos are also important in astrophysics and in cosmology. They are ubiquitous, and promise to be an important tool in astronomy if only we can detect them more e ciently. For all these reasons, neutrino physics promises to be a very active field in the near future.

For their important discoveries, Davis and Koshiba were awarded the Nobel Prize for Physics in 2002.

(a)60

(GeV)

(b)60

(GeV)

Figure 9.5: A schematic plot for the energy variation of the three inverse coupling constants 4π/gi2 as a function of energy. (a) Without supersymmetry. (b) With supersymmetry.

they appear to be di erent because gY < gI < gS at the present energy. However, we learned in Sec. 9.5.5 that coupling constants run with the energy scale µ, so it is conceivable that all coupling strengths become equal at some high energy µGUT. If that happens, the three di erent forces may simply be three di erent manifestations of a single force. A theory of that kind, uniting the di erent Standard Model forces and particles, is known as a grand unified theory, or GUT for short. The GUT group must contain the SM group SU(3)C × SU(2)L × U(1)Y as a subgroup.

Let us first look at the possible unification of gS(µ) and gI(µ), which for later convenience we will re-label as g3(µ) and g2(µ), respectively. Both of them decrease with increasing µ, but since there are eight gluons and only three W ’s, g3(µ) decreases faster than g2(µ). With g3 > g2 at the present energy, these two are getting closer together at higher µ’s (see Fig. 9.5, but note that it is plotted in 1/g2, not g), so they will intersect at some large µGUT. The strengths of the electroweak and the strong forces are then the same for r c/µGUT .

What about the hypercharge force? gY is smaller than g2 and g3 at current energies, and being an abelian theory, it increases with µ. So at some high µ it will

374 Elementary Particles and Forces

catch up with g2(µ) and/or g3(µ). If it should intersect with both of them at µGUT, then all the three strengths become the same at that energy. Does it do that?

It actually does not, but there is a problem of normalization. In a gauge theory, the gauge particle couples universally to a conserved quantum number, with some strength g. However, there is an arbitrariness in the normalization of the quantum number, which is reflectd in an arbitrariness in the strength g. Let us explain what we mean by looking at the coupling of W 0.

W 0 couples to the third component of isospin through a term proportional to gI I3. However, we could have used I3 = aI3 as the additive quantum number instead, in which case the coupling strength would have to be gI = gI/a in order to preserve gI I3 = gI I3, and hence the same interaction. Without specifying the normalization of the quantum number, there is no way that di erent coupling strengths can be meaningfully compared.

For SU(2), one uses I3 by convention, not I3. Note that the sum of I32 over the two components of the (fundamental) doublet is 2 × (12 )2 = 12 .

The conventional normalization for SU(n) is a generalization of this. Let Qi be any of the n − 1 additive quantum numbers of SU(n). We require each of them to be normalized like I3 of SU(2), namely, the sum of Q2i over the n members of a fundamental multiplet be equal to 12 .

We shall see below that the correct normalization of the hypercharge quantum number in a GUT is not Y/2, but (Y/2) 3/5. This means that the correct coupling

When this idea of unification was first proposed, data were not very precise, and indeed these three coupling constants seem to meet at a common point. With the precise data available nowadays, they no longer do, as seen in the sketch in Fig. 9.5a. However, with the insertion of supersymmetry, they once again meet at a common µGUT 1016 GeV, as sketched in Fig. 9.5b. This gives an impetus to the possible correctness of supersymmetric grand unified theories.

We shall discuss supersymmetry in the next subsection. Briefly, it proposes a doubling of the existing particles, with opposite statistics. The new ‘supersymmetric particles’ are heavy, with a mass believed to be of order of 1 TeV (1012 eV). As far as the running of the coupling constants is concerned, these new particles provide additional shielding that serves to decrease the slope of every line, as can be seen by comparing Fig. 9.5b with 9.5a.

In the rest of this subsection, we will discuss GUT by ignoring supersymmetry. What the latter does is to add additional particles to the spectrum, but it does not invalidate the discussion in the rest of this subsection.

The SM group SU(3)C × SU(2)L × U(1)Y has rank 4. Namely, it contains four additive quantum numbers (I3, Y , and two additive quantum numbers from color). Since a GUT group must contain the SM group, its rank must be four or more. There is a rank 4 candidate group, SU(5), and a rank 5 candidate group, SO(10),