The creator of the neutrino is testing and teasing us. Moshe Gai The Oscillating Neutrino We do not know . . . [if] neutrinos are massive or massless. We do not know if the potentially massive neutrinos are Majorana or Dirac, and we do not know if these neutrinos can oscillate among flavours. An introduction to neutrino . . In short, there is a great deal we do not know about neutrinos. Jeremy Bernstein, 1984. masses and mixings

Richard Slansky, Stuart Raby, Terry Goldman, and Gerry Garvey as told to Necia Grant Cooper C

Looks left-handed. No—right-handed?!*

he neutrino, the theoretical construct The primer that follows explains why this of sixty years ago, has acquired a strange behavior would fit in with theoretical Tpresence in both physics and cosmology. expectations and how oscillations could reveal It is both actor and probe. It explains numerous neutrino masses no matter how small. It also mysteries of the observable world. Yet every new introduces questions that will become relevant. characteristic it reveals opens up more questions Why are neutrino masses so small? Do the very about its true nature. light neutrinos have very heavy relatives that For decades, these bits of matter have been make their masses small and give us hints of the described as massless, left-handed particles: new physics predicted by the Grand Unified left-handed because they were always “spinning” Theories? Are neutrinos their own antiparticles? counterclockwise in the manner of a left-handed Do neutrinos have very light sterile relatives corkscrew. But new evidence implies that neutrinos that provide a hiding place from all interactions? have very tiny masses and can spin in either Physicists continue to chase after neutrinos, direction. Remarkably, the new data also suggest and every time these ghostly particles are caught, that neutrinos might oscillate, or periodically they seem to point toward new challenges present themselves as one of several different types. and new possibilities.

*This neutrino must have mass.

8 Los Alamos Science Number 25 1997 Number 25 1997 Los Alamos Science he Oscillating Neutrino The Oscillating Neutrino

After Reines and Cowan detected Neutrinos in the strong decays are 10–16 second and First Family Second Family Third Family he neutrino in the late 1950s, particle Standard Model 10–23 second, respectively. hysics went through a spectacular It is precisely this lack of interaction Up u Charm c Top t flowering that culminated in the formu- The Standard Model identifies strength that makes the neutrino so Electric charge ϭ ϩ2/3. Electric charge ϭ ϩ2/3. Electric charge ϭ ϩ2/3. ation of the Standard Model. This twelve building blocks of matter (see elusive. For not only does the weak Protons have two up quarks; Is heavier than the u. Is heavier than the c. model incorporates all that is known Figure 1), six quarks and six leptons force create neutrinos, often through neutrons have one. bout the subatomic world. It identifies (and their respective antiparticles). The beta decay, but it also mediates the Mass 3 MeV/c2. Mass Ϸ 1,500 MeV/c2. Mass 175,000 MeV/c2. he most elementary constituents of quarks are the building blocks that have only processes that can absorb them. Ϸ Ϸ matter, the elusive neutrino being fractional electric charge and interact The intimate connection between Down d Strange s Bottom b mong them, and then describes all primarily through the strong nuclear the weak force and the neutrino has Quarks he ways in which these elementary force, also called the color force. Color sometimes made their separate proper- Electric charge ϭ Ϫ1/3. Electric charge ϭ Ϫ1/3. Electric charge ϭ Ϫ1/3. Protons have one down quark; Is heavier than the d. Is heavier than the s. onstituents can interact with and binds quarks together to form the ties difficult to sort out. In fact, the neutrons have two. ansform among each other. Awesome proton, the neutron, all nuclei, and all theory that the neutrino is massless and n scope, this theory provides a consis- the other hadrons (strongly interacting left-handed (and the antineutrino right- Mass Ϸ 6 MeV/c2. Mass Ϸ 170 MeV/c2. Mass Ϸ 4,500 MeV/c2. ent picture of every realm of the particles). The charged leptons are the handed) was invented to explain why hysical world: from the hot, dense building blocks that interact through the weak force violates the symmetry arly universe resulting from the the other three forces of nature (weak, known as parity, also called right-left, Electron e Muon ␮ Tau ␶ ig Bang to the thermonuclear furnace electromagnetic, and gravity) but never or mirror, symmetry. If the weak force Electric charge ϭ Ϫ1. Electric charge ϭ Ϫ1. Electric charge ϭ Ϫ1. t the center of the Sun, from through the strong force. As a result, conserved parity, any weak process and Is responsible for electrical Is heavier than the e. Is heavier than the ␮. henomena at the smallest, subatomic leptons are never bound inside the its mirror image would be equally and chemical reactions. istance scales accessible at particle nucleus by the strong force. The leptons likely. Instead, in 1956, C. S. Wu and Mass ϭ 0.511 MeV/c2. Mass ϭ 105 MeV/c2. Mass ϭ 1,782 MeV/c2. ccelerators to those at the farthest include the electron, the heavier coworkers observed a striking asymme- eaches visible through the Hubble “electron-like” muon and tau, and these try in the beta decay of cobalt-60 (see ␯ ␯ Leptons Electron Neutrino ␯e Muon Neutrino ␮ Tau Neutrino ␶ elescope. The same forces and sym- three particles’ neutral partners: the box “Parity Nonconservation and metries and the same set of elementary the electron neutrino, the muon the Two-Component Neutrino” on page Electric charge ϭ 0. Electric charge ϭ 0. Electric charge ϭ 0. Is paired with electrons by the Is paired with muons by the Not yet seen directly. uilding blocks seem sufficient to neutrino, and the tau neutrino. Among 32.) The asymmetry suggested that all weak force. Billions fly weak force. Assumed to be paired with escribe the underlying physics of these twelve constituents, only the neu- the antineutrinos emitted in the decay through us every second. the tau by the weak force. ll phenomena observed so far. trinos are nearly or exactly massless. had right helicity,1 that is, they were Mass ϭ 0 (assumed). Mass ϭ 0 (assumed). Mass ϭ 0 (assumed). But for over two decades, ever Dubbed “the little neutral ones” “spinning” like right-handed corkscrews ince the Standard Model was initially because they have no electric charge, (rotating clockwise around their direc- ormulated, expectations of “physics neutrinos interact with matter only tion of motion). But in a universe with Anti-up ෆu Anticharm cෆ Antitop ෆt

eyond the Standard Model” have been through the weak force and gravity. right-left symmetry, an equal number Antidown ෆd Antistrange ෆs Antibottom ෆb lmost palpable among those familiar Recall that the weak force creates of antineutrinos should have been with the model’s details: The theory neutrinos through beta decay (see the spinning counterclockwise, like left- Positron eϩ Antimuon ␮ϩ Antitau ␶ϩ

ust has far too many arbitrary parame- box “Beta Decay and the Missing Ener- handed corkscrews. The fact that only Antiparticles Electron antineutrino ␯ ␯ ෆe Muon antineutrino ෆ␮ Tau antineutrino ␯ෆ␶ ers and mysterious relationships to be gy” on page 7). In that particular weak right helicity was observed is an he final one. Now, after many years of decay process, a neutron, either free or example of “maximal” parity violation. earching, the first hard evidence for in a nucleus, transforms into a proton, ew physics may be at hand. The new and two leptons are created: an electron Figure 1. Building Blocks of Matter in the Standard Model hysics—nonzero neutrino masses and (or “beta” particle) and an electron 1Helicity is identical to handedness (or chirality) The elementary building blocks of lepton pair. Apart from the neutrinos, cosmic rays (mostly protons) with mixing” among the neutrinos from antineutrino. More generally, the weak for massless neutrinos and nearly identical to matter in the Standard Model are six which are massless, the particles in the molecules in the earth’s atmosphere. handedness for particles traveling near the speed ifferent families—has long been force is the force of transmutation, able of light. For that reason, helicity is sometimes quarks and six leptons, each carrying an second and third families are more The first hint that there are particles nticipated because it parallels the to transform one type, or “flavor,” of loosely referred to as “handedness.” For massive intrinsic spin of 1/2. The first family con- massive than their counterparts in the beyond the first family came in 1937 ehavior seen among quarks. But still, quark into another or one flavor of lep- particles, however, the two quantities are quite tains one quark pair—the up and first family. They are also unstable and with the discovery of the muon. The top different. Massive particles must exist in right- is quite exciting because it both ton into another. It is also the “weak- and left-helicity and in right- and left-handed the down—and one lepton pair—the only stick around for tiny fractions of a quark, the heaviest member of the third ffirms the central concepts of the est” known force (apart from gravity), states. As illustrated in the box on page 32 and electron and the electron neutrino. second because the weak force allows family, was not seen until 1995. And so tandard Model and appears to point about a hundred million times weaker the cartoon on page 29, helicity is the projection These four particles make up the them to decay into less massive far, the tau neutrino has not been of the spin along the direction of motion. It can oward the most popular extensions, than electromagnetism at “low” ener- be measured directly, but its value depends on ordinary matter that is found on Earth particles. The more massive versions of detected directly. Nevertheless, the three he Grand Unified Theories. gies, which means that it acts a hundred the frame from which it is viewed. In contrast, and throughout most of the immediate the quarks and leptons are created in families are so similar in structure that handedness is a relativistically invariant quantity, This article introduces the neutrino million times more slowly. For exam- universe. In particular, the proton is very high energy processes at the center some of their members were anticipated but it is not a constant of the motion for a free n the context of the Standard Model ple, unstable particles decay through particle and cannot be measured directly. Never- made of the quark triplet duu, and the of stars and galaxies, in high-energy long before they were observed. nd explains how the new data on the weak force in times on the order of theless, handedness is the quantity that describes neutron is made of the quark triplet udd. accelerators, and at about 30 kilometers All particles have corresponding the properties of the weak force and of the –8 The second and third families are also above the surface of the earth through antiparticles (listed last in this figure) eutrinos relate and suggest extensions 10 second, whereas the characteristic particle states that interact through the weak o that theory. times for electromagnetic decays and force and have definite weak charges. composed of one quark pair and one the collision of very high energy with opposite charge.

0 Los Alamos Science Number 25 1997 Number 25 1997 Los Alamos Science he Oscillating Neutrino The Oscillating Neutrino

Parity Nonconservation and the Massless Two-Component Neutrino

The helicity of a particle relates its intrinsic spin to its direction of motion. momentum is measured by the average value of the dot product s иp, (c) Maximum Parity Violation in the Cobalt-60 Experiment All quarks and leptons, including the neutrino, carry 1/2 unit of which changes sign under a parity inversion and therefore must be (a) Four Statespp of a Spin-1/2 Particle intrinsic angular momentum s, or spin (measured in units of h–). Spin is zero if parity is conserved in a given process. The nearly perfect quantized, and for a spin-1/2 particle, it has two values relative to any correlation of the nuclear spins and the electron momenta in Magnetic selected axis of quantization, which we choose to call the z-axis. The the cobalt experiment was an example of maximal parity violation. field Right- s Left- helicity helicity spin is often represented by a pseudovector (red arrow) that points up particle particle (sz ϭ 1/2 ) or down (sz ϭ Ϫ1/2) along the axis of quantization, depending To explain the violation, Lee and Yang assumed that the antineutrino 60 s on whether the particle is spinning clockwise or counterclockwise around was always emitted with right helicity. As shown in (d), the decay Co nuclei with that axis when it is viewed from below. Helicity uses the direction of decreases the nuclear spin by one unit. Aligning the spins of the spins polarized p p Ð along magnetic e (1 ؍ motion, or the momentum p, as the axis of quantization, where helicity electron and the antineutrino along the nuclear spin (1/2 ؉1/2 s field is defined as ␭ ϭ s ؒ p/p ϭ Ϯ1/2. will make up for this decrease. If the antineutrino always has right eÐ Electrons emitted helicity (momentum and spin aligned), the electron will have to be Right- Left- eÐ in lower half plane helicity helicity As shown in (a), spin-1/2 particles usually have four independent states: emitted with left helicity (momentum opposite to spin) and in the antiparticle santiparticle the particle with right or left helicity and the antiparticle with right or left direction opposite to the nuclear spin, which is just what was helicity. A particle has right helicity (␭ ϭ 1/2) if its spin and momentum observed in the cobalt experiment. Yang and Lee formalized this interpretation in the theory point in the same direction. It has left helicity (␭ ϭ Ϫ1/2) if its spin and of the two-component neutrino (1957), which postulates that the neutrino comes in only two

Spin aligned Spin opposite momentum point in opposite directions. The mirror image of a right- forms, a left-helicity (␭ ϭ Ϫ1/2) particle and a right-helicity (␭ ϭ 1/2) antiparticle. But definite with momentum to momentum helicity particle is a left-helicity particle, as shown in (b). (Note that, being helicity has a profound consequence. To have left helicity in all coordinate systems moving a pseudovector, s does not change direction under spatial inversions. Like with constant velocity relative to each other, as required by special relativity, the left-helicity p = Particle momentum = Spin pseudovector total angular momentum J and orbital angular momentum l, it transforms neutrino must be traveling at the speed of light. Otherwise, one could imagine observing the s ,as r ؋ p does.) Until the 1950s, it was taken for granted that the laws of particle from a coordinate system that is moving faster than the neutrino. As one zipped past physics were invariant under a mirror reflection or an inversion of spatial the neutrino’s momentum would appear to be reversed, while its spin direction would remain coordinates (also called parity inversion). If parity were conserved, a unchanged. The neutrino would then appear to have right helicity! So, helicity remains spin-1/2 particle would exist in both left- and right-helicity states. independent of the reference frame only if the neutrino moves at (b) Mirror Reflection of a Right-Helicity Particle the speed of light. But then the neutrino must be a massless (d) Explanation of Cobalt-60 Experiment But in June of 1956, two young physicists, C. N. Yang and T. D. Lee, particle. Helicity then becomes identical to the relativistically ν Right suggested that the weak force might violate parity conservation, and invariant quantity known as “handedness,” so the neutrino is a e helicity they outlined several types of experiments that could test their left-handed massless particle. hypothesis. Six months later, C. S. Wu reported the results of one such + axis

experiment. Wu aligned the spins of cobalt-60 nuclei along an external The theory of the two-component massless neutrino fits nicely with z- Ð Left magnetic field and measured the directions of the electrons emitted by the Gell-Mann and Feynman formulation (1958) of the left-handed e helicity those nuclei in beta decay: weak force (also known as the VϪA theory, for vector current minus Jz 5 = 4 + 1/2 + 1/2 Mirror image has axial vector current, a form that violates parity maximally). In this left helicity 60 60 Ϫ Co → Νi28 ϩ e ϩ ␯ . theory, the weak force picks out the left-handed components of 27 ෆe 60 60 Ð ν particles and the right-handed components of antiparticles. Since Co Ni + e L + eR Particle has 27 28 right helicity The electrons were almost always emitted in the direction opposite to the neutrino interacts only through the weak force, the two missing the nuclear spins, as shown in (c). If parity were conserved, there components of the neutrino (the right-handed particle and the should be no correlation between the spins and the momenta of the left-handed antiparticle) would never be “seen” and would be electrons emitted in the decay. A correlation between spin and superfluous—unless the neutrino had mass.

The results of the cobalt experi- itself was soon recognized to violate could have a small mass. and the Search for Neutrino Mass” on family; and the tau and its neutrino families. For example, the muon and ment were formalized in the theory of parity maximally because it acts Nonetheless, the original theory page 86). The assumption of massless the tau family (refer again to Figure 1). the muon neutrino can transmute into the two-component massless neutrino, on only the left-handed states of of the massless, left-handed neutrino neutrinos, however, has a consequence Each lepton family is part of a much each other through the weak force (no according to which the antineutrino is all quarks and leptons, whether was included in the “minimal” in the minimal Standard Model: It larger family that also includes the change in the muon-family number), always right-handed (or has right they have mass or not. In other Standard Model, primarily because implies that lepton-family number is quarks and the respective antiparticles but the muon cannot decay directly into helicity), the neutrino is always left- words, left-handedness is an intrinsic there was no evidence to the contrary. conserved. Each lepton family consists of the leptons and quarks. an electron. Instead, a member of the handed (or has left helicity), and the property of the weak force and All direct measurements of neutrino of a lepton pair. The electron and its Conserving lepton-family number muon family (the muon neutrino) must neutrino is a massless particle (see not necessarily of the neutrino. masses have yielded only upper limits neutrino constitute the electron family; means preserving strict boundaries also be produced during muon decay. the box above). But the weak force Thus, in principle, the neutrino (see the article “Tritium Beta Decay the muon and its neutrino the muon between the electron, muon, and tau Similarly, a tau cannot decay directly

2 Los Alamos Science Number 25 1997 Number 25 1997 Los Alamos Science he Oscillating Neutrino The Oscillating Neutrino

Figure 2. The Electromagnetic Force nos whose interactions are unobservable rotational invariance, and it has the tion between the gauge particle and the charges before a reaction equals the he electromagnetic force is transmitted through the exchange of the photon, the in the detectors being used. As shown in profound consequence that the total quark or lepton actually causes one of sum of the charges after the reaction. auge particle for the electromagnetic field. later sections, this oscillation from one angular momentum of an isolated sys- the internal rotations defined by the flavor to another can happen only if the tem is conserved and therefore never local gauge symmetry, that is, the emis- The Strong Force. The local gauge ) Electron scattering. Two elec- Initial state Final state different neutrino types have different changes. Similarly, if a system is invari- sion or absorption of a gauge particle symmetry for the strong force is called ons (straight arrows) interact with masses, so measurement of oscillation is ant under time translations, its total causes that quark or lepton to transform color symmetry. The color charge has Electron Ð Ð ach other through the exchange e e Electron proof that neutrinos have mass. energy is conserved. If a system is into another member of the same multi- three distinct aspects that, for conve- a virtual photon (wiggly line). Positive results from the LSND oscil- invariant under spatial translations, its plet. To give a geometrical analogy, if nience, are labeled red, green, and blue he direction of time is from left to lation experiment have therefore caused total linear momentum is conserved. the world were perfectly symmetrical (no relation to real colors is intended). ght. The dots represent interaction γ Photon a stir in the physics community. These In addition to these space and time and the quarks and leptons in Figure 1 The gauge particles are called gluons, ertices, where one electron emits results could explain the “solar-neutrino symmetries, the Standard Model has were like the faces of a cube, then the the quarks can carry any of the three photon and the other electron puzzle” (the apparent deficit in the num- certain powerful internal symmetries, action of the gauge particles would be color charges, and two colored quarks bsorbs it. ber of solar neutrinos) and could also called local gauge symmetries, that to rotate, or transform, one face (quark interact and change color through the Ð Electron eÐ e Electron have impact on other topics in astro- define both the charges of the quarks or lepton) into another. exchange of one of the eight colored physics and cosmology that involve large and leptons and the specific nature of gluons. Like the gauge symmetry for numbers of neutrinos. On a more the forces between them. Just as cubic The Electromagnetic Force. In the electromagnetic force, the gauge ) Electron-positron annihilation abstract note, nonzero neutrino masses symmetry implies the existence of four the Standard Model, each force (strong, symmetry for the strong force implies nd creation. When the diagram Initial state Final state and oscillations among flavors would corners, six faces, and a group of rota- weak, and electromagnetic) is associat- that the gluons are massless and that ove is rotated by 90°, it represents parallel the properties and behaviors seen tions that interchange the position of ed with its own local gauge symmetry, the total color charge is conserved. Electron eÐ eϩPositron electron and positron that annihi- among the quarks and would thus point the cube’s faces and corners, the inter- which, in turn, defines a set of charges Because the gluons carry color, and are e at the left interaction vertex to toward a greater symmetry between nal symmetries of the Standard Model and a set of gauge bosons that are thus like electrically charged photons, come a virtual photon, and then quarks and leptons than now exists in imply that (1) the quarks and leptons the “carriers” or mediators of the force the strong force is highly nonlinear and Photon e virtual photon converts back to the Standard Model. They might even fall into certain groups or particle mul- between the charged particles. The elec- has some very bizarre properties. One γ electron and positron at the right point toward a more encompassing and tiplets, (2) the charges of the particles tromagnetic force is the simplest to is that quarks can never appear individ- eraction vertex. Note that, when unifying symmetry that has been antici- in each multiplet are related in a defi- describe. Figure 2(a) shows that two ually, and another is that all observable arrow points backward in time, pated in the Grand Unified Theories, in nite way, and (3) there is a group of electrons, or any particles carrying elec- states of quarks and antiquarks epresents the antiparticle moving Positron eϩ eÐ Electron which quarks and leptons are different internal rotations that transform one tric charge, interact by the exchange of (protons, neutrons, pions, and so forth) ward in time. aspects of the same field and the strong, member of each multiplet into other a photon, the gauge boson for the elec- are colorless bound states, that is, weak, and electromagnetic forces are due members of that same multiplet. tromagnetic field. The exchange process they have no net color charge. ) The interaction vertex. All processes involving the electromagnetic force can be built up to a single symmetry. But there is much more. Local gauge can be pictured as a game of catch: In the discussions that follow, we om the basic interaction vertex. In the left vertex, an electron emits or absorbs a symmetries are those in which the mag- One electron emits (throws) a photon, can ignore the strong force because lep- hoton, and in the right vertex, a photon turns into an electron-positron pair or vice versa. nitude of the transformation can vary in the other electron absorbs (catches) it, tons do not carry the color charge and ϩ Gauge Symmetries in space and time. If the results of experi- and the net result is that the two parti- the part of the Standard Model that Ð Ð e Positron Electron e e Electron the Standard Model ments are to stay invariant under such cles repel, or scatter from, each other. describes the color interactions of the transformations (which is what symme- The classical electromagnetic field that quarks (known as quantum chromody- Photon γ The Standard Model is built almost try means), gauge particles must exist explains how particles can interact at namics) will not be affected by new entirely from symmetry principles, and that transmit or mediate the forces a distance is thus replaced by the data on neutrino masses and mixings. γ Photon Electron emits Photon turns into an those principles have enormous predic- between the quarks and leptons. One exchange of a gauge particle.2 The or absorbs a photon. electron-positron pair, or the pair annihilates tive power. Symmetry means an invari- quark or lepton emits a gauge particle, photon, of course, exists as an indepen- The Weak Force. The Standard Model into a photon. eÐ Electron ance of the laws of physics under some and another quark or lepton absorbs it. dent particle and can itself transform identifies two local gauge symmetries group of transformations. And in the Through this exchange, each “feels” the into a particle-antiparticle pair, most for the weak force and, therefore, two nto a muon or an electron unless a tau solar- and atmospheric-neutrino experi- formalism of quantum field theory, the force of the other. Further, the interac- often an electron-positron pair. types of weak charges (weak isotopic eutrino is also produced. Finally, con- ments, suggest that muon neutrinos can invariance implies the existence of a Figure 2(b) illustrates this process, and charge and weak hypercharge). As a ervation of lepton-family number periodically change into electron conserved quantity. One example is the Figure 2(c) shows the basic interaction consequence, there are two types of 0 means that an electron neutrino cannot neutrinos, and vice versa, as they travel group of rotations. We take for granted, 2The exchange of the photon does not change the vertex. The local symmetry implies that gauge particles, the W and the Z hange into a neutrino from another through the Sun or even through empty and know from high-precision measure- identity of the charged particle; it only rotates all possible interactions involving bosons, that carry the weak force amily, or vice versa. These predictions space. One consequence would be that ments, that space has no preferred the phase of the quantum field that describes the the photon and electrically charged between particles with weak charges. charged particle. The point of the local symmetry f the minimal Standard Model have electron neutrinos arriving at Earth from direction and that we can rotate an iso- is that the phase is not observable. That phase particles can be built up from this The neutrino, although electrically eld up to increasingly precise tests. the center of the Sun would appear to lated system (a group of atoms, a solar rotation is compensated for by the photon field, basic interaction vertex. neutral, carries both weak isotopic Recent evidence, however, is chang- be too few in number although, in fact, system, a galaxy) about any axis (or and thus the interaction Lagrangian is invariant Finally, the local gauge symmetry charge and weak hypercharge and thus under phase rotations at every space-time point. ng this picture. Data from the neutrino the right total number would be present. rotate the coordinates we use to The local gauge (or phase) symmetry of electro- holds only if the photon is identically interacts with matter through the scillation experiment at Los Alamos, Some would be “invisible” because they describe that system) and not change magnetism is a local unitary symmetry in one massless, and the symmetry guarantees exchange of either the W or the Z0. dimension, U(1). The symmetry implies electric nown as LSND (for liquid scintillator would have temporarily changed into the laws of physics observed by current conservation at every point as well as that electric charge is a conserved Let us first consider the processes eutrino detector), as well as from another flavor—into muon or tau neutri- that system. This property is called global charge conservation. quantity, that is, the sum of the electric mediated by the W. This gauge boson

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Figure 3. Beta Decay and Other Processes Mediated by the W comes in two forms, the Wϩ and the Figure 4. Electron Neutrino–Electron Scattering he W is the charged gauge particle of the weak force, so processes mediated by the W involve the exchange of one unit of electric WϪ. Each carries one unit of electric The electron and the electron neutrino can interact through the exchange of either the harge. Quarks and leptons therefore change their identities through the emission or absorption of the W. In all the processes charge (plus or minus, respectively), so W, shown in (a), or the Z0, shown in (b). hown here, the arrow of time is from left to right, and an arrow pointing backward represents an antiparticle moving forward in that when a particle carrying the weak me. The arrow on the W indicates the flow of electric charge. Note also that in each of these processes, electric charge is isotopic charge emits or absorbs a W, it Initial state Final state onserved at every step. gains or loses one unit of electric (and weak isotopic) charge. The particle Ð ν (a) Charged-current scattering. Initial state Final state Initial state Final state Electron e e Electron thereby changes its identity. Figure 3 neutrino An electron and an electron d d Ð illustrates neutron beta decay, muon neutrino interact through Neutron u u Proton Muon µ νµ Muon neutrino d u beta decay, inverse beta decay, and exchange of a WϪ. The electron electron or positron capture, all of WÐ transmutes into an electron WÐ WϪ which are processes mediated by the W. neutrino as it emits a WϪ, and the eϪ Electron The transmutation of the down quark electron neutrino transmutes into eÐ Electron Ϫ → Ϫ an electron as it absorbs a WϪ. ␮ ␯␮ϩ e ϩ ෆ␯e into the up quark through emission of → Ϫ Ϫ n p ϩ e ϩ ෆ␯e ν Electron the W is the origin of the transmutation ν e ν eÐ Electron e Electron antineutrino of a neutron into a proton in ordinary Electron e antineutrino neutrino beta decay. In inverse beta decay, the a) Neutron beta decay. A neutron decays to a proton when a d quark in (b) Muon beta decay. This process is exactly analogous to the process used by Reines and Cowan to Initial State Final State he neutron emits a WϪ and transmutes into a u quark. Like the photon, beta decay of the neutron. The muon transforms into a muon detect the electron antineutrino, an up Ϫ Ϫ Ϫ he W can decay into a particle and an antiparticle, but here the particle neutrino as it emits a W ; the W decays into an electron and quark transmutes into a down quark as it Electron eÐ eÐ Electron (b) Neutral-current scattering. s the electron, and the antiparticle is the electron antineutrino. an electron antineutrino. emits a Wϩ, and an electron antineutrino The electron and electron transmutes into a positron as it absorbs neutrino interact through that Wϩ. Because of the exchange of exchange of the Z0. No charge Initial state Final state Initial state Final state 0 electric charge, the processes involving Z is transferred, and the particles d d d d the exchange of the W are called maintain their identities. Proton u u Neutron Proton u u Neutron “charged-current” weak processes. They u d u d are to be contrasted with the “neutral- current” processes mediated by the Z0, Electron ν ν Electron ␯ ϩ p → n ϩ e ϩ Wϩ eϪϩ p → n ϩ ␯ ϩ e e ෆe e in which no electric charge is neutrino neutrino W exchanged. Note that this picture of the weak force, in which particles interact at through the charged-current weak inter- Consequently, most quarks and leptons ν Electron ν eϩPositron Electron eÐ e Electron a distance through the exchange of the action. In neutral-current scattering, carry both types of weak charge as well e neutrino antineutrino W, modifies Fermi’s original current-cur- the electron neutrino emits the Z0, and as electric charge and can interact rent theory of beta decay, in which two the electron absorbs it. The two particles through exchange of the photon, the W, c) Inverse beta decay. An electron antineutrino interacts with a proton (d) Electron capture. This process is similar to inverse beta decay, currents interacted at a point (see the scatter from each other, but each main- or the Z0. For the neutrino, however, y exchanging a Wϩ. The u quark emits a Wϩ and transmutes to a d except that an electron interacts with the proton. The electron trans- box “Fermi’s Theory of Beta Decay tains its identity as in electromagnetic the specific sum that equals the electric uark (thus the proton turns into a neutron). The electron antineutrino mutes into an electron neutrino as it absorbs the Wϩ. and Neutrino Processes” on page 8). scattering. All neutrino types can charge (and couples to the photon) is ansmutes into a positron as it absorbs the Wϩ. The distance over which the W is interact with electrons through neutral- zero, so that the neutrino is electrically µÐ exchanged is very short, on the order of current scattering, but only electron neutral. (The electroweak theory, which u → d µÐ Ϫ → ν u d ␮ ␯␮ µ 10Ϫ16 centimeter, which is substantially neutrinos can interact with electrons describes the electromagnetic and weak u Ð W less than the diameter of a proton. through charged-current scattering. forces, unifies the description of the ϩ W ϩ W The scattering of electron neutrinos That additional interaction may be photon and the Z0 in a complicated way ϩ by electrons is a purely leptonic reaction important in enhancing the oscillation that will not be discussed in this article.) W that illustrates both charged-current and of electron neutrinos that exit the Sun Now, let us consider the particle neutral-current modes (see Figure 4). In (see the article “MSW” on page 156). multiplets that are consistent with the d charged-current scattering, the electron It is not coincidental that neutral- local symmetries of the weak force. ν Ϫ Ϫ µ emits a W and loses one unit of nega- current scattering resembles electro- Figure 5 lists the quarks and the lep- Ð e → ␯ ν e e e tive electric charge to become an elec- magnetic scattering. One of the great tons, along with their weak and electric (e) Basic interaction vertices of the charged-current tron neutrino. At the other end of this successes of the Standard Model was to charges. These particles fall naturally weak force. All the processes illustrated above can be built from exchange, the electron neutrino absorbs show that the weak force and the elec- into three families (columns) consisting eÐ variations of the interaction vertices shown here. These are the WϪ and gains one unit of negative tromagnetic force are related. The two of a pair of quarks and a pair of lep- analogous to the vertices shown in Figure 2 for the electromag- charge to become an electron. The ini- types of weak charges, when added tons. Each pair is a doublet whose Ð W Ð W netic force, except here the gauge particle, the W, carries one tial and final particles are the same, but together in a specific linear combina- members transform into each other ν e unit of electric charge. each has been transmuted into the other tion, are equal to the electric charge. under the rotations of the local symme-

6 Los Alamos Science Number 25 1997 Number 25 1997 Los Alamos Science he Oscillating Neutrino The Oscillating Neutrino

(a) Weak Doublets First Family Second Family Third Family try group (called weak isospin)3. The try is apparent in them. Also, the local and leptons are expected in all theories Q I w Q w Q w 3 I 3 I 3 red arrows indicate the transmutation of gauge symmetry specific to electromag- in which these particles are relatives and one member into the other through netic interactions is not affected at all can transmute into each other. 1 2 1 2 1 ϩ2 ϩÐ uЈ ϩÐ ϩÐ cЈ ϩÐ ϩÐ tЈ 3 2 3 2 3 2 absorption or emission of the W. Thus, by the symmetry breaking, and the members of a weak isospin doublet are electric charge is always conserved. Quarks Quarks like the two faces of a coin, and inter- These complications notwithstanding, The Mysteries of Masses 1ϪÐ 1ϪÐ d Ј 1ϪϪ 1ϪÐ sЈ 1ϪÐ 1ϪÐ bЈ 3 2 3 2 3 2 action with the W flips the coin from each of the forces in the Standard Model and Families in one face to the other. (Note that, for is derived from a local gauge symmetry; the Standard Model each weak particle doublet, there is a the three forces, therefore, look similar

1 ␯e νµ corresponding weak antiparticle dou- in that they act through the exchange of Figures 1 and 5 reflect two different 0 ϩÐ 0 1ϩÐ 0 1ϩÐ ντ 2 2 2 blet. Because the charge-changing weak gauge bosons. The reliance on local ways of defining and placing particles Leptons Leptons force is left-handed, the particle gauge symmetries has worked so well in families: The families in Figure 1 µ e 1 τ doublets include only the left-handed that many theorists have tried to extend contain particles with definite mass (the 1Ϫ 1ϪÐ 1Ϫ 1ϪÐ 1Ϫ ϪÐ 2 2 2 components of the particles, whereas this idea even further. They are trying unprimed quarks), whereas those in the antiparticle doublets include only the to find a local gauge symmetry that Figure 5 contain particles defined by = Electric charge w = Weak isotopic charge = Transmutation by the W I 3 right-handed components of the antipar- combines into one all the separate sym- the local gauge symmetries (the primed Q ticles. That technicality becomes impor- metries associated with the strong and quarks). These local gauge symmetries (b) Charged-Current Weak-Interaction Vertices tant in the later discussion of mass.) electroweak forces. Such theories predict provide the guiding principles in the First Family Second Family Third Family Alternatively, if the W is emitted and that the quarks and leptons within a construction of the Standard Model u' d' s' t' b' not absorbed by another weakly charged family will fall into one multiplet, one and in most extensions to it; therefore, particle, it decays into one member of set of particles that transform into each the weak states described in Figure 5 c' the doublet and the antiparticle of the other through the gauge particles associ- offer a fundamental starting point W other member of the doublet. This ated with the local symmetry. This in shaping our understanding of occurs, for example, in beta decay (refer effort is called grand unification and is the fundamental particles. ϩ ϩ ϩ W to Figure 3). The WϪ is not absorbed the basis of the Grand Unified Theories. If we were to ignore the masses of W but decays to an electron and an It is remarkable that theorists antici- the particles and focus on the symme- Ϫ→ Ϫ ν electron antineutrino (W e ϩ ෆ␯ ). pated not only the structure of the tries, each family would look like a ν eϪ νµ µϪ τ ␶Ϫ e e Likewise, the Wϩ can decay to a electroweak force, but also the existence carbon copy of the other two. In other positron (antielectron) and an electron of the charmed quark (the partner to words, each particle would have a ϩ→ ϩ neutrino (W e ϩ ␯e). the strange quark) and later the top and “clone” in each of the other two fami- This brief introduction to the forces the bottom quarks by identifying the lies that has identical weak and electric ϩ ϩ ϩ W W W associated with and derived from the correct local gauge symmetry for charges and that has a partner with local gauge symmetries needs one cru- the weak force and then predicting that which it forms a doublet under the cial addition. The local gauge symme- all quarks and leptons form doublets weak force. The quark clones are u′, c′, tries of the weak force are not exact under that symmetry. and t′ and their weak partners d′, s′, and symmetries of nature, and one sign of The symmetry of the weak force was b′, respectively. The lepton clones are igure 5. The “Weak” States—Particles Defined by the Local Symmetries of the Weak Force the symmetry breaking is that, unlike not immediately apparent from experi- e, ␮, and ␶ and their weak partners ␯e, he particle doublets, defined by the doublet transforms into the other member the W means that the muon and the the photon and the gluons, which must ment for several reasons. For example, ␯␮, and ␯␶, respectively. eak isospin symmetry of the weak force, by absorbing or emitting the W, the muon neutrino or the tau and the tau neu- be massless to preserve the local gauge the charm, top, and bottom quarks and In fact, if the local gauge symme- re listed in (a) along with their electric gauge particle for the charged-current trino could replace the electron and its symmetry, the W and the Z are very the tau particle are very heavy. They tries of the weak force were exact, the w harge Q and weak isotopic charge I3 . weak interaction. Only the left-handed neutrino wherever the latter pair appears massive, weighing about 100 times were not observed at low energies, and quarks and leptons would all be mass- he particles fall into three families, each particles (or right-handed antiparticles) in the charge-changing weak processes the mass of the proton. More precisely, thus half of all family members were less. There is no way to include in the ontaining a quark weak isospin doublet carry the weak isotopic charge and are in Figures 3 and 4. (Whether those the mechanism that gives mass to the not known to exist. Also, the physical theory a “mass” term that remains nd a lepton weak isospin doublet. (For members of the doublet. In the quark sec- processes actually occur with the heavier particles breaks the weak symmetries quarks listed in Figure 1 are not identi- invariant under those local symmetries. ach weak particle doublet, there is also tor, the u′ quark transforms into the d′ leptons depends on the available energy.) and, in certain situations, causes the cal with the members of the quark dou- (The general features of mass terms for corresponding weak antiparticle doublet quark and vice versa, the c′ quark trans- The similarities among the weak isospin weak charges not to be conserved. blets listed in Figure 5. In the next spin-1/2 particles are described in the hat is not shown.) The quarks in each forms into the s′ quark and vice versa, pairs extend to the electric-charge assign- However, processes mediated by the section, we examine what is known sidebar “Neutrino Masses” on page 64.) oublet have been labeled with primes, u′ and the t′ quark transforms into the b′ ments as well. In each quark doublet, one weak gauge particles do conserve the about the differences between those two In reality, particles do have mass, and nd d′ for example, to indicate that the quark and vice versa. In the lepton sec- member has electric charge ؉ 2/3; the weak charges, and the original symme- sets of quark states because there is a thus the Standard Model contains a eak quark states are distinct from the tor, the electron and the electron neutrino other, ؊ 1/3. In each lepton doublet, one strong possibility that leptons may be symmetry-violating mechanism known uark states shown in Figure 1. (The dis- transform into each other through interac- member has charge zero, and the other 3Weak isospin symmetry is an example of described by two sets of states analo- as the Higgs mechanism. This mecha- nction will be made clear in the text.) tion with the W, as do the muon and has charge ؊1. The weak-isotopic-charge the special unitary symmetry in two dimensions, gous to the quark sets. In that case, neu- nism was specifically introduced into SU(2). Rotational symmetry, which leads to s indicated by the red arrows and also muon neutrino and the tau and tau assignments are likewise maintained from the allowed states of angular momentum, is trino oscillations would be predicted. the Standard Model to explain the xpanded in (b), one member of the neutrino. This universal interaction with family to family. another example of SU(2) symmetry. These analogous properties of quarks masses of the weak gauge particles,

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Families of Mass States Families of Weak States Lepton Mass States Lepton Weak States

First Second Third First Second Third First Second Third First Second Third

u Ј Ј tЈ ν ν ν ν ν ν 1 2 3 e µ τ c t c Quarks u d s d' sЈ bЈ e µ τ e µ τ b

ν ν ν ν ν ν Figure 7. Lepton Mass States and Weak States for Nonzero Mixing among the Leptons e µ τ e µ τ If neutrinos have mass and there is mixing among the leptons as there is among the quarks, all the mixing can be placed among Leptons the neutrinos, the neutral components of the weak doublets. Compare these lepton states with those in Figure 6. Although there is no mixing among the leptons in the Standard Model, present oscillation data suggest that such mixing may indeed occur. e µ τ e µ τ However, the pattern of mixing among the leptons is an open question. This figure suggests one possible pattern (shown by the color mixtures), which involves mainly the second and third families.

the upper half of the doublets are the weak doublets “interact” with the Why is there mixing among the equivalent: u = u′, c = c′, and t = t′. It Higgs background and acquire mass, quarks and not among the leptons? igure 6. A Comparison of Mass which must be very large to account the different quark states. Experiment should be stressed, however, that no the resulting states of definite mass do In the Standard Model, this difference States and Weak States in for the short range and the reduced shows that the quark states of definite matter which way one views the mix- not conserve the weak charge and thus follows directly from the assumption he Standard Model strength of the weak force relative to mass (shown in Figure 1) are not the ing, the quark states that transmute into break the symmetry. Indeed, they cease that all three neutrinos have the same he mass states (colored squares at left) the electromagnetic force. same as the quark states that make up each other through the action of the W being the states in the weak doublets. mass, namely, zero. The mathematical nd the weak states (colored squares at The same general mechanism is the weak doublets. (Recall that the (red arrows) are always the members of In the most general version of this argument is given in the sidebar ght) are two alternative descriptions of assumed to explain the masses of the quark weak states have been labeled the weak doublets. symmetry-breaking, mass-generating “Family Mixing and the Origin of he spin-1/2 particles of the Standard quarks and leptons, but the theory has with primes.) The weak force seems to The mixing that results from the scheme, the particles that acquire defi- Mass” on page 72. Model. Here, the first, second, and third so many undetermined constants that have a kind of skewed vision that pro- nonalignment between mass and weak nite masses through interaction with But as we said earlier, there is no amilies of weak states are represented experiment rather than theory is required duces and acts on quarks that are mix- states is a natural outcome of the the Higgs background are mixtures of fundamental principle that keeps the y colors: the greens, the purples, and to determine the masses. A theory of tures of the mass states from the differ- symmetry-breaking mechanism through the weak states from the different weak neutrinos massless. If they have small he blues, respectively. By convention, masses for spin-1/2 particles has yet to ent families. Equivalently, the which particles acquire mass in the families. Indeed, the quarks in the Stan- masses and acquire those masses ll mixing among the quarks is placed in be found. Whatever the solution, it must symmetry-breaking mechanism that Standard Model. According to the dard Model follow this most general through the Higgs mechanism, the mass he lower half of the mass state quark give different masses to the clones in gives particles their masses mixes the Higgs mechanism, the ground state, or scheme. The Higgs mechanism thus states would likely be mixtures of the oublets. Thus, the mass states d, s, and each family, because as can be seen in quark clones in the weak families to lowest-energy state, has no physical causes the mismatch between the quark weak states. The lepton mass states are shown as mixtures of the particular going from left to right along any row in create mass states. particles (it is called “the vacuum”), but weak states and mass states. would then change to look like those in olors that represent the quark weak Figure 1, the three families form a mass Figure 6 stresses this point. Each it contains an everpresent background Since the leptons also acquire mass Figure 7, in which the neutrino mass ′ tates d, s′, and b′. For example, the hierarchy from light to heavy. That is a family of weak states is denoted by a of virtual, spin-zero Higgs particles. through the Higgs mechanism, one states ␯1, ␯2, and ␯3 are related to the mass state d is mostly green but tantalizing pattern with no explanation. different color (green, purple, and That background interacts with the might expect to find a similar type of three weak states ␯e, ␯␮, and ␯␶ by a set ontains a purple stripe whose area There are more mysteries surrounding blue), and the mass states are shown as quarks, leptons, and gauge bosons and mixing among the lepton weak states of mixing parameters analogous to epresents the fraction of d in the weak the states defined by the weak symme- mixtures of weak states (mixed colors). provides a “drag” on them, which we and mass states. So far, experiments those relating the quark weak states tate s′, and so forth. Most of the quark tries (those shown in Figure 5). Why do Areas of color represent the fraction of observe as rest mass. The Higgs parti- have not confirmed that expectation, to the quark mass states. mixing occurs between the first and quarks and leptons fall naturally into dis- a mass state that is in a particular weak cles are weak doublets, and the back- and the Standard Model holds that the Mixing among the leptons would econd families. The mass and weak tinct families? Are these two types of state. Notice that most of the quark ground of virtual Higgs particles, by lepton mass states and weak states are allow processes that violate lepton- tates for the quarks in the upper half of particles related to each other in some mixing occurs between the first two definition, has a nonzero value of weak essentially identical. The weak force family number, but because neutrinos u′, way that is not yet apparent but that is families. The exact amounts of mixing charge. When the quarks and leptons in always appears to act on the weak have such small masses, we would ؍ he doublets are equivalent: u t′. In the Standard Model, anticipated in the Grand Unified cannot be derived from theory; instead, doublets within a family, and there is no expect most of those processes to be ؍ c′, and t ؍ 4 here is no mixing among the leptons, Theories? Why are there three different they are determined experimentally and The amounts of mixing determined from experi- mixing of weak states through the Higgs barely detectable. In fact, in a particular ment become the numbers in the famous CKM nd so the lepton weak states and mass families with exactly the same properties included in the Standard Model as matrix (named after Cabibbo, Kobayashi, and mechanism in the lepton sector. Conse- range of masses and mixings, the only tates are identical. under the weak force? And why do they arbitrary parameters.4 Notice also that, Maskawa), the unitary matrix that rotates the quently, one can define a quantity called example of lepton-family mixing that have different masses? Here, the Grand by convention, all the mixing is placed complete set of quark mass states into the com- lepton-family number that is conserved is accessible to measurement is neutrino plete set of quark weak states or vice versa. Unified Theories are no guide at all. in the lower half of the quark doublets by all weak interactions involving the oscillation, the spontaneous periodic A related mystery is the one men- (the d, s, and b quarks are mixtures 5The freedom to put all the mixing in one-half of leptons. (Lepton-family numbers and change from one weak family to ′ ′ ′ 5 a weak isospin doublet depends on the fact that tioned at the beginning of this of d , s , and b ). Therefore, the weak the weak force always acts between the two the corresponding conservation laws are another as the neutrino propagates section—the “nonalignment” between and the mass states for the quarks in members of a weak doublet. discussed later in this article.) freely through space.

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Look where we have arrived. We are Mass states Weak states Neutron Decay Lambda Decay aying that mixing among the leptons is First Second First Second Ϫ ν Ϫ ν n ϩ e ϩ e ⌳ p ϩ e ϩ e natural extension of the Standard (udd ) (usd ) Model if neutrinos have mass and that u d = cos␪ d' Ϫ sin␪ s' s= sin␪ d' ϩ cos␪ s' he most likely place to observe the C C C C c mixing is in the peculiar manifestation (uud ) (uud ) f quantum mechanics known as u' Ϫ ν Ϫ ν eutrino oscillation. Furthermore, d ϩ e ϩ e p ϩ e ϩ e c' p ince oscillations can only occur if the s eutrino types have different masses, Initial state Final state Initial state Final state irect observation of neutrino oscilla- d' u u u u ′ ons would reveal the relative sizes of d cos␪c Ϫsin␪c d d′ cos␪c sin␪c s' d Neutron d d Proton Lambda d d Proton he neutrino masses. No wonder that ϭ ϭ s sin␪ cos␪ s′ ′ hysicists have been searching for this ΂ ΃ ΂ c c΃ ΂ ΃ ΂s ΃ ΂Ϫsin␪c cos␪c΃ ΂ s΃ d u s u henomenon for well over two decades. mass mixing weak weak mixing mass We will turn to the theory and states matrix states Ϫ Ϫ states matrix states cos␪ W e Electron sin␪ W e Electron etection of neutrino oscillations C C nd examine how two types of ν Electron ν Electron nformation—neutrino masses and the Figure 8. Two-Family Mixing P(n → p ϩ eϪ ϩ ␯ ) ∝ cos2␪ e P(⌳ → p ϩ eϪϩ ␯ ) ∝ sin2␪ e s ෆe c antineutrino ෆe c antineutrino mount of mixing across families—can among the Quarks s' e determined from oscillation data. The quark weak states and mass states The neutron decay amplitude is proportional The lambda decay amplitude is proportional

But first, we will backtrack to the are like two alternative sets of unit vectors sin␪C to cos␪c, the amplitude of the state d′ in the to sin␪c, the amplitude of the state d′ in the uarks and explain how mixing works. in a plane (see diagram at right) that are d quark. The decay probability is proportional s quark. The lambda decay probability is cos␪ C

related to each other by the rotation C d' to the square of the amplitude: proportional to the square of the amplitude. ␪

through an angle ␪ . In this analogy, one c C cos ␪

C ␪ Mixing among the Quarks mixing matrix is just a rotation matrix that sin takes, say, the mass coordinates d and s ␪C Figure 9. Neutron and Lambda Beta Decay in the Two-Family Picture Consider ordinary neutron beta into the weak-force coordinates d′ and s′; d In beta decay, the neutron transforms into a proton through the transition d → u, and the lambda transforms into a proton through ecay and suppose we had no idea of its inverse is the rotation through the the transition s → u. However, in both cases, the W acts between members of the quark weak isospin doublets in the first family, → he difference between the weak states angle Ϫ␪c that takes the weak coordinates that is, the W causes the transition d′ u. So, only the fraction of the d in the state d′ takes part in neutron decay, and only the nd the mass states. A neutron trans- into the mass coordinates. fraction of the s in the state d′ takes part in lambda decay. The multicolored lines for d and s show their fractional content of d’ orms into a proton, and an electron (green) and s’ (purple). nd an electron antineutrino are created identical in the two processes. states. The mathematics of this mixing ′ ′ ′ n the decay process, Where did the missing strength of is interesting not only for tracking and d are an alternative description, wise be linear superpositions of |d͘ = cos␪c|d ͘ Ϫ sin␪c|s ͘ ; ′ ′ the weak force go? It turns out to be down the missing 4 percent, but also and the two sets of states are like two different neutrino mass states, and |s͘ = sin␪c|d ͘ ϩ cos␪c|s ͘ . (3) → Ϫ n p ϩ e ϩ ෆ␯e . (1) “hiding” in the beta decay of the lamb- because it has the same form as the independent sets of orthogonal unit vec- those mass states, or matter waves, da particle (⌳): mixing that causes neutrino oscillations. tors in a plane that are related to each can generate the interference patterns Figure 9 also illustrates that the he neutron is made of the triplet of For simplicity, we will consider mixing other by a rotation through the angle ␪c, that we call oscillations. transition amplitude for a neutron to → Ϫ uark mass states udd, and the proton is ⌳ p ϩ e ϩ ෆ␯e . (2) between the first two families of quarks also called the mixing angle. Thus, the But first, let us track down the turn into a proton (that is, for a d quark made of the triplet of quark mass states only, which accounts for most of the weak states can be described as linear 4 percent decrease in the expected rate to turn into a u quark) is proportional to ud. At the quark level, the change of a The lambda (uds) differs from mixing among the quarks anyway. combinations of the mass states, and of neutron beta decay. Figure 8 also cos␪c, or the amplitude of the d quark eutron to a proton looks like the trans- the neutron (udd) by having a strange Figure 8 shows the quark weak conversely, the mass states can be shows the weak quark doublets that that is in the state d′. Similarly, the mutation of a down quark to an up quark replace a down quark. The states and the quark mass states in the described as linear combinations of transform into each other through inter- transition amplitude for the lambda uark d → u (refer to Figure 3a). lambda decays to a proton because two-family picture. Underneath the the weak states. action with the W in the two-family to turn into a proton (that is, for an s However, when the strength (effective the strange quark transforms into an families of mass states, the 2 ϫ 2 The phenomenon of mixing, while picture. The weak doublets are not (u, quark to change into a u quark) is → ′) oupling) of the force responsible for up quark, s u. Lambda beta decay rotation matrix is shown, which rotates perhaps nonintuitive, emerges naturally d) and (c, s), but rather (u, d and (c, proportional to sin␪c, the amplitude eutron beta decay is measured, it is is thus analogous to neutron beta decay, the weak states d′ and s′ into the mass from the fundamental tenet of quantum s′). Now, consider the beta decay of of the s quark that is in the state d′. ound to be 4 percent smaller than the and the sum of the strengths for lambda states d and s; the inverse transforma- mechanics that particles have wavelike the lambda and the neutron. As shown The rate of neutron beta decay is trength of the force responsible for and neutron beta decays equals the tion is shown under the families properties. Like sound and light waves, in Figure 9, both decays involve the proportional to the square of that transi- muon beta decay (refer to Figure 3b). strength for muon beta decay. of weak states. In this quantum matter waves, or quantum mechanical transition d′ → u. The d quark in tion amplitude and is thus proportional 2 But these are just two different Why is this so? The answer is mechanical world, the quark mass states, can add together to form a the neutron and the s quark in the to cos ␪c. The rate of lambda beta 2 xamples of the charged-current mixing—the fact that the quark mass states s and d are one complete descrip- coherent linear superposition of waves. lambda are mass states that contain decay is proportional to sin ␪c. weak force, and theory says that states that appear in the neutron and the tion of the quarks with electric charge We will see later that the neutrinos a fraction of d′. The compositions The sum of the rates for the two he strength of the force should be lambda are mixtures of the quark weak Q ϭ Ϫ1/3. The quark weak states s′ produced in weak processes may like- of these mass states are given by processes equals the rate for the transi-

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µϪ µϩ CKM CKM CKM CKM Neutrino Source Neutrino Detector W W d d d K Ϫ 0 u,c,t u,c,t K 0 u,c,t K 0 u,c,t W νµ νµ s d s s Time W W d Wϩ Particle CKM CKM Antiparticle CKM CKM Particle u Inverse muon decay πϪ s ν ϩ µ ϩ N µ ϩ X Lepton number Ϫ1 ϩ 0 ϭ Ϫ1 ϩ 0 Pion decay u d igure 10. Oscillation of the Neutral Kaons πϪ µϪ ν ν ϩ N µϪ ϩ X' 0 0 ϩ µ Forbidden by µ Ϫ X(n+1, p-1) he neutral kaon K (sෆd) can transform into its antiparticle Kෆ (sd) and back again, in each case through the four weak- ෆ Lepton total-lepton-number Lepton N(n, p) teraction vertices shown above. The CKM matrix at each vertex indicates that the transitions mediated by the W are between number 0 = ϩ1 Ϫ1 conservation number Ϫ1 ϩ 0 Þ ϩ1 ϩ 0 embers of the weak doublets, and they can proceed only because the quark mass states in the neutral kaons contain mixtures the weak states d′, s′, and b′. Figure 11. Test of Lepton-Number Conservation At left is a neutrino source consisting of muon antineutrinos (L = ؊1) from pion decay. If total lepton number is conserved, then as .(on of d′ into u. That rate is the same between the quark weak and mass states the number of antileptons) and the other shown in the figure, those antineutrinos should interact with matter through inverse muon decay and produce antimuons (L = ؊1 or transitions between all weak results in the oscillation of the neutral for individual-lepton-family number (the They should never produce muons because that reaction would change the total lepton number by two units. Shown in the figure are oublets, including the leptonic kaon into its antiparticle, the oscillation number of leptons minus the number of the lepton numbers for pion decay and inverse muon decay as well as the reaction forbidden by total-lepton-number conservation. → ansition ␮ ␯␮ in muon beta decay of one neutrino flavor into another is antileptons in a particular lepton family). hown in Figure 3(b). possible only if there is mixing between Although these laws can be viewed as The mixing angle ␪ for these first lepton weak and mass states. predictions of the Standard Model, they The positive pion will decay to the c Table I. Lepton Numbers and Lepton-Family Numbers wo families is called the Cabibbo were deduced empirically a decade antimuon and the negative pion to ngle, and it has been determined from before the Standard Model was formu- the muon because electric charge must xperiment. The 4 percent decrease in Nonmixing among Leptons lated. Let us review the relevant leptonic be conserved. The law of total-lepton- Lepton Electron-Family Muon-Family Tau-Family he rate of neutron beta decay relative and Lepton-Number reactions and methods of interpretation number conservation says that the Particle Number Number Number Number o muon beta decay provides a measure Conservation Laws because the same reactions are now number of leptons minus the number of L L L L 2 2 e ␮ ␶ f that angle: 1 Ϫ cos ␪c ϭ sin ␪c Ϸ being used to detect neutrino oscillations antileptons must not change in any .04. And that decrease is made up for To recap what we discussed earlier, and to search for the consequences of reaction. To formalize this law, every eϪ ϩ1 ϩ1 0 0 y the rate of lambda beta decay. The in the usual version of the Standard nonzero neutrino masses. particle is assigned a lepton number L. ␯e ϩ1 ϩ1 0 0 2 measured value for sin ␪c is 0.22. Model, there is no mixing among By convention, the negatively charged In the Standard Model, the mixing the leptons. Because the three neutrinos Conservation of Total Lepton leptons are called leptons and assigned eϩ Ϫ1 Ϫ1 0 0

etween the quark weak and mass states are assumed to have the same mass Number. The primary sources of a lepton number of ϩ1, and their ෆ␯e Ϫ1 Ϫ1 0 0 ccurs among the three families, not just (namely, zero), the lepton version of neutrinos in cosmic-ray- and accelera- positively charged counterparts are wo, and the amounts of mixing the CKM mixing matrix for quarks is tor-based neutrino experiments are pion called antileptons and are assigned a ␮Ϫ ϩ1 0 ϩ1 0 6 ppear in the famous CKM matrix, the the identity matrix. Thus, the mass and muon decays. Pions come in three lepton number of Ϫ1 (see Table I). ␯␮ ϩ1 0 ϩ1 0 ϫ 3 unitary mixing matrix for the states and weak states are equivalent, charge states, the ␲ϩ, ␲Ϫ, and ␲0. Because quarks are not leptons, they are hree quark families that is analogous to and there is no mechanism to produce Shortly after they are produced assigned a lepton number of zero. ␮ϩ Ϫ1 0 Ϫ1 0

he 2 ϫ 2 rotation matrix in Figure 8. In reactions that will cross family lines. through the strong force, the Since the pion is also not a lepton ෆ␯␮ Ϫ1 0 Ϫ1 0 he Standard Model, the mismatch As with the quarks, the weak force charged pions decay into muons (lepton number L ϭ 0), its decay must etween quark mass states and weak always acts between the members through the weak force: produce one lepton and one antilepton ␶Ϫ ϩ1 0 0 ϩ1 ates is responsible for all processes in of a weak doublet and simply (L ϭ 1 Ϫ 1 ϭ 0). Thus a ␯␮ (lepton) is ␯␶ ϩ1 0 0 ϩ1 ϩ → ϩ ϩ which quarks transmute across family transforms a muon into a muon ␲ ␮ ϩ ␯␮ ; created with the ␮ (antilepton), or a Ϫ ϩ nes. Among those processes is the neutrino and vice versa, or allows (4) ෆ␯␮ (antilepton) is created with the ␮ ␶ Ϫ1 0 0 Ϫ1 0 Ϫ → Ϫ scillation between the neutral kaon, K similar transformations for the other ␲ ␮ ϩ ෆ␯␮ . (lepton). Conservation of total lepton ෆ␯␶ Ϫ1 0 0 Ϫ1 0 ), ෆ sෆd and its antiparticle, K (ෆsd).The lepton families. A further assumption number can easily be checked in all the aons periodically change from particle in the Standard Model is that, although processes shown in Figures 3 and 4. 6The pion is a massive spin-0 particle made of o antiparticle during free flight in space. electrically neutral, the left-handed quark-antiquark pairs from the first family. It is How can one prove that the neutrino interaction of those neutrinos with mat- Indeed, these tests have been performed, igure 10 shows how oscillations can neutrino and the right-handed antineu- a carrier, or mediator, of the residual strong force and antineutrino have different lepton ter. As shown in Figure 11, the antineu- and conservation of total lepton number ome about as the quark mass states trino are distinct particles and cannot that binds neutrons and protons inside nuclei. numbers? How can one show that, for trino from ␲Ϫ decay has lepton number holds to a very high level of precision. Pions are produced, or “boiled off,” in great ϩ omposing the kaons interact through transmute into each other. numbers when nuclei are bombarded by ener- example, the neutrino from ␲ decay Ϫ1 if it produces an antimuon (L ϭ (However, if neutrinos have a small he W. The quarks transmute across fam- These theoretical assumptions lead getic protons. Yukawa predicted the existence has lepton number +1, like the muon, Ϫ1)—and never a muon—when interact- nonzero mass and furthermore if they y lines because they are mass states, directly to two types of lepton-number of this particle in the 1930s. When the muon, whereas the antineutrino from ␲Ϫ decay ing with matter. Likewise, the neutrino acquire that mass through what is called which is slightly less massive than the pion, ϩ ach a mixture of weak states from all conservation laws: one for total lepton was discovered in cosmic rays in 1937, it was at has lepton number Ϫ1, like the from ␲ decay has lepton number ϩ1 if a Majorana mass term, neutrinos hree weak families. Just as the mixing number (the number of leptons minus first mistakenly identified as Yukawa’s particle. antimuon? The test requires detecting the it produces a muon—never an antimuon. would be their own antiparticles. They

4 Los Alamos Science Number 25 1997 Number 25 1997 Los Alamos Science he Oscillating Neutrino The Oscillating Neutrino

igure 12. Is the Muon Neutrino (a) Initial state Final state he Same Particle as the Electron ␮Ϫ → ␯ ϩ eϪ ϩ ␯ . (6) Table II. Decays Forbidden by Lepton-Family-Number Conservation Laws ␮ ෆe Neutrino? Muon µϪ ν Neutrino ϩ ϩ rdinary muon decay is shown in (a). Likewise, in ␲ϩ decay, the neutrino ␮ → e ϩ ␥ t one weak-interaction vertex, a muon created with the antimuon is a muon ␮ ansmutes into a muon neutrino and neutrino (␲ϩ → ␮ϩ ϩ ␯ ), not an ϩ → eϩ ϩ eϪ ϩ eϩ WϪ ␮ mits a W؊, and at the second, the W؊ eϪ Electron electron neutrino. Thus, a second ecays into an electron and an electron lepton family was thought to exist. ␮Ϫ ϩ N(n, p) → eϪ ϩ N(n, p) ntineutrino. Two neutrinos are ν The conjecture of two neutrino e Electron antineutrino roduced, one associated with the muon flavors was tested by Leon Lederman, ␮Ϫ ϩ N(n, p) → eϩ ϩ N(n ϩ 2, p Ϫ 2) nd the other with the electron. (b) If the Mel Schwartz, and Jack Steinberger, muon neutrino were the same as the who designed an ingenious experi- ϩ → ϩ ␮ e ϩ ෆ␯e ϩ ␯␮ ectron neutrino, then the muon could ment—analogous to the one illustrated ecay to an electron through two weak- in Figure 11—at the Brookhaven nteraction vertices. At one vertex, the (b) Initial state Final state 30-giga-electron-volt (GeV) proton Table I). These laws are analogous to (averaged over the experimental energy muon transforms into a neutrino and accelerator. As in most accelerator- the conservation laws of total lepton and distances). Neutrino ν ؊ mits a W ; at the second vertex, that Muon µϪ Electron eϪ neutrino experiments, a pulsed beam of number except that they apply separately ame neutrino absorbs a W؊ and trans- protons is directed at a target, where to the electron, electron neutrino, and orms into an electron. To conserve WϪ WϪ they produce a myriad of pions that their antiparticles on the one hand and Neutrino Oscillations nergy and momentum, the (virtual) W؊ rapidly decay into muons and neutrinos. to the muon, muon neutrino, and their adiates a photon. Thus, muon decay In this case, the experimenters found antiparticles on the other. The first suggestion that free roduces an electron and a gamma ray, a way to tailor a narrow beam of high- To conserve muon-family number, neutrinos traveling through space might ut no neutrinos are emitted. In other energy neutrinos from a much wider a muon can turn into a muon oscillate, that is, periodically change ؊ → ؊ ords, the process µ e ϩ ␥ could Gamma γ distribution. They allowed these high- neutrino—never into a particle with from one neutrino type to another, was ccur if the muon neutrino were the energy neutrinos to pass through a huge a muon number of zero. Similarly, to made in 1957 by Bruno Pontecorvo. ame as the electron neutrino. spark chamber containing 10 tons of conserve electron number, an electron Gell-Mann and Pais had just shown aluminum plates in parallel stacks can turn into an electron neutrino; how quantum mechanical interference 0 might induce, at some low rate, outlined in Fermi’s theory of beta exchange of a virtual neutrino that separated by narrow gaps. A neutrino it cannot turn into a particle with an would allow the neutral kaon K (sෆd ) 0 eactions that would change total decay seemed to be at work, and so couples to both the electron and the entering the spark chamber could electron number of zero. Once the tau, and its antiparticle Kෆ (ෆsd) to oscillate epton number. This possibility will be the mechanism of muon decay was muon. In other words, the muon trans- interact with an aluminum nucleus the charged lepton of the third family, back and forth because the quark mass iscussed later in the text.) believed to be entirely analogous to mutes into a neutrino, and then that and produce a high-energy muon or was discovered, the tau neutrino was states are mixtures of weak states. that of neutron beta decay. same neutrino transmutes into an electron. Either one would leave an assumed to exist, and tau-family number Pontecorvo noted very briefly that, if Conservation of Lepton-Family At that time, the local symmetry of electron. Because there are three ionization track in the gas between and its conservation were postulated. the neutrino had mass and if total lep- Number. One might also wonder how the weak force was not known, but interaction vertices in the diagram for the plates, and if the plates were At the beginning of this section, we ton number were not conserved, the t was shown that the muon neutrino is Fermi’s theory did place particles in ␮Ϫ → eϪ ϩ ␥ , two weak and one charged, they would discharge along stated that strict separation between the neutrino could imitate the neutral kaon, eally distinct from the electron neutrino pairs that transformed into each other electromagnetic, the rate for this second that track and create a trail of bright lepton families is implied by the gauge oscillating between particle and antipar- nd that distinct lepton families are under the weak force. It was therefore mode would be small but still observ- sparks that could easily be photo- symmetry of the weak force, combined ticle as it travels through empty space. nder the weak force. Those discoveries assumed that the weak force trans- able, about 10Ϫ5 of the total decay rate graphed. The experiment produced with the assumption that the three This possibility would have implied ame from studies of muon decay. formed the muon into a neutral particle of the muon. This decay mode, a total of 29 spark-chamber neutrinos are massless. But this that the neutrino is a massive Majorana n the late 1940s, the muon was of some kind, perhaps the neutrino, and however, has never been observed. The photographs containing long, straight assumption always seemed to rest on particle with no definite distinction bserved to decay into an electron emit- that, to conserve charge, an electron and MEGA (muon to electron plus gamma) tracks that started from within the spark shaky ground. More important, new between particle and antiparticle forms. ed with a spectrum of energies. As in an antineutrino were produced as in experiment, currently nearing comple- chamber and were characteristic of an forces could exist, even weaker than Although very interesting and still rdinary beta decay, a spectrum of ordinary neutron beta decay: tion at Los Alamos, has put the most energetic muon. The erratic, staggered the weak force, that have yet to be seen relevant today, Pontecorvo’s suggestion lectron energies rather than a single stringent upper limit on the rate of this tracks that would be produced by the but that allow leptons to transmute was not explored in 1957 because Lee → Ϫ11 nergy means that the decay must yield ␮ ␯ ϩ e ϩ ෆ␯ e . (5) process so far. It is less than 4 ϫ 10 much lighter electron were essentially across family lines. Consequently, there and Yang’s theory of the massless two- hree particles in the final state. of the total muon-decay rate. absent. Thus, the neutrino produced in have been many searches for various component neutrino was just gaining However, only the electron revealed Then, in the 1950s, theorists consid- The absence of ␮Ϫ → eϪ ϩ ␥ ␲ϩ decay could transform into a muon “forbidden” reactions such as those acceptance. This theory helped explain ts presence, so the two unidentified ered the possibility that a massive is a clue that there are two neutrino but not into an electron. listed in Table II. Searches for viola- why parity was maximally violated in articles (␮ → e ϩ ? ϩ ?) had to be gauge boson (like the W) mediated flavors—one strictly associated with the These results supported the idea of tions of the Standard Model have nuclear beta decay. The existence of a lectrically neutral. It was also the weak force, in which case the muon electron; the other, with the muon. In two independent neutrino flavors and mostly reported null results. The excep- left-handed neutrino, distinct from the bserved that the rates of muon decay could decay to an electron through the muon beta decay, for example, a muon led the way for establishing separate tion is the LSND experiment, which right-handed antineutrino by having the nd muon capture by nuclei were very two processes shown in Figure 12. transforms into a muon neutrino, and conservation laws for two new quantum reports that muon antineutrinos can opposite lepton number, was a crucial imilar to the rates for beta decay and The latter process involves not only the an electron and its antineutrino are numbers, muon-family number and oscillate into electron antineutrinos postulate (see the box “Parity Noncon- lectron capture. The same weak force exchange of a virtual W but also the created to conserve charge: electron-family number (refer again to with a probability of about 0.3 percent servation and the Massless Two-

6 Los Alamos Science Number 25 1997 Number 25 1997 Los Alamos Science he Oscillating Neutrino The Oscillating Neutrino

Component Neutrino” on page 32). dependence predicted by the MSW family-number and muon-family- Figure 13. Three Types of Evidence for Neutrino Oscillations n that theory, particle-antiparticle effect (see the articles “Exorcising number conservation laws, the electron (a) Solar neutrinos—a disappearance experiment. The flux of electron neutrinos produced scillations could not occur. Ghosts” on page 136 and “MSW” antineutrino was detected through its in the Sun’s core was measured in large underground detectors and found to be lower than on page 156). charged-current interaction with matter, expected. The “disappearance” could be explained by the oscillation of the electron neutrino olar Neutrinos. In 1963, after that is, through inverse beta decay. into another flavor. Lederman, Steinberger, and Schwartz Atmospheric Neutrinos. In 1992, Recently, members of the LSND howed that there were two distinct another neutrino deficit was seen—this collaboration reported a second positive flavors of neutrino, the idea of oscilla- time in the ratio of muon neutrinos to result. This time, they searched for the Sun on between electron neutrinos and electron neutrinos produced at the top oscillation of muon neutrinos rather Earth muon neutrinos surfaced for the first of the earth’s atmosphere. When high- than muon antineutrinos. The muon me. This possibility requires mixing energy cosmic rays, mostly protons, neutrinos are only produced during pion Solar cross the lepton families as well as strike nuclei in the upper atmosphere, decay-in-flight, before the pions reach Underground Primary neutrino source core 8 ν onzero neutrino masses. In 1969, it they produce pions and muons, which the beam stop. Therefore, these neutri- ~10 kilometers e detector was decided that the idea of neutrino then decay through the weak force and nos have a higher average energy than p + p D + e+ + ν scillation was worth testing. The Sun produce muon and electron neutrinos. the muon antineutrinos measured in the e s known to drench us with low-energy The atmospheric neutrinos have very earlier experiment. The muon neutrinos lectron neutrinos that are produced in high energies, ranging from hundreds were observed to turn into electron Other sources of neutrinos: he thermonuclear furnace at its core, of million electron volts (MeV) to tens neutrinos at a rate consistent with the Cosmic-ray Ð 7 7 ν s shown in Figure 13(a). By using of giga-electron-volts, depending on the rate for antineutrino oscillation reported e + Be Li + e shower 8 4 + ν tandard astrophysics models about energy of the incident cosmic ray and earlier. Since the two experiments B 2 He + e + e tellar processes and the observed value on how this energy is shared among the involved different neutrino energies and π0 f the Sun’s luminosity, theorists can fragments of the initial reaction. As different reactions to detect the π+ redict the size of the neutrino flux. But shown in Figure 13(b), the decay of oscillations, the two results are indeed (b) Atmospheric neutrinos—a disappearance experiment. Collisions between high-energy + measurements of the solar-neutrino flux pions to muons followed by the decay independent. The fact that the two ~30 kilometers µ resent an intriguing puzzle: A signifi- of muons to electrons produces two results confirm one another is therefore protons and nuclei in the upper atmosphere can ant fraction of those electron neutrinos muon neutrinos for every electron neu- most significant. The complete story of create high-energy pions. The decay of those pparently disappear before reaching trino. But the measured ratio of these LSND can be found in the article pions followed by the decay of the resulting + ur terrestrial detectors. Ray Davis two types is much smaller (see the arti- “A Thousand Eyes” on page 92. muons produces twice as many muon-type e made the first observation of a neutrino cle “The Evidence for Oscillations” on Each type of experiment shown in neutrinos (blue) as electron-type neutrinos ν ν µ hortfall at the Homestake Mine in page 116). The oscillation of muon Figure 13, when interpreted as an (red). But underground neutrino detectors e νµ outh Dakota, and all experiments neutrinos into tau neutrinos appears to oscillation experiment, yields informa- designed to measure both types see a much ince have confirmed it. Today, the be the simplest explanation. tion about the oscillation amplitude and smaller ratio than 2 to 1. The oscillation of most plausible explanation of the solar- wavelength. One can therefore deduce muon neutrinos into tau neutrinos could Atmospheric neutrino source eutrino puzzle lies in the oscillation of Accelerator Neutrinos. The lone information about the sizes of neutrino explain that deficit. + + lectron neutrinos into other types of accelerator-based experiment with masses and lepton-family mixing para- π µ + νµ eutrinos. Although the measured short- evidence for neutrino oscillations is meters. The specific relationships are + ν ν e + e + µ all is large and the expected amplitude LSND. This experiment uses the high- explained in the next section. πÐ µÐ ν or neutrino oscillations in a vacuum intensity proton beam from the linear + µ Ð ν ν s small, neutrino oscillations can accelerator at the Los Alamos Neutron Underground e + e + µ till explain the shortfall through Science Center (LANSCE) to generate The Mechanics of Oscillation ν ν ν ν e, e, µ, µ he MSW effect. an intense source of neutrinos with detector Named after Mikheyev, Smirnov, and average energies of about 50 MeV. Oscillation, or the spontaneous peri- Wolfenstein, the MSW effect describes In 1995, the LSND collaboration odic change from one neutrino mass (c) LSND—an appearance experiment. Positive pions decay at rest into positive muons, ow electron neutrinos, through their reported positive signs of neutrino state to another, is a spectacular exam- which then decay into muon antineutrinos, positrons, and electron neutrinos. Negative pions nteractions with electrons in solar mat- oscillations. An excess of 22 electron ple of quantum mechanics. A neutrino decay and produce electron antineutrinos, but that rate is almost negligible. A giant liquid- er, can dramatically increase their antineutrino events over background produced through the weak force in, scintillator neutrino detector located 30 meters downstream looks for the appearance of ntrinsic oscillation probability as they was observed. They were interpreted as say, muon decay, is described as the electron antineutrinos as the signal that the muon antineutrinos have oscillated into that flavor. ravel from the solar core to the surface. evidence for the oscillation of muon sum of two matter waves. As the This matter enhancement of neutrino antineutrinos, into electron antineutri- neutrino travels through space (and ν ν ν ν detector Muons and electrons Neutrinos e, µ, and µ e scillations varies with neutrino energy nos (see Figure 13c). The muon anti- depending on which masses are Pions nd matter density. The next generation neutrinos had been produced at the measured), these matter waves interfere Proton f solar-neutrino experiments is specifi- accelerator target through antimuon with each other constructively or de- beam ally designed to explore whether the decay-at-rest. As in the experiments structively. For example, the interfer- Water Copper beam stop target lectron neutrino deficit has the energy described earlier to study electron- ence causes first the disappearance and 30 meters

8 Los Alamos Science Number 25 1997 Number 25 1997 Los Alamos Science The Oscillating Neutrino The Oscillating Neutrino

igure 14. Neutrino Oscillations in Mass states Weak states action of the weak force, but rather one another. In quantum mechanics, the interaction properties as opposed to the he Two-Family Context First Second First Second through the pattern of interference that time evolution of a state is determined free-propagation characteristics of mass a) Neutrino mass states and weak states. develops as the different mass states by its energy, and the energies of the and momentum. So, in an individual he weak states ␯ and ␯ are shown as ν ν ν ν composing the original neutrino state mass states are simply given by measurement of an event, there are only e ␮ 1 2 e µ olor mixtures of the mass states ␯1 (yellow) evolve in time. two possibilities: to detect the muon 2 2 2 4 nd ␯2 (red), and the mixing matrix that To see how the oscillation depends Ek ϭ ͙pෆෆcෆෆϩෆmkෆෆc , (8) neutrino or the electron neutrino, but otates ␯ and ␯ into ␯ and ␯ is shown ν on the masses of the different neutrino not some linear combination. Thus, 1 2 e ␮ ν 2 elow the weak states. Each set of states is µ sinθ mass states as well as the mixing angles where p is the momentum of the what is relevant for an experiment is so represented as a set of unit vectors in a cosθ ν ν cosθ sinθ ν between the lepton families, we limit neutrinos and m (k = 1, 2) is the mass the probability that the muon neutrino e e 1 k θ (ν ) = (Ϫ sin θ cosθ ) ( ν ) ane. The two sets are rotated by an angle θ µ 2 the discussion to the first two families of the states ␯ and ␯ , respectively. remains a muon neutrino at a distance x θ 1 2 relative to each other. cos (␯ → ␯ ) sin and assign the mixing to the electron Note that, if the particle is at rest, this from its origin, P ␮ ␮ , or the proba- θ ν 1 neutrino and the muon neutrino (the is just the famous energy relation of bility that the muon neutrino has trans- b) Time evolution of the muon neutrino. halves of the lepton weak doublets with Einstein’s special relativity, E ϭ mc2. formed into an electron neutrino, w → -as a specific I3 ϭ 1/2, as shown in Figure 5). In quantum mechanics, the time evolu- P(␯␮ ␯e). The box “Derivation of Neu 0 ؍ he ␯␮ is produced at t Pure νµ Pure νµ Pure νµ near combination of mass states: Instead of expressing the mass states in tion of each mass component ␯k is trino Oscillations” on the next page -Ϫsin␪ ␯1 ؉ cos␪ ␯2. The amplitude of ν terms of the weak states, as was done obtained by multiplying that component shows how to calculate these probabili ؍ ␮ 2 – ach mass state is shown oscillating in time in Equation (3), we can use the alter- by the phase factor exp[Ϫi(Ek/h)t], ties from the time-evolved state. ith a frequency determined by the energy θ nate point of view and express the and thus the time evolution of the muon The results are cos θ that mass state. The energies of the two neutrino weak states |␯e͘ and |␯␮͘ as neutrino is given by

sin ␲x ates are different because their masses Ϫ (␯ →␯ ) 2 2 Ϫ linear combinations of the neutrino P ϭ 1Ϫsin 2␪ sin (ᎏᎏ) (10) ␮ ␮ ␭ re different, m Þ m . Each time the two ν mass states |␯ and |␯ with masses m |␯ (t) ϭ Ϫsin ␪ exp[Ϫi(E /h–)t]|␯ osc 1 2 1 1͘ 2͘ 1 ␮ ͘ 1 1͘ ass states return to the original phase and m , respectively (where we have – and 2 ϩcos ␪ exp[Ϫi(E2/h)t]|␯2͘ (9) .(they compose a pure assumed that m1 and m2 are not equal ,0 ؍ elationship at t 0 ␲x . At other times, the two mass states have Time, t Figure 14(a) illustrates this point of as discussed in the box “Derivation of P(␯ →␯ ) ϭ sin22␪ sin2(ᎏᎏ) , (11) ␮ ␮ e ␭ different phase relationship and can be view. It shows how the weak states and Neutrino Oscillations” on the next page. osc ought of as a mixture of ␯ and ␯ . ␮ e Source mass states are like alternate sets of Because the two states |␯1͘ and |␯2͘ where ␪ is the mixing angle defined ν ν ν unit vectors in a plane that are related have different masses, they also have above, x is measured in meters, and ν νe ν νe ν νe ν c) Neutrino oscillation. Because the two µ µ µ µ µ µ µ to each other by a rotation through an different energies (E1 is not equal ␭osc is the oscillation length given in

ass components interfere with each other, ) angle ␪. The rotation, or mixing, yields to E2), and the two components evolve meters. The oscillation length (the dis- µ 1 e probability of finding a muon neutrino ν sin22θ the following relationships: with different phases. tance between two probability maxima

purple) oscillates with distance from Figure 14(b) plots the wavelike or two probability minima) varies with µ ν e source. The probability of finding an ( |␯ ͘ ϭ cos␪Խ␯1͘ ϩ sin␪|␯2͘ ; behavior of each of the mass compo- the energy of the neutrino E (in P 2.5 Eν e ␯ ectron neutrino in its place also oscillates, λ = osc 2 (7) nents (red and yellow) and shows how million electron volts), and it also ) + ∆m nd in the two-family approximation, the e |␯ ϭ Ϫsin␪ ␯ ϩ cos␪|␯ . the relative phase of the two depends on the squared mass difference ν ␮͘ Խ 1͘ 2͘ um of the probabilities is always 1. The components varies periodically in time. (in electron volts squared):

µ

avelength of this oscillation ␭ increases ν The mixing angle ␪ is the lepton At t ϭ 0, the two components add up osc ( s the masses of the two neutrinos get P 0 analog of the Cabibbo mixing angle for to a pure muon neutrino (a pure weak ␭ ϭ 2.5E /⌬m2 , (12) Distance, x = ct osc ␯ oser in value. the quarks. If ␪ is small, then cos␪ is state), and their relative phase is ␲. As ν ν ν ν close to 1, and the electron neutrino is their relative phase advances in time, the The two probabilities in Equations (10) Probability that µ has become e Probability that µ is still µ mostly made of the state with mass m1, mass components add up to some linear and (11) oscillate with distance x from whereas the muon neutrino is mostly combination of a muon neutrino |␯␮͘ the source, as shown in Figure 14(c). en the reappearance of the original In the examples of quark mixing ments, the neutrinos always start and made of the state with mass m2. If the and an electron neutrino |␯e͘, and when To summarize, a muon neutrino pro- pe of neutrino. The interference can described earlier, the quarks within the end as pure weak states. They are mixing angle is maximal (that is, ␪ ϭ the relative phase has advanced by 2␲, duced at t ϭ 0 travels through space at

ccur only if the two matter waves have composite particles (proton, neutron, typically created through weak-force ␲ր4, so that cos␪ ϭ sin ␪ ϭ 1/͙2ෆ, ) the components add back up to a muon almost the speed of light c. As time fferent masses. Thus, the mechanics lambda) start and end as pure mass processes of pion decay and muon each weak state has equal amounts of neutrino. The relative phase oscillates passes, the probability of finding the → oscillation start from the assumption states, and the fact that they are mix- decay, and they are typically detected the two mass states. with a definite period, or wavelength, muon neutrino P(␯␮ ␯␮) decreases at the lepton weak and mass states tures of weak states shows up through through inverse beta decay and inverse To see how oscillations can occur, that depends on the difference in the below unity to a minimum value of e not the same and that one set is the action of the weak force. When a muon decay, weak processes in which we must describe the time evolution energies of the two mass components, 1 Ϫ sin22␪ and then increases back to omposed of mixtures of the other set neutron decays through the weak force the neutrinos are transmuted back to of a free neutrino. Consider a muon or equivalently, the squared mass unity. This variation has a periodicity 2 2 2 a manner entirely analogous to and the d quark transforms into a u, their charged lepton partners. Between neutrino produced by the weak force at differences, ⌬m ϭ m1 Ϫ m2 . over a characteristic length ␭osc Х cT, e descriptions of the quark weak and only a measurement of the decay rate the point of creation and the point of t ϭ 0. It is a linear combination of two In quantum mechanics, observations where T is the period of neutrino oscil- ass states in Figure 8. In other words, reflects the degree to which a d quark is detection, they propagate freely, and if mass states, or matter waves, that are, pick out the particle rather than the lation. The oscillation length varies ere must be mixing among the leptons composed of the weak state d′. In they oscillate into a weak state from a by the convention in Equation (7) wave aspects of matter, and in the case inversely with ⌬m2. The probability there is among the quarks. contrast, in neutrino oscillation experi- different family, it is not through the exactly 180 degrees out of phase with of neutrinos, they pick out the weak- of finding an electron neutrino in place

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Derivation of Neutrino Oscillations ∆ m2 (eV2) 102 101 1 10Ð1 10Ð2 10Ð3 10Ð4 10Ð5 10Ð6 10Ð7 10Ð8 10Ð9 10Ð10 10Ð11 Some simple algebra can show how neutrino oscillation effects depend on the mass difference of the neutrino mass eigenstates. Con- ν sider the simplified case of just two neutrino flavors. We express the quantum mechanical wave function for a muon neutrino produced νµ (Accelerator) e (Solar/MSW) at t ϭ 0 as a mixture of the mass eigenstates |␯ ͘ and |␯ ͘ with masses m and m , respectively. ν 1 2 1 2 e (Reactor) ν ν ν ν ν µ, e, µ, e (Atmospheric) e (Solar) |␯␮(0)͘ ϭ|␯␮͘ ϭ Ϫ sin␪ |␯1͘ ϩ cos␪ |␯2͘ ,

10Ð2 10Ð1 1 10 102 103 104 105 106 107 108 109 1010 1011 where an electron neutrino is given by |␯e͘ ϭ cos␪ |␯1͘ ϩ sin␪ |␯2͘ and the angle ␪ characterizes the extent of mixing of the mass eigen- states in the weak-interaction eigenstates. It is called the mixing angle. (For more than two flavors, there are more mixing angles as well Ð1 ␭osc/2.5E (m MeV ) as charge-conjugation and parity, CP, violating phases.) At a later time t, the wave function is Figure 15. Accessible Ranges of ⌬m2

|␯␮(t)͘ ϭ Ϫ sin␪ exp(ϪiE1t)|␯1͘ ϩ cos␪ exp(ϪiE2t)|␯2͘ , Neutrino energies are specific to the source, and source-to-detector distances also vary with the source. The ratio of these two vari- ables determines the range of values for ⌬m2 that neutrino oscillation experiments can measure using each source. These ranges

where the mass eigenstates propagate as free particles and E1 and E2 are the energies of those states |␯1͘ and |␯2͘, respectively. (We are labeled with the source and the neutrinos produced by that source. Two ranges are given for solar-neutrino experiments. One – 2 are working in units for which h ϭ c ϭ 1.) For relativistic neutrinos (E␯ ϾϾ m), we can approximate E1 and E2 by assumes that the MSW effect enhances oscillations, in which case, the range of ⌬m is determined in part by the electron density of matter in the Sun. The other assumes no matter enhancement. 2 2 1/2 2 Ek ϭ (p ϩ mk ) Х p + mk /2p , → 2 of the muon neutrino P(␯␮ ␯e) also make quantitative determinations of Figure 15 shows the regions of ⌬m where we are assuming that the two mass states have the same momentum. After substituting these energies, the wave function at time oscillates as a function of distance from the neutrino masses and mixing angles. (and its inverse, ␭osc/2.5E␯) that can be t becomes the source and has the same wavelength Among the quarks, the amount of probed with the neutrinos from reactors, ␭osc. That probability has a maximum mixing is small and occurs primarily accelerators, the upper atmosphere, and 2 2 2 |␯␮(t)͘ ϭ exp͓Ϫit (p ϩ m1 /2E␯)͔ ͓Ϫ sin␪ |␯1͘ ϩ cos␪ |␯2͘ exp(i⌬m t/2E␯)͔ , value of sin 2␪. These formulas show between the first two families. It is nat- the Sun. Variations in neutrino energies explicitly that, if neutrinos oscillate ural to assume the same should hold and source-to-detector distances make 2 2 2 where ⌬m ϭ m1 Ϫ m2 and E␯ Х p. Since these neutrinos are traveling almost at the speed of light, we can replace t by x/c ϭ x, where between family types, neutrinos must for the leptons, although theory pro- each type of experiment sensitive to a → x is the distance from the source of muon neutrinos. Let us now calculate P(␯␮ ␯e), which is defined as the probability of observing a have nonzero masses and the neutrino vides no such restriction. Consequently, different range of values. The largest ␯e at x, given that a ␯␮ was produced at the origin x ϭ 0. The probability is the absolute square of the amplitude ͗␯e|␯␮(t)͘. Using the weak states are not states of definite neutrino oscillation experiments have mass difference accessed by solar- orthonormality relation ͗␯i|␯j͘ ϭ ␦ij, we can compute the probability mass but rather mixtures of mass states. traditionally been interpreted in the neutrino experiments (assuming the Although we have restricted the two-family context. Applying the two- MSW effect) is below the lowest value → 2 2 P(␯␮ ␯e) ϭ |cos␪ sin␪ (1 Ϫ exp(i⌬m t/2E␯) analysis to mixing between two fami- family formalism to each experiment accessed by other experiments. Given lies, there is every reason to expect allows one to derive a range of possible the electron densities in the Sun and 2 2 2 2 2 ϭ sin 2␪ sin ΂⌬m xր4E␯΃ that, if mixing takes place among the values for ⌬m and a range for sin 2␪, the energies of solar neutrinos leptons, it would occur among all three where ␪ is the mixing angle between (1 to 10 MeV), MSW enhancement can 2 2 2 1.27⌬m x ϭ sin 2␪ sin ᎏᎏ , families and that there would be a the two families. Input to the interpreta- take place only for very small values ΂ E␯ ΃ mixing matrix for the leptons analogous tion includes the neutrino energies in a of ⌬m2 from 10Ϫ4 to 10Ϫ9 eV2, with a to the CKM matrix for the quarks. particular experiment, the distance from favored value on the order of 10Ϫ5 eV2. 2 where ⌬m is measured in electron volts squared, x is in meters, and E␯ is in million electron volts, and the factor of 1.27 derives from The three-flavor mixing problem is source to detector, the expected So, if neutrino oscillation is the explana- → working in these units. P(␯␮ ␯␮) is the probability of observing a ␯␮ at x, given that a ␯␮ is produced at x ϭ 0. This probability can be more difficult, but it boils down to neutrino flux, and the measured flux or tion behind the solar- and atmospheric- computed explicitly, or by the conservation of probability, it is carrying out the analysis, which is probability. In a disappearance neutrino deficits as well as the LSND a technical problem. experiment, one measures P(␯ → ␯ ), appearance measurements, the two- 1.27⌬m2x i i → 2 2 P(␯␮ ␯␮) ϭ 1 Ϫ sin 2␪ sin ᎏᎏ . the probability of finding the original family analysis must be extended to ΂ E␯ ΃ neutrino flavor ␯i, where i ϭ e, ␮, ␶. three families. All the data supporting It is often useful to define an oscillation length, ␭osc for these probabilities, which, as shown in Figure 14(c), equals the distance between Interpreting In an appearance experiment, one mea- neutrino oscillations are reviewed the maxima (or the minima). Note that the spatial period of sin2x is one-half that of sin x, and so the oscillation length is defined by the Oscillation Experiments sures the probability of finding a flavor in the article “The Evidence for → following equation: different from the original P(␯i ␯j), Oscillations” on page 116. Most extensions of the Standard where i Þ j. We will give one simple example of 1.27⌬m2x ␲x P(␯ → ␯ ) ϭ sin22␪ sin2 ϭ sin22␪ sin2 , Model tell us to expect mixing among The only definite constraints on neu- a model that fits the oscillation data ␮ e ᎏᎏE ᎏ ΂ ␯ ΃ ΂ ΃ leptons in analogy with mixing among trino masses are the following upper consistently, but there are many other where quarks. But so far, those theories make limits: ␯e < 10 electron volts (eV), such models with no way to choose no quantitative predictions on masses derived from tritium beta decay, among them. This simple model ␲E 2.5E ␭ ϭ ␯ Ϸ ␯ . and mixing angles. Thus, neutrino ␯ < 170 kilo-electron-volts (keV), assumes the traditional mass hierarchy, osc ᎏᎏ1.27⌬m2 ᎏ⌬m2 ␮ oscillation experiments have a twofold derived from pion decay, and ␯␶ m1 Ͻ m2 Ͻ m3. But the first two masses, purpose: first to establish convincing < 24 MeV, derived from tau decay. So, m1 and m2,, are assumed to be very evidence for oscillations and then to the field is wide open for exploration. nearly identical and therefore almost

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Initial state Final state qually distant from the third mass m3: oscillate directly to muon neutrinos with masses, and therefore the basic descrip- other extensions to the Standard Model no involvement of tau neutrinos. tion of the neutrino would have to be Gamma γ suggest that the familiar neutrinos are 2 Ϫ5 2 Although the intrinsic amplitude for this altered. The neutrino might be a Dirac Majorana particles and that they have ⌬m12 Ϸ 10 (eV) 2 2 process is very small (small mixing particle and parallel the Dirac electron very heavy relatives that reduce their ⌬m23 Ϸ 0.3 (eV) (12) angle ␪12), the amplitude is enhanced by in having four independent states— masses through, what is sometimes 2 2 WϪ Ϫ ⌬m13 Ϸ 0.3 (eV) . the MSW effect. Solar experiments are right-handed and left-handed particle W called, the seesaw mechanism 2 thus a measure of ⌬m12. That mass dif- states, ␯R and ␯L, and right-handed and × (explained later in this article). Ϫ ν ν Ϫ Thus, there are two distinct oscillation ference is quite small, corresponding to left-handed antiparticle states, ␯ and µ µ e e ෆR Muon Electron engths differing by 4 orders of a long oscillation length, and it therefore ෆ␯L. To complete this set of four, two Handedness versus Helicity. To elabo- magnitude. Since the upper limit on does not contribute to the LSND results. new neutrino states would have to be Interaction with rate further on these issues, we must he electron neutrino mass is 10 eV, all Finally, atmospheric-neutrino oscilla- added to the Standard Model: the right- Higgs bosons consider the esoteric concept of hand- eutrinos in this model would be very tions are explained by muon neutrinos handed neutrino ␯R and the left-handed edness, a two-valued quantity Figure 16. Example of Lepton-Family Mixing ght and nearly degenerate in mass. oscillating into tau neutrinos, a pathway antineutrino ෆ␯L. The new states would related in a nontrivial way to helicity. 2 The model also assumes that the mixing dominated by ⌬m23 and a large mixing be “sterile” in the sense that they would If neutrinos have mass and lepton-family number is not conserved, a muon neutrino ␯␮ Helicity and handedness are identical ngle between the second and third angle. This consistent set of mixing not interact through the weak force emitted at the first weak-interaction vertex could become an electron neutrino ␯e for massless particles and almost amilies is close to the maximum value angles and mass differences for the (or any other known force except through interaction with the Higgs background and be transmuted into an electron e؊ identical for massive particles, those .f ␲/4, whereas the mixing angles neutrinos was outlined by Cardall and gravity), and they would be included in at the second vertex. Thus, the reaction ␮؊→ e؊ϩ ␥ could proceed if mixing occurred traveling close to the speed of light etween the first two families and the Fuller (1996). The specifics of their the theory only as necessary ingredients across lepton families. But the concept of handedness is first and third families are quite small. solution are not as important as the to give the Dirac neutrino a mass. crucial because (1) the weak force of Inverse Beta Decay Note that this mixing pattern is quite fact that neutrino oscillations could Those sterile neutrino states, the W distinguishes between different Proton Neutron nlike the CKM matrix for the quarks, explain the results coming from however, could differ in mass from the values of handedness and (2) the origin n which the mixing angle for the solar, atmospheric, and accelerator ordinary neutrino states that couple to Electron of particle masses and the fundamental econd and third families is very small.) neutrino experiments. the W, in which case the ordinary neu- eϪ differences between Dirac and Both LSND and solar-neutrino trinos could oscillate into those sterile, W ϩ Majorana neutrinos also involve the xperiments measure the oscillation of noninteracting forms. That possibility concept of handedness. he muon neutrino to the electron neu- What If Neutrinos Have Mass? could have an impact in various × ν Figure 18 displays the helicity and ν R → L rino P(␯␮ ␯e), or vice versa, so one astrophysical and cosmological con- ϩ handedness states of the electron and ϩ e might naively assume that both measure As data accumulate and the evidence texts, and conversely, cosmological W Positron the massless electron neutrino as they ⌬m2 , the difference between neutrino for oscillations grows stronger, it is arguments would place limits on appear in the Standard Model. Helicity 12 Majorana mass term masses in the first and second families. appropriate to examine the implications the existence of such sterile neutrinos. (interaction with Higgs is easy to describe. It is the polariza- But the LSND results for ⌬m2 differ by of lepton mixing. In terms of weak- On the other hand, the neutrino Proton Neutron background flips handedness, tion, or projection, of a particle’s intrin- changing particle to antiparticle) t least 4 orders of magnitude from the interaction physics, individual-lepton- might be a Majorana particle, which, by Electron Capture sic spin along its direction of motion. olar results. How can the two be rec- family number would no longer be definition, has just two particle states. There are two such states: spin along nciled? The resolution comes about strictly conserved, and the forbidden The two observed states (left-handed Figure 17. Example of Lepton-Number Nonconservation the direction of motion (right helicity, ecause mixing occurs among the three processes listed earlier could occur. neutrino ␯L and right-handed antineutri- If neutrinos are Majorana particles, a left-handed neutrino emitted in electron capture or motion like a right-handed could become a right-handed antineutrino and create a positron through inverse beta amilies. Then, three oscillatory terms Figure 16 illustrates how the oscillation no ෆ␯R) would be the full set. But they corkscrew) and spin opposite to the → an contribute to P(␯␮ ␯e), one with of a muon neutrino into an electron would have a new property that would decay. Such a process would change lepton number by two units. Notice that the left- direction of motion (left helicity, or n oscillation length determined by neutrino would facilitate the process make them freaks in the pantheon of handed neutrino flips its handedness through interaction with the Higgs background. motion like a left-handed corkscrew). 2 Ϫ → Ϫ ⌬m12 and two others with oscillation ␮ e ϩ ␥. Unfortunately, the pre- elementary spin-1/2 particles—they This example of lepton-number violation should be compared with the example of A particle can be produced in a state of 2 2 engths determined by ⌬m13 and ⌬m23, dicted rate for the process in Figure 16, could transform into each other and, in lepton-number conservation in Figure 11. definite helicity, and because angular espectively. in which the mixing occurs through effect, would be their own antiparticles. momentum is conserved, that state can The source-to-detector distance neutrino interactions with the Higgs As a result, the weak force could trans- become a positron. Such particle- should one care one way or the other? be measured directly. The problem is 30 meters), combined with the background, is far below the limit of form an electron into a left-handed antiparticle transitions would violate the First, if neutrinos were Majorana that, for particles with mass, helicity is eutrino energies, makes LSND sensi- detectability, about 10Ϫ40 times the electron neutrino, as usual, but then that law of total-lepton-number conservation particles, there would be no new low- not a relativistically invariant quantity: ve to the two terms whose oscillations rate of ordinary muon decay.7 More left-handed neutrino could later appear as well as individual-family-number mass neutrino states, and the number As shown in the cartoon on page 57, if 2 2 re determined by ⌬m13 and ⌬m23. generally, lepton-family mixing through as a right-handed antineutrino and conservation (see Figure 17). They of mass and mixing-angle parameters neutrinos have mass, then their helicity ↔ Those ␯␮ ␯e oscillations take place interaction with the Higgs bosons interact through the weak force to would also make possible a new type of in the theory would be highly restricted. can change with the reference frame. ndirectly through ␯␶. These “indirect would parallel the mixing seen among beta decay known as neutrinoless dou- This, in turn, would put strong In contrast, handedness (also called scillations” do not contribute to the quarks and lend further support to the ble beta decay. Unfortunately, that constraints on any theoretical fit to chirality), although harder to define 7Perhaps new forces, such as those expected in olar-neutrino deficit because the wave- idea presented in the Grand Unified supersymmetric theories, also cause transitions process may be the only measurable the neutrino oscillation data. Second, without using the Dirac equation for 2 2 engths determined by ⌬m13 and ⌬m23 Theories that quarks and leptons are between families and contribute to the process sign that the neutrino is a Majorana the difference between Majorana spin-1/2 particles, provides a relativisti- re too large. The resulting oscillation close relatives. ␮ → e ϩ ␥. For a discussion of how new forces rather than a Dirac particle. and Dirac neutrinos is directly related cally invariant description of a could contribute to muon decay, see the article annot be amplified by the MSW effect. In terms of the neutrino itself, oscil- “The Nature of Muon Decay and Physics beyond If it is so hard to tell the types of to how neutrinos acquire mass. particle’s spin states. There are two nstead, the solar electron neutrinos lations would imply nonzero neutrino the Standard Model” on page 128. neutrinos apart at low energies, why Grand Unified Theories and most independent handedness states for

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pin-1/2 particles—left and right. A decomposed into two independent handedness without changing its left-handed, we mean that they pick out urely left-handed state has Nx ϭ L, a components, left-handed and right- helicity. Nevertheless, because it is the left-handed components of particles C urely right-handed state has Nxϭ R, handed, and this decomposition does relativistically invariant, handedness is and the right-handed components of nd like lepton number and electric not change with the reference frame. an essential quantity for describing the antiparticles. The correct Standard harge, a particle’s handedness is The confusing thing about handed- properties of the weak force and the Model description of the weak isospin ndependent of the reference frame ness is that it is not a constant of the origin of particle masses, as well as par- doublets, therefore, includes handedness om which it is viewed. Further, a motion; a spin-1/2 particle traveling ticle properties. For example, when we labels. In the first family, the left- article, massless or massive, can be through space can change its say that interactions involving the W are handed electron eLand the left-handed electron neutrino ␯eL form a weak isospin doublet. The right-handed Particle Antiparticle Figure 18. Helicity and Handedness electron e exists but does not interact Looks like a left-handed corkscrew. No—like a right-handed corkscrew! 1 R —1 (Ϫ— —1 Ϫ—1 Helicity Right ( 2 ) Left 2 ) Right ( 2 ) Left ( 2 ) (a) The Standard Model. The four helicity with the W, and it is therefore called a and the four handedness (chirality) states of weak isospin singlet (that is, it is a the electron are illustrated here. The strikers scalar quantity under the weak isospin Table III. First-Family Weak States and Electroweak Charges between these states indicate that each symmetry, and its weak isotopic charge in the Standard Model Electron handedness state can be written as a linear is zero). The right-handed electron Weak states combination of helicity states. The neutrino does couple electromagnetically to Particle Particle Isotopic Weak Electric Handedness e e e e has only two states, and because it is the photon and weakly to the Z0. R L R L Number Handedness States Charge Hypercharge Charge massless, its helicity is identical to its The right-handed (right-helicity) neu- w w N Nx I3 Y Q 1 (Ϫ — ) —1 handedness. Recall that the spin can be trino ␯ does not exist in the Standard Helicity Left 2 Right ( 2 ) eR QUARKS represented by a pseudovector (red arrow) Model, but if included, it would be a ϩ1 L u ϩ1/2 ϩ1/3 ϩ2/3 Massless and that its direction relative to the weak isospin singlet and thus sterile in L neutrino ϩ1 L ΂dL΃ Ϫ1/2 ϩ1/3 Ϫ1/3 states momentum determines helicity. the sense already described. Similarly, because the weak force picks out the Ϫ1 R u Ϫ1/2 Ϫ1/3 Ϫ2/3 ν ν right-handed components of antiparti- ෆR Handedness L R cles, the right-handed positron e and Ϫ1 R dෆ ϩ1/2 Ϫ1/3 ϩ1/3 ෆR ΂ R΃ the right-handed electron antineutrino Particle Antiparticle ␯ form a weak isospin doublet. The ෆeR ϩ1 R uR 0 ϩ4/3 ϩ2/3 1 —1 —1 1 Helicity Right (—) Left (Ϫ 2 ) Right ( ) Left (Ϫ—) 2 2 2 left-handed positron ෆe is a weak L Ϫ1 L ෆuL 0 Ϫ4/3 Ϫ2/3 isospin singlet and has no weak isotopic ϩ1 R dR 0 Ϫ2/3 Ϫ1/3 (b) Neutrinos with mass. The states of the charge. Table III lists the weak isospin Dirac Ϫ1 L ෆd 0 ϩ2/3 ϩ1/3 states Dirac neutrino versus those of the Majorana doublets, the weak isospin singlets, and L neutrino are shown. Like the electron, the their charges for the first family of weak LEPTONS Dirac neutrino has four helicity and four states in the Standard Model. The Handedness ν ν ν ν 0 ϩ1 L e Ϫ1/2 Ϫ1 Ϫ1 R L R L chirality states. The Majorana neutrino has charges of the Higgs doublet h and L only two handedness and two helicity states. ϩ ϩ1 L ␯ ϩ1/2 Ϫ1 0 1 h are also listed. (The Higgs doublet is ΂ L΃ Left (Ϫ—) Right (—1 ) Helicity 2 2 Further, no clear distinction exists between discussed in the sidebar “Neutrino particle and antiparticle. Masses” on page 64.) Ϫ1 R e ϩ1/2 ϩ1 ϩ1 ෆR Majorana Handedness is also a crucial concept Ϫ1 R ␯ Ϫ1/2 ϩ1 0 ΂ෆR΃ states for the discussion of neutrinos masses. It can be shown mathematically that ϩ1 R eR 0 Ϫ2 Ϫ1 Handedness ν ν any interaction or mechanism that gives L R Ϫ1 L e 0 ϩ2 ϩ1 spin-1/2 particles a nonzero rest mass ෆL ϩ1 R ␯ 0 0 0 must connect particles of different R Initial Final Ϫ1 L ␯ 0 0 0 ν ν (c) Effects of mass terms on particle handedness, that is, the interaction must ෆL R × L states. Mass terms always flip the annihilate a particle of one handedness handedness of a particle. The Dirac mass and create a particle of the opposite HIGGS BOSONS or Conserves Dirac term conserves lepton number or particle handedness. Thus, a particle with mass 0 0 hϩ ϩ1/2 ϩ 1 ϩ1 ν ν lepton number R × L number, whereas a Majorana mass term may switch between right- and left- 0 0 ΂h0΃ Ϫ1/2 ϩ 1 0 × = Action of the changes particle into antiparticle as it handed states as it travels through space, mass term changes the handedness. A Majorana mass changing its handedness by two units, ν ν w L R Changes term is allowed only for neutral particles. whereas a massless particle undergoes w Y Majorana × lepton number R ϭ Right-handed Q ϭ I3 ϩ ᎏ The mass term(s) of the neutrino could be no such transformation and maintains 2 by 2 units L ϭ Left-handed Particle Antiparticle Majorana, Dirac, or both combined. both its handedness and helicity.

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Although the mathematical definition Initial state Final state each other under the weak force. that again involves the Higgs back- (a) Electron Mass f handedness is beyond this discus- ν The difference in the properties ground, but it acts between the two on, we can get a more concrete idea + µ of a Dirac versus Majorana neutrino neutrino states already available in the eL eR Pion u W × y seeing the purely left-handed and πϩ has to do with the way in which the Standard Model, changing ␯L into ෆ␯R. d urely right-handed states of, say, µϩ neutrino acquires its mass. We already The Majorana mass term changes both (Higgs background) he electron written in terms of helicity ϩ ϩ said that, whatever mechanism gives a handedness and lepton number by two π µ ϩ νµ ates |e␭͘: spin-1/2 particle its mass, it must change units. Since it changes a neutrino state that particle’s handedness by two into an antineutrino state, the distinc- (b) Neutrino Mass |e ∝ e ϩ m/E |e , units, from left to right or vice versa. tion between particle and antiparticle L͘ Խ Ϫ1/2͘ 1/2͘ p 1. Effective Majorana mass term (13) 1 Figure 20(a) illustrates how the electron, becomes blurred. The neutrino ν Jz = sz = Ϫ— |e ͘ ∝ Խe ͘ Ϫ m/E |e ͘ , µ 2 a Dirac particle with four states, becomes a Majorana particle, or its R 1/2 Ϫ1/2 o o The massless, left-handed acquires its mass in the Standard Model. own antiparticle. This option requires where m is the mass of the particle, E is neutrino is in a state of The interaction is between the Higgs no new neutrino states. The particular left helicity. s energy, and ␭ ϭ s ؒ p/p is the s background (this is the Higgs mecha- term shown in Figure 20(b.1) is called elicity (with right- and left-handed Pion u nism that gives mass to all particles in an “effective” theory, good only at rojections of 1/2 and Ϫ1/2, axis πϩ the Standard Model) and the electron. low energies because, like Fermi’s orig- × z- d ν ν s µϩ L R espectively). These formulas show that, J = s = —1 Called a Dirac mass term, this inter- inal theory of beta decay, it gives physi- z z 2 a particle is massless (m ϭ 0), helicity action annihilates the state eL and cally inconsistent answers at nd handedness are identical. And, if a The right-handed antimuon creates the state e , or it annihilates e high energies. R ෆR 2. Dirac mass term eft-handed particle is relativistic, or must be produced in a state of and creates e . In each case, the mass Figure 20(b.2) pictures the second left helicity. ෆL ν ν aveling at nearly the speed of light p term changes the handedness by two approach: introducing a Dirac mass L × R m very much less than E), it is mostly units, as required for any mass term. term for the neutrino analogous to that Angular 1 1 Jz = 0 Jz = Ϫ — ϩ — = 0 n a state of left helicity; similarly, a momentum 2 2 But it preserves the particle’s electric shown in Figure 20(a) for the electron. ght-handed particle traveling at charge and lepton number because a It would change ␯L into ␯R and ෆ␯R into

elativistic speeds is mostly in a state particle state remains a particle state ෆ␯L. In other words, it would f right helicity. Handedness and and an antiparticle state remains an conserve lepton number. The Dirac 3. Seesaw mass term (Dirac plus Majorana) elicity are very much related, yet Figure 19. Pion Decay and Helicity versus Handedness antiparticle state. mass term requires the introduction o o < Through the weak force, it decays at rest into a µϩ coupling strength such that the mass < φo> as an intrinsic spin of zero, so for charged antilepton that is in the state Dirac or Majorana particle. Either way, the interactions that might be added of the electron neutrino is at least Nambu-Goldstone he decay of a pion at rest to conserve of left helicity can take part in the there would be two neutrino states: ␯L to the Standard Model to give mass 50,000 times smaller than the mass of boson (from Higgs triplet) 8 oth angular and linear momentum, decay. Being proportional to m/E, and ෆ␯R. Each would travel at the speed to these neutral particles. the electron. But the electron and the he spins and momenta of the two the left-helicity fraction is much of light, and each would maintain its The first is a Majorana mass term electron neutrino are part of the same eptons must point in opposite direc- larger for the antimuon than for the handedness (and its helicity) independent weak doublet, and there seems to be no Figure 20. Neutrino Mass Terms ons (see Figure 19). In other words, positron. Since the decay rate is of the observer’s reference frame. Either reason why they should have such The figures above illustrate the mecha- he two leptons must be in the same proportional to the square of that way, the weak isospin doublets would be 8These extensions are explained more fully in the enormously different interaction nisms for giving the neutrino its mass. sidebar “Neutrino Masses—How to Add Them to elicity state. But the decay process fraction, the pion decays into an (␯ , e ) and (␯ , e ) as defined in strengths to the Higgs bosons. In each case, the ؋ represents the effect eL L ෆeR ෆR the Standard Model” on page 64. They provide the 4 ccurs through the left-handed weak antimuon about 10 times more Table III, and the members within each simplest way of including nonzero neutrino masses In 1979, without introducing an arbi- from the Higgs background. The direction orce and therefore produces a right- frequently than into a positron! weak doublet would transform into while preserving the local gauge symmetries. trarily small coupling strength to the of time is from left to right.

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Higgs bosons, Murray Gell-Mann, Figure 20(b.4) shows one last possi- the first neutron transforms into a proton Initial state Final state Figure 21. Neutrinoless Double ierre Ramond, and Richard Slansky bility for adding neutrino masses to the and emits a W, that W produces a right- (a) N(n, p) N(n, p) N(n Ϫ 2, p ϩ 2) ϩ 2eϪ N(n Ϫ 2, p ϩ 2) Beta Decay nvented a model that yields very Standard Model. No new neutrino handed antineutrino and an electron, as (a) The exchange of a virtual Majorana mall neutrino masses. As explained components are added to the Standard usual. Then that right-handed antineu- neutrino allows double beta decay to n “Neutrino Masses,” the two neutrino Model. Instead, the neutrino is postu- trino switches to a left-handed neutrino Nuclear physics occur without the emission of any neutri- nos. A right-handed Majorana antineutrino ates ␯R and ෆ␯L that must be added lated to be a two-component Majorana through the interaction that gives the o the theory to form the Dirac particle that acquires mass by coupling neutrino its Majorana mass. Finally, this is emitted (along with an electron) from W Ϫ mass term could themselves be coupled to a new type of Higgs boson, one that left-handed neutrino then interacts with the weak vertex at left. Its handedness Ϫ o form a Majorana mass term. has three charge states and is a triplet in the second W (emitted when the second W flips as it propagates through the interac- hat term could also be added to a weak isospin space. Thus, introducing neutron transforms into a proton), and tion with the Higgs background, and the × ν ( m) L he theory without violating any a new type of Higgs boson allows the left-handed electron neutrino is Ϫ right-handed antineutrino becomes a left- ν ϭ Majorana neutrino (ν ) e ymmetry principle. neutrino masses to be added. This last transformed into a left-handed electron. m m R handed Majorana neutrino. In its left- ν ν Electron ϭ ( ) ϩ ( ) Further, it could be assumed that the possibility has several interesting The neutrino is never seen; it is a virtual m R m L handed form, this particle has the correct Mass term oefficient M of the Majorana mass term consequences. Total-lepton-number particle exchanged between the two Ws flips chirality handedness to be absorbed at the weak Ϫ very large. If the theory contains both conservation is not explicitly violated that are emitted when the two neutrons e vertex at right and then transformed into Electron Dirac mass term and this Majorana by the addition of a Majorana mass change into two protons simultaneously. an electron. Thus, two electrons are emit- mass term, then the four components of term. Instead, the new Higgs boson is The net result is that two neutrons in a ted as the nucleus increases its positive he neutrino would no longer be states assumed to have a nonzero vacuum nucleus turn into two protons and two (b) electric charge by two units. (b) The spec- f definite mass m determined by the value; the resulting Higgs background electrons are emitted. In this process, Double beta decay Double beta decay trum of the total energy carried by two oefficient of the Dirac mass term. spontaneously breaks lepton-number the total charge is conserved, but the with neutrinos without neutrinos electrons from neutrinoless double beta nstead, the four components would split conservation and gives a Majorana mass number of leptons has changed from decay is just a single line because the nto two Majorana neutrinos, each made to the neutrino. A consequence of this zero to two. Also, because no neutrinos two electrons always carry off all p of two components. One neutrino spontaneous (or vacuum) breaking of the are emitted, the two electrons will the available energy (a heavy nucleus would have a very small mass, equal to lepton-number symmetry is the existence always share all the available energy absorbs momentum but, essentially, 2 no energy). In contrast, the electrons m /M from the mass term in of a massless scalar particle known as a released in the decay, and thus the sum Number of Electrons igure 20(b.3); the second neutrino Nambu-Goldstone boson. This massless of their energies has a single value, the from ordinary double beta decay share would have a very large mass, approxi- boson could be produced in a new form single spike in Figure 21(a), rather than the available energy with the two electron mately equal to M. The very light of neutrinoless double beta decay. a spectrum of values as in ordinary antineutrinos emitted in the decay. Majorana neutrino would mostly be the double beta decay. eft-handed neutrino that couples to Neutrinoless Double Beta Decay. The The rate of neutrinoless double beta Sum of Both Electron Energies (MeV) he W, and the very heavy neutrino one process that should be within the decay is proportional to an effective would mostly be a right-handed neutrino limits of detectability and would mass that is a complicated sum over the hat does not couple to the W. Similarly, exhibit the unmistakable mark of a three neutrino masses. This sum Implications of Neutrino Mass the three types. If individual neutrino of these sterile neutrinos in the cosmic he very light antineutrino would be Majorana neutrino is neutrinoless dou- involves the intrinsic charge-conjugation for Astrophysics, Cosmology, masses are on the order of a few elec- soup could shift the delicate balance mostly the original right-handed ble beta decay. In double beta decay, and parity properties of the neutrinos and Particle Physics tron volts, their sum would add up to a of ingredients needed to predict the ntineutrino that couples to the W, and two neutrons in a nucleus transform (CP parities), and the resulting phases significant fraction of the mass of the observed primordial abundances of heli- he very heavy antineutrino would be into two protons almost simultaneously multiplying each mass can lead to can- If neutrino masses and oscillations universe—not enough mass to close the um and other light elements up through mostly a left-handed antineutrino that and bring the nucleus to a stable config- cellations such that the effective mass are real, they can have an impact on universe and have it collapse back on lithium. As a result, nucleosynthesis cal- oes not couple to the W. uration with an increase in electric is smaller than any of the individual astrophysics and cosmology, and, itself (that would require the average culations place stringent limits on sterile This so-called seesaw mechanism in charge of ϩ2. This process occurs in masses of the neutrinos. At present, conversely, astrophysics and cosmology mass of the three neutrinos to be 30 eV), neutrinos, ruling out significant portions which the Dirac mass m is reduced by a “even-even” nuclei, those containing the experimental upper limit on the will place constraints on the masses of but at smaller values, it could have influ- in the ⌬m2 – sin2␪ plane for the mixing actor of m/M through the introduction even numbers of protons and neutrons. effective mass is about 2 eV. neutrinos and on the number or types enced the expansion of matter after between ordinary and sterile neutrinos. f a large Majorana mass term has Like single beta decay, double beta Finally, if the neutrino acquires mass of neutrinos. Neutrinos are very weakly the Big Bang and helped produce the Oscillation could also alter the een used in many extensions to the decay occurs through the interaction of through the vacuum value of a Higgs coupled to matter. At energies of superlarge-scale filigree pattern of picture of the neutrino as the driver tandard Model to explain why a nucleus with the W. In the “ordinary” triplet, as discussed above, a massless 1 MeV, a neutrino interacts 1020 times galaxies and galactic clusters that of supernova explosions (see the eutrino masses are small. The large process shown in Figure 21(a), the Nambu-Goldstone boson would be less often than a photon. To have any extends as far as today’s telescopes can article “Neutrinos and Supernovae” on Majorana mass M is often associated nucleus emits two electrons and two emitted along with the two electrons of impact at all, they must be present in see. (See the article “Dark Matter and page 164). Electron neutrinos, the with some new, weak gauge force that antineutrinos. Figure 21(b) shows, how- the neutrinoless double beta decay. The extraordinary numbers. One such Massive Neutrinos” on page 180. ) primary drivers, might be lost or gained perates at a very high energy (mass) ever, that if the neutrino is a Majorana presence of the massless boson would “place” is the universe itself. Neutrinos Neutrino oscillations, too, may be an from the region that powers the cale dictated by the mass of a new, particle, the same process can occur lead to a definite energy spectrum for left over from the Big Bang fill the uni- important ingredient in making the uni- explosion, depending on the oscillation ery heavy gauge boson. The net result without the emission of any neutrinos— the emitted electrons that would distin- verse and outnumber protons and neu- verse as we know it. For example, the length and, again, on whether sterile f this approach is that the neutrino hence the name of neutrinoless double guish this form of double beta decay trons by a billion to one. On average, neutrinos we know might oscillate into neutrinos exist. Neutrino oscillations een at low energies is predicted to be beta decay. The weak force has not from either ordinary double beta decay the universe contains about 300 neutri- sterile neutrinos, those which have no and the enhancement of those oscilla- mostly a Majorana particle! changed its character. Indeed, when or neutrinoless double beta decay. nos per cubic centimeter, 100 of each of weak interactions at all. The presence tions through interactions with matter

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may also be the only way to create the electromagnetic forces become one at ␮Ϫ → eϪ + ␥, and they typically Further Reading Richard (Dick) Slansky was born in California. Having earned eutron-rich environment that is the unification scale, these theories con- provide a candidate for the cold dark a B.A. from Harvard and a Ph.D. in physics from the University of California at Berkeley (in 1967), Slanksy became postdoctoral Bernstein, J. 1969. The Elusive Neutrino. United bsolutely required for the synthesis of strain the strengths of the strong, weak, matter that may be needed to explain fellow at Caltech and later assistant professor in the Physics States Atomic Energy Commission–Division he elements heavier than iron. And to and electromagnetic couplings to be the observed large-scale structures Department at Yale University. In 1974, he joined Los Alamos as of Technical Information. Library of Congress ecap the earlier discussion, oscillation equal at that scale. Thus, one can put the and large-scale motions of the a staff member in the newly established Elementary Particles and Catalog Card Number: 71-601702. Field Theory Group in the Theoretical (T) Division. At the Labo- om one neutrino type into another measured values of the weak- and elec- luminous matter. ratory, his interests turned to unified theories of electroweak and might explain why neutrino physicists tromagnetic-coupling strengths into the Cardall, C. Y., and G. M. Fuller. 1996. Can a strong interactions—proton decay, neutrino masses, and applying ave been measuring a shortfall in the framework of the Grand Unified “Natural” Three-Generation Neutrino Mixing group theory to these unified theories. Slansky was also active in Scheme Satisfy Everything? Physical Review D atio of muon neutrinos to electron neu- Theories and predict the strong-coupling Superstrings and Conclusions string theory, and he co-authored a two-volume book on the high- 53: 4421. est weight representations of Kac-Moody algebras. Eventually, he inos produced by cosmic rays in the strength and the scale of unification. worked on numerous interdisciplinary issues and was appointed pper atmosphere. Matter-enhanced In the Grand Unified Theories that To tie up our discussion, we will Fermi, E. 1933. Tentative Theory of Beta- adjunct professor of physics at the University of California at Irvine. Slansky then became a eutrino oscillations in the electron-rich include a new symmetry, called super- mention superstring theory, one Radiation. In Strachan, C. 1969. The Theory Laboratory Fellow and a Fellow of the American Physical Society and the American Association for nvironment of the Sun might explain symmetry, the prediction for the strong possible truly unified theory that of Beta-Decay. New York: Pergamon Press. the Advancement of Science. In 1993, after having been T-Division leader at Los Alamos for four years, Slanksy was reappointed as T-Division Director, a position he currently holds. why physicists observe a shortfall in coupling agrees with all the available includes not only the electroweak and Feynman, R. P., and M. Gell-Mann. 1958. he flux of electron neutrinos that are data, and the grand unification scale strong interactions, but also gravity in Theory of the Fermi Interaction. Physical Stuart Raby was born in Bronx, New York, and has been a roduced by thermonuclear fusion turns out to be on order of 1016 GeV. the sense of a quantum mechanical Review 109 (1): 193. physics professor at the Ohio State University in Columbus, Ohio, since 1989. He is married to Elaine M. Raby and is a proud parent rocesses in the core of the Sun. (For comparison, the proton mass theory of Einstein’s general theory of of Eric and Liat. Raby received his B.Sc. from the University of 2 Halzen, F., and A. D. Martin. 1984. Quarks & Ϸ 1 GeV/c , and the largest accessible relativity. Although not yet a full- Leptons: An Introductory Course in Modern Rochester in 1969 and an M.Sc. and a Ph.D. from Tel Aviv Uni- Grand Unified Theories. On a more energies at the new accelerator being fledged theory, superstrings have Particle Physics. New York: John Wiley & versity in 1973 and 1976, respectively. He was a postdoctoral fellow at Cornell and Stanford Universities as well as a visiting bstract note, the existence of neutrino planned in Europe will be a few times enjoyed significant recent progress. Sons, Inc. professor at the University of Michigan before joining the Labora- 3 masses and mixing will extend the 10 GeV.) These supersymmetric theo- At “low energies” (although they are tory as a staff member in the Elementary Particles and Field Kayser, B., F. Gibrat-Debu, and F. Perrier. 1989. lose parallel already observed between ries also predict relations between the very high compared with current Theory Group, whose leader he became in 1985. His research The Physics of Massive Neutrinos. World uarks and leptons and, for that reason, masses of the charged quarks and lep- accelerator energies), superstring interests include physics beyond the Standard Model, the origin of Scientific Lecture Notes in Physics Vol. 25. fermion masses, flavor violation, and consequences for laboratory may well add fuel to the ongoing tons, and these relations are also well theories reduce to models with large Singapore: World Scientific. experiments, cosmology, and astrophysics. Raby is a leading earch for a theory that unifies the satisfied. Neutrino masses are typically gauge symmetries that may unify the advocate for supersymmetric Grand Unified Theories of nature. rong, weak, and electromagnetic not as constrained as charged fermion electroweak and strong interactions, Lee, T. D., and C. N. Yang. 1957. Parity These theories provide the simplest explanation for the experimentally measured unification of Nonconservation and a Two-Component orces. Attempts to explain the pattern masses because the neutrino sector con- along with other undiscovered interac- the strong, weak, and electromagnetic charges and for the observed spectrum of fermion masses and Theory of the Neutrino. Physical Review mixing angles. They also make testable predictions for flavor violation and nucleon decay processes. f charges and masses of quarks and tains the possibility of very heavy (as in tions of nature. Although superstrings 105 (5): 1671. eptons within a single weak family the seesaw) Majorana masses. are insufficiently formulated to predict T. (Terry) Goldman received a B.Sc. (Hons.) in physics and columns in Figure 5) lead naturally to The proton, which is the most stable the parameters of the Grand Unified Particle Physics. 1984. Los Alamos Science. mathematics in 1968 from the University of Manitoba and an A.M. and a Ph.D. in physics from Harvard in 1969 and 1973, n extension of the Standard Model particle we know, is typically unstable Theories, the suggestive link between Summer/Fall. Number 11. Los Alamos National Laboratory document respectively. Goldman was a Woodrow Wilson Fellow from nown as the Grand Unified Theories. in the Grand Unified Theories and has the two makes us pay close attention LA-UR-84-2100. 1968 to 1969 and a National Research Council of Canada postdoc- n these theories, the local gauge sym- a lifetime set by the grand unification to the Grand Unified Theories, even in toral fellow at the Stanford Linear Accelerator from 1973 until metries of the weak, strong, and elec- scale. Supersymmetric Grand Unified the absence of direct experimental Riordan, M. 1987. The Hunting of the Quark: A 1975, when he joined the Laboratory’s Theoretical Division as a postdoctoral fellow. In 1978, Goldman became a staff member in omagnetic forces are subsumed under Theories predict that the dominant evidence for them. On a less ambitious True Story of Modern Physics. New York: Simon and Schuster/Touchstone. the Medium Energy Theory Group at Los Alamos. From 1978 to larger local gauge symmetry. That decay mode for the proton is plane, experimental values for neutrino 1980 he was on leave from the Laboratory as a senior research + arger symmetry becomes apparent only p → K + ␯ . The cumulative evidence masses and mixing angles would Strachan, C. 1969. The Theory of Beta-Decay. fellow at the California Institute of Technology, and during the ෆ academic year 1982–1983, he was a visiting associate professor New York: Pergamon Press. t the enormous energies and tiny dis- collected over the next five years at constrain the parameters of the Grand at the University of California, Santa Cruz. In 1991, Goldman ance scales known as the unification super-Kamiokande will be sensitive Unified Theories—particularly when became group leader in T-5. His research interests are in the areas of electroweak interactions and the Sutton, C. 1992. Spaceship Neutrino. New York: cale. At that scale, the strong, weak, to this decay mode with a predicted there is a better understanding of the quark structure of nuclei. Goldman is a member of the American Physical Society. Cambridge University Press. nd electromagnetic forces become uni- lifetime on the order of 1033 years. origin of mass and mixing. Gerry Garvey was born in New York City. He earned a B.S. fied into one force, and the quarks and Finally, supersymmetric Grand Unified No one yet understands why mass Weinberg, S. 1995. The Quantum Theory of in physics from Fairfield University and, in 1962, a Ph.D. in eptons within a family become mem- Theories require new particle states, states and weak states differ or, even Fields. Volume II. New York: Cambridge physics from Yale University. He then served as a faculty ers of a particle multiplet that trans- some of which may be observed at with experimental data on hand, why University Press. member at Yale and Princeton Universities, becoming a full professor in 1969. After being an Alfred Sloan Foundation orm into each other under the unified high-energy accelerators, specifically, the pattern of mixing for quarks is as Fellow from 1967 to 1969, he spent a year at the Clarendon orce, just as the members of each at the new Large Hadron Collider we observe it. Why there should be Laboratory in Great Britain. In 1976, he became Director of the weak isospin doublet transform into at CERN scheduled for completion three repetitive families is likewise Acknowledgment Physics Division at Argonne National Laboratory and in 1979 ach other under interaction with the W. in 2002, at the Fermilab Tevatron (an mysterious. If we are to develop a was named Associate Laboratory Director for Physical Science. From 1979 through 1984, Garvey was a professor at the Univer- N. Cooper would like to thank Peter Herczeg The Grand Unified Theories provide 1,800-GeV machine) following its unified theory combining the quark sity of Chicago, returning in 1980 to full-time research as a for many illuminating conversations and natural explanation for the different upgrade in 1999, and at the Hadron and lepton families, we need to solve senior scientist at Argonne. In 1984, he joined Los Alamos as for inspiring greater clarity and precision in director of LAMPF. In 1990, Garvey stepped down from that harges (electric, weak, and strong) for Electron Ring Accelerator at DESY these unknowns. Neutrino masses and understanding the neutrino. articles in a family. In addition, these (Deutsches Elektronen Synchrotron). mixings are among the few uncharted position to become a Laboratory Senior Fellow, a position he currently holds. From 1994 through 1996, he served as the Assistant Director for Physical Science and heories make several successful predic- These new states can lead to observ- realms that may provide important Engineering in the White House Office of Science and Technology. Garvey is an active member of the ons. Since the strong, weak, and able lepton family mixing such as clues to this puzzle. ■ American Physical Society, having been chairman and councilor for the Division of Nuclear Physics.

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of spatial coordinates) has the property of interchanging the two states eR and eL. Neutrino Masses What about the neutrino? The right-handed neutrino has never been observed, How to add them to the Standard Model and it is not known whether that particle state and the left-handed antineutrino c exist. In the Standard Model, the field ␯e , which would create those states, is not Stuart Raby and Richard Slansky included. Instead, the neutrino is associated with only two types of ripples (particle states) and is defined by a single field ␯e:

␯ annihilates a left-handed electron neutrino ␯ or creates a right-handed he Standard Model includes a set of particles—the quarks and leptons e eL electron antineutrino ␯ . —and their interactions. The quarks and leptons are spin-1/2 particles, or ෆeR fermions. They fall into three families that differ only in the masses of the T The left-handed electron neutrino has fermion number N = +1, and the right- member particles. The origin of those masses is one of the greatest unsolved handed electron antineutrino has fermion number N = Ϫ1. This description of the mysteries of particle physics. The greatest success of the Standard Model is the neutrino is not invariant under the parity operation. Parity interchanges left-handed description of the forces of nature in terms of local symmetries. The three families and right-handed particles, but we just said that, in the Standard Model, the right- of quarks and leptons transform identically under these local symmetries, and thus handed neutrino does not exist. The left-handedness of the neutrino mimics the they have identical strong, weak, and electromagnetic interactions. left-handedness of the charged-current weak interactions. In other words, the W In the Standard Model, quarks and leptons are assumed to obtain their masses gauge boson, which mediates all weak charge- in the same way that the W and Z0 bosons obtain theirs: through interactions with changing processes, acts only on the fields e Table I. Lepton Charges the mysterious Higgs boson (named the “God Particle” by Leon Lederman). But and ␯ . The interaction with the W transforms Yw before we write down some simple formulas that describe the interactions of quarks e w ϩ I3 ᎏᎏ ؍ the left-handed neutrino into the left-handed Q and leptons with the Higgs boson, let us define some notation. 2 electron and vice versa (e ↔ ␯ ) or the right- L eL NNParticle States I w Yw Q Defining the Lepton Fields. For every elementary particle, we associate a field handed antineutrino into the right-handed x 3 residing in space and time. Ripples in these fields describe the motions of these positron and vice versa (␯ ↔ e ). Thus, ෆeR ෆR ϩ1LeLϪ1/2 Ϫ1 Ϫ1 particles. A quantum mechanical description of the fields, which allows one to we say that the fields e and ␯e, or the particles ϩ1L΂␯L΃ϩ1/2 Ϫ10 describe multiparticle systems, makes each field a quantum mechanical operator eL and ␯eL, are a weak isospin doublet under that can create particles out of the ground state—called the vacuum. The act of the weak interactions. Ϫ1R eϩ1/2 ϩ1 ϩ1 creating one or more particles in the vacuum is equivalent to describing a system These lepton fields carry two types of weak ෆR Ϫ1R␯Ϫ1/2 ϩ10 w ΂ෆR΃ in which one or more ripples in the fabric of the field move through space-time. charge: The weak isotopic charge I3 couples Let us now discuss the simple system of one family of leptons. To be specific, them to the W and the Z0, and the weak ϩ1R e 0Ϫ2 Ϫ1 we will call the particles in this family the electron and the electron neutrino. hypercharge Yw couples them to the Z0. (The Z0 R Ϫ1Le 0ϩ2 ϩ1 The electron field describes four types of ripples (or particles). We label these four is the neutral gauge boson that mediates neutral- ෆL ϩ1R ␯ 00 0 types by two quantum charges called fermion number N and handedness, or current weak interactions.) Electric charge Q is R Ϫ1Lෆ␯L 00 0 chirality, Nx. For the electron field, the particle state with fermion number N = +1 related to the two weak charges through the w w is the electron, and the particle state with N = Ϫ1 is the antielectron (or positron). equation Q ϭ I3 ϩ Y /2. Table I lists the weak Each of these states comes as right-handed, Nx = R, and left-handed, Nx =L. charges for the particle states defined by the Handedness is a Lorentz invariant quantity that is related in a nontrivial way to three fields e, ␯ , and ec. Note that the particle states e and e defined by e R ෆL helicity, the projection of the spin s in the direction of the momentum p. (For a the field ec do not couple to the W and have no weak isotopic charge. The field discussion of handedness versus helicity, see “The Oscillating Neutrino” on page 28.) and the particle states are thus called weak isotopic singlets. However, e and e R ෆL In relativistic quantum field theory, the right-handed and left-handed electron do carry weak hypercharge and electric charge and therefore couple to the Z0 and the right-handed and left-handed antielectron can be defined in terms of two and the photon. c c fields denoted by e and e , where each field is a Weyl two-component left-handed Likewise, the field ␯e and its neutrino states ␯R and ෆ␯L would be isotopic spinor. The compositions of the fields are such that singlets with no coupling to the W. But unlike their electron counterparts, they w w must be electrically neutral (Q ϭ I3 ϩ Y /2 ϭ 0), which implies they cannot 0 e annihilates a left-handed electron eL or creates a right-handed have weak hypercharge. Thus, they would not couple to the W, the Z , or the positron e , and photon. Having no interactions and, therefore, not being measurable, they ෆR are called sterile neutrinos and are not included in the Standard Model. However, ec annihilates a left-handed positron e or creates a right-handed if the left-handed neutrino has mass, it may oscillate into a sterile right-handed ෆL electron eR. neutrino, a possibility that could be invoked in trying to give consistency to all the data on neutrino oscillations. These fields are complex, and for the action of the Hermitian conjugate fields e† and ec†, just interchange the words annihilate and create above. For example, The Origin of Electron Mass in the Standard Model. What is mass? Mass is e† creates a left-handed electron or annihilates a right-handed positron. Hence, the the inertial energy of a particle. It is the energy a particle has when at rest and the fields e and ec and their complex conjugates can create or annihilate all the possi- measure of the resistance to an applied force according to Newton’s law F = ma. ble excitations of the physical electron. Note that parity (defined as the inversion A massless particle cannot exist at rest; it must always move at the speed of light.

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A fermion (spin-1/2 particle) with mass has an additional constraint. It must exist for the neutrino that would mirror the mass term for the electron. It would in both right-handed and left-handed states because the only field operators that have the form yield a nonzero mass for fermions are bilinear products of fields that flip the parti- c cle’s handedness. For example, in the two-component notation introduced above, m␯␯e ␯e . (5) the standard, or Dirac, mass term in the Lagrangian for free electrons is given by* c But, as we said above, the field ␯e is not included in the Standard Model c mee e . (1) because, so far, weak-interaction experiments have not required it. The neutrino, though, has no electric charge, which makes it possible to write down a mass term This fermion mass operator annihilates a left-handed electron and creates a right- from the existing neutrino field ␯e with the form handed electron in its place. The mass term does not change the charge of 1 the particle, so we say that it conserves electric charge. Also, because this mass ᎏ␮ ␯ ␯ . (6) 2 ␯ e e term does not change a particle into an antiparticle, we say that it conserves fermion number N. However, the weak isospin symmetry forbids such a mass (Note that m␯ and ␮␯ refer just to the electron neutrinos, but similar masses can operator because it is not an invariant under that symmetry. (The field e is a be defined for the ␮ and ␶ neutrinos.) The mass operator in (6) annihilates a left- member of a weak isotopic doublet, whereas the field ec is a weak isotopic singlet, handed neutrino and creates a right-handed antineutrino, which means that it is a so that the product of the two is not a singlet as it should be to preserve the weak Majorana mass term. Any mass term that changes a particle to an antiparticle is isospin symmetry.) But the electron does have mass. We seem to be in a bind. called a Majorana mass term. In changing a neutrino to an antineutrino, this term The Standard Model solves this problem: the electron and electron neutrino violates fermion number N, changing it by two units. It is a legitimate mass term fields are postulated to interact with the spin-zero Higgs field h0 (the God particle). in that it changes handedness in the right way to yield a nonzero rest mass, and it The field h0 is one member of a weak isospin doublet whose second member is conserves electric charge because the neutrino is electrically neutral. Nevertheless, h+. The superscripts denote the electric charge of the state annihilated by each it is not included in the Standard Model because it violates the weak symmetry in field (see Table III on page 57 for the other quantum number of the two fields). two ways: It is not invariant under the weak isospin symmetry, and it changes the The field h0 plays a special role in the Standard Model. Its ground state is not a weak hypercharge by two units. We conclude that, in the minimal Standard Model, c vacuum state empty of particles, but it has a nonzero mean value, much like a which does not include ␯e and contains only the Higgs doublet mentioned above, Bose-Einstein condensate. This nonzero value, written as the vacuum expectation there is no way to give mass to the neutrinos if fermion number is conserved. 0 0 value <0|h |0> ϵ = v/͙ෆ2 is the putative “origin of mass.” (The “mystery” of Two consequences follow directly from the result that neutrino masses are mass then becomes the origin of the Higgs boson and its nonzero vacuum value.) identically zero in the minimal Standard Model. First, the weak eigenstates and The interaction between the Higgs fields and the electron and electron neutrino the mass eigenstates of the leptons are equivalent, and therefore individual-lepton- is given by family number (electron number, muon number, and tau number) are conserved c + † 0 † ␭ee ΂␯e(h ) ϩ e(h ) ΃ , (2) (for the proof, see “Family Mixing and the Origin of Mass” on page 72). Thus, the Standard Model forbids such processes as where ␭e is called a Yukawa coupling constant and describes the strength of the coupling between the Higgs field and the electron. The Higgs field is a weak ␮ϩ → e+ ϩ ␥ , or (7) isospin doublet, so the term in parentheses is an inner product of two doublets, making an invariant quantity under the weak isospin symmetry. Since it also con- ␮ϩ → e+ ϩ eϩ ϩ eϪ . (8) serves weak hypercharge, it preserves the symmetries of the Standard Model. 0 0 Because the mean value of h in the vacuum is ϭ v/͙2ෆthe, operator in Similarly, the proposed process of neutrino oscillation, which may recently have (2) contributes a term to the Standard Model of the form been observed, is forbidden. Second, total lepton number, equal to the sum of individual-family-lepton numbers, is also conserved, and the process of 0 c c ␭ee e ϭ (␭ev/͙ෆ2)e e . (3) neutrinoless double beta decay is forbidden. The converse is also true: If individual-lepton-number violation is observed, In other words, as the electron moves through the vacuum, it constantly feels the or if the LSND results on neutrino oscillation are confirmed, then either of those interaction with the Higgs field in the vacuum. But (3) is a fermion mass operator experiments could claim the discovery of nonzero neutrino masses and thus of exactly analogous to the Dirac mass operator in (1), except that here the electron new physics beyond the Standard Model. rest mass is given by Adding Neutrino Masses to the Standard Model. What could this new me ϭ ␭ev/͙ෆ2 . (4) physics be? There are several simple extensions to the Standard Model that could yield nonzero neutrino masses without changing the local symmetry of We see that, in the Standard Model, electron mass comes from the Yukawa the weak interactions. interaction of the electron with the Higgs background. The simplest extension would be to add no new fields but just a new “effective” interaction with the Higgs field: Why Neutrinos Are Massless in the Minimal Standard Model. What about the 1 0 ϩ 2 The addition of the Hermitian conjugate is assumed in neutrino? Because the neutrino has spin 1/2, its mass operator must also change ᎏ (h ␯e Ϫ h e) . (9) M l equations if the operator is not explicitly Hermitian. handedness if it is to yield a nonzero value. We could introduce a Dirac mass term effective

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This effective interaction is invariant under the local symmetries and yields a Standard Model, including the strong interactions, they observed that, if one c Majorana mass term equal to introduced the right-handed neutrino field ␯e into the Standard Model to form a Dirac mass term, one could also add a Majorana mass term of the form 1 0 2 ᎏ ␯e␯e , (10) M 1 c c effective ᎏM␯ ␯ (14) 2 e e c and a value for the neutrino mass without violating the local symmetries of the Standard Model (as stated above, ␯e 2 has no weak charge and is thus an invariant under the local symmetry). Further, if 22 v ␮␯ ϭ ᎏ ϭ ᎏ . (11) M were large enough, the mass of the left-handed neutrino would be small enough M M effective effective to satisfy the experimental bounds. To see how this reduction occurs, we write the operators for both the Dirac This mass term, as all fermion mass terms, changes handedness from left to mass term and the Majorana mass term: right, but it violates the fermion number N listed in Table I. The term Meffective 0 + c 1 c c must be large so that the mass of the neutrino be small. The new term in (9) is ᏸ ϭ ␭ (h ␯ Ϫ h e)␯ ϩ ᎏM␯ ␯ ϩ other terms . (15) mass ␯ e e 2 e e called “effective” because it can only be used to compute the physics at energies 2 well below Meffectivec , just as Fermi’s “effective” theory of beta decay yields Here we are assuming that ␭␯ Х ␭e. These additions to the Lagrangian yield the 2 valid approximations to weak processes only at energies well below MWc , where following mass terms: MW is the mass of the W. (Outside their specified energy ranges, “effective” theo- c 1 c c ries are, in technical language, nonrenormalizable and yield infinite values for ᏸ ϭ m ␯ ␯ ϩ ᎏM␯ ␯ , (16) ␯e mass ␯e e e 2 e e finite quantities.) Thus, the mass term in (9) implicitly introduces a new scale of physics, in which new particles with masses on the order of M presumably where m is the Dirac mass defined in (13), except that now we assume ␭ ␭ , effective ␯e ␯ Х e play a role. Below that energy scale, (9) describes the effects of the seesaw in which case m ␭ v/ 2ෆIn. other words, the Dirac neutrino mass is about ␯e Х e ͙ mechanism for generating small neutrino masses (see below as well as the box equal to the electron mass (or some other fermion mass in the first family). “The Seesaw Mechanism at Low Energies” on page 71). The two neutrino mass terms may be rewritten as a matrix, frequently referred to as the mass matrix: A Dirac Mass Term. Another extension would be to introduce a right-handed c neutrino field ␯i , one for each neutrino flavor i (i ϭ e, ␮, ␶), where, for example, 0 m␯ ␯e c e the right-handed field for the electron neutrino is defined such that 1/2(␯e ␯e ) . (17) m M ΂ ␯ c΃ ΂ ␯ ΃ e c e ␯e annihilates a left-handed electron antineutrino ෆ␯eL and creates a right- c handed electron neutrino ␯eR. It is clear that the fields ␯e and ␯e do not describe states of definite mass, or mass eigenstates, but rather the two fields are mixed by the interaction with the Higgs We could then define an interaction with the Higgs field exactly analogous to field. Diagonalizing this matrix yields the masses of the physical neutrinos. the interaction in (3) that gives electrons their mass: [The expressions in (16) and Equation (17) are equivalent. The proof requires more detail than is presented here.] One mass is very small: ␭ ␯ c(␯h0 Ϫ eh+) . (12) 2 ␯ e m␯ ␮ Ϸ ᎏ e . (18) light M Again, because the Higgs field h0 has a nonzero vacuum expectation value, the interaction in (12) would give the neutrino a Dirac mass It is the Dirac mass reduced by ratio m /M that gave this mechanism its ␯e name—the “seesaw.” The second mass is very large: ␭␯v m␯ ϭ ᎏ . (13) ͙2ෆ ␮heavy Ϸ M . (19) But why are neutrino masses much smaller than the masses of their charged lepton weak partners? Specifically, why is m␯ ϽϽ me? The electron mass is The fields corresponding to these masses are given by 500,000 eV, whereas from experiment, the electron neutrino mass is known to be m␯ less than 10 eV. The only explanation within the context of the interaction above ␯ Ϸ ϩ ᎏ e ␯ c Ϸ ␯ , (20) light ␯e M e e is that the strength of the Yukawa coupling to the Higgs field is much greater for ΂ ΃ and the electron than for the electron neutrino, that is, ␭ Ͼ 5 ϫ 104␭ . But this is not e ␯ m␯ an explanation; it just parametrizes the obvious. ␯ Ϸ c Ϫ ᎏ e ␯ Ϸ ␯ c . (21) heavy ␯e ΂ M ΃ e e The Seesaw Mechanism and Majorana Neutrinos. The first real model of why Both fields define Majorana particles, that is, particles that are their own antiparti- neutrino masses are very much smaller than the masses of their lepton partners cles, and total-lepton-number conservation can be violated in processes involving was provided by Murray Gell-Mann, Pierre Ramond, and Richard Slansky. these neutrinos. The light neutrino would correspond to the neutrino we see in the Motivated by a class of theories that attempt to unify the interactions of the

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weak processes observed so far, and is essentially the left-handed neutrino field ␯e. The Seesaw Mechanism at Low Energies c The right-handed neutrino field ␯e would not be observed directly at low energies. Its effect in the low-energy theory would only be visible as an effective neutrino The seesaw mechanism for neutrino masses defines a new scale of nature given c mass operator, like the operator in (9), which would give the neutrino a very small by M, the mass associated with the heavy right-handed neutrino ␯e. Since M is mass and would signal the presence of a new scale of physics on the order of postulated to be very large, well above the energies accessible through experiment, 2 Meffective ϭ 2M/␭␯ (see the box “The Seesaw Mechanism at Low Energies” on it is interesting that the “effective” neutrino mass operator in (11) approximates the facing page). the seesaw terms in (15) at energies below M. To show this, we consider the (a) Effective neutrino mass term effective operator 1 0 ϩ 2 o o A New Higgs Isospin Triplet. Another possibility is that there are no right- ᎏ (h ␯e Ϫ h e) . h h M handed neutrinos, but there is, instead, a new set of Higgs-type bosons ␾ that effective 0 + ++ come in three varieties —␾ , ␾ , ␾ — and transform as a triplet under the When the Higgs vacuum expectation value is accounted for, this operator yields 1

local weak isospin symmetry. The superscript denotes the electric charge of the nonrenormalizable mass term in diagram (a) and a Majorana mass given by Meffective each boson. Using this Higgs triplet, we can introduce the interaction v2 ␮␯ ϭ ᎏ ν ν 0 + ++ Meffective e e ␭m(␯␯␾ ϩ ␯e␾ ϩ ee␾ ) , (22) In the seesaw mechanism, the light neutrino acquires its mass through the which is consistent with all Standard Model symmetries. If, in analogy with h0, the exchange of the heavy neutrino, as shown in diagram (b). Diagram (b), which is Higgs field ␾0 has a nonzero vacuum expectation value <␾0> ϭ v , the neutrino approximated by diagram (a) at energies below Mc2, is a renormalizable mass (b) Seesaw mass term for m the light neutrino would also have a Majorana mass given by term that involves both Dirac and Majorana masses. It yields a neutrino mass 2 m o o 0 ␯e v h h ␮␯ ϭ ␭m<␾ > ϭ ␭mvm , (23) ␮light ϭ ᎏ with m␯ ϵ ␭␯ ᎏ . M e e ͙2ෆ λν λν where this fermion mass is a Majorana mass. In a theory with a Higgs triplet, Equating the values for ␮␯ and ␮light, we obtain the relation between M and e e the Higgs doublet is still necessary. In fact, in order to preserve the observed ratio M : νc effective e of strengths of neutral- to charged-current interactions (equal to 1 Ϯ .01), the 2 1 (␭␯ ) 1 vacuum expectation value v must be much smaller than in (3). Also, such a ϭ e . ν ν m ᎏ ᎏ e M e M 2M theory has a massless Nambu-Goldstone boson ␾ due to the spontaneous breaking effective of total lepton number, and it allows the process At energies below MW, the mass of the W boson, a similar type of relationship → ␯␮ ␯e ϩ ␾ . (24) exists between Fermi’s “effective” theory shown in diagram (c) and the W-boson (c) Fermi’s current-current exchange processes shown in diagram (d). The exchange processes are defined by interaction Apart from the effective interaction in Equation (9), the other extensions we the gauge theory of the charged-current weak interactions. Fermi’s theory is a ν discussed introduce new states. Each makes predictions that can be tested. nonrenormalizable current-current interaction of the form e e The Higgs triplet extension is the largest departure from the Standard Model. G † F ␮ W G The seesaw mechanism is less intrusive than the Higgs triplet. In general, its only ᏸFermi ϭ ᎏ J W J␮ , F 2ෆ low-energy consequence is an arbitrary Majorana mass term for the three neutrino ͙ 2 e species given by where the weak current for the neutrino-electron doublet is given by ν e ␮ † ␮ ␮ i ␮ijvivj , where i, j = e, ␮, ␶ . (25) JW ϭ 2␯e ෆ␴ e and ෆ␴ ϭ (1, Ϫ␴ ) ,

A general mass matrix such as the one in (25) would lead to lepton- and the Fermi constant GF defines the strength of the effective interaction in (d) Weak charged-current family-number violating processes, CP (charge-conjugation/parity) violation, diagram (c), as well as a new mass/energy scale of nature. The experimentally gauge interaction Ϫ5 Ϫ2 and neutrino oscillations. This simple hypothesis will be tested by present observed value is GF ϭ 1.66 ϫ 10 GeV . Equating the low-energy limit of or proposed experiments. diagram (c) with that of diagram (d) yields the formula ν - e e On a final note, the new scale M in (15) can be very large. It may be associated g G g2 with the proposed grand unification scale for strong, weak, and electromagnetic F ϭ , 2 2 ᎏ2 ᎏ8M 2 interactions, which is predicted to occur at energies on the order of 1016 GeV. ͙ෆ W If so, neutrino masses and mixings can give us information about the physics at where g is the weak isospin coupling constant in the charged-current weak 1 W 2 this enormous energy scale. There is also the exciting possibility that, through a Lagrangian given by MW sequence of interactions that violate CP, lepton-number, and baryon-number c 2 ␮ϩ g ϩ ␮ g ␮† g conservation, the decay of the very heavy right-handed neutrino ␯ in the hot, ᏸweak ϭ ϪMW W W␮ෆϩ ᎏ W␮ J W ϩ ᎏ W␮ෆJ W . 2 2ෆ 2 2ෆ early universe generates the observed baryon number of the universe, that is, ͙ ͙ 2 2 e- ν the presence of matter as opposed to antimatter. ■ This Lagrangian neglects the kinetic term for the W, which is a valid e approximation at energies much less than the W boson mass.

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exception of neutrino oscillations. If oscillations are confirmed, the mixing Family Mixing and the Origin of Mass angles measured in the neutrino experiments will become part of a CKM mixing The difference between weak eigenstates and mass eigenstates matrix for the leptons. This sidebar derives the form of the CKM matrix and shows how it reflects Stuart Raby the difference between the rotation matrices for the up-type quarks (Q ϭϩ2/3) and those for their weak partners, the down-type quarks (Q ϭϪ1/3). This he Standard Model of elementary particle physics contains two disjoint difference causes the family mixing in weak-interaction processes and is an sectors. The gauge sector describes the interactions of quarks and leptons example of the way in which the Higgs sector breaks the weak symmetry. We T(fermions, or spin-1/2 particles) with the spin-1 gauge bosons that mediate will also show that, because the neutrino masses are assumed to be degenerate the strong, weak, and electromagnetic forces. This sector has great aesthetic appeal (namely, zero), in the Standard Model, the rotation matrices for the neutrinos can because the interactions are derived from local gauge symmetries. Also, the three be defined as identical to those for their weak partners, and therefore the CKM families of quarks and leptons transform identically under those local symmetries matrix for the leptons is the identity matrix. Thus, in the minimal Standard Model, and thus have the same basic strong, weak, and electromagnetic interactions. in which neutrinos are massless, no family mixing can occur among the leptons, The Higgs sector describes the interactions of the quarks and leptons with the and individual-lepton-family number is conserved. spin-0 Higgs bosons h+ and h0. This sector is somewhat ad hoc and contains many This discussion attributes the origin of mixing to the mismatch between weak free parameters. The Higgs bosons were originally introduced to break the weak eigenstates and mass eigenstates caused by the Higgs sector. A more fundamental isospin gauge symmetry of the weak interactions by giving mass to the weak understanding of mixing would require understanding the origin of fermion masses gauge bosons, the W and the Z0. The W and the Z0 must be very heavy to explain and the reason for certain symmetries, or approximate symmetries, to hold in why the weak force is so weak. But in the Standard Model, interactions with those nature. For example, a fundamental theory of fermion masses would have to Higgs bosons are also responsible for giving nonzero masses to the three families explain why muon-family number is conserved, or only approximately conserved. 0 0 of quarks and leptons. Those interactions must yield different masses for the parti- It would also have to explain why the K – Kෆ mixing amplitude is on the order 2 cles from different families and must cause the quarks from different families to of GF and not larger. The small amount of family mixing observed in nature mix, as observed in experiment. But neither the nine masses for the quarks and puts severe constraints on any theory of fermion masses. Developing such a theory charged leptons nor the four parameters that specify the mixing of quarks across is an outstanding problem in particle physics, but it may require a significant families are determined by any fundamental principle contained in the Standard extension of the Standard Model. Model. Instead, those thirteen parameters are determined from low-energy experi- To discuss mixing as it appears in the Standard Model, it is necessary to explic- ments and are matched to the free parameters in the Standard Model Lagrangian. itly write down the parts of the Standard Model Lagrangian that contain the By definition, weak eigenstates are the members of the weak isospin doublets Yukawa interactions between the fermions and the Higgs bosons (responsible for that transform into each other through interaction with the W boson (see Figure 5 fermion masses) and the weak gauge interaction between the fermions and the W on page 38). Mass eigenstates are states of definite mass created by the interaction boson (responsible for charge-changing processes such as beta decay). But first, with Higgs bosons. Those states describe freely propagating particles that are iden- we must define some notation. As in the sidebar “Neutrino Masses” on page 64, tified in detectors by their electric charge, mass, and spin quantum numbers. Since we describe the fermion states by two-component left-handed Weyl spinors. c c c the Higgs interactions cause the quark weak eigenstates to mix with each other, Specifically, we have the fields ui, di, ui , di , ei, ␯i , and ei , where the family the resulting mass eigenstates are not identical to the weak eigenstates. index i runs from one to three. The ui are the fields for the three up-type quarks u, Each set of eigenstates provides a description of the three families of quarks, c, and t with electric charge Q ϭϩ2/3, the di are the fields for the three down- and the two descriptions are related to each other by a set of unitary rotations. type quarks d, s, and b with Q ϭϪ1/3, the ei stand for the three charged leptons Most experimentalists are accustomed to seeing the Standard Model written in e, ␮, and ␶ with Q ϭϪ1, and the ␯i stand for the three neutrinos ␯e, ␯␮, and ␯␶ c the mass eigenstate basis because the quarks of definite mass are the ingredients with Q ϭ 0. The fields ui and ui , for example, are defined as follows: of protons, neutrons, and other metastable particles that the experimentalists measure. In the mass eigenstate basis, the Higgs interactions are diagonal, and ui annihilates the left-handed up-type quark uL and creates the right-handed

the mixing across families appears in the gauge sector. In other words, the unitary up-type antiquark ෆuR in family i, and rotations connecting the mass eigenstate basis to the weak eigenstate basis appear c in the gauge interactions. Those rotation matrices could, in principle, appear in all ui annihilates the left-handed up-type antiquark uෆL and creates the right-handed the gauge interactions of quarks and leptons; but they do not. The Standard Model up-type quark uR in family i. symmetries cause the rotation matrices to appear only in the quark charge- † c† changing currents that couple to the W boson. To describe the Hermitian conjugate fields ui and ui , interchange the words c The specific product of rotation matrices that appears in the weak charge- annihilate and create used above. Thus ui, ui , and their Hermitian conjugates changing currents is just what we call the CKM matrix, the unitary 3 ϫ 3 mixing describe the creation and annihilation of all the states of the up-type quarks. matrix deduced by Cabibbo, Kobayashi, and Maskawa. The elements in the The down-type quark fields and the charged lepton fields are similarly defined.

CKM matrix have been determined by measuring, for example, the strengths For the neutrinos, only the fields ␯i containing the states ␯L and ෆ␯R are observed; c of the strangeness-changing processes, in which a strange quark from the second the fields ␯i are not included in the Standard Model. In other words, the Standard family of mass states transforms into an up quark from the first family. So far, Model includes right-handed charged leptons, but it has no right-handed neutrinos family mixing has not been observed among the leptons, with the possible (or left-handed antineutrinos).

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0 The Weak Eigenstate Basis. We begin by defining the theory in terms of the density, whereas the spatial component is the flux. Similarly, J3 is the weak iso- weak eigenstates denoted by the subscript 0 and the color red. Specifically, the topic charge density. It contains terms of the form f † f , which are number opera- 0 0 – weak gauge coupling to the W is given by tors Nf that count the number of f particles minus the number of f antiparticles g present. When this density is integrated over all space, it yields the weak isotopic ϩ ␮ Ϫ ␮† w ᏸweak ϭ ϩᎏ (W␮ J ϩ W␮ J ) , (1) charge I3 : ͙2ෆ ␮ 0 3 w where the charge-raising weak current J is defined as ͵J3 (x)d x ϭ I3 .

␮ † ␮ † ␮ J ϭ Α u0i ෆ␴ d0i ϩ ␯0i ෆ␴ e0i , (2) Now, let us consider the Higgs sector. The fermion fields interact with the Higgs i weak isospin doublet (h+, h0) through the Yukawa interactions given by The Pauli Matrices for Spin-1/2 and the charge-lowering current J␮† is defined as Particles c 0 + c + † 0 † c + † 0 † ␮† † ␮ † ␮ ᏸYukawa ϭ Αu 0i (Yup )ij [u0j h Ϫ d0j h ] ϩ d 0i (Ydown)ij [u0j (h ) + d0j (h ) ] ϩ e 0i (Ylepton)ij [␯0j (h ) + e0j (h ) ] , The Pauli spin matrices generate all J ϭ Α d0i ෆ␴ u0i ϩ e0i ෆ␴ ␯0i . (3) i,j rotations of spin-1/2 particles. i Spin-1/2 particles have only two The constant g in Equation (1) specifies the strength of the weak interactions, and where Yup, Ydown, and Ylepton are the complex 3 ϫ 3 Yukawa matrices that give ␮ j j possible spin projections along, say the ෆ␴ is a four-component space-time vector given by (1, Ϫ␴ ), where the ␴ are the strengths of the interactions between the fermions and the Higgs bosons. ϩ1/2, and the standard Pauli spin matrices for spin-1/2 particles with j ϭ x, y, z, the spatial Because the Higgs fields form a weak isospin doublet, each expression in brackets ؍ the 3-axis: spin up, or s3 spin down, or s3 ϭ Ϫ1/2. ⌻he step-up directions. These 2 ϫ 2 matrices act on the spin components of the spin-1/2 fields is an inner product of two weak doublets, making an isospin singlet. Thus, each operator ␴ϩ raises spin down to spin and are totally independent of the family index i. Each term in the charge-raising term in the Lagrangian is invariant under the local weak isospin symmetry since The charge-raising weak Ϫ c up, the step-down operator ␴ and charge-lowering currents connects states from the same family, which means the conjugate fields (for example, u 0) are weak singlets. The lepton terms in interaction in the first family lowers spin up to spin down, and ␴ the weak interactions in Equation (1) are diagonal in the weak eigenstate basis. In Equation (5) are introduced in the sidebar “Neutrino Masses” (page 64), where ϩ ␮ ϩ † ␮ 3 (W␮ J )first family ϭ W␮ u0 ␴ෆ d0 . gives the value of the spin projection fact, those interactions define the weak eigenstates. masses are shown to arise directly from the Yukawa interactions because h0 has a † ␮ 0 d along the 3-axis. The basis set for To understand the action of the currents, consider the first term, u0 ෆ␴ d0, in nonzero vacuum expectation value = v/͙ෆ2 that causes each type of fermion the spin quantized along the 3-axis the charge-raising current J␮. It annihilates a left-handed down quark and creates a to feel an everpresent interaction. These interactions yield mass terms given by → u is given by left-handed up quark (d0L u0L) and, thereby, raises the electric charge by one ϩ Ϫ → c 0 c 0† c 0† 1 0 unit. Electric charge is conserved because the W field creates a W (see top dia- ᏸYukawa ᏸmass ϭ u 0i (Yup )ij u0j ϩ d 0i (Ydown)ij d0j ϩ e 0i (Ylepton)ij e0j . (6) ␮ ΂ 0΃ and ΂ 1΃ , gram at right). The first term in the charge-lowering current J † does the reverse: † ␮ c d0 ෆ␴ u0 annihilates a left-handed up quark and creates a left-handed down quark Notice that each term in ᏸmass contains a product of two fermion fields f0 f0, W → and the matrices are given by (u0L d0L) and, thereby, lowers the electric charge by one unit; at the same time, which, by definition, annihilates a left-handed fermion and creates a right-handed Ϫ ϩ 0 1 0 Ϫi the W field creates a W (see bottom diagram at right). Thus, the members of fermion. Thus, these Yukawa interactions flip the handedness of fermions, a pre- ␴1 ϭ ␴2 ϭ ΂ 1 0΃ ΂ i 0 ΃ each pair u0i and d0i transform into each other under the action of the charge- requisite for giving nonzero masses to the fermions. These terms resemble the A down quark changes to an up raising and charge-lowering weak currents and therefore are, by definition, a weak Dirac mass terms introduced in the sidebar “Neutrino Masses,” except that the quark with the emission of a WϪ. 1 0 isospin doublet. The quark doublets are (u0, d0), (c0, s0), and (t0, b0), and the lep- matrices Yup, Ydown, and Ylepton are not diagonal. Thus, in the weak eigenstate ␴3 ϭ ΂ 0 Ϫ1 ΃ . ton doublets are (␯e0, e0), (␯␮0, ␮0), and (␯␶0, ␶0). The first member of the doublet basis, the masses and the mixing across families occur in the Higgs sector. w w has weak isotopic charge I3 ϭ ϩ1/2, and the second member has I3 ϭ Ϫ1/2. Defining the matrices ␴Ϯ as Finally, note that J␮ and J␮† are left-handed currents. They contain only the The Mass Eigenstate Basis and the Higgs Sector. Let us examine the theory in c fermion fields f0 and not the fermion fields f0 , which means that they create and the mass eigenstate basis. We find this basis by diagonalizing the Yukawa Ϯ 1 1 2 ␴ ϭ ᎏᎏ΂␴ Ϯ i␴ ΃ , annihilate only left-handed fermions f (and right-handed antifermions fෆ . The matrices in the mass terms of Equation (6). In general, each Yukawa matrix is The charge-lowering weak 2 0L 0R) right-handed fermions f0R (and left-handed antifermions fෆ0L) are simply impervi- diagonalized by two unitary 3 ϫ 3 transformation matrices. For example, interaction in the first family c ^ one arrives at the following ous to the charge-changing weak interactions, and therefore, the f are weak the diagonal Yukawa matrix for the up quarks Y is given by Ϫ ␮† Ϫ † ␮ 0 up (W␮ J )first family ϭ W␮ d0 ␴ෆ u0 . commutation relations: isotopic singlets. They are invariant under the weak isospin transformations. ^ R L† (7) u d Weak isospin symmetry, like strong isospin symmetry from nuclear physics and Yup ϭ Vu YupVu , [␴3, ␴ Ϯ] ϭ Ϯ 2␴ Ϯ , and the symmetry of rotations, is an SU(2) symmetry, which means that there are three ϩ Ϫ 3 R c [␴ , ␴ ] ϭ ␴ . generators of the group of weak isospin symmetry transformations. Those genera- where matrix Vu acts on the right-handed up-type quarks in the fields u 0, and L tors have the same commutation relations as the Pauli spin matrices. (The Pauli matrix Vu acts on the left-handed up-type quarks in u0. The diagonal elements ␮ ^ + matrices, shown at left, generate all the rotations of spin-1/2 particles.) The J and of Yup are (␭u, ␭c, ␭t ), the Yukawa interaction strengths for all the up-type quarks: ␮† ϩ ^ ^ J are the raising and lowering generators of weak isospin analogous to ␴ and the up, charm, and top, respectively. Matrices Ydown and Ylepton are similarly Ϫ ␮ c W ␴ . The generator analogous to 1/2␴3 is J3 given by diagonalized. If u0 and u 0 are the fields in the weak eigenstate, the fields in the mass eigenstate, uc and u, are defined by the unitary transformations An up quark changes to a down ␮ † ␮ † ␮ † ␮ † ␮ ϩ J3 ϭ1/2 Α u0i ෆ␴ u0i Ϫ d0i ෆ␴ d0i ϩ ␯0i ␴ෆ ␯0i Ϫ e0i ␴ෆ e0i , (4) quark with the emission of a W . i c c R L† u 0 ϭ u Vu and u0 ϭ Vu u . (8) and the time components of these three currents obey the commutation relations 0 0† 0 † † [J , J ] ϭ 2J3 . In general, the time component of a current is the charge Since the Vs are unitary transformations, V V ϭ VV ϭ ⌱, we also have

4 Los Alamos Science Number 25 1997 Number 25 1997 Los Alamos Science he Oscillating Neutrino The Oscillating Neutrino

c c R † L L u = u 0 Vu and u = Vu u0 . difference between the rotation matrices for the up-type quarks Vu and those for L the down-type quarks Vd . It is that difference that determines the amount of In this new mass basis, ᏸmass in Equation (6) takes the form family mixing in weak-interaction processes. For that reason, all the mixing can be placed in either the up-type or down-type quarks, and by convention, the CKM c ^i 0 c ^i 0 c ^i 0 ᏸmass ϭ Α u i Yup ui ϩ di Ydown di ϩ e i Ylepton ei matrix places all the mixing in the down-type quarks. The weak eigenstates for the i down-type quarks are often defined as d′: c ^ i c ^ i c ^ i ϭ Α u i Mup ui ϩ di Mdown di ϩ e i Mlepton ei , (9) i d′ = V d ϭ VL V L† d ϭ VL d , (14) ^ i ^ i CKM u d u 0 where the matrices M ϭ Y vր͙ෆ2 are diagonal, and the diagonal elements are just the masses of the fermions. In particular, we can write out the three terms for the in which case, the up-type weak partners to d′ become u′: up-type quarks u, c, and t: ′ L u ϭ Vu u0 ϵ u . c ^ i c 0 c 0 c 0 Α u i M up ui ϭ ␭u u u ϩ ␭c c c ϩ ␭t t t i When all the mixing is placed in the down-type quarks, the weak eigenstates for c c c ϭ ␭u v/͙ෆ2 u u ϩ ␭c v/͙ෆ2 c c ϩ ␭t v/͙ෆ2 t t the up-type quarks are the same as the mass eigenstates. (We could just as easily place the mixing in the up-type quarks by defining a set of fields u′ given in c c c ϭ mu u u ϩ mc c c ϩ mt t t , (10) terms of the mass eigenstates u and VCKM.) Independent of any convention, the weak currents J␮ couple quark mass eigenstates from different families. The form with the masses of the up, charm, and top quarks given by of the CKM matrix shows that, from the Higgs perspective, the up-type and down-type quarks look different. It is this mismatch that causes the mixing across mu ϭ ␭uvր͙ෆ2 , mc ϭ ␭cvր͙ෆ2 , and mt ϭ ␭tvր͙ෆ2 . quark families. If the rotation matrices for the up-type and down-type left-handed L L quarks were the same, that is, if Vu ϭ Vd , the CKM matrix would be the Thus, the Higgs sector defines the mass eigenstate basis, and the diagonal elements identity matrix, and there would be no family mixing in weak-interaction of the mass matrices are the particle masses. processes. The existence of the CKM matrix is thus another example of the way in which the mass sector (through the Higgs mechanism) breaks the weak isospin Mixing in the Mass Eigenstate Basis. Now, let us write the weak gauge interac- symmetry. It also breaks nuclear isospin symmetry (the symmetry between tion with the W in the mass eigenstate. Recall that up-type and down-type quarks), which acts symmetrically on left-handed and g right-handed quarks. ϩ ␮ Ϫ ␮† R ᏸweak ϭ ϩ ᎏ (W␮ J ϩ W␮ J ) , Note that the mixing matrices V associated with the right-handed fermions 2ෆ ͙ do not enter into the Standard Model. They do, however, become relevant in but to write the charge-raising weak current J␮ in the mass eigenstate, we extensions of the Standard Model, such as supersymmetric or left-right-symmetric substitute Equation (8) into Equation (2), models, and they can add to family-number violating processes. Finally, we note that, because the neutrinos are assumed to be massless in the ␮ † ␮ † ␮ J ϭ Α u0i ෆ␴ d0i ϩ ␯0i ෆ␴ e0i Standard Model, there is no mixing matrix for the leptons. In general, the leptonic i analog to the CKM matrix has the form † L ␮ L† † ␮ ϭ Α ui (Vu )ikෆ␴ (Vd )kjdj ϩ ␯i ෆ␴ ei i,k,j L L† Vlepton ϭ V␯ Ve . † ␮ † ␮ ϭ Αui ෆ␴ (VCKM)ijdj ϩ ␯i ෆ␴ ei , (11) i,j But we are free to choose any basis for the neutrinos because they all have the and to rewrite the charge-lowering current J␮†, we substitute Equation (8) into same mass. By choosing the rotation matrix for the neutrinos to be the same as L L Equation (3): that for the charged leptons V␯ ϭ Ve , we have

␮† † ␮ † ␮ L† L† J ϭ Α d0i ෆ␴ u0i ϩ e0i ෆ␴ ␯0i ␯0 ϭ Ve ␯ and e0 ϭ Ve e . i † L ␮ L† † ␮ ϭ Α di (Vd )ikෆ␴ (Vu )kjuj ϩ ei ෆ␴ ␯i The leptonic part of, for example, the charge-raising current is i,k,j † ␮ † † ␮ † ␮ † L ␮ L† † ␮ ϭ Α di ෆ␴ (VCKM) ij uj ϩ ei ෆ␴ ei , (12) Α␯0i ෆ␴ e0i ϭ Α ␯i (Ve )ijෆ␴ (Ve )jkek ϭ Α␯i ෆ␴ ei , i,j i i,k,j i and the leptonic analog of the CKM matrix is the identity matrix. This choice L L† (13) where VCKM ϭ Vu Vd . of eigenstate would not be possible, however, if neutrinos have different masses. On the contrary, the neutrinos would have a well-defined mass eigenstate and there Thus, the charge-raising and charge-lowering quark currents are not diagonal would likely be a leptonic CKM matrix different from the identity matrix. It is this in the mass eigenstate basis. Instead, they contain the complex 3 ϫ 3 mixing leptonic mixing matrix that would be responsible for neutrino oscillations as well → ■ matrix VCKM. This matrix would be the identity matrix were it not for the as for family-number violating processes such as ␮ e ϩ ␥.

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