The Enigmatic NeutrinoDon Lincoln and Tia Miceli Citation: The Physics Teacher 53, 331 (2015); doi: 10.1119/1.4928345 View online: http://dx.doi.org/10.1119/1.4928345 View Table of Contents: http://scitation.aip.org/content/aapt/journal/tpt/53/6?ver=pdfcov Published by the American Association of Physics Teachers Articles you may be interested in Response to Brown and Kumar Phys. Teach. 53, 324 (2015); 10.1119/1.4928341 Response to “Learning from the Starry Message” Phys. Teach. 53, 324 (2015); 10.1119/1.4928340 Origins of Newton's First Law Phys. Teach. 53, 80 (2015); 10.1119/1.4905802 The International History, Philosophy and Science Teaching (IPHST) group Phys. Teach. 48, 207 (2010); 10.1119/1.3317470 2009 Distinguished Service Citations Awarded to Alan Gibson, David Maiullo, Bruce Mason, Mary Winn,and Mel Steinberg Phys. Teach. 47, 408 (2009); 10.1119/1.3225495

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DOI: 10.1119/1.4928345 The Physics Teacher ◆ Vol. 53, September 2015 331

tritium atom at rest has energy, E, given by Einstein’s E = mc2. Through energy conservation, this energy must be present in its final state, thought at the time to be a helium-3 atom and an electron. Since the combined mass of the helium-3 and electron is less than the mass of tritium, the remaining energy is transformed into kinetic energy for the final state particles. Further, since momentum is likewise conserved, the helium-3 atom and the electron must have equal and opposite momen-tum. These combined facts imply that the energy spectrum of the emitted electron should be monochromatic, with a value of 18.59 keV (see Fig. 1).

Instead, Chadwick saw that the electron was emitted with a variety of energies and not the expected unique value. Was this indicative of the failure of energy conservation in the quantum realm? For 16 years after Chadwick’s measure-ment, the physics community entertained the prospect that conservation of energy fails for nuclear processes. Given the observed universality of energy conservation, this possibility was regarded with widespread distaste.

An innovative solution was offered in December of 1930, at a conference on radioactivity in Zurich. Wolfgang Pauli proposed a solution that he dubbed “The Desperate Remedy.” He proposed that if an unobserved particle is also emitted in beta decay that has a mass of about equal to or less than the mass of the electron, then the law of energy conservation could be rescued. He called this particle the neutron.1 By in-troducing a third particle into the final state, the energy could be shared in a variety of ways between the three particles. One would expect a continuous distribution for the final par-ticles’ energy depending on the angle at which each particle emerges.

In 1933, Enrico Fermi used Pauli’s particle, which Fermi renamed the neutrino (“little neutral one” in Italian) in a very predictive model of beta decay. His model explained the rates observed in both beta decay and muon decay. Fermi’s theory works well until the energies involved reach ~100 GeV, millions of times more energy than in tritium beta decay, at

The Enigmatic NeutrinoDon Lincoln, Fermilab, Batavia, ILTia Miceli, New Mexico State University, Department of Physics, Las Cruces, NM

Through a century of work, physicists have refined a model to describe all fundamental particles, the forces they share, and their interactions on a microscopic

scale. This masterpiece of science is called the Standard Model. While this theory is incredibly powerful, we know of at least one particle that exhibits behaviors that are outside of its scope and remain unexplained. These particles are called neutrinos and they are the enigmatic ghosts of the quantum world. Interacting only via the weak nuclear force, literally billions of them pass through you undetected every second. While we understand that particular spooky behavior, we do not understand in any fundamental way how it is that neutri-nos can literally change their identity, much as if a house cat could turn into a lion and then a tiger before transitioning back into a house cat again.

Neutrinos have a long history of confounding researchers, making them uncertain about the law of conservation of en-ergy in the microrealm and showing them that some particles have no mirror reflection, just to mention a couple of the neu-trino’s surprises. The neutrino continues to mystify physicists, and the study of their properties is a vibrant and ongoing area of research.

The radioactive birth of the neutrinoOur story begins in the 1910s when radioactivity was the

hot topic of study in physics and chemistry. In 1914, Sir James Chadwick was studying the decay of tritium into helium-3 plus an electron (3H→ 3He + e–). An interaction where a neutron changes to a proton and emits an electron is called beta decay. This nomenclature reflects early experimenters’ ignorance that the particle in beta decay was actually an elec-tron. While subsequent research clarified the situation, the name “beta decay” is still used to describe a nuclear process in which an electron is emitted.

Chadwick and his colleagues measured the momentum of the emerging electron in beta decay. If a particle at rest decays into two daughter particles, we would expect from the law of energy conservation that the momentum and kinetic energy of the daughter particles should be completely constrained. Chadwick’s experiment was precise enough that he could accurately measure the energy of the emitted electron. The re-sult was puzzling. In radioactive beta decay experiments, the law of energy conservation seemed to be violated.

The rationale is the fol-lowing. Before the decay, the

Endpoint ofspectrum

Expectedelectronenergy

Observed spectrum of energies

Energy

Num

ber o

f ele

ctro

ns

Fig. 1. Wolfgang Pauli in 1930 proposed a desperate remedy, a new neutral particle to salvage con-servation energy. (Photo by Samuel Goudsmit, courtesy, AIP Emilio Segre Visual Archives, Goudsmit Collection.)

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332 The Physics Teacher ◆ Vol. 53, September 2015

spin is a quantized amount of angular momentum that is in-trinsic to a particle.

It was taken for granted that physical processes which ap-pear in a mirror (a parity transformation) occur as commonly as the unmirrored process. For example, electrons scatter with equal probability, independent of whether their spin is left- or right-handed.

In 1956, Chien-Shiung Wu showed experimentally that neutrinos, and processes involving neutrinos (specifically the weak nuclear force), do have preference of left versus right. In her experiment, she inserted a sample of cobalt-60 atoms in a magnetic field, which caused their spin to be aligned with the magnetic field direction. She then waited for the cobalt-60 to radioactively decay to nickel-60 via emission of an electron and an antineutrino (60Co → 60Ni + e – + v–) [Fig. 3(a)]. Naïve expectations suggest that the electron would be emitted both parallel and antiparallel to the nuclear spin with equal prob-ability. However, she found that the electron was always emit-ted in the direction opposite of the nuclear spin, never in the same direction [Fig. 3(b)].

Thus the force responsible for beta decay did not treat right-handed and left-handed particles symmetrically. In fact, in her experiment, parity was maximally violated, lead-ing to the broader conclusion that neutrinos are left-handed and all antineutrinos are right-handed.

Wu’s findings led to our current understanding of the weak force, in which only left-handed particles and right-

which point the predicted rates of interaction become unre-alistically high. At such a point, the theory breaks down and a more realistic model is needed, what we now call electroweak theory, which is the modern theory describing neutrino in-teractions.

To catch a neutrinoIn 1956, at South Carolina’s Savannah River nuclear reac-

tor, Clyde Cowan and Frederick Reines eagerly awaited pulses of light indicating that a neutrino had interacted in their detector.2 The neutrinos, or rather the antineutrinos (v–), pro-duced by the nuclear reactor come from the fission processes in which a neutron decays: n° → p+ + e – + v–. It was estimated that the reactor’s antineutrino flux at the detector would be 1013 /(cm2 . s).

Cowan and Reines called their experiment “Project Polter-geist,” since they were hunting for experimental verification of Pauli’s elusive and ghost-like particle. Project Poltergeist was composed of two tanks of water doped with cadmium chloride, which sandwiched a layer of liquid scintillator. As the antineutrino came into the water tanks, it could hit a pro-ton and transmute the proton into a neutron and emit a posi-tron—this was the inverse beta decay reaction for which they were looking: v – + p+ → n°+ e+. As the positron flew into their detector, they looked for a flash of light as it immediately an-nihilated with an electron in the tank to produce gamma rays, followed shortly thereafter by another gamma ray burst as the neutron was absorbed by the cadmium.

As Cowan and Reines saw the electronic blips of their photodetectors, indicating that they had detected an anti-neutrino, they readied a telegraph to Pauli (shown in Fig. 2). In 1995, Reines, the surviving member of this ghost-busting pair, was awarded the Nobel Prize for detection of the (anti)neutrino.

Neutrinos are weak left-handersTo explain why neutrinos have no reflection, we need to

introduce the concept of spin. Spin calls to mind a twirling ball that has some angular momentum. In particle physics,

Fig. 2. Wolfgang Pauli was notified by telegram of the discovery of his predicted particle. (Credit, Pauli Archive, CERN)

Fig. 3. (a) In Wu’s experiment, cobalt-60 decayed into nickel-60, an electron and an antineutrino. The cobalt nucleus has a spin of 5 and the nickel nucleus has a spin of 4. Accordingly the remaining unit of spin is carried by the two leptons, which must be aligned. (b) While both decays shown here have the leptons’ spin aligned as required, the right-most process is never observed, indicating that neutrinos are only left-handed and antineutrinos are always right-handed. This observation demonstrated that processes involving the weak nuclear force do not have parity symmetry. (Credit, From chapter 7, The Quantum Frontier, in the suggested reading.)

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The Physics Teacher ◆ Vol. 53, September 2015 333

in equal quantities. However, what was observed was that only muons were produced. Given that the beam of neutrinos was produced from muon interactions, this implied that the neutrino retained a “muon-affiliated identity.” This experi-ment demonstrated that there existed at least two types of neutrinos, one electron-like (ve) and one muon-like (vm). For their discovery of the muon neutrino, Lederman, Schwartz, and Steinberger shared the 1988 Nobel Prize.

The Sun and the lost neutrinosIn the 1960s, John Bahcall was trying to calculate what

types of nuclear processes occurred in solar fusion. He predicted that the reaction H+ + H+ → He2++ v + (other) generates 7 31010 neutrinos/(cm2 .s) on Earth—or almost 100 billion solar neutrinos passing through your thumbnail every second. He was hoping that this was enough neutrinos to compete with Project Poltergeist, which studied neutrinos from nuclear fission. He teamed up with experimentalist Ray Davis, who was determined to detect neutrinos originating from the Sun. They needed a new experimental design to detect these neutrinos since they were searching for electron neutrinos versus Project Poltergeist’s electron antineutrinos.

Nearly a mile underground in the Homestake Gold Mine in South Dakota, Ray Davis and his team built a tank to hold 380,000 liters of perchloroethylene (i.e., dry cleaning fluid). The hope was that placing their detector so far underground would shield it from cosmic rays and that the neutrinos from the Sun would freely stream through the Earth. As a neutrino from the Sun came into the detector, it would undergo the reaction v + Cl → Ar + e–.

Periodically helium gas was bubbled through the perchlo-rethylene, which captured the produced argon atoms, and the argon atoms were then counted. The tank contained of order 1030 atoms and over the course of a running period of a week or so, the expectation was that about 10 argon atoms would be produced. However, the number of atoms of argon that were detected were about a third the rate predicted by Bahcall. Given the tiny number of atoms involved, the expla-

handed antiparticles participate (see Fig. 4). This result irretrievably broke the idea of parity (P) as a fundamental symmetry and, by extension, also a symmetry of matter and antimatter (called C, for charge conjugation). However theo-rists quickly devised a replacement symmetry (CP). For any observed process, if we both flip the parity of the particle [i.e., replace (x, y, z) → (–x, –y, –z)] and replace all particles with antiparticles (and vice versa), the new configuration will also be observed, as illustrated in Fig. 5.

Surprises from the second generation: Muon neutrinos

In 1962, at Brookhaven National Laboratory in New York, Leon Lederman, Melvin Schwartz, and Jack Steinberger led an experiment3 to search for another flavor of neutrino that was hypothesized based on the fact that muons were never seen to decay into an electron and photon. If there were not something unique about muons compared to electrons, this decay should have occurred. Because it didn’t (and because of the linkages between neutrinos and electrons), it seemed plausible that there might be two distinct flavors of neutrinos, one associated with electrons and one with muons.

The first ever man-made neutrino beam was made using Brookhaven’s Alternating Gradient Synchrotron. Protons were smashed into a metal target and a mixture of particles emerged from the collision. These were focused to produce a beam of charged pions, which were given time to decay into muons and neutrinos. The decay product muons were removed by allowing them to decay and stopping them with shielding, thereby producing a beam of neutrinos primarily composed of neutrinos from muon interactions.

The researchers looked for interactions induced by this beam of neutrinos. If there was only a single kind of neutrino, these interactions should have produced electrons and muons

Fig. 5. For any given weak interaction, the mirrored (parity-flipped) image is not allowed by nature. However, if matter is substituted for antimatter and vice versa, the parity-flipped image is allowed by nature.

Fig. 4. Chien-Shiung Wu discovered in 1956 that neutrinos are left-handed. (Courtesy, wikimedia.org/wikipedia/commons.)

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334 The Physics Teacher ◆ Vol. 53, September 2015

Atmospheric neutrinos predict that there should be two muon-type neutrinos for every electron-type. Yet experi-ments similar to the Homestake experiment observed a ratio of one to one, which is clearly inconsistent with predictions. Physicists now had two neutrino mysteries to solve and people wondered if there was a common origin. Italian-born physicist Bruno Pontecorvo postulated that perhaps neutrino identities were not fixed and that the two types of neutrinos could morph (or oscillate) into one another.

The second generation and the SunIn 1983, a scientific team based at Tokyo University built

the Kamioka Nucleon Decay Experiment (KamiokaNDE) to search for proton decay as a test of the validity of the Standard Model. Although KamiokaNDE was built to look for this re-action, it was never found and instead the experiment placed the world leading limits on proton decay, bolstering the valid-ity of the Standard Model (with the result that the lifetime of the proton was measured to be >1031 years, 21 orders of mag-nitude larger than the age of the Universe!). This was later in-creased to >1033 years after KamiokaNDE was upgraded and enlarged to Super-KamiokaNDE (Super-K) in 1996.5 (The most modern limits are about >1034 years.)

While Super-K did not find proton decay, it turned out to be a superlative neutrino detector, of both solar and at-mospheric neutrinos. Super-K’s initial analyses confirmed earlier studies, which identified a mystery in the population of these neutrinos. However, the Super-K detector had an edge; it could detect the direction from which the neutrinos originated. By looking at the neutrinos originating from cosmic rays hitting the atmosphere overhead (about 10 miles distance), to neutrinos originating on the other side of Earth (about 8000 miles distance), researchers were able to see that the deficits were reduced for neutrinos produced nearby and higher for those produced farther away. Announced in 1998, these measurements are considered the first definitive proof of neutrino oscillations, although a subsequent measurement was needed to complete the picture.

Solar answers from the third generation: Tau neutrinos

In 2000, the tau flavor neutrino was discovered by Fermi-lab’s DONUT experiment. (The tau lepton is a third cousin of the charged lepton family, which includes electrons and muons.) This would prove to be the last piece for the solar neutrino puzzle. The Sudbury Neutrino Observatory (SNO), in Sudbury, Ontario, was able to detect all three types of solar neutrinos (electron, muon, and tau). In 2001, the SNO ex-periment announced that the integrated flux of all three fla-vors were in complete agreement with Bahcall’s solar fusion calculations.6 In short, Bahcall had correctly calculated the neutrino flux from the Sun, but the neutrinos were changing their identity as they traveled to Earth.

The theory of neutrino oscillation is given in Appendix A, but essentially it says that a population of neutrinos of a single

nation could simply have been a poorly understood detec-tion efficiency. However, Davis was able to inject precisely controlled amounts of argon and he demonstrated that he could extract what he expected. The experiment seemed to be sound. Researchers then turned their attention to Bahcall’s predictions, in which the neutrino flux was proportional to the 25th power of the temperature of the core of the Sun. With such sensitivity, it seemed quite possible that a small experimental uncertainty could be amplified to the observed discrepancy. However, as was the case for Davis’ experiment, the calculations and supporting astronomical measurements seemed similarly sound. There was definitely a mystery to be solved. Subsequent experiments confirmed the discrepancy. The solar electron neutrino flux at Earth is about 1/3 that predicted.

While the “solar neutrino problem,” as it was called, could possibly have originated in a misunderstanding of the solar production of neutrinos, a similar discrepancy was seen in at-mospheric neutrinos. Atmospheric neutrinos are created as a consequence of cosmic ray protons from space hitting Earth’s atmosphere (which consists of protons and neutrons). High- energy proton/proton or proton/neutron collisions copiously produce charged pions. As was mentioned in the description of the experiment by Lederman et al., charged pions decay into muons and muon-type neutrinos. Muons then decay into an electron, an electron-type neutrino and a muon-type neutrino (π+ → vμ + μ+ → vμ + e+ + v–μ + ve).4

Neutrinos and the Matter-Antimatter Imbalance

Neutrinos may provide a clue to one of the most fundamental questions in nature: how is it that the Universe is made solely of matter, and not antimatter? The Big Bang hypothesis has it that in the beginning there was only energy, then that energy converted to equal parts of matter and antimatter. If that’s true, where did all of the antimatter go?

Since the weak force distinguishes between matter and antimatter neutrinos, this suggests that neutrino studies may explain how particles were favored over antiparticles as the Uni-verse evolved. The first step in determining if neutrinos could be fully or partly responsible for the Universe’s matter-antimatter imbalance is in neutrino oscillations. Current experiments are in a race with each other to see if the neutrino oscillation pa-rameters differ depending if they start with a neutrino beam or an antineutrino beam. In Japan, the Tokai to Kamioka (T2K) ex-periment is using Super-K as a neutrino detector for a beam of neutrinos produced 300 km away in Tokai. In the United States, Fermi National Laboratory has the NOvA experiment. NOvA uses a neutrino beam produced at Fermilab, and sent 800 km through the Earth to a mine in Ash River, Minnesota. At CERN, a beam of neutrinos is shot from Switzerland to the Grand Sasso Laboratory in Italy (CNGS), located 732 km away. The competi-tion is quite fierce.

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The Physics Teacher ◆ Vol. 53, September 2015 335

– Majorana or Dirac?A Majorana particle is one in which the matter and an-

timatter version of the particle is the same. Particles whose particle and antiparticle are distinguishable are called Di-rac particles. For example a photon’s antiparticle is also the photon, so the photon is a Majorana particle. If the neutrino turns out to be Majorana also, it would be the first fundamen-tal fermion discovered to have this property.

There are many experiments testing if neutrinos are Ma-jorana. They look for a process called neutrinoless double beta decay. In such a process, the neutrinos emerging from each beta decay annihilate each other immediately. In their detectors, the experiments look for two electrons produced with the absence of any neutrinos. No such process has yet been observed. Current experiments exploring the question include: KamLand-Zen, EXO, XMASS, and MAJORANA. In addition, experiments at the Large Hadron Collider are also pursuing analyses with this goal.

Determining if the neutrino is Majorana or Dirac would help guide us to determining if the neutrino masses follow the normal or inverted hierarchy. Also, if the neutrino turns out to be Majorana, then these neutrinoless double beta decay experiments could also estimate the mass of the neutrino. For example, the EXO-200 experiment showed that if the neu-trino is Majorana, then the average neutrino mass is less than 140-380 meV.11

– Are there more neutrinos?Neutrinos we observe, for instance from the Wu experi-

ment, always have the axis of their spin pointed opposite of the neutrinos’ momentum (left-handed helicity), while anti-neutrinos always appear to have their spin axis aligned with their momentum (right-handed helicity). The only way that a particle will always be found with a certain helicity is if it trav-els at the speed of light. If a particle is traveling slower than the speed of light, then we can boost ourselves to another reference frame where the particle is traveling the opposite direction, but with unchanged spin. Since the oscillation re-sults in 1998 from Super KamiokaNDE, we have known that neutrinos have a small mass. Thus the neutrinos that we have found experimentally do not fit into the Standard Model of particle physics, which requires that neutrinos are massless and are always left-handed, while the antineutrinos are al-ways right-handed.

Since the neutrino has been found to have mass, the ques-tion remains, “Where are the right-handed neutrinos and left-handed antineutrinos?” An extension of the Standard Model, developed to resolve this question, postulates new neutrinos that are much more massive and only interact through gravity, not the weak force. Such a neutrino is called a “sterile neutrino.” These are described as being part of the “seesaw mechanism” that relates the masses of the flavored neutrinos we know about, and either these theoretical sterile neutrinos or massive right-handed neutrinos. In the seesaw model, as the flavored neutrinos become lighter, the sterile or

identity (say, muon-type neutrinos) can oscillate into the other types. The parameters that determine the mix of neutri-nos at a distant detector depend on the distance between the source and the detector, the energy of the initial neutrinos, and the difference between the masses of the various neutrino types. As discussed in the appendix, the observed flavors of neutrinos do not have uniquely determined masses. Further, what is important in neutrino oscillation is not the mass of the neutrinos, but the mass difference. Thus Super-K, SNO, and subsequent accelerator-based experiments merely estab-lished a mass difference. The absolute values of the masses of the neutrinos remain unknown.

Outstanding mysteriesWhile the history of neutrino research has been full of

surprise discoveries, the enigmatic particle hasn’t revealed all its mysteries just yet. These mysteries are broad, ranging from the actual mass of neutrinos, to differences in the matter and antimatter versions of the neutrino, to whether the three known flavors are the end of the story.

– What are the masses of the neutrinos?Neutrinos oscillate in flavor because they have mass.7As

described in the appendix, a neutrino of a particular flavor state (i.e., electron vs muon vs tau) has a probability of having one of the three distinct mass states, v1, v2, v3. A neutrino of a particular mass state has a probability of having one of the three different flavors states, ve, vμ, vτ. The oscillation in fla-vor depends not on the mass of any particular neutrino type, but rather on the mass-squared difference between the fla-vors. Experiments have determined that neutrinos v1 and v2 are similar in mass, with v1 being the heavier of the two. We don’t know if v3 is much higher in mass (“normal hierarchy”) or much lower in mass (“inverted hierarchy”). The average neutrino mass is also not known, but a wide variety of experi-ments, from astronomical surveys to beta decay experiments, are setting limits.

Cosmological observations using the cosmic microwave background, supernovae data, and the Lyman-alpha forest (hydrogen absorption in the spectra of distant galaxies) pre-dict that the sum of the neutrino masses must be less than 0.3 eV.8 Other cosmological observations of gravitational lensing (the bending of the path of light emitted by distant objects by the warp of spacetime due to massive foreground objects) also limits the neutrino mass below 1.5 eV.9

The Karlsruhe Tritium Neutrino Experiment (KATRIN) in Germany is taking data to make a very precise measure-ment of the electron energy spectrum from beta decay. By determining the maximum possible energy of the emerging electron, scientists will know the minimum energy possible for the emerging antineutrino. The thought is that the mini-mum energy will be composed of all mass energy, and zero kinetic energy, thereby providing a measurement of the elec-tron antineutrino mass.9,10

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336 The Physics Teacher ◆ Vol. 53, September 2015

tested by MicroBooNE. The short-baseline program and the long-baseline programs are complementary, as they measure neutrino oscillations at different neutrino beam energies and different distances. Both are needed to understand how these strange particles behave.

From the ongoing and future neutrino experiments, we should expect surprises, and we will see history unfold as we discover new details of the most elusive particle ever. Soon we hope to know the mass of the neutrinos from KATRIN and astronomical surveys. We will find out if they are their own antiparticle from a set of independent neutrinoless double beta decay experiments. We may discover new neutrinos at MicroBooNE. Finally, we may learn something about the Universe’s imbalance of matter and antimatter from the CNGS, T2K, and NOvA experiments.

Research institutes around the world are pursuing the mysteries of the neutrino. Fermilab, America’s premier par-ticle physics institution, has elected to focus on neutrino studies as its flagship domestic research program for the next two decades. Neutrinos have long surprised researchers and it is very likely that they will continue to teach us something unexpected about the Universe.

Appendix A: Neutrino Masses and Flavor Oscillations

The theory of neutrino oscillation is tied deeply in the the-ory of quantum mechanics, and it serves as a simple and prac-tical example of how to use the time-dependent Schrödinger equation. As with many quantum problems, it has counter-intuitive components. One such example is the fact that neu-trinos can change flavor is intimately related to them having a nonzero mass. In addition, there is a mass basis (v1, v2, v3) that is distinct from the detectable flavor basis (ve, vμ, vτ). For example, the electron neutrino is a quantum mechani-cal admixture of the three mass components. To get a feeling for this behavior, let’s consider the simplified two-neutrino model, ignoring for the moment the third neutrino.

We start with the solution to Schrödinger’s equation for a

right-handed neutrinos become heavier. There may also be multiple sterile neutrinos with different masses.

Sterile neutrinos could explain unexpected experimental results. The MiniBooNE detector, an oil Cherenkov detector, at the United States’ Fermi National Accelerator Laboratory saw an unexpected increase in the number of low-energy electron flavor neutrinos oscillating from a beam of muon neutrinos. A possible explanation of this excess could be the existence of massive, sterile neutrinos.

A follow-up experiment, MicroBooNE, will make mea-surements from the same neutrino beam, but will have better detector sensitivity to determine the nature of this excess. MicroBooNE utilizes liquid argon as the detection medium, which will allow it to distinguish better between electrons and photons, which was the main limitation of the technology used in the MiniBooNE experiment.

MicroBooNE can measure the energy deposition along a track leading to an electromagnetic shower, whereas Mini-BooNE could only measure the final energy deposition of the electromagnetic shower. An electromagnetic shower arises from either an electron or photon traveling through the elec-tron clouds surrounding atoms. An electromagnetic shower is the resulting cascade of electrons, positrons, and photons. Before a photon creates an electromagnetic shower, it will convert to an electron and positron and deposit more energy per unit length as compared to a single electron. By measur-ing this track leading to the electromagnetic shower, Micro-BooNE can differentiate between electrons and photons.

Neutrino adventures continueThere are many cutting-edge detectors preparing to take

data or being planned right now to discover the remaining mysteries of neutrinos. The MicroBooNE detector at Fer-milab will investigate the nature of the MiniBooNE excess. Fermilab also has an extensive short-baseline neutrino oscil-lation program planned (~1 km) and an international long-baseline neutrino oscillation experiment now dubbed DUNE (~ 1300 km), the longest one yet (see Fig. 6). Both programs will use the liquid argon detection technology currently being

Fig. 6. The Deep Underground Neutrino Experiment (DUNE) is an international effort to understand more about how these strange particles behave. (Courtesy, Fermilab.)

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The Physics Teacher ◆ Vol. 53, September 2015 337

In neutrino physics we define the quantity: mass difference squared, ∆m2

12 = m21 – m2

2.So the probability for an electron flavor neutrino oscillat-

ing to a muon flavor neutrino in the two-neutrino model of mixing is

(11)

It is common to express this equation in a special form, in which ∆m2

12 is in units of eV2, L in km, and E in GeV,

(12)

In Eq. (12), we see that the probability of starting with an electron flavor neutrino of energy E and detecting it as a mu-on flavor neutrino at distance L is dependent on the neutrino masses, and the angle q, which relates the flavor eigenbasis to the mass eigenbasis.

Because we have experimentally observed neutrino flavor oscillations, we know that the neutrino masses are nonzero. However, from oscillation experiments we can’t determine the absolute neutrino mass, we can only measure the mass difference squared. This means that the neutrino masses m1, m2, and m3 may be in two possible configurations (m1 > m2 > m3) or (m3 > m1 > m2).12

References1. This name is misleading from a historical perspective. Chad-

wick discovered the particle we now call a neutron about two years later, with properties (other than its charge) quite differ-ent from Pauli’s proposed particle.

2. “The Reines-Cowan Experiments: Detecting the Poltergeist,” Los Alamos Science 25 (1997); http://library.lanl.gov/cgi-bin/getfile?00326606.pdf .

3. L. Lederman, “The two-neutrino experiment,” Sci. Am. 208 (3), 60–70 (March 1963); http://www.scientificamerican.com/article/the-two-neutrino-experiment/.

4. Note that negatively charged muons and electrons are matter, while the positive ones are antimatter. Further, the line over the neutrino indicates antimatter. Finally, while the decay of nega-tively charged muons is shown here, the decay of positively charged muons is identical, except that all matter particles are replaced by antimatter and vice versa.

5. J. Bahcall, “The solar-neutrino problem,” Sci. Am. 262, 26–33 (1990); http://www.scientificamerican.com/article/the-solar-neutrino-problem/.

6. A. B. McDonald, J. R. Klein, D. L. Wark, “Solving the solar neutrino problem,” Sci. Am. 288 (4), 40–49 (April 2003); http://www.scientificamerican.com/article/solving-the-solar- neutrin/.

7. Eugene Hecht, “On morphing neutrinos and why they must have mass,” Phys. Teach. 41, 164 (2003).

8. Ariel Goobar, Steen Hannestad, Edvard Mortsell, and Huitzu Tu, “The neutrino mass bound from WMAP-3, the baryon acoustic peak, the SNLS supernovae and the Lyman-alpha for-est,” J. Cosmol. Astropart. Phys. 0606,019 (2006); arXiv:astro-ph/0602155v2.

9. S. Hannestad, “Neutrino physics from precision cosmology,” Prog. Part. Nucl. Phys. 65, 185-208 (2010); arXiv: hep-ph/1007.0658.

free particle: . (1)

In bra-ket notation, this is:

(2)where

neutrino flavors.

If we assume that the flavor eigenstates (|ve and |vμ ) can be written as a linear combination of the mass eigenstates (|v1 and |v2 ) with a mixing angle q, then

|ve = cos q | v1 + sin q |v2 |vμ = –sin q| v1 + cos q |v2 .

(3)

To compute the probability of a |ve at (x,t) = (0,0) oscillating to a |vμ at some arbitrary (x,t) later, we follow the usual pre-scription of quantum mechanics by taking the bra of the final state and ket of the initial state and squaring it.

P(|ve → |vμ ) = | vμ (x,t)|ve (0,0) |2 . (4)

Substituting Eq. (3) and doing some algebra, we get:

(5)

To clearly see the flavor oscillation dependence on mass, we explicitly evaluate j1 – j2.

(6)

We simplify this by asking for the solution at a particular place and time and assuming that the neutrinos travel at nearly the speed of light, c. In this regime, x = ct = L, where L is the distance from a neutrino source to the detection point.

(7)

Let’s further assume that during the oscillation, momentum remains unchanged so that p1 = p2. Then we would have

(8)

We now write the relativistic energy-momentum relation for Note that if the momentum is much greater than the mass, we can use a binomial expansion. We can simplify this further by recalling that p1 = p2 = p, and fur-ther yet since p >> mc, so p ~ E/c and therefore

(9)

Substituting Eq. (9) into Eq. (8) yields

(10)

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338 The Physics Teacher ◆ Vol. 53, September 2015

10. http://www.katrin.kit.edu/.11. M. Auger et al., “Search for neutrinoless double-beta decay

in 136Xe with EXO-200,” Phys. Rev. Lett. 109, 032505 (2012); arXiv:1205.5608v2 [hep-ex].

12. We also had to consider the four other cases (m2 > m1 > m3), (m3 > m2 > m1), (m1 > m3 > m2), and (m2 > m3 > m1). The first two are ruled out from measuring solar neutrino oscillations, and the last two are ruled out from measuring atmospheric neutrino oscillations.

Suggested Reading1. Frank Close, Neutrino (Oxford University Press, 2012).2. doi:10.1038/nature.2014.14752: http://www.scientific

american.com/article/cosmic-mismatch-hints-at-the- existence-of-a-sterile-neutrino1/.

3. M. Hirsch, H. Pas, W. Porod, “Ghostly beacons of new physics,” Sci. Am. 308 (4), 41–47(April 2013). Article titled “Neutrino experiments light the way to new physics,” online at http://www.scientificamerican.com/article/neutrino-experiments-light-way-new-physics/.

4. D. Setton, “Neutrinos: Ghosts of the universe,” Discover Magazine 35 (7), 30 (Sept. 2014); http://discovermagazine.com/2014/sept/9-ghosts-of-the-universe.

5. Neutrino Physics: http://www.symmetrymagazine.org/article/june-2015/how-do-you-solve-a-puzzle-like-neutrinos.

6. D. Lincoln, The Quantum Frontier: From Quarks to the Cosmos (Johns Hopkins University Press, 2012, revised).

Don Lincoln is a senior researcher at Fermilab and an adjunct profes-sor at the University of Notre Dame. He is a member of the CERN CMS collaboration and has co-authored over 800 papers. He is also an avid popularizer of frontier physics and has written several books on science for the public, most recently The Large Hadron Collider: The Extraordinary Story of the Higgs Boson and Other Things That Will Blow Your Mind. You can follow him on Facebook (www.facebook.com/Dr.Don.Lincoln), where he interacts with the public. lincoln@fnal.gov

Tia Miceli is a postdoctoral scholar for New Mexico State University. She is a member of Fermilab’s MicroBooNE collaboration, where she uses neutrinos to probe protons and neutrons. She completed her graduate work in 2013 on CERN’s CMS experiment at University of California, Davis, where she searched for extra dimensions. She has authored or co-authored over 300 scientific papers. She is also a frequent host on the Titanium Physicist Podcast (http://titaniumphysicists.brachiolopemedia.com/) and enjoys giving public physics lectures. miceli@nmsu.edu

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