Neutrinos Have Mass:
Experimental Evidence for Neutrino Oscillations

Science Spectra Magazine
Number 19, January 2000

Neutrinos can oscillate back and forth between different "flavors" only if they have mass. Vernon Barger of the University of Wisconsin and John Learned of the University of Hawaii describe experiments that reveal such oscillations and discuss the far-reaching implications of the results.

Neutrinos, ubiquitous elementary particles of a ghostly nature, have been a source of fascination ever since the distinguished theorist Wolfgang Pauli proposed their existence in 1930. Our bodies are constantly bombarded by neutrinos that pass freely through us without our notice. Each second, about 10¹⁴ neutrinos from the sun and 10³ neutrinos made by cosmic rays in the Earth's atmosphere pass through each of us. Within each of us at any moment are about 10⁷ neutrino relics of the Big Bang. To quote from the famous poem "Cosmic Gall" by John Updike:

Neutrinos, they are very small
They have no charge and have no mass
And do not interact at all.
The earth is just a salty ball
To them, through which they simply pass,
Like dust maids down a drafty hall
Or photons through a sheet of glass...

However, Updike's poem is not quite accurate. First, neutrinos do interact, if only weakly and thus rarely. Although a neutrino from the sun would have to travel on average through more than a light-year of lead before it interacts, the amazingly high solar neutrino flux makes it possible to observe the sun in neutrino "light". Also, neutrinos with higher energies interact with matter more often, making them easier to observe and, as we discuss later in this article, this permits detection of neutrinos made by high-energy cosmic rays impinging upon the atmosphere. Second, as Updike could not have known, new experimental evidence that we will discuss in this article indicates that neutrinos do in fact have some mass after all.

Pauli postulated the neutrino as a desperate intellectual gamble to save the laws of conservation of energy, momentum and angular momentum, which appeared to be violated in a process called nuclear beta decay, where a radioactive nucleus emits an electron as it transmutes to a daughter nucleus. In particular, conservation of energy and momentum requires that in beta decays of a given kind of nucleus, the kinetic energy of the emitted electron should always be the same, but instead it is observed to be anything from zero to some maximum ("endpoint") value. Proposing that an unobserved neutral particle (nowadays called the electron anti-neutrino) is emitted along with the electron and shares the available energy saved the conservation laws, but caused Pauli despair about hypothesizing a particle he thought undetectable.

In 1934, Enrico Fermi incorporated Pauli's particle in his famous theory of beta decay and named it the neutrino, which means "little neutral one" in Italian. In Fermi's theory all processes involving the neutron, proton, electron and neutrino (or their anti-particles) are described by the same basic interaction (and by the same strength of interaction, the eponymous Fermi constant). Nuclear beta decay results from the decay of a neutron through the process shown in Figure 1a, resulting in a proton, an electron and an electron anti-neutrino. This decay implies the existence of a scattering process termed "inverse beta decay" (Figure 1b), in which the collision of a proton with an electron anti-neutrino converts the proton into a neutron and the anti-neutrino into a positron (anti-electron). That these processes are related can be seen by re-arranging the "chemical" equation describing beta decay (n –> p + e^- + ^–v), remembering that transfer of the electron from the right to the left-hand-side of the equation changes its "sign", i.e. it becomes its anti-particle, the positron e⁺.

In 1955, Fred Reines and Clyde Cowan, Jr. astounded the physics world by directly detecting the elusive anti-neutrino through this "inverse beta decay" process. Their experiment was carried out at the Savannah River reactor in South Carolina, which provided a tremendous flux of 10¹³ anti-neutrinos per second per square centimeter. Their detector consisted of tanks of water (to provide a proton target) in which cadmium chloride was dissolved (to capture the final neutrons). They observed gamma rays from the annihilation of the antineutrino-produced positron and the de-excitation of the cadmium nucleus after capture of the produced neutron. Thus was Pauli's conjectured particle shown to be real!

Another neutrino discovery soon followed. In 1962, Leon Lederman, Melvin Schwartz, Jack Steinberger and collaborators demonstrated in an experiment at the Brookhaven National Laboratory on Long Island that there was a second type of neutrino (v_mu) associated with a heavy sibling of the electron known as the muon. The third neutrino (v_tau), associated with a still heavier sibling of the electron known as the tau, has been inferred via the missing energy and momentum in tau decay. The meaning here of "associated" is that in a weak interaction an electron, muon or tau (called flavors of charged lepton) transmutes into the correspondingly named neutrino — v_e , v_mu or v_tau — which are called flavors of neutrino. Conversely, a neutrino that in a collision transmutes into, for example, a muon must have been a muon neutrino. From more recent experiments at the Large Electron Positron Collider in Geneva, Switzerland, which counted the number of neutrino types in decays of a particle called the neutral weak boson Z⁰, we now know that there are indeed no more than these three light neutrino flavors (and nobody knows why).

The fundamental question of whether neutrinos have mass went unanswered for decades. In the absence of compelling theoretical reasons why they should have mass, the Standard Model of particle physics was formulated under the premise of no mass. Experiments with steadily improving precision have been carried out to determine or put a bound on the anti-electron-neutrino (^–v_e) mass from the energy distribution (spectrum) of the electrons emitted in tritium beta decay. If the neutrino has finite mass, its rest mass energy (E=mc²) will reduce the energy available to the electron. The mass of the neutrino may be inferred from the energy spectrum of the electron near the endpoint. The most recent of these extremely delicate experiments (at Mainz, Germany and Troitsk, Russia) find a neutrino mass upper limit of about 3 electron volts (in energy units, E=mc²). For comparison, the electron mass is 511,000 electron volts, so the neutrino mass is tiny on the scale of the electron mass.

Recent breakthroughs in probing small neutrino masses are associated with observations that indicate that the flavor of a neutrino oscillates. The idea for this phenomenon goes back to the Japanese theorists Maki, Nakagawa and Sakata in 1962, and the Italian physicist Pontecorvo in 1968. The cause of oscillations is that the neutrino quantum states of definite flavors are linear superpositions ("mixtures") of the neutrino quantum states of definite masses. Conversely, a neutrino of definite mass is a mixture of flavors. If we designate the three mass states as v₁, v₂, and v₃, with masses m₁, m₂, and m₃, then in a simplified case of the mixing of two neutrinos, the v_mu and v_tau flavor states are related to states v₂ and v₃ by

where q is the "mixing angle", a constant of nature. The mass states v₂ and v₃, however, evolve in time at a rate that depends on their masses; if m₂ is not equal to m₃, the mass states evolve at different rates. Thus, a neutrino that was born as a v_mu will not remain a v_mu, rather it will evolve into a mixture of v_mu and n_tau. The probability that, after traveling a distance L , the neutrino, born as a v_mu, is instead a v_tau, is given by the formula

where E is the neutrino's energy and d_m² = m₃² - m₂² is the difference of the squares of the masses of mass states v₂ and v₃. (It has been assumed that E is large compared to m₂² or m₃²; a generally valid formula can be given, but is of no interest in view of the difficulty of detecting neutrinos of very low energy.) The probability that a v_mu remains a v_mu is of course

Thus these flavor probabilities oscillate with distance traveled, at a rate proportional to dm²/E , and with an amplitude dependent on the mixing angle q. These oscillations are illustrated in Figure 2. Note that if all neutrino masses were zero, the dm² term would vanish and no flavor oscillations would occur.

The most convincing present evidence for neutrino oscillations comes from neutrinos born as a result of cosmic ray collisions in the Earth's atmosphere (see Figure 3). Based on reliable calculations using known interaction and decay rates, we expected twice as many muon neutrinos of about 1 GeV as electron neutrinos of the same energy; hence, since the cross sections for inverse beta decay collisions are nearly equal for 1 GeV muon neutrinos and 1 GeV electron neutrinos, we also expected twice as many muons as electrons to be produced in neutrino collisions. To everyone's surprise, when the first measurements of atmospheric neutrinos were made in underground detectors (one in Ohio called IMB and later at Kamiokande in Japan) over a decade ago, the numbers of muon and electron neutrinos were found to be comparable. Oscillations were proposed as an explanation by us and our collaborators and independently by a Japanese group. At that time the measurements could not determine whether there was a deficit of muons or an excess of electrons, so no firm conclusions could be reached.

The advent of the Super Kamiokande detector under Mount Ikena in the Japanese Alps in 1996 has revolutionized the experimental study of atmospheric and solar neutrinos. The detector, illustrated in Figure 4, consists of a mammoth, double-layered, 50,000-ton tank of ultra-pure water. When a neutrino interacts within the tank, a cone of light is generated by the electron or muon resulting from the interaction. The tank is instrumented on the outer surface with 11,146 20-inch diameter photomultiplier tubes that record the light signals. The spatial and temporal pattern of the light intensity hitting the detectors that surround the water allows particle identity, energy and direction to be determined. Almost immediately the new data strengthened the earlier indications of a low muon-to-electron ratio.

However, the major breakthrough of the experiment was the measurement of the dependence of the flux on the zenith angle — the angle between straight downwards and the neutrino arrival direction. The ratio of the measured v_e flux to its expected value was found to be independent of the zenith angle, but the v_mu flux showed a statistically compelling deficit that increased with zenith angle, and hence with path length L. The SuperK distribution of atmospheric muon neutrinos plotted versus the reconstructed L/E ratio in Figure 5 is well described by muon-neutrino oscillations with dm² = 3.5x10^–3 eV² and a mixing angle of close to 45°, which means maximal mixing (sin²2q = 1). The observed distribution varies smoothly with L/E because the oscillation ripples have been smeared out by the relatively poor experimental measurements of L and E.

The final confirmation of the oscillation interpretation will be the direct observation of the sinusoidal dependence on L/E. Trans-Earth experiments are underway to attempt this test but they are far from easy. There are also some variations on the simplest oscillation explanation that cannot be definitely excluded, but they are theoretically unlikely. In any case, they all involve neutrino mass. For example, we have shown in collaboration with Sandip Pakvasa and Tom Weiler that a decaying neutrino could mimic the observed L/E dependence. A decaying neutrino would have to be massive and would be even more astounding than oscillations.

The "solar neutrino problem", an even more long-standing neutrino puzzle, refers to a deficit of observed solar neutrinos compared to predictions of solar models, mostly those of John Bahcall and collaborators (see "Cosmic Phantoms" by R. L. Hahn in Science Spectra Issue 1, p. 48, 1995). Solar neutrinos have been measured in radiochemical experiments (³⁷Cl and ⁷¹Ga) and in direct directional counting by the Kamioka and SuperK experiments. All show a deficit of 30 to 60 percent compared to expectations. Numerous studies and reviews of the solar model over many years strongly indicate that the problem lies with the neutrinos, and the most plausible interpretation is that electron-neutrinos produced in the core of the sun by nuclear reactions oscillate to other flavors on their way out of the sun and to the Earth. These oscillation effects may be enhanced by the refractive index due to the neutrino-electron interactions in dense regions of the sun (the so-called Mikheyev-Smirnov-Wolfenstein mechanism), for which dm² needs to be of order 10^–5 eV². Another possibility is that the oscillations occur mainly outside the sun, with an oscillation length comparable to the Earth-sun distance. This latter scenario was pursued by one of the authors (VB) with Roger Phillips and Kerry Whisnant and also by Sheldon Glashow and Larry Krauss; the vacuum oscillation interpretation requires dm² of approximately 10^–10 eV² and large mixing.

Frustratingly, the present solar data are not yet adequate to distinguish between matter and vacuum oscillations. Indeed there are at present competing hints for the several possible solutions, and no single "smoking gun". In any case, whichever solution proves correct, the betting at present strongly favors some type of oscillations to solve the solar neutrino problem. Further SuperK data, and a number of other experiments in progress or in preparation, should settle the issue in a few years.

(click the picture above to enlarge)

Though oscillation experiments determine only mass-squared differences, if oscillations of three neutrinos are indeed the correct explanation of the atmospheric and solar neutrino anomalies, we can infer from the atmospheric dm² = m₃² – m₂² that the highest mass m₃ is above 0.03 eV. We can also infer from the beta decay endpoint measurements that the masses are less than about 3 eV. Further progress in determining neutrino masses requires considerable ingenuity. If neutrinos are their own anti-particles, then a very rare nuclear decay into two electrons and no neutrinos (so-called neutrinoless double-beta decay) could occur and give information on neutrino masses. While this process has not been observed, the upper limits resulting from searches for it give interesting constraints on the neutrino spectrum. Trans-Earth neutrino oscillation experiments could provide information on the sign of dm² through interactions of the neutrinos with matter. Such experiments will require long baselines between the neutrino source and the detector, such as between an accelerator in the U.S. and a detector in Europe or vice versa.

There are also tantalizing results from an accelerator experiment at Los Alamos, which claims muon-neutrino to electron-neutrino oscillations at a dm² scale of 1 eV², but with very small mixing. If this result is confirmed, then it, taken together with the atmospheric and solar evidence for quite different oscillation parameters, would present a considerable conundrum. The favored way out invokes a sterile neutrino with no weak interactions, an unwanted complication to be sure. Are we being led by the data, as Pauli was, to postulate yet another new particle species? An upcoming experiment at the Fermilab accelerator near Chicago will confirm or disprove the Los Alamos results in a few years.

Cosmology is a promising alternative avenue for learning about neutrino mass. Neutrinos of electron-volt masses would make a significant contribution to the mass density of the universe and would affect the large-scale structures of galaxies. Theoretical studies indicate that large-scale structure surveys together with precision measurements of anisotropies in the temperature of the cosmic background radiation may be sensitive to neutrino mass if the sum of the three neutrino masses exceeds 0.3 eV. Upcoming satellite experiments in the next five years could thus yield exciting results on neutrinos

In oscillation experiments we observe only mass differences (or in fact mass-squared differences), but the neutrino mass sum could be very much larger than the differences (e.g. if three neutrinos all had very nearly the same mass of a few eV). However, cosmologists do not at present favor the currently indicated "light" neutrinos to solve the much publicized "dark matter problem"; yet neutrinos may still make a significant contribution to the mix of dark matter. Neutrinos may also play a key role in the origin of the one-part-in-a-billion excess of matter over anti-matter which arose during the Big Bang (and thus accounts for our being here to wonder about this fact!). One may also note that even the seemingly tiny neutrino masses associated with present lower mass limits still sum to about as much of the mass of the universe as all the visible stars!

Particle physics has moved beyond the Standard Model in the neutrino sector. The masses of the fundamental charged fermions (the six quarks — up, down, charmed, strange, top, and bottom — and the three charged leptons — electron, muon and tau) are all on a very different scale from the neutrinos, by about ten orders of magnitude or so. In fact it appears that the neutrino masses lie approximately as far below the other fermion masses as the latter are below the energy scale where the strong, electromagnetic and weak interactions have the same couplings (the Grand Unified scale). Also, the apparent maximal mixing of the muon neutrino has no counterpart among quarks. How to fit these new discoveries into a theoretical framework for elementary particles and cosmology is now a challenge that fuels the excitement in the field, and leads to a paradigm shift in our understanding of the universe on large and small scales. Whatever the long term results, Pauli's ghostly particle is a universal and deeply important component of the universe in which we live and where we marvel at our new understanding of neutrinos.

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