For the last few years we have had experimental verification that neutrinos have a mass, although a very small one. Understanding in depth why this is so requires some background in quantum mechanics, the most non-intuitive scientific theory ever seriously proposed, and yet at the same time the one which, in its refinement as quantum electrodynamics, has given us the most accurate prediction ever made for an experimentally measured value. (A well-known quip attributed to David Moser refers to quantum mechanics as "the dreams stuff is made of.") We won't attempt to provide that background here, but will only try to give a flavor of the argument.
The evidence that neutrinos have mass
The argument relies on two facts. First, because the second- and third-generation versions of the electron (the muon and tau) are so unstable and short-lived, it is a near certainty that the Sun produces only one kind of neutrino: the electron neutrino. Second, if neutrinos were all massless (or for that matter, if they all had exactly the same mass), it would be impossible for a given type of neutrino to become a different type (for instance, for an electron neutrino to become a muon neutrino).
Prior to the SNO experiment, all neutrino detectors that had ever attempted to measure the rate at which the Sun made neutrinos were getting values between 1/3 and 2/3 of the theoretical predictions. One might at first think, "No big deal, the predictions must be wrong." But while there are many fine details involved in making such predictions, to a first order the number of neutrinos that come out of the Sun has to be directly proportional to the amount of light it produces. The proportionality constant comes from a precisely known fact about nuclear fusion: for every four protons converted to a helium nucleus, two neutrinos and 26.7 MeV of energy are released. (One MeV, or mega-electronvolt, is about twice the amount of energy that could be obtained if the mass of a single electron were somehow completely converted into energy.) The discrepancy between theory and experiment was so severe, one of the hypotheses invoked to explain it was that there is a small black hole at the center of the Sun!
Prior to SNO, all solar neutrino experiments had been able to detect only electron neutrinos. The genius of SNO is that the detector was based on a novel material: heavy water, in which each molecule contains one or two atoms of deuterium. Deuterium is a heavy version of hydrogen whose nucleus, a "deuteron," consists of both a proton and a neutron. All kinds of neutrinos can react with deuterium by simply splitting apart the deuteron, giving a free proton and free neutron. This is an example of a neutral-current (NC) reaction. But only electron neutrinos can convert the neutron into a proton, incidentally also producing a spare electron (a charged-current, CC, reaction). For further redundancy, as a consistency check SNO also made use of a third reaction (neutrino-electron elastic scattering). Since the elastic scattering may occur via both NC and CC mechanisms, it can occur with all three types of neutrino, but with a known, increased probability for electron neutrinos. The three different reactions can be distinguished by the characteristic patterns of light that they produce in the detector.
What SNO found is that when all three types of neutrinos can be detected, the neutrino rates match the predictions for solar neutrinos. On the other hand, if only CC reactions are considered, the rates of only electron neutrinos match the results obtained by most previous experiments (read on to see why I emphasize "most"). But electron neutrinos can only turn into the other two types if neutrinos have mass! Q.E.D.
From a quantum mechanical point of view, it is worth noting that an electron neutrino does not suddenly and randomly become a muon or tau neutrino. In most of its journey, the flavor (electron, muon or tau) of the neutrino is in what physicists call an indeterminate state, meaning that it is not necessarily any particular one of these. Instead, its probability of acting as if it were an electron neutrino instead of a different type is what varies as it travels: starting at 100% when it is first produced in the Sun, and then oscillating between a minimum of ~14% and maximum of 100%. This quantity is referred to as the "electron neutrino survival probability," P(νe → νe) or just Pee for short. As you might guess, the probability that the neutrino instead acts as a muon or tau neutrino is 100% minus Pee. (It happens that the probabilities that it acts as a muon neutrino vs. tau neutrino are just about equal: each varies between 0% and about 43%, in phase.)
The time period of the oscillation (or, since neutrinos are traveling at almost exactly the speed of light, equivalently the wavelength) depends on the neutrino energy. For the solar neutrinos observed by SNO, which have energies of several MeV, the oscillation wavelengths are on the order of a couple hundred miles or less. But neutrinos are produced everywhere in the core of the Sun, a volume that is tens of thousands of miles across. Our Earth-based detectors therefore see neutrinos at every point in their oscillation cycles. The crests and troughs of the different probability waves cancel out as seen from Earth, and so one might guess that a "typical" solar neutrino at the energies observed by SNO has a uniform probability of about Pee = (100% + 14%) / 2 = 57% of acting as an electron neutrino.
Because there is an additional complication, one would be wrong. That complication is the Mikheyev-Smirnov-Wolfenstein (MSW) effect, which results in what are known as "matter oscillations." In very dense matter such as that found at the center of the Sun, neutrino oscillations work differently, because the probability that a neutrino interacts with the surrounding matter is no longer negligible. High-energy solar neutrinos such as those seen by SNO instead act like electron neutrinos with a probability of about 31%. It is only low-energy neutrinos that behave according to the "vacuum oscillations" giving a 57% electron neutrino survival probability. So neutrino experiments such as SNO that have a high energy threshold see electron neutrinos with a 31% survival probability, whereas experiments that can see nearly all neutrinos (99% of which are low-energy), such as the GALLEX and SAGE radiochemical experiments, obtain the 57% value.
In particular, Borexino will be the first real-time neutrino detector able to observe the low-energy neutrinos expected to behave this way. Unlike the radiochemical experiments, it will be able to distinguish between higher and lower energy neutrinos, and therefore to make a direct observation of the transition between vacuum and matter oscillations.
Actual neutrino masses
The mathematics works out in such a way that the neutrino oscillation probabilities are dependent only on the differences between the squares of masses of two neutrino types. The actual masses of the various neutrinos cannot be determined solely based on oscillations; solar neutrino detectors such as SNO and Borexino are unable to help with this. One needs to do a direct measurement or a calculation based on other types of observation.
The current most promising approaches are to consider the maximum masses that neutrinos could have without affecting the formation of structure in the universe just after the Big Bang; and to do a direct measurement of the energy spectra produced in certain low-energy radioactive decays, in which the law of conservation of energy implies that the neutrino mass should distort the shape of the spectrum at the endpoint. The current limits from cosmological considerations are less than about 0.5 eV (one millionth of the electron mass!) for the sum of the masses of all three neutrino types. The known values of the mass-squared differences imply that the heaviest neutrino type cannot be less massive than about 0.05 eV. Thus the heaviest neutrino must be between about a ten-millionth and a millionth of the electron mass. The lightest neutrino could in principle be massless, although this would be surprising.
Although we have glossed over it until now, it needs to be pointed out (at the risk of further confusion) that the electron, muon, and tau neutrinos do not have well-defined masses. The things that do have masses, two of which we just referred to as the lightest and heaviest neutrinos, are the neutrino "mass eigenstates," symbolized as ν1, ν2 and ν3. Each of the three flavors of neutrino, electron, muon or tau, is on the other hand a "flavor eigenstate," symbolized as νe, νμ and ντ. The fact that the mass eigenstates and flavor eigenstates are not the same (a fact that does not result in an ab initio way from principles of quantum mechanics or the Standard Model, but has only been empirically ascertained) is what leads to the phenomenon of neutrino oscillation.
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