P(L) = 1 - A sin² (L/l)

for the oscillating probability that a neutrino born as a n_e is still a n_e after a journey of length L through vacuum.

The characteristic vacuum oscillation length in kilometers is given by

l = E/1.27Dm²,

where E is the neutrino's energy in GeV and Dm² is the difference between the squared masses of the two relevant mass eigenstates in eV². The amplitude of the probability oscillation is

A = sin²2q,

where the "mixing angle" q is the angle by which the state n_e is rotated from the lighter of the two solar-neutrino mass states in the Hilbert space spanned by the mass eigenstates.

Because there are, as yet, no data good enough to reveal the very small contribution of the third neutrino mass eigenstate to n_e, it is customary to summarize solar-neutrino oscillation results in terms of the two parameters q and Dm2. Prior to the Kamland result, the region of parameter space favored by the existing solar-neutrino data was the so-called large-mixing-angle (LMA) solution, with q ranging from about 25° to 40° and Dm² near 5 x 10^-5 eV².

However, several alternatives to the LMA solution had also survived the experimental gauntlet with statistically respectable, albeit larger, chi-squares. Some, like the LMA solution, had large mixing angles; but they had much smaller Dm², ranging from 10^-7 eV² all the way down to 5 x 10^-12 eV². Another solution still in play last year gave a q of only about 1°. So small a mixing angle would suit the theoretical prejudice that neutrino mixing angles ought not to be much bigger than those that describe the mixing of quark flavors among the quark-mass eigenstates. There were also several solutions that
incorporated a hypothetical fourth neutrino mass eigenstate that would be "sterile" in the sense that it did not participate in the usual weak interactions.

The highest energy of neutrinos produced in the solar core is only 20 MeV. The third neutrino mass eigenstate, though it contributes little, if anything, to solar-neutrino oscillation, is important for the well-attested oscillation of GeV muon neutrinos produced in the atmosphere by cosmic rays. For atmospheric n_m oscillation, the relevant Dm² is a few times 10^-3 eV² (see Physics Today, August 1998, page 17).

Figure 2

The principal goal of the Kamland experiment is to test and refine the LMA solar-neutrino solution with controlled terrestrial antineutrino sources free of astrophysical complications. The heart of the detector is a kiloton of liquid scintillator in a transparent spherical vessel monitored by a surrounding array of almost 2000 large photomultiplier tubes. It sits, shielded from cosmic-ray muons by a kilometer of rock overhead, in the same cavern of the Kamioka mine that previously housed the first-generation Kamiokande water-Ùerenkov solar-neutrino detector. The Japan-US Kamland collaboration, headed by Atsuto Suzuki of Tohuko University in Sendai, began building the detector in 1998.

The beta decay of fission fragments in a power reactor produces a continuous, penetrating flux of electron antineutrinos. The 22 reactor sites in Japan and Korea that contribute significant _e flux at the Kamland detector range in distance from 80 to 900 km, with the strongest concentration between 160 and 180 km. If the neutrino oscillation parameters do indeed lie within the LMA region, a significant fraction of the reactor antineutrinos will change flavor along the way and become invisible to the detector. Kamland detects arriving neutrinos when they instigate inverse beta decay,

_e + p ® e⁺ + n,

in the liquid scintillator. And that reaction can only be accomplished by electron antineutrinos.

The energy distribution of the neutrinos recorded by Kamland peaks at about 4 MeV. For a neutrino of that energy, the LMA solution gives an oscillation length l of about 100 km. That's why the Kamland complex is so well suited to probe the LMA region of the parameter space. All of the alternative solutions to the solar-neutrino data have much smaller Dm² and thus correspondingly longer l. Therefore, only the LMA solution predicts that Kamland will record a significant shortfall of reactor neutrinos. And previous reactor-neutrino detectors, none of which was farther than a kilometer from its source reactor, could only have seen a shortfall if Dm² were much larger than 10^-4 eV².

"When Kamland was first proposed," recalls Stuart Freedman (University of Calfornia, Berkeley), a spokesman for the collaboration's US contingent, "we had a hard time selling it to the community, because most of them were betting on the small-mixing-angle solution, to which Kamland wouldn't be sensitive."

In the first five months of data taking, the Kamland group found a total of 53 _e events above an estimated background of only one event. That's only 61% of the 87 events one would expect in the absence of neutrino oscillation. But it's in good agreement with what the LMA solution predicts. The figure on page 14 shows the Kamland shortfall, together with the null results from the earlier reactor-neutrino detectors.

The estimated background of only about one event in the final sample of 54 is impressively low by the standards of neutrino detectors in underground settings rife with radioactive contaminants. It points up a considerable advantage of looking for reactor antineutrinos rather than solar neutrinos. Each inverse beta-decay reaction produces not just one, but a correlated pair of signals that makes it much easier to spot spurious events. First, the slowing and annihilation of the positron produces a "prompt" scintillation signal whose intensity provides a measure of the incident neutrino's energy. Then, typically a few hundred microseconds later, the neutron can be captured by a proton to create a deuteron and a telltale 2.2 MeV photon.

Figure 3

One learns about the oscillation parameters not simply from the overall fraction of neutrinos that have become invisible to the detector, but also from the energy dependence of the shortfall. The figure above shows the distribution of prompt positron-annihilation energies for the 54 events in the final sample. To good approximation, the prompt energy measured by the phototubes is the incident neutrino's energy minus a kinematic correction of 0.8 MeV.

The Kamland group's best-fit oscillation parameters from the observed shortfall and energy distribution are q = 45° and Dm² = 6.9 x 10^-5 eV². The 32 events in the plot with prompt energies below 2.6 MeV were excluded from the analysis because, below this cutoff, antineutrinos from radioactive uranium and thorium in Earth's crust contribute significantly to the tally of inverse-beta-decay events. At the 95% confidence level, the Kamland data already exclude all the previously surviving alternatives to the LMA solution.

The posting of the first Kamland results on the Web triggered a torrent of analyses by the theorists. (One wag commented that the appearance of five of those papers hours before the Kamland paper constituted evidence of causality violation.) The first order of business was to extract the oscillation parameters from a global fit to all the existing solar-neutrino data together with the Kamland results. A typical fit of this kind, carried out by John Bahcall (Institute for Advanced Study) and coworkers,² is shown in the lower figure on page 15. "The Kamland results decisively shrink the allowed range of the solar-neutrino parameters," says Bahcall. "And they limit a possible sterile component to less than 9%."

The Kamland result is also good evidence that antineutrinos and neutrinos share the same oscillation parameters, as required by fundamental theory. But the theory does allow the neutrino mixing matrix to include a complex phase that might engender a subtle neutrino-antineutrino asymmetry (leptonic CP violation) strong enough to explain the upsetting of the matter-antimatter balance in the early cosmos. (See the article by Helen Quinn in Physics Today, February 2003, page 30.) The triumph of the LMA solution raises the prospect that future long-baseline neutrino experiments may find evidence for leptonic CP violation.

For neutrinos coming from the solar core, the LMA solution implies that vacuum oscillation on the journey to Earth plays second fiddle to an irreversible flavor change that takes place in high-density regions of the Sun: the Mikheyev-Smirnov-Wolfenstein (MSW) effect. If vacuum oscillation with a l very much shorter than our distance from the Sun were the dominant process, the energy dependence of the solar-neutrino shortfall would be almost completely washed out. But that's not the case; the solar-neutrino detectors do see a clear energy dependence.

Because of its association with the electron, a n_e passing through matter feels an extra interaction energy, proportional to the ambient electron density, beyond the matter-interaction energy common to all the neutrino flavors. In 1986, Stanislav Mikheyev and Alexei Smirnov at the Institute for Nuclear Research in Moscow, using a formalism developed by Lincoln Wolfenstein (Carnegie Mellon University), pointed out that this extra energy term should produce a flavor metamorphosis when a n_e produced in the core passes through a critical-electron-density region of the Sun. The critical density, and thus the distance from the core at which it's encountered, depends on the neutrino's energy. When the neutrino finally emerges from the Sun it is, to good approximation, in a coherent superposition of flavor states that constitutes a mass eigenstate. A pure mass eigenstate would experience no vacuum oscillation on the rest of the journey to Earth. There is now good evidence that this mass eigenstate, the heavier of the two whose splitting is given by Dm², is roughly an equal superposition of all three neutrino flavors. The MSW mechanism was originally invoked to explain how a small mixing angle might cause a large solar-neutrino shortfall.

"I'm sometimes asked whether our first Kamland result is a discovery or just a confirmation of what we already believed," says Giorgio Gratta (Stanford University), the US contingent's other spokesman. "I like to compare it to the first creation of spectral lines in the laboratory in the 19th century. Frauenhofer had already found lines in the solar spectrum. But until they were also made on Earth, one couldn't be sure that they were more than just something that happened only in stars."

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© 2003 American Institute of Physics