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MSW-A Possible Solution to the Solar MSW MSW mk. Thus, 2 1.0 mk p2 1 m2 } E 5 Ïwwwww < p 1 , k k 2p MSWMSWMSWMSW where k 5 1, 2. Because m Þ m , and 1 2 0.75 hence E1 Þ E2, the relative phase A possible solution to the solar-neutrino problem between |n1l and |n2l will change as |n(t)l evolves with time. At an arbitrary time t, S. Peter Rosen the neutrino has evolved to the state 0.5 Probability 2iE t 2iE t |n(t)l5cos u e 1 |n1l 1 sin u e 2 |n2l ver since 1968, when Ray Davis neutrino within a certain energy range are created as coherent, linear superpo- 0.25 and his colleagues reported their can make a dramatic change to a sitions of the mass states and are the In general, this linear combination Efirst observations of solar neutri- different flavor even though its intrinsic analogues of the independent planes of of |n1l and |n2l is neither a pure os, there have been continuing reports in vacuo probability for doing so may polarization. Because the phase of each electron neutrino state |n l nor a pure e 0 f a deficit in the number of electron be very small (Figure 1). mass state |nkl depends on the mass mk muon neutrino state |nml. Instead, it is eutrinos arriving at Earth compared (k 5 1, 2, 3), each state evolves with a a linear superposition of both states. 0 0.2 0.4 0.6 0.8 1.0 with the number predicted by the stan- different phase, and therefore the rela- Quantum mechanics then tells us that, Distance/Solar Radius ard solar model. In vacuo neutrino Neutrinos that Travel tive phase between the states changes after travelling a distance of x meters, scillations, in which electron neutrinos in a Vacuum with time. Quite distinct from our |n(t)l could be detected as a muon neu- Figure 1. The Leopard Changes Its Spots ave a certain probability of transform- analogy with ordinary light, this change trino. That transformation probability, These curves represent the probability that a neutrino, after travelling through matter, → ng into muon and/or tau neutrinos (or In vacuo neutrino oscillations can can lead to the appearance of different denoted as P(ne nm), is given by would be detected as either an electron neutrino (blue), muon neutrino (red), or tau hange flavor) as they travel between occur if the neutrino flavor states are neutrino flavors. neutrino (yellow). A neutrino is “born” in the core of the sun as the superposition of → 2 he Sun and Earth, were often viewed “coherent,” linear superpositions of To keep things simple, in this article P(ne nm) 5 knm|n (t)l mass states, in a combination that corresponds to an electron neutrino. In the specific s one way to explain that deficit. neutrino mass states. Coherent means we shall consider only two neutrino model used to generate these curves, the superposition changes through the MSW 2 2 To date, there have been four that the phases of two or more of the mass states: |n1l and |n2l. The electron- 5 sin 2u sin (px/l). resonance effect after the neutrino has traversed about 15 percent of the solar radius. ifferent experiments with varying mass states are correlated, so that the and muon-neutrino flavor states would The neutrino has only a 75 percent chance of being detected as an electron neutrino, ensitivities to different parts of the relative phase between the states then be written as (Even massive neutrinos would be and a 25 percent chance of being detected as a tau neutrino. By the time it flees the olar-neutrino energy spectrum. All leads to an interference term in the highly relativistic and would travel at sun, the neutrino is most likely to be detected as a muon neutrino. ave reported a significant reduction calculation of quantum probabilities |nel 5 cos u |n1l 1 sin u |n2l , and nearly the speed of light c. We have n the neutrino flux. The combined and expectation values. Quantum inter- |nml 52sin u |n1l 1 cos u |n2l . made the approximation t < x/c 5 x in ata from them suggests that only ference between the mass states our units. See the box “Derivation of intrinsic mixing angle is small, then the phase of each polarization component eutrinos with energies between 1 and becomes an integral part of the The angle u characterizes the amount of Neutrino Oscillation Length” on page oscillation probability will always be evolves differently. When polarized 0 million electron volts (MeV) are neutrino’s description. mixing between the mass states and is 161.) Because of the term sin2(px/l), small, independent of neutrino energy. light passes through a birefringent eriously depleted. It is unlikely that in In certain respects, this phase phe- known as the intrinsic mixing angle. If the probability oscillates with distance material, the relative phase between acuo oscillations alone could produce nomenon parallels the relationship u is small, the electron neutrino consists from the source. The parameter l is the polarization states changes, and the hose results. Instead, the data suggest between circular and plane polarization primarily of the state |n1l and has only called the oscillation length and is Neutrinos that Travel plane of polarization rotates. hat a resonance phenomenon might be for ordinary light. The left and right a small admixture of |n2l, whereas the given in meters by through Matter A similar phenomenon applies to the t work, one in which the probability states of circular, polarized light emerge muon neutrino would be dominated by neutrino flavor states as they pass pEn or a neutrino to transform into another most naturally from Maxwell equations. |n l and would have only a small l 5 } , The work of Wolfenstein, and then through matter. In the standard elec- 2 1.27Dm2 flavor is high only over a limited region All other states can be expressed as amount of |n1l. Mikheyev and Smirnov, showed that the troweak model, all neutrinos interact f the solar-neutrino energy spectrum. linear superpositions of those two states. We assume that at t 5 0, the neutrino where En is the neutrino energy in oscillation probability could increase with up quarks, down quarks, and elec- The MSW effect provides us with In particular, plane, polarized light is a is created in the distinct superposition million electron volts and dramatically because of an additional trons through neutral currents (the ust such an energy-dependent phenom- superposition of equal amounts of two that corresponds to the electron neutrino: phase shift that occurs when neutrinos exchange of neutral Z0 bosons). All fla- 2 2 2 non. Named after Lincoln Wolfenstein, circular states that have a constant rela- Dm 5 m2 2 m1 travel through matter. The origin of vor states see a refractive index n that is who formulated the underlying physics, tive-phase difference. Changing the |n(0)l 5 |nel 5 cos u |n1l 1 sin u |n2l . this phase shift can be understood by a function of the neutral-current forward- nd S. P. Mikheyev and Alexei relative phase of the states rotates the is approximately the mass difference analogy with a well-known phenome- scattering amplitude fnc, the density of mirnov, who recognized its impor- plane of polarization. When the neutrino travels through a between the muon neutrino and the non: When light travels through a the electrons Ne, and the momentum p: ance for the Sun, the MSW effect is an Similarly, we think of the neutrino vacuum, each mass-state component electron neutrino measured in electron material, it sees a refractive index 2pNe legant application of quantum mechan- mass states |n1l,|n2l, and |n3l, with dis- |nk l evolves with its own phase factor volts squared, assuming a small, intrinsic because of coherent forward scattering nnc 5 1 1 }2 fnc . cs that explains how neutrino oscilla- tinct masses m , m , and m , as the exp(2iE t). (We are working in units in mixing angle. (The factor of 1.27 in the from the constituents of the medium. p 1 2 3 _ k ons can be enhanced by the medium analogues of the circularly polarized which h 5 c 5 1.). We assume each denominator allows l to be expressed in Birefringent materials have different Electron neutrinos, and only electron hrough which the neutrinos travel. At states. The weak-interaction states, or mass state has the same momentum p, meters. It derives in part from hidden refractive indices for independent, linear neutrinos, can also interact with elec- _ he right density of matter, an electron the flavor neutrinos, |nel, |nml, and |ntl, which is much greater than the masses factors of h and c.) Note that, if the polarizations of the light, and so the trons through charged currents in a 56 Los Alamos Science Number 25 1997 Number 25 1997 Los Alamos Science MSW MSW 1.0 with distance and, as always, is accom- of transforming into a neutrino of a 1.0 panied by interference phenomena. The different flavor, even if its intrinsic Dm2 = 5 x 1026 interference can be totally destructive, in vacuo probability for doing so might sin22u = 0.007 totally constructive, or somewhere be very small. This enhancement is pp 0.8 2 ) 7 sin 2u = 0.007 in between, depending upon relative called the MSW effect. e Be conditions. n The equation for the MSW probabil- e al Probality n ( w viv 0.6 ity is derived in a heuristic fashion in Oscillation Enhancement ms the box on page 161.
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