MSW MSW
mk. Thus, 1.0 m2 2 2 k E ϭ ͙pෆෆෆϩෆmෆϷ p ϩ ᎏ , k k 2p MSWMSWMSWMSW where k ϭ 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 |1͘ and |2͘ will change as |(t)͘ evolves with time. At an arbitrary time t, S. Peter Rosen the neutrino has evolved to the state 0.5 Probability ϪiE t ϪiE t |(t)͘ϭcos e 1 |1͘ ϩ sin e 2 |2͘ 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 |1͘ and |2͘ 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 | ͘ nor a pure e 0 f a deficit in the number of electron be very small (Figure 1). mass state |k͘ depends on the mass mk muon neutrino state |͘. Instead, it is eutrinos arriving at Earth compared (k ϭ 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 |(t)͘ 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(e ), 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(e ) ϭ ͗| (t)͘ 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: |1͘ and |2͘. The electron- ϭ sin 2 sin (x/). 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 |e͘ ϭ cos |1͘ ϩ sin |2͘ , and nearly the speed of light c. We have n the neutrino flux. The combined and expectation values. Quantum inter- |͘ ϭϪsin |1͘ ϩ cos |2͘ . made the approximation t Ϸ x/c ϭ 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 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(x/), 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 is small, the electron neutrino consists from the source. The parameter is the polarization states changes, and the hose results. Instead, the data suggest between circular and plane polarization primarily of the state |1͘ 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 |2͘, 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 E or a neutrino to transform into another most naturally from Maxwell equations. | and would have only a small ϭ , The work of Wolfenstein, and then through matter. In the standard elec- 2͘ ᎏ 2 1.27⌬m flavor is high only over a limited region All other states can be expressed as amount of |1͘. 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 ϭ 0, the neutrino where E 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- ⌬m ϭ m2 Ϫ 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 |(0)͘ ϭ |e͘ ϭ cos |1͘ ϩ sin |2͘ . 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 2Ne legant application of quantum mechan- mass states |1͘,|2͘, and |3͘, with dis- |k ͘ evolves with its own phase factor volts squared, assuming a small, intrinsic because of coherent forward scattering nnc ϭ 1 ϩ ᎏ 2 fnc . cs that explains how neutrino oscilla- tinct masses m , m , and m , as the exp(ϪiE 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 ϭ c ϭ 1.). We assume each denominator allows 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, |e͘, |͘, and |͘, 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 ⌬m2 = 5 x 10Ϫ6 interference can be totally destructive, in vacuo probability for doing so might sin22 = 0.007 totally constructive, or somewhere be very small. This enhancement is pp 0.8 2 ) 7 sin 2 = 0.007 in between, depending upon relative called the MSW effect. e Be conditions.
The equation for the MSW probabil- e al Probality ( w viv 0.6 ity is derived in a heuristic fashion in Oscillation Enhancement ms
the box on page 161. Here, we simply in the Sun P pep 0.5 ino Sur state the results. The MSW probability th for an electron neutrino to transform The density of the Sun is not con- 0.4 into a muon neutrino is stant but decreases monotonically from xW about 150 grams per cubic centimeter 8B P ( → ) ϭ sin22 sin2 (g/cm3) at the center to 0.1 g/cm3 at Electron-Neutr MSW e m ᎏ , a radius of 700,000 kilometers. In Probability at Ear 0.2 where other words, the density decreases from 2 50 times the density of terrestrial rocks 2 sin 2 sin 2m ϭ ᎏ ᎏ2 , to one-tenth the density of water. For 0 W this range of densities, the value of the 0.0 104 105 106 107 108 2 2 2 W ϭ sin 2 ϩ (D Ϫ cos 2) , and parameter D varies from a few tens 0.1 1.0 10.0 E/∆m2 (MeV/eV2) near the core to a few hundredths near Neutrino Energy (MeV) 2E D ϭ ͙2ෆG N . the edge. It passes through 1 at some F e ᎏ⌬m2 intermediate point. Large values of D Figure 3. MSW Solution to the Solar-Neutrino Problem → ) igure 2. The MSW Survival-Probability Curve for the Sun Note the similarity between this expres- lead to very small values for the The MSW survival probability PMSW(e e (red dashed line) has been plotted as a he probability that an electron neutrino born in the core of the Sun will emerge from sion for the MSW probability and the effective mixing angle and damp out function of neutrino energy and superimposed over a simplified picture of the solar- → he Sun as an electron neutrino is called the survival probability Ps(e e). (The sur- in vacuo oscillation probability. The neutrino oscillations, whereas small neutrino spectrum. Solar neutrinos with energies that fall within the gray-shaded region → 2 7 8 val probability is equal to 1 ؊ P(e ).) It is plotted as a function of E /⌬m , in vacuo mixing angle is replaced by values essentially leave the in vacuo (those from the Be, pep, and B reactions) have a low survival probability, and thus hich is essentially the in vacuo oscillation length . The mixing angle was chosen an effective mixing angle m that oscillation unchanged. Values near 1 have a high probability of oscillating into muon and/or tau neutrinos as they flee the and calculation of the curve takes into account the density depends on the matter oscillation term give rise to the maximum enhancement solar core. If the MSW effect occurs in the Sun, experiments that detect only electron ,0.007 ؍ uch that sin22 rofile of the Sun. For a range of oscillation length values between 105 and (through the parameter D). When a (we are assuming that is small). neutrinos would measure a reduced flux from those reactions. An experiment such as 06 MeV/eV2, the probability that an electron neutrino remains an electron neutrino is neutrino travels in vacuum, the electron An electron neutrino born in the core the one planned at the Sudbury Neutrino Observatory, which is sensitive to electron, ery small. In our two-state model, the neutrino would oscillate into a muon neutrino. density is zero, and hence D is equal of the Sun will start its journey to the muon, and tau flavors, should detect the full flux. to zero, so that W 2 ϭ 1. Thus, the edge without oscillating—until it MSW probability reduces to the reaches the region where D is of the volts squared (eV2), mass differences are harge-changing process mediated by state develops a phase exp[ip(n Ϫ 1)x] in vacuo probability. order of 1. In this region, it will under- 7 ϫ 106 he Wϩ boson. The refractive index due to the index of refraction. For When a neutrino travels through go the oscillation enhancement, and the 2.5 ϭ ᎏ ᎏ cos 2 . 10Ϫ4 Ն ⌬m2 Ն 10Ϫ9 eV2 . een by electron neutrinos, therefore, muon or tau neutrinos, that phase is matter, however, W 2 can become less probability that it will remain an elec- e as an additional term ncc given by exp [ip(nnc Ϫ 1)x], while electron than 1. This is the MSW resonance tron neutrino will decrease significantly. The in vacuo oscillation length is mea- To study this range of mass differences neutrinos have a phase given by effect. The oscillation probability In Figure 1, this region is observed at a sured in meters for a neutrino with terrestrial neutrinos, we would 2Ne n ϭ f . exp ͓ip(n ϩ n Ϫ 1)x͔. Substituting increases with the resonance condition distance of 0.2 solar radius. At larger with given momentum and mass para- need intense sources of extremely low cc ᎏp2 cc nc cc in the expressions for nnc and ncc leads given by D ϭ cos 2. At that point, solar radii, D is much smaller, and the meters. The scale of the right-hand side energy neutrinos, well below 1 MeV. 2 2 2 4 he charged-current forward-scattering to the expressions W ϭ sin 2 and sin 2m ϭ 1, and the electron neutrino’s survival probability is 1.8 ϫ 10 kilometers, which is compa- Those sources do not exist, and there- mplitude fcc in this term is proportional oscillation probability reaches a maxi- will oscillate around a relatively small rable with Earth’s diameter. Given the fore solar neutrinos are the only means i2xN f /p o the weak-interaction coupling con- e e nc for and mum. This resonance condition is inde- value with an amplitude corresponding range of solar densities, the MSW effect available to us. ant GF (the Fermi constant) times the pendent of the size of the intrinsic mix- to the in vacuo mixing angle. could occur if were in the range i(2N f /p ϩ ͙ෆ2G N )x eutrino momentum e e nc F e for e. ing angle , but it requires matching We can estimate the range of or the properties of the material with the ⌬m2 for which the MSW effect is 104 Յ Յ 108 meters , The Most Favored Solution GFp fcc = ᎏ , The additional term ͙2ෆGFNe in the neutrino oscillation length through important. We rewrite the enhancement to the Solar-Neutrino Problem 2ෆ ͙ phase of the electron neutrino is called the relation condition in terms of the solar density as seen in Figure 2. 2 2 o that ncc takes the form the matter oscillation term. This term e measured in grams per cubic Because ϭ E/1.27⌬m The value of ⌬m determines how cos 2 . causes the relative phase between the ͙ෆ2G N ϭ 1.2 ᎏ ᎏ centimeter and then multiply by and because solar-neutrino energies the curve for the survival probability ͙2ෆGFNe F e ncc ϭ ᎏᎏ . electron and muon states to change Avogadro’s number. For the neutrino vary from a fraction of 1 MeV to overlays the spectrum of solar neutri- p with distance. Hence, the interference When this matching occurs, a energy measured in million electron about 10 MeV, the MSW effect can nos. Decreasing ⌬m2 moves the curve In travelling a distance x, each flavor between the two states also changes neutrino will have a high probability volts and ⌬m2 measured in electron occur in the Sun if the squared to the left, while increasing it moves
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S. Peter Rosen, a native of London, earned his he curve to the right. For values of Heuristic Derivation of the MSW Effect B.S. in mathematics and an M.A. and Ph.D. in (for the students in us all) oughly ⌬m2 ϭ 5 ϫ 10Ϫ6 eV2 and theoretical physics from Merton College, Oxford, n22 ϭ 0.007, we find that pp neutri- in 1954 and 1957, respectively. Rosen came to the United States as a research associate at Washing- os survive as electron neutrinos most For simplicity, we shall consider only oscillations between electron and muon ton University, and from 1959 through 1961 he 7 f the time, while Be neutrinos are worked as scientist for the Midwestern Universi- neutrinos. The neutrino mass states |1͘ and |2͘ are assumed to have distinct lmost completely converted to muon ties Research Association at Madison, Wisconsin. masses m1 and m2, respectively. We define the neutrino flavor states |e͘ and eutrinos (Figure 3). This appears to In 1961, he was awarded a NATO Fellowship to | ͘ in terms of two mass states: the Clarendon Laboratory at Oxford. Rosen e a good description of data measured returned to the United States as a Professor of y the current generation of solar- Physics at Purdue University and later served as |e͘ ϭ cos |1͘ ϩ sin |2͘ , and (1a) senior theoretical physicist for the High Energy eutrino experiments. (See the article |͘ ϭ Ϫsin |1͘ ϩ cos |2͘ . (1b) Exorcising Ghosts” on page 136.) Physics Program of the U.S. Energy Research and Development Administration. Some of Rosen’s Next-generation experiments, such additional appointments include program associate We further assume that an electron neutrino is born at time t = 0. That neutrino s the one planned at the Sudbury for theoretical physics with the National Science will evolve in time as a superposition of states with time-dependent coefficients. Neutrino Observatory, are designed Foundation, chairman of the U.S. Department of The neutrino can be described by either mass states or flavor states: Energy’s Technical Assessment Panel for Proton o determine whether oscillations Decay and of Universities Research Association, Inc., Board of Overseers at Fermi National Accelera- o other neutrino states do indeed tor Laboratory. Rosen first came to Los Alamos in 1977 as a visiting staff member and later became |(t)͘ ϭ a1(t) |1͘ ϩ a2(t) |2͘ ϭ ae(t) |e͘ ϩ a(t) |͘ , (2) ccur and whether the MSW effect or associate division leader for Nuclear and Particle Physics of the Theoretical Division. In 1990, he n vacuo oscillations solve the solar- accepted the position of dean of science and professor of physics at the University of Texas, Arlington, where and remained in that position until January 1997, when he was appointed associate director for High eutrino problem. Energy and Nuclear Physics at the Department of Energy in Washington, D.C. He is a fellow There is one final, but very interest- of the American Physical Society and of the American Association for the Advancement of Science. ae(t) ϭ a1(t) cos ϩ a2(t) sin , and (3a) ng, comment. Our planet Earth may a(t) ϭ Ϫa1(t) sin ϩ a2(t) cos . (3b) lay a unique role in the study of the MSW effect. There turns out to be a Further Reading We will also be using the inverse of Equations (1) and (3): well-defined range of mixing angles Rosen, S. P., and J. M. Gelb. 1986. Mikheyev- nd mass differences for which the |1͘ ϭ cos |e͘ Ϫ sin |͘ , and (4a) Smirnov-Wolfenstein Enhancement of nhancement density is less than |2͘ ϭ sin |e͘ ϩ cos |͘ ; (4b) 3 Oscillations as a Possible Solution to the 5 g/cm . This density occurs in both Solar-Neutrino Problem. Physical Review D he Sun and Earth, and thus neutrinos 34 (4): 969. a1(t) ϭ ae(t) cos Ϫ a(t) sin , and (4c) hat are converted from electron neutri- a2(t) ϭ ae(t) sin ϩ a(t) cos . (4d) os to muon neutrinos in the Sun may Wolfenstein, L. 1978. Neutrino Scintillations in Matter. Physical Review D 17 (9): 2369. e reconverted to electron neutrinos In general, the time development of the neutrino states described in when they pass through Earth. Equation (2) has a phase that depends on both the momentum and the ener- This effect would be seen as a sig- gy of the neutrino. For example, an electron neutrino evolves as ificant increase in the solar-neutrino ipиxϪiE t ipиxϪiE t gnal at night, when Earth is between |e(t)͘ ϭ cos e 1 |1͘ ϩ sin e 2 |2͘ . (5) he Sun and the detector. Observation _ f such a “day-night” effect would be We work in units in which h ϭ c ϭ 1. Let us first consider the evolution of |(t)͘ n unambiguous and definitive proof of as a superposition of mass eigenstates during an infinitesimal time ⌬t. We he MSW effect and of neutrino mass. assume a common momentum for each mass state, so that only the differ- would also be nature’s tip-of-the-hat ence between the energies of the mass states (due to the difference in the
o the insightful Lincoln Wolfenstein, neutrino masses) governs the time development of the state. With p ϾϾ mk , who once observed that “…for neutri- we can approximate the energy as os, the Sun shines at night!” ■ 2 2 2 Ek ϭ͙pෆෆෆϩෆmkෆ Ϸ p ϩ mk /2p ϭ p ϩ Mk , (6)
2 where Mk ϭ mk /2p (k ϭ 1, 2). The neutrino evolves in time ⌬t as
ϪiE ⌬t ϪiE ⌬t |(tϩ⌬t)͘ ϭ a1(tϩ⌬t)e 1 |1͘ ϩ a2(tϩ⌬t)e 2 |2͘ ϪiM ⌬t ϪiM ⌬t Ϸ a1(tϩ⌬t) e 1 |1͘ ϩ a2(tϩ⌬t)e 2 |2͘ . (7)
We have dropped the overall phase factor of exp(Ϫip⌬t) in Equation (7) because it has no bearing on the final result. With the help of Equations (4a) and (4b), we can write Equation (7) in the flavor basis:
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ϪiM ⌬t ϪiM ⌬t |(tϩ⌬t)͘ ϭ [a1(tϩ⌬t)e 1 cos ϩ a2(tϩ⌬t)e 2 sin ] |e͘ The eigenvalues of the matrix H are given by ϪiM ⌬t ϪiM ⌬t ϩ [Ϫa1(tϩ⌬t)e 1 sin ϩ a2(tϩ⌬t)e 2 cos ] |͘ . (8) 2 2 ϩ ⌴1 ϩ ⌴2 ͙ෆෆෆϩෆ(ෆM2ෆෆϪෆM1ෆ)ෆෆϪෆ2ෆෆ(M2ෆෆϪෆM1ෆ)cෆoෆsෆෆ2ෆ ϭ ϯ . (16) 1,2 ᎏᎏ 2 ᎏᎏᎏᎏᎏ 2 We next consider the neutrino as a superposition of flavor states: Equation (15) can then be solved: ei 1t Ϫ ei 2t |(t)͘ ϭ ae(t) |e͘ ϩ a(t) |͘ . (9) 2 1 ei 2t Ϫ ei 1t A(t) ϭ ᎏᎏ Ϫ I ϩ H A(0) , (17) ΄ 2 1 ᎏᎏ Ϫ ΅ Because only electron neutrinos interact via charged currents, the two flavor states 2 1 have different forward-scattering amplitudes, and each sees a different effective refrac- where I is the identity matrix. At time t ϭ 0, the beam consists only of electron neutrinos. Thus, a (0) ϭ 1, and a (0) ϭ 0 so that tive index in matter. We assume that the change in the probability amplitudes ae(t) and e a(t) during an infinitesimal time ⌬t can be expressed as a simple phase shift that is ei 1t Ϫ ei 2t ei 2t Ϫ ei 1t 2 1 2 2 proportional to the refractive index: ae(t) ϭ ᎏᎏ ϩ ᎏᎏ (M1cos + M2sin ϩ ) ,(18a) 2 Ϫ 1 2 Ϫ 1
i t i t ae(tϩ⌬t) Ϸ ae(t) exp [ip (nnc ϩ ncc Ϫ 1)⌬t] ϭ ae(t)exp [i( ϩ )⌬t , and (10a) e 2 Ϫ e 1 a(t) ϭ ᎏᎏ (M2 Ϫ M1)cos sin . (18b) 2 Ϫ 1 a(tϩ⌬t) Ϸ a(t) exp [ip (nnc Ϫ 1)⌬t] ϭ a(t)exp [i ⌬t] , (10b)
The probability of detecting a muon neutrino after a time t is given by where ϭ (2Ne /p)fnc and ϭ ͙ෆ2GF Ne. The latter relation is the matter oscillation term. We have also used ⌬x Ϸ ⌬t. The neutrino state, therefore, evolves as → 2 2 2 PMSW(e ) ϭ ͉Ό͉(t)͉ ϭ ͉ae(t)Ό͉e͘ ϩ a(t)Ό͉͉͘ ϭ ͉a(t)͉ (19)
i( ϩ)⌬t i ⌬t |(tϩ⌬t)͘ ϭ ae(t)e |e͘ ϩ a(t)e |͘ so that i⌬t 2 2 2 ϭ ae(t)e |e͘ ϩ a(t) |͘ , (11) (M2 Ϫ M1) cos sin P ( → ) ϭ 2(1 Ϫ cos( Ϫ )t) MSW e ᎏᎏᎏ( Ϫ )2 2 1 where again we have dropped the overall phase factor of exp(i⌬t) because it does not 2 1 affect the final result. Equations (8) and (11) are expressions for |(t ϩ ⌬t)͘. Equating (M Ϫ M )2sin22 ( Ϫ ) 2 1 2 2 1 the coefficients of |e͘ and |͘ results in a set of coupled equations: ϭ ᎏᎏᎏ Ϫ )2 sin ᎏᎏ t . (20) ( 2 1 2 ϪiM ⌬t ϪiM ⌬t i( ϩ)⌬t a1(tϩ⌬t)e 1 cos ϩ a2(tϩ⌬t)e 2 sin ϭ ae(t)e , (12a) By substituting in the expressions for 1, 2, M1, M2, we have ϪiM ⌬t ϪiM ⌬t i ⌬t Ϫa1(tϩ⌬t)e 1 sin ϩ a2(tϩ⌬t)e 2 cos ϭ a(t)e . (12b) 2 ( Ϫ )2 ϭ (M Ϫ M )2 sin22 ϩ Ϫ cos2 (21a) 2 1 2 1 ᎏᎏM Ϫ M Both sides of Equations (12a) and (12b) are expanded to first order in ⌬t, ΄ 2 1 ΅ ⌬m2 ⌬m2 [a1(t) ϩ a˙1(t)⌬t Ϫ ia1(t)M1⌬t]cos ϩ [a2(t) ϩ a˙2(t)⌬t Ϫ ia2(t)M2⌬t]sin ϭ ae(t)(1ϩi⌬t) (13a) (M2 Ϫ M1) ϭ ᎏ Ϸ ᎏ . (21b) 2p 2E
Recalling that x ϭ t, and ϭ ͙2ෆGF Ne,we arrive at the MSW probability for Ϫ[a1(t) ϩ a˙1(t)⌬t Ϫ ia1(t)M ⌬t]sin ϩ [a2(t) ϩ a˙2(t)⌬t Ϫ ia2(t)M ⌬t]cos ϭa(t) , (13b) 1 2 an electron neutrino to oscillate into a muon neutrino: where a dot indicates the time derivative. Equations (4c) and (4d) are used to express . . sin22 xW a (t),a (t), a (t), anda (t) in terms of a (t), aи (t), a (t), and aи (t). Following more alge- P ( → ) ϭ sin2 ᎏ , (22) 1 1 2 2 e e MSW e ᎏW2 braic operations, 2 2 2E 2 W ϭ sin 2 ϩ ͙2ෆGF Ne ᎏ 2 Ϫ cos 2 , (23) 2 2 ⌬m Ϫia˙.e(t) ϭ [M1cos ϩ M2sin ϩ ]ae(t) ϩ (M2 Ϫ M1)cos sin a(t) , (14a) where is the in vacuo oscillation length, Ϫa˙ (t) ϭ (M Ϫ M )cos sin (t) ϩ [M sin2 ϩ M cos2]a (t) . (14b) 2 1 ae 1 2 2E = 2 , (24) ᎏ⌬m2 These expressions can be cast in a Schrödinger-like equation for a column matrix A consisting of the probabilitydA amplitudes a (t) and a (t): and ⌬ m2 ϭ m2 Ϫ m2 is required to be nonzero. If the numerical values of the e 2 _ 1 hidden factors of h and c are included, the expression for the oscillation length becomes ϭ E/1.27⌬m2. ■ Ϫ i ᎏ ϭ HA , (15) dt
2 2 M1cos ϩ M2sin ϩ (M2 Ϫ M1)cos sin ae(t) where H ϭ (M Ϫ M )cos sin sin2 ϩ M cos2 and A ϭ 2 1 M1 2 a(t)
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