 Neutrino oscillation

Neutrino oscillation is a quantum mechanical phenomenon predicted by Bruno Pontecorvo^{[1]} whereby a neutrino created with a specific lepton flavor (electron, muon or tau) can later be measured to have a different flavor. The probability of measuring a particular flavor for a neutrino varies periodically as it propagates. Neutrino oscillation is of theoretical and experimental interest since observation of the phenomenon implies that the neutrino has a nonzero mass, which is not part of the original Standard Model of particle physics.
Contents
Observations
A great deal of evidence for neutrino oscillation has been collected from many sources, over a wide range of neutrino energies and with many different detector technologies.^{[citation needed]}
Solar neutrino oscillation
The first experiment to detect the effects of neutrino oscillation was Ray Davis's Homestake Experiment in the late 1960s, in which he observed a deficit in the flux of solar neutrinos with respect to the prediction of the Standard Solar Model, using a chlorinebased detector. This gave rise to the Solar neutrino problem. Many subsequent radiochemical and water Cherenkov detectors confirmed the deficit, but neutrino oscillation was not conclusively identified as the source of the deficit until the Sudbury Neutrino Observatory provided clear evidence of neutrino flavor change in 2001.
Solar neutrinos have energies below 20 MeV and travel an astronomical unit between the source in the Sun and detector on the Earth. At energies above 5 MeV, solar neutrino oscillation actually takes place in the Sun through a resonance known as the MSW effect, a different process from the vacuum oscillation described later in this article.
Atmospheric neutrino oscillation
Large detectors such as IMB, MACRO, and Kamiokande II observed a deficit in the ratio of the flux of muon to electron flavor atmospheric neutrinos (see muon decay). The Super Kamiokande experiment provided a very precise measurement of neutrino oscillation in an energy range of hundreds of MeV to a few TeV, and with a baseline of the diameter of the Earth.
Reactor neutrino oscillation
Many experiments have searched for oscillation of electron antineutrinos produced at nuclear reactors. A high precision observation of reactor neutrino oscillation has been made by the KamLAND experiment since 2002. Neutrinos produced in nuclear reactors have energies similar to solar neutrinos, a few MeV. The baselines of these experiments have ranged from tens of meters to over 100 km.
Beam neutrino oscillation
Neutrino beams produced at a particle accelerator offer the greatest control over the neutrinos being studied. Many experiments have taken place which study the same neutrino oscillations which take place in atmospheric neutrino oscillation, using neutrinos with a few GeV of energy and several hundred km baselines. The MINOS experiment recently announced that it observes consistency with the results of the K2K and SuperK experiments.
The controversial observation of beam neutrino oscillation at the LSND experiment in 2006 was tested by MiniBooNE. Results from MiniBooNE appeared in Spring 2007, and appeared to contradict the findings of the LSND experiment.
On 31 May 2010, the INFN and CERN announced^{[2]} having observed a tau particle in a muon neutrino beam in the OPERA detector located at Gran Sasso, 730 km away from the neutrino source in Geneva.
The upcoming T2K experiment will direct a neutrino beam through 295 km of earth, and will measure the parameter θ_{13}. The experiment is scheduled to begin in 2009 and uses the SuperK detector. NOνA is a similar effort. This detector will use the same beam as MINOS and will have a baseline of 810 km.
Theory
Neutrino oscillation arises from a mixture between the flavor and mass eigenstates of neutrinos. That is, the three neutrino states that interact with the charged leptons in weak interactions are each a different superposition of the three neutrino states of definite mass. Neutrinos are created in weak decays and reactions in their flavor eigenstates^{[nb 1]}. As a neutrino propagates through space, the quantum mechanical phases of the three mass states advance at slightly different rates due to the slight differences in the neutrino masses. This results in a changing mixture of mass states as the neutrino travels, but a different mixture of mass states corresponds to a different mixture of flavor states. So a neutrino born as, say, an electron neutrino will be some mixture of electron, mu, and tau neutrino after traveling some distance. Since the quantum mechanical phase advances in a periodic fashion, after some distance the state will nearly return to the original mixture, and the neutrino will be again mostly electron neutrino. The electron flavor content of the neutrino will then continue to oscillate as long as the quantum mechanical state maintains coherence. It is because the mass differences between the neutrinos are small that the coherence length for neutrino oscillation is so long, making this microscopic quantum effect observable over macroscopic distances.
Pontecorvo–Maki–Nakagawa–Sakata matrix
Main article: Pontecorvo–Maki–Nakagawa–Sakata matrixThe idea of neutrino oscillation was first put forward in 1957 by Bruno Pontecorvo, who proposed that neutrinoantineutrino transitions may occur in analogy with neutral kaon mixing.^{[1]} Although such matterantimatter oscillation has not been observed, this idea formed the conceptual foundation for the quantitative theory of neutrino flavor oscillation, which was first developed by Maki, Nakagawa, and Sakata in 1962^{[4]} and further elaborated by Pontecorvo in 1967.^{[5]} One year later the solar neutrino deficit was first observed,^{[6]} and that was followed by the famous paper of Gribov and Pontecorvo published in 1969 titled "Neutrino astronomy and lepton charge".^{[7]}
The unitary transformation relating the flavor and mass eigenbases can be written
 ,
where
 is a neutrino with definite flavor. α = e (electron), μ (muon) or τ (tauon).
 is a neutrino with definite mass m_{i}, i = 1, 2, 3.
 The asterisk ( ^{*} ) represents a complex conjugate. For antineutrinos, the complex conjugate should be dropped from the second equation, and added to the first.
U_{αi} represents the Pontecorvo–Maki–Nakagawa–Sakata matrix (also called the PMNS matrix, lepton mixing matrix, or sometimes simply the MNS matrix). It is the analogue of the CKM matrix describing the analogous mixing of quarks. If this matrix were the identity matrix, then the flavor eigenstates would be the same as the mass eigenstates. However, experiment shows that it is not.
When the standard three neutrino theory is considered, the matrix is 3×3. If only two neutrinos are considered, a 2×2 matrix is used. If one or more sterile neutrinos are added (see later) it is 4×4 or larger. In the 3×3 form, it is given by:^{[8]}
where c_{ij} = cosθ_{ij} and s_{ij} = sinθ_{ij}. The phase factors α_{1} and α_{2} are physically meaningful only if neutrinos are Majorana particles — i.e. if the neutrino is identical to its antineutrino (whether or not they are is unknown) — and do not enter into oscillation phenomena regardless. If neutrinoless double beta decay occurs, these factors influence its rate. The phase factor δ is nonzero only if neutrino oscillation violates CP symmetry. This is expected, but not yet observed experimentally. If experiment shows this 3×3 matrix to be not unitary, a sterile neutrino or some other new physics is required.
Propagation and interference
Since are mass eigenstates, their propagation can be described by plane wave solutions of the form
where
 quantities are expressed in natural units
 E_{i} is the energy of the masseigenstate i,
 t is the time from the start of the propagation,
 is the threedimensional momentum,
 is the current position of the particle relative to its starting position
In the ultrarelativistic limit, , we can approximate the energy as
This limit applies to all practical (currently observed) neutrinos, since their masses are less than 1 eV and their energies are at least 1 MeV, so the Lorentz factor γ is greater than 10^{6} in all cases. Using also t ≈ L, where L is the distance traveled and also dropping the phase factors, the wavefunction becomes:
Eigenstates with different masses propagate at different speeds. The heavier ones lag behind while the lighter ones pull ahead. Since the mass eigenstates are combinations of flavor eigenstates, this difference in speed causes interference between the corresponding flavor components of each mass eigenstate. Constructive interference causes it to be possible to observe a neutrino created with a given flavor to change its flavor during its propagation. The probability that a neutrino originally of flavor α will later be observed as having flavor β is
This is more conveniently written as
where . The phase that is responsible for oscillation is often written as (with c and restored)^{[9]}
where 1.267 is unitless. In this form, it is convenient to plug in the oscillation parameters since:
 The mass differences, Δm^{2}, are known to be on the order of 1×10^{−4} eV^{2}
 Oscillation distances, L, in modern experiments are on the order of kilometers
 Neutrino energies, E, in modern experiments are typically on order of MeV or GeV.
If there is no CPviolation (δ is zero), then the second sum is zero.
Two neutrino case
The above formula is correct for any number of neutrino generations. Writing it explicitly in terms of mixing angles is extremely cumbersome if there are more than two neutrinos that participate in mixing. Fortunately, there are several cases in which only two neutrinos participate significantly. In this case, it is sufficient to consider the mixing matrix
Then the probability of a neutrino changing its flavor is
Or, using SI units and the convention introduced above
This formula is often appropriate for discussing the transition ν_{μ} ↔ ν_{τ} in atmospheric mixing, since the electron neutrino plays almost no role in this case. It is also appropriate for the solar case of ν_{e} ↔ ν_{x}, where ν_{x} is a superposition of ν_{μ} and ν_{τ}. These approximations are possible because the mixing angle θ_{13} is very small and because two of the mass states are very close in mass compared to the third.
Classical analogue of neutrino oscillation
The basic physics behind neutrino oscillation can be found in any system of coupled harmonic oscillators. A simple example is a system of two pendulums connected by a weak spring (a spring with a small spring constant). The first pendulum is set in motion by the experimenter while the second begins at rest. Over time, the second pendulum begins to swing under the influence of the spring, while the first pendulum's amplitude decreases as it loses energy to the second. Eventually all of the system's energy is transferred to the second pendulum and the first is at rest. The process then reverses. The energy oscillates between the two pendulums repeatedly until it is lost to friction.
The behavior of this system can be understood by looking at its normal modes of oscillation. If the two pendulums are identical then one normal mode consists of both pendulums swinging in the same direction with a constant distance between them, while the other consists of the pendulums swinging in opposite (mirror image) directions. These normal modes have (slightly) different frequencies because the second involves the (weak) spring while the first does not. The initial state of the twopendulum system is a combination of both normal modes. Over time, these normal modes drift out of phase, and this is seen as a transfer of motion from the first pendulum to the second.
When the pendulums are not identical the analysis is slightly more complicated. In the smallangle approximation, the potential energy of a single pendulum system is , where g is the standard gravity, L is the length of the pendulum, m is the mass of the pendulum bob, and x is the horizontal displacement of the pendulum bob. As an isolated system the pendulum is a harmonic oscillator with a frequency of . The potential energy of a spring is where k is the spring constant and x is the displacement. With a mass attached it oscillates with a period of . With two pendulums (labeled a and b) of equal mass but possibly unequal lengths and connected by a spring, the total potential energy is
This is a quadratic form in x_{a} and x_{b}, which can also be written as a matrix product:
The 2×2 matrix is real symmetric and so (by the spectral theorem) it is "orthogonally diagonalizable". That is, there is an angle θ such that if we define
then
where λ_{1} and λ_{2} are the eigenvalues of the matrix. The variables x_{1} and x_{2} describe normal modes which oscillate with frequencies of and . When the two pendulums are identical (L_{a} = L_{b}), θ is 45°.
The description of the system in terms of the two pendulums (a and b) is analogous to the flavor basis of neutrinos. These are the parameters that are most easily produced and detected (in the case of neutrinos, by weak interactions involving the W boson). The description in terms of normal modes is analogous to the mass basis of neutrinos. These modes do not interact with each other when the system is free of outside influence. The angle θ is analogous to the Cabibbo angle (though that angle applies to quarks rather than neutrinos).
When the number of oscillators (particles) is increased to three, the orthogonal matrix can no longer be described by a single angle; instead, three are required (Euler angles). Furthermore, in the quantum case, the matrices may be complex. This requires the introduction of complex phases in addition to the rotation angles, which are associated with CP violation but do not influence the observable effects of neutrino oscillation.
Theory, graphically
Two neutrino probabilities in vacuum
In the approximation where only two neutrinos participate in the oscillation, the probability of oscillation follows a simple pattern:
The blue curve shows the probability of the original neutrino retaining its identity. The red curve shows the probability of conversion to the other neutrino. The maximum probability of conversion is equal to sin^{2}2θ. The frequency of the oscillation is controlled by Δm^{2}.
Three neutrino probabilities
If three neutrinos are considered, the probability for each neutrino to appear is somewhat complex. Here are shown the probabilities for each initial flavor, with one plot showing a long range to display the slow "solar" oscillation and the other zoomed in to display the fast "atmospheric" oscillation. The oscillation parameters used here are consistent with current measurements, but since some parameters are still quite uncertain, these graphs are only qualitatively correct in some aspects. These values were used:
 sin^{2}2θ_{13} = 0.10 (If it turns out to be much smaller or zero, the small wiggles shown here will be much smaller or nonexistent, respectively.)
 sin^{2}2θ_{23} = 0.97 (It may turn out to be exactly one.)
 sin^{2}2θ_{12} = 0.861.
 δ = 0 (If it is actually large, these probabilities will be somewhat distorted and different for neutrinos and antineutrinos.)
 Δm2
12 = 7.59×10^{−5} eV^{2}.  Δm2
32 ≈ Δm2
13 = 2.32×10^{−3} eV^{2}.  Normal mass hierarchy.
Observed values of oscillation parameters
 sin^{2}(θ_{13}) < 0.032 at 95% confidence level (θ_{13} < 10.3°)^{[8]}
 tan^{2}(θ_{12}) = 0.45+0.09
−0.07. This corresponds to θ_{12} ≡ θ_{sol} = 33.9+2.4
−2.2° ("sol" stands for solar)^{[10]}  sin^{2}(2θ_{23}) = 1+0
−0.1, corresponding to θ_{23} ≡ θ_{atm} = 45±7° ("atm" stands for atmospheric)^{[11]}  Δm2
21 ≡ Δm2
sol = 8.0+0.6
−0.4×10^{−5} eV^{2}^{[10]}  Δm2
31 ≈ Δm2
32 ≡ Δm2
atm = 2.43+0.13
−0.13×10^{−3} eV^{2}^{[12]}  δ, α_{1}, α_{2}, and the sign of Δm2
32 are currently unknown
Solar neutrino experiments combined with KamLAND have measured the socalled solar parameters Δm2
sol and sin^{2}θ_{sol}. Atmospheric neutrino experiments such as SuperKamiokande together with the K2K and MINOS long baseline accelerator neutrino experiment have determined the socalled atmospheric parameters Δm2
atm and sin^{2}θ_{atm}.For atmospheric neutrinos (where the relevant difference of masses is about Δm^{2} = 2.4×10^{−3} eV^{2} and the typical energies are ~1 GeV), oscillations become visible for neutrinos traveling several hundred km, which means neutrinos that reach the detector from below the horizon.
From atmospheric and solar neutrino oscillation experiments, it is known that two mixing angles of the MNS matrix are large and the third is smaller. This is in sharp contrast to the CKM matrix in which all three angles are small and hierarchically decreasing. Nothing is known about the CPviolating phase of the MNS matrix.
If the neutrino mass proves to be of Majorana type (making the neutrino its own antiparticle), it is possible that the MNS matrix has more than one phase.
Since experiments observing neutrino oscillation measure the squared mass difference and not absolute mass, one can claim that the lightest neutrino mass is exactly zero, without contradicting observations. This is however regarded as unlikely by theorists.
Origins of neutrino mass
The question of how neutrino masses arise has not been answered conclusively. In the Standard Model of particle physics, fermions only have mass because of interactions with the Higgs field (see Higgs boson). These interactions involve both left and righthanded versions of the fermion (see chirality). However, only lefthanded neutrinos have been observed so far.
Neutrinos may have another source of mass through the Majorana mass term. This type of mass applies for electricallyneutral particles since otherwise it would allow particles to turn into antiparticles, which would violate conservation of electric charge.
The smallest modification to the Standard Model, which only has lefthanded neutrinos, is to allow these lefthanded neutrinos to have Majorana masses. The problem with this is that the neutrino masses are implausibly smaller than the rest of the known particles (at least 500,000 times smaller than the mass of an electron), which, while it does not invalidate the theory, is not very satisfactory.
The next simplest addition would be to add righthanded neutrinos into the Standard Model, which interact with the lefthanded neutrinos and the Higgs field in an analogous way to the rest of the fermions. These new neutrinos would interact with the other fermions solely in this way, so are not phenomenologically excluded. The problem of the disparity of the mass scales remains.
Seesaw mechanism
Main article: Seesaw mechanismThe most popular conjectured solution currently is the seesaw mechanism, where righthanded neutrinos with very large Majorana masses are added. If the righthanded neutrinos are very heavy, they induce a very small mass for the lefthanded neutrinos, which is proportional to the inverse of the heavy mass.
If it is assumed that the neutrinos interact with the Higgs field with approximately the same strengths as the charged fermions do, the heavy mass should be close to the GUT scale. Note that, in the Standard Model there is just one fundamental mass scale (which can be taken as the scale of SU(2)_{L} × U(1)_{Y} breaking) and all masses (such as the electron or the mass of the Z boson) have to originate from this one.
There are other varieties of seesaw and currently it is not clear which, if any, nature has chosen.^{[13]}
The apparently innocent addition of righthanded neutrinos has the effect of adding new mass scales, completely unrelated to the mass scale of the Standard Model. Thus, heavy righthanded neutrinos look to be the first real glimpse of physics beyond the Standard Model. It is interesting to note that righthanded neutrinos can help to explain the origin of matter through a mechanism known as leptogenesis.
Other sources
There are alternative ways to modify the standard model that are similar to the addition of heavy righthanded neutrinos (e.g., the addition of new scalars or fermions in triplet states) and other modifications that are less similar (e.g., neutrino masses from loop effects and/or from suppressed couplings). One example of the last type of models is provided by certain versions supersymmetric extensions of the standard model of fundamental interactions, where R parity is not a symmetry. There, the exchange of supersymmetric particles such as squarks and sleptons can break the lepton number and lead to neutrino masses. These interactions are normally excluded from theories as they come from a class of interactions that lead to unacceptably rapid proton decay if they are all included. These models have little predictive power and are not able to provide a cold dark matter candidate but they are considered interesting since they would be compatible with new observable signals in particle colliders.
See also
 MSW effect
 Majoron
 Neutral kaon mixing
Notes
 ^ More formally, the neutrinos are emitted in an entangled state with the other bodies in the decay or reaction, and the mixed state is properly described by a density matrix. However, for all practical situations, the other particles in the decay may be well localized in time and space (e.g. to within a nuclear distance), leaving their momentum with a large spread. When these partner states are projected out, the neutrino is left in a state that for all intents and purposes behaves as the simple superposition of mass states described here. See ^{[3]} for more information.
References
 ^ ^{a} ^{b} B. Pontecorvo (1957). "Mesonium and antimesonium". Zh. Eksp. Teor. Fiz. 33: 549–551. reproduced and translated in Sov. Phys. JETP 6: 429. 1957.
 ^ http://press.web.cern.ch/press/PressReleases/Releases2010/PR08.10E.html
 ^ Andrew G. Cohen, Sheldon L. Glashow, and Zoltan Ligeti (2009). "Disentangling neutrino oscillations". Physics Letters B 678: 191. Bibcode 2009PhLB..678..191C. doi:10.1016/j.physletb.2009.06.020.
 ^ Z. Maki, M. Nakagawa, and S. Sakata (1962). "Remarks on the Unified Model of Elementary Particles". Progress of Theoretical Physics 28: 870. Bibcode 1962PThPh..28..870M. doi:10.1143/PTP.28.870.
 ^ B. Pontecorvo (1967). "Neutrino Experiments and the Problem of Conservation of Leptonic Charge". Zh. Eksp. Teor. Fiz. 53: 1717. reproduced and translated in Sov. Phys. JETP 26: 984. 1968. Bibcode 1968JETP...26..984P.
 ^ Raymond Davis Jr., Don S. Harmer, and Kenneth C. Hoffman (1968). "Search for Neutrinos from the Sun". Physical Review Letters 20: 1205. Bibcode 1968PhRvL..20.1205D. doi:10.1103/PhysRevLett.20.1205.
 ^ V. Gribov and B. Pontecorvo (1969). "Neutrino astronomy and lepton charge". Physics Letters B 28: 493. Bibcode 1969PhLB...28..493G. doi:10.1016/03702693(69)905255.
 ^ ^{a} ^{b} S. Eidelman et al. (2004). "Particle Data Group  The Review of Particle Physics". Physics Letters B 592 (1). doi:10.1016/j.physletb.2004.06.001. http://pdg.lbl.gov. Chapter 15: Neutrino mass, mixing, and flavor change. Revised September 2005.
 ^ Minako Honda; Yee Kao; Naotoshi Okamura; Tatsu Takeuchi (2006). "A Simple Parameterization of Matter Effects on Neutrino Oscillations". arXiv:hepph/0602115 [hepph].
 ^ ^{a} ^{b} B. Aharmim et al. (2005). "Electron energy spectra, fluxes, and daynight asymmetries of ^8B solar neutrinos from measurements with NaCl dissolved in the heavywater detector at the Sudbury Neutrino Observatory". Physical Review C 72 (5): 055502. arXiv:nuclex/0502021. Bibcode 2005PhRvC..72e5502A. doi:10.1103/PhysRevC.72.055502.
 ^ Y. Ashie at al. (2004). "Evidence for an oscillatory signature in atmospheric neutrino oscillation". Physical Review Letters 93 (10): 101801. arXiv:hepex/0404034. Bibcode 2004PhRvL..93j1801A. doi:10.1103/PhysRevLett.93.101801.
 ^ P. Adamson et al. (2008). "Measurement of Neutrino Oscillations with the MINOS Detectors in the NuMI Beam". Physical Review Letters 101 (13): 131802. Bibcode 2008PhRvL.101m1802A. doi:10.1103/PhysRevLett.101.131802. PMID 18851439.
 ^ J. W. F. Valle (2006). "Neutrino physics overview". Journal of Physics: Conference Series 56 (1): 473. arXiv:hepph/0608101. doi:10.1088/17426596/53/1/031.
Further reading
 Maltoni; Schwetz; Tortola; Valle (2004). "Status of global fits to neutrino oscillations". New Journal of Physics 6: 122–122. arXiv:hepph/0405172. Bibcode 2004NJPh....6..122M. doi:10.1088/13672630/6/1/122.
 GonzalezGarcia; Nir (2002). "Neutrino Masses and Mixing: Evidence and Implications". Reviews of Modern Physics 75 (2): 345–402. arXiv:hepph/0202058. Bibcode 2003RvMP...75..345G. doi:10.1103/RevModPhys.75.345.
External links
 Maury Goodman, "The Neutrino Oscillation Industry" (2006). (Provides links to many other neutrino oscillation websites.)
 Review Articles on arxiv.org
Categories: Leptons
 Standard Model
 Electroweak theory
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