# Neutrino oscillations - Wikipedia

## Transcript Of Neutrino oscillations - Wikipedia

Neutrino oscillations - Wikipedia, the free encyclopedia

http://en.wikipedia.org/wiki/Neutrino_oscillations

Neutrino oscillations

From Wikipedia, the free encyclopedia

Neutrino oscillations are a quantum mechanical phenomenon predicted by Bruno Pontecorvo 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 non-zero mass, which is not part of the original Standard Model of particle physics.

A zero mass particle would have to travel at the speed of light. At the speed of light, time stands still so no change (and, therefore, no oscillation) is possible. Therefore if particles change, they must have mass.

Contents

1 Observations 1.1 Solar neutrino oscillation 1.2 Atmospheric neutrino oscillation 1.3 Reactor neutrino oscillations 1.4 Beam neutrino oscillations 1.5 Decay oscillations - GSI anomaly

2 Theory 2.1 Classical analogue of neutrino oscillation 2.2 Pontecorvo–Maki–Nakagawa–Sakata matrix 2.3 Propagation and interference 2.4 Two neutrino case

3 Theory, graphically 3.1 Two neutrino probabilities in vacuum 3.2 Three neutrino probabilities

4 Observed values of oscillation parameters 5 Origins of neutrino mass

5.1 Seesaw mechanism 5.2 Other sources

1 of 15

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http://en.wikipedia.org/wiki/Neutrino_oscillations

6 See also 7 Notes 8 References 9 External links

Observations

A great deal of evidence for neutrino oscillations has been collected from many sources, over a wide range of neutrino energies and with many different detector technologies.

Solar neutrino oscillation

The first experiment to detect the effects of neutrino oscillations was Ray Davis's Homestake Experiment in the late 1960s, in which he observed a deficit in the flux of solar neutrinos using a chlorine-based detector. This gave rise to the Solar neutrino problem. Many subsequent radiochemical and water Cherenkov detectors confirmed the deficit, but neutrino oscillations were 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 oscillations 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 high precision measurement of neutrino oscillations in an energy range of hundreds of MeV to a few TeV, and with a baseline of the diameter of the Earth.

Reactor neutrino oscillations

Many experiments have searched for oscillations of electron anti-neutrinos 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 oscillations

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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 Super-K 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[1] 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 θ13. The experiment is scheduled to begin in 2009 and uses the Super-K detector. NOνA is a similar effort. This detector will use the same beam as MINOS and will have a baseline of 810 km.

Decay oscillations - GSI anomaly

A study of the decay of radioactive hydrogen-like Praseodymium-140 and Promethium-142 at GSI nuclear physics laboratory in Darmstadt, Germany in 2008 revealed an oscillation superimposed on the otherwise exponential decay of the atomic nuclei. It has been suggested that this oscillation is related to neutrino oscillations[2][3]. However, the theoretical description of this effect in terms of neutrino oscillations has been a source of much debate and there is no clear consensus on whether it is viable or not[4]. Independent experiments are needed to assess if the effect is real and if the process can be used as a supplementary observation method of neutrino oscillations.

Theory

Classical analogue of neutrino oscillation

3 of 15

09.10.2010 г. 20:28

Neutrino oscillations - Wikipedia, the free encyclopedia

http://en.wikipedia.org/wiki/Neutrino_oscillations

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 two-pendulum 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.

Spring-coupled pendulums Time evolution of the pendulums Lower frequency normal mode Higher frequency normal mode

When the pendulums are not identical the analysis is slightly more complicated. In the small-angle 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

two pendulums (labeled a and b) of equal mass but possibly unequal lengths and connected by a spring, the total potential energy is

. With

This is a quadratic form in xa and xb, 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

4 of 15

09.10.2010 г. 20:28

Neutrino oscillations - Wikipedia, the free encyclopedia

then

http://en.wikipedia.org/wiki/Neutrino_oscillations

where λ1 and λ2 are the eigenvalues of the matrix. The variables x1 and x2 describe normal modes which oscillate with frequencies of

and

. When the two pendulums are identical (La = Lb), 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.

Pontecorvo–Maki–Nakagawa–Sakata matrix

Main article: Pontecorvo–Maki–Nakagawa–Sakata matrix

The idea of neutrino oscillations was put forward in 1957 by Bruno Pontecorvo, in analogy with a similar phenomenon observed in the neutral kaon system. The quantitative theory described below was developed by him in 1967. One year later the solar neutrino deficit was first observed, that was followed by the famous paper of Gribov and Pontecorvo published in 1969 titled "Neutrino astronomy and lepton charge".

Solar and atmospheric neutrino experiments have shown that neutrino oscillations are due to a mismatch between the flavor and mass eigenstates of neutrinos. The relationship between these eigenstates is given by

,

where

is a neutrino with definite flavor. α = e (electron), µ (muon) or τ (tauon). is a neutrino with definite mass. i = 1, 2, 3.

5 of 15

09.10.2010 г. 20:28

Neutrino oscillations - Wikipedia, the free encyclopedia

http://en.wikipedia.org/wiki/Neutrino_oscillations

The asterisk ( * ) represents a complex conjugate. For antineutrinos, the complex conjugate should be dropped from the first equation, and added to the second.

Uαi 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:[5]

where cij = cosθij and sij = sinθij. The phase factors α1 and α2 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 non-zero only if neutrino oscillation violates CP symmetry. This is expected, but not yet observed experimentally. If experiment shows this 3x3 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

6 of 15

09.10.2010 г. 20:28

Neutrino oscillations - Wikipedia, the free encyclopedia

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quantities are expressed in natural units

Ei is the energy of the mass-eigenstate i, t is the time from the start of the propagation,

is the 3-dimensional 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 1eV and their energies are at least 1MeV, so the Lorentz factor γ is greater than 106 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)[6]

7 of 15

09.10.2010 г. 20:28

Neutrino oscillations - Wikipedia, the free encyclopedia

http://en.wikipedia.org/wiki/Neutrino_oscillations

,

where 1.267 is unitless. In this form, it is convenient to plug in the oscillation parameters since: The mass differences, ∆m2, are known to be on the order of 1eV2 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 CP-violation (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 νµ ↔ ντ in atmospheric mixing, since the electron neutrino plays almost no role in this case. It is also appropriate for the solar case of νe ↔ νx, where νx is a superposition of νµ and ντ. These approximations are possible because the mixing angle θ13 is very small and because two of the mass states are very close in mass compared to the third.

Theory, graphically

8 of 15

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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 sin22θ. The frequency of the oscillation is controlled by ∆m2.

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:

sin22θ13 = 0.08. (If it turns out to be much smaller or zero, the small wiggles shown here will be much smaller or non-existent, respectively.) sin22θ23 = 0.95. (It may turn out to be exactly one.) sin22θ12 = 0.86.

9 of 15

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δ = 0. (If it is actually large, these probabilities will be somewhat distorted and different for neutrinos and antineutrinos.)

. .

Electron neutrino oscillations, long range. Here and in the following diagrams black means electron neutrino, blue means

muon neutrino and red means tau neutrino.

Electron neutrino oscillations, short range

10 of 15

09.10.2010 г. 20:28

http://en.wikipedia.org/wiki/Neutrino_oscillations

Neutrino oscillations

From Wikipedia, the free encyclopedia

Neutrino oscillations are a quantum mechanical phenomenon predicted by Bruno Pontecorvo 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 non-zero mass, which is not part of the original Standard Model of particle physics.

A zero mass particle would have to travel at the speed of light. At the speed of light, time stands still so no change (and, therefore, no oscillation) is possible. Therefore if particles change, they must have mass.

Contents

1 Observations 1.1 Solar neutrino oscillation 1.2 Atmospheric neutrino oscillation 1.3 Reactor neutrino oscillations 1.4 Beam neutrino oscillations 1.5 Decay oscillations - GSI anomaly

2 Theory 2.1 Classical analogue of neutrino oscillation 2.2 Pontecorvo–Maki–Nakagawa–Sakata matrix 2.3 Propagation and interference 2.4 Two neutrino case

3 Theory, graphically 3.1 Two neutrino probabilities in vacuum 3.2 Three neutrino probabilities

4 Observed values of oscillation parameters 5 Origins of neutrino mass

5.1 Seesaw mechanism 5.2 Other sources

1 of 15

09.10.2010 г. 20:28

Neutrino oscillations - Wikipedia, the free encyclopedia

http://en.wikipedia.org/wiki/Neutrino_oscillations

6 See also 7 Notes 8 References 9 External links

Observations

A great deal of evidence for neutrino oscillations has been collected from many sources, over a wide range of neutrino energies and with many different detector technologies.

Solar neutrino oscillation

The first experiment to detect the effects of neutrino oscillations was Ray Davis's Homestake Experiment in the late 1960s, in which he observed a deficit in the flux of solar neutrinos using a chlorine-based detector. This gave rise to the Solar neutrino problem. Many subsequent radiochemical and water Cherenkov detectors confirmed the deficit, but neutrino oscillations were 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 oscillations 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 high precision measurement of neutrino oscillations in an energy range of hundreds of MeV to a few TeV, and with a baseline of the diameter of the Earth.

Reactor neutrino oscillations

Many experiments have searched for oscillations of electron anti-neutrinos 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 oscillations

2 of 15

09.10.2010 г. 20:28

Neutrino oscillations - Wikipedia, the free encyclopedia

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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 Super-K 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[1] 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 θ13. The experiment is scheduled to begin in 2009 and uses the Super-K detector. NOνA is a similar effort. This detector will use the same beam as MINOS and will have a baseline of 810 km.

Decay oscillations - GSI anomaly

A study of the decay of radioactive hydrogen-like Praseodymium-140 and Promethium-142 at GSI nuclear physics laboratory in Darmstadt, Germany in 2008 revealed an oscillation superimposed on the otherwise exponential decay of the atomic nuclei. It has been suggested that this oscillation is related to neutrino oscillations[2][3]. However, the theoretical description of this effect in terms of neutrino oscillations has been a source of much debate and there is no clear consensus on whether it is viable or not[4]. Independent experiments are needed to assess if the effect is real and if the process can be used as a supplementary observation method of neutrino oscillations.

Theory

Classical analogue of neutrino oscillation

3 of 15

09.10.2010 г. 20:28

Neutrino oscillations - Wikipedia, the free encyclopedia

http://en.wikipedia.org/wiki/Neutrino_oscillations

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 two-pendulum 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.

Spring-coupled pendulums Time evolution of the pendulums Lower frequency normal mode Higher frequency normal mode

When the pendulums are not identical the analysis is slightly more complicated. In the small-angle 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

two pendulums (labeled a and b) of equal mass but possibly unequal lengths and connected by a spring, the total potential energy is

. With

This is a quadratic form in xa and xb, 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

4 of 15

09.10.2010 г. 20:28

Neutrino oscillations - Wikipedia, the free encyclopedia

then

http://en.wikipedia.org/wiki/Neutrino_oscillations

where λ1 and λ2 are the eigenvalues of the matrix. The variables x1 and x2 describe normal modes which oscillate with frequencies of

and

. When the two pendulums are identical (La = Lb), 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.

Pontecorvo–Maki–Nakagawa–Sakata matrix

Main article: Pontecorvo–Maki–Nakagawa–Sakata matrix

The idea of neutrino oscillations was put forward in 1957 by Bruno Pontecorvo, in analogy with a similar phenomenon observed in the neutral kaon system. The quantitative theory described below was developed by him in 1967. One year later the solar neutrino deficit was first observed, that was followed by the famous paper of Gribov and Pontecorvo published in 1969 titled "Neutrino astronomy and lepton charge".

Solar and atmospheric neutrino experiments have shown that neutrino oscillations are due to a mismatch between the flavor and mass eigenstates of neutrinos. The relationship between these eigenstates is given by

,

where

is a neutrino with definite flavor. α = e (electron), µ (muon) or τ (tauon). is a neutrino with definite mass. i = 1, 2, 3.

5 of 15

09.10.2010 г. 20:28

Neutrino oscillations - Wikipedia, the free encyclopedia

http://en.wikipedia.org/wiki/Neutrino_oscillations

The asterisk ( * ) represents a complex conjugate. For antineutrinos, the complex conjugate should be dropped from the first equation, and added to the second.

Uαi 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:[5]

where cij = cosθij and sij = sinθij. The phase factors α1 and α2 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 non-zero only if neutrino oscillation violates CP symmetry. This is expected, but not yet observed experimentally. If experiment shows this 3x3 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

6 of 15

09.10.2010 г. 20:28

Neutrino oscillations - Wikipedia, the free encyclopedia

http://en.wikipedia.org/wiki/Neutrino_oscillations

quantities are expressed in natural units

Ei is the energy of the mass-eigenstate i, t is the time from the start of the propagation,

is the 3-dimensional 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 1eV and their energies are at least 1MeV, so the Lorentz factor γ is greater than 106 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)[6]

7 of 15

09.10.2010 г. 20:28

Neutrino oscillations - Wikipedia, the free encyclopedia

http://en.wikipedia.org/wiki/Neutrino_oscillations

,

where 1.267 is unitless. In this form, it is convenient to plug in the oscillation parameters since: The mass differences, ∆m2, are known to be on the order of 1eV2 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 CP-violation (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 νµ ↔ ντ in atmospheric mixing, since the electron neutrino plays almost no role in this case. It is also appropriate for the solar case of νe ↔ νx, where νx is a superposition of νµ and ντ. These approximations are possible because the mixing angle θ13 is very small and because two of the mass states are very close in mass compared to the third.

Theory, graphically

8 of 15

09.10.2010 г. 20:28

Neutrino oscillations - Wikipedia, the free encyclopedia

http://en.wikipedia.org/wiki/Neutrino_oscillations

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 sin22θ. The frequency of the oscillation is controlled by ∆m2.

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:

sin22θ13 = 0.08. (If it turns out to be much smaller or zero, the small wiggles shown here will be much smaller or non-existent, respectively.) sin22θ23 = 0.95. (It may turn out to be exactly one.) sin22θ12 = 0.86.

9 of 15

09.10.2010 г. 20:28

Neutrino oscillations - Wikipedia, the free encyclopedia

http://en.wikipedia.org/wiki/Neutrino_oscillations

δ = 0. (If it is actually large, these probabilities will be somewhat distorted and different for neutrinos and antineutrinos.)

. .

Electron neutrino oscillations, long range. Here and in the following diagrams black means electron neutrino, blue means

muon neutrino and red means tau neutrino.

Electron neutrino oscillations, short range

10 of 15

09.10.2010 г. 20:28