Neutrino oscillation

Neutrino oscillation is a quantum mechanical phenomenon 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 as 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.

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 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 Cerenkov detectors confirmed the deficit, but neutrino oscillations weren't conclusively identified as the source of the deficit until the Sudbury Neutrino Observatory provided clear evidence of neutrino flavor change.

Solar neutrinos have energies below 20 MeV and travel an astronomical unit between the source and detector. 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. 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 radius of the Earth.

Reactor neutrino oscillations

Many experiments searched for oscillations of electron anti-neutrinos produced at nuclear reactors, and a high precision observation of reactor neutrino oscillation has been made by the KamLAND experiment. Neutrinos produced in nuclear reactors have energies similar to solar neutrinos (few MeV) and short baselines (tens of meters, or tens of km in the case of KamLAND).

Beam neutrino oscillations

Neutrinos 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 MINOS result has not yet been published in a peer reviewed journal but it is expected that their results will be published soon.

The controversial observation of beam neutrino oscillation at the LSND experiment is currently being tested by MiniBooNE, results from MiniBooNE are expected in the summer of 2006.

Theory

Maki-Nakagawa-Sakata matrix

It is generally accepted that neutrino oscillations are due to a mismatch between the flavor and mass eigenstates of neutrinos. The relationship between these eigenstates is given by

$\left|\nu _{\alpha }\right\rangle =\sum _{i}U_{\alpha i}^{*}\left|\nu _{i}\right\rangle \,$

$\left|\nu _{i}\right\rangle =\sum _{\alpha }U_{\alpha i}\left|\nu _{\alpha }\right\rangle$,

where

• $\left|\nu _{\alpha }\right\rangle$ is a neutrino with definite flavor. α = e (electron), $\mu$ (muon) or $\tau$ (tau).
• $\left|\nu _{i}\right\rangle$ is a neutrino with definite mass. i = 1, 2, 3.
• ${}^{*}$ represents a complex conjugate (for antineutrinos, the complex conjugate should be dropped from the first equation, and added to the second).

$U_{\alpha i}$ represents the Maki-Nakagawa-Sakata matrix (also called the "MNS matrix", "neutrino mixing matrix", or sometimes "PMNS matrix" to include Pontecorvo). It is the equivalent of the CKM matrix for 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 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:

$U_{\alpha i}={\begin{bmatrix}u_{e1}&u_{e2}&u_{e3}\\u_{\mu 1}&u_{\mu 2}&u_{\mu 3}\\u_{\tau 1}&u_{\tau 2}&u_{\tau 3}\end{bmatrix}}={\begin{bmatrix}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta }\\-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta }&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta }&s_{23}c_{13}\\s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta }&-c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta }&c_{23}s_{13}\end{bmatrix}}\times diag(e^{i\alpha _{1}/2},e^{i\alpha _{2}/2},1)$,

where $s_{12}=\sin \theta _{12}$, $c_{12}=\cos \theta _{12}$, etc. The phase factors α₁ and α₂ are non-zero only if neutrinos are Majorana particles and do not enter into oscillation phenomena regardless. They affect neutrinoless double beta decay. 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.

Mass differences

Since $\left|\nu _{i}\right\rangle$ are mass eigenstates (i.e. obey Einstein's relativistic energy-momentum equation), their propagation can be described by plane wave solutions of the form

$\exp(-i(Et-{\vec {p}}\cdot {\vec {x}})/\hbar )$,

where

• $E$ is the energy of the particle,
• $t$ is the time from the start of the propagation,
• ${\vec {p}}$ is the 3-dimensional momentum,
• ${\vec {x}}$ is the current position of the particle relative to its starting position
• $\hbar$ is the reduced Planck constant.

The energy depends on the mass $m$ by

$E={\sqrt {|{\vec {p}}\,|^{2}c^{2}+m^{2}c^{4}}}\approx |{\vec {p}}|c+m^{2}c^{3}/(2|{\vec {p}}|)$,

where $c$ is the speed of light.

In all practical applications, the last term — the interesting one — is small in comparison to the first one (that is, one considers the so-called ultrarelativistic limit).

If the mass eigenstates each have different masses, then they will have different energies due to the last term in previous equation, hence they will different frequencies (as the frequency is the coefficient of time in the plane-wave function). The different frequencies will interfere with each other, which will create different ratios of the mass eigenfunctions in the mixing matrix for each flavour eigenstate. This means that a neutrino created with a given flavor can change its flavor during its propagation.

The phase that is responsible for oscillation is the difference between two of the mass eigenstates, $\Delta m^{2}$, is often written as

${\frac {\Delta m^{2}\,c^{3}}{2\hbar |{\vec {p}}|}}\times t\approx 2.54\times {\frac {\Delta m^{2}}{{\rm {eV}}^{2}}}{\frac {L}{\rm {km}}}{\frac {\rm {GeV}}{E}}$,

where

• The mass difference $\Delta m^{2}$ between the two mass eigenstates is given in electronvolts,
• $L$ is the distance between the source and the detector in kilometers,
• The energy $E$ is given in GeV.

Observed values

• $\sin ^{2}(2\theta _{13})<0.2_{}^{}$, corresponding to $\Delta m_{\rm {atm}}^{2}=2.0\cdot 10^{-3}eV^{2}$ at 90% confidence level and $\theta _{13}<13^{\circ }$. .
• $\tan ^{2}(\theta _{12})=0.45_{-0.07}^{+0.09}$. This corresponds to $\theta _{12}=\theta _{\rm {sol}}={33.9^{\circ }}_{-2.2^{\circ }}^{+2.4^{\circ }}$ ("sol" stands for solar) .
• $\sin ^{2}(2\theta _{23})=1_{-0.1}^{+0}$, corresponding to $\theta _{23}=\theta _{\rm {atm}}=45\pm 7^{\circ }$ ("atm" for atmospheric)
• $\Delta m_{21}^{2}=\Delta m_{\rm {sol}}^{2}=8.0_{-0.4}^{+0.6}\cdot 10^{-5}{\rm {eV}}^{2}$ .

• $\Delta m_{31}^{2}=\Delta m_{\rm {atm}}^{2}=2.4_{-0.5}^{+0.6}\cdot 10^{-3}{\rm {eV}}^{2}$
• $\alpha ={\frac {\Delta m_{\rm {sol}}^{2}}{\Delta m_{\rm {atm}}^{2}}}$. $\alpha <1$ and survival probabilities (e.g. $P({\bar {\nu }}_{e}\rightarrow {\bar {\nu }}_{e})$ ) are often expanded in $\alpha$.
• $\delta =\mathrm {unknown}$

Solar neutrino experiments combined with KamLAND have measured the so-called solar parameters $\Delta m_{\rm {sol}}^{2}$ and $\sin ^{2}\theta _{\rm {sol}}$. Atmospheric neutrino experiments such as Super-Kamiokande together with the K2K first long baseline accelerator neutrino experiment have determined the so-called atmospheric parameters $\Delta m_{\rm {atm}}^{2}$ and $\sin ^{2}2\theta _{\rm {atm}}$. An additional experiment MINOS is expected to reduce the experimental errors significantly thereby increasing precision. (from the Double Chooz Letter of Intent)

For atmospheric neutrinos (where the relevant difference of masses is about $\Delta m^{2}=2.5\times 10^{-3}{\mbox{ eV}}^{2}$ and the typical energies are $E\approx 1\,{\mbox{ GeV}}$), oscillations become visible for neutrinos travelling 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 CP-violating 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.

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 right-handed versions of the fermion (see Chirality (physics)). However, only left-handed neutrinos have been observed so far.

Neutrinos may have another source of mass through the Majorana equation. This mechanism only applies for electrically-charged particles since otherwise it would allow particles to turn into anti-particles, which would violate conservation of electric charge.

The smallest modification to the Standard Model, which only has left-handed neutrinos, is to allow these left-handed 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 right-handed neutrinos into the Standard Model, which interact with the left-handed 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. Still, the problem of the disparity of the mass scales remains.

See-saw mechanism

Main article: Seesaw mechanism

The most popular solution currently is the seesaw mechanism, where right-handed neutrinos with very large Majorana masses are added. If the right-handed neutrinos are very heavy, they induce a very small mass for the left-handed 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 strength as electrons do (which is quite reasonable as neutrinos and electrons/muons/tau leptons are associated with each other in the same way as up and down quarks are associated with each other), 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 $SU(2)_{L}\times U(1)_{Y}$ breaking) and all masses (such as the electron or the mass of the Z boson) have to originate from this one.

The apparently innocent addition of right handed neutrinos has the effect of adding new mass scales, completely unrelated to the mass scale of the Standard Model. Thus, heavy right handed neutrinos look to be the first real glimpse of physics beyond the Standard Model. It is interesting to note that right handed neutrinos can help to explain the origin of matter through a mechanism known as leptogenesis.

Other sources

There are other ideas for the origin of neutrino mass, such as R-parity violating supersymmetry, which proposes that the masses for the neutrinos come from interactions with squarks and sleptons, rather than the Higgs field. However, 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), do not help to understand why neutrinos are so light and are not able to provide a cold dark matter candidate. Still, these theories have not been ruled out yet.

References

• M.C. Gonzalez-Garcia, Y. Nir, "A review of evidence of neutrino masses and the implications", Reviews of Modern Physics 75 (2003) p.345-402.
• Maury Goodman, "The Neutrino Oscillation Industry" (2005). (Provides links to many other neutrino oscillation websites).
• US DOE

See also

• MSW effect

External links

• All Neutrino data in one plot