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. More specifically, 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 nonzero neutrino mass.

1 Evidence for neutrino oscillations
2 The effect of neutrino masses
3 Origins of neutrino mass
4 Neutrino parameters and their values
5 Neutrino experiments
- 5.1 Measuring Θ13
6 See also
7 References

Evidence for neutrino oscillations

The discrepancy between the amount of electron neutrinos predicted to reach Earth from the Sun by models of Solar fusion and the actual amount detected (see solar neutrino problem) was the initial experimental evidence for neutrino oscillations. Further evidence was also provided by experiments measuring the neutrino flux from the upper atmosphere (where they are produced by cosmic rays), nuclear reactors and particle accelerators.

The effect of neutrino masses

It is widely accepted in the particle physics community that the oscillations are due to the different flavors of neutrino having different masses. If a neutrino is produced as a flavor eigenstate, and flavor eigenstates are not mass eigenstates, then we should write the neutrino wavefunction as a superposition of various mass eigenstates:

$\left| \nu_{a} \right\rangle = \sum_{i} U_{ai}^{*} \left| \nu_{i} \right\rangle\,$

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Here, $\left| \nu_{a} \right\rangle$ is a neutrino with given flavor (a=electron, muon or tau), $U a i$ are the coefficients of the neutrino mixing matrix, and $*$ means complex conjugate (for antineutrinos, the complex conjugate should be dropped). Since $\left| \nu_{i} \right\rangle$ are mass eigenstates (i.e. obey Einstein's relativistic energy-momentum equation) during their propagation they acquire the characteristic phase of plane wave solutions:

$\exp( -2 \pi i ( E t - \vec{p} \cdot \vec{x}) / h ).\,$

where we consider a neutrino component with a definite 3-dimensional momentum. The key remark is that the energy depends on the mass:

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

Note that, 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). Thus, if the mass eigenstates have different masses, then they will have different energies (due to the last term in previous equation) and thus different frequencies (as the frequency is the coefficient of time in the plane-wave function). If they have different frequencies, then they will interfere in a manner that will produce different ratios of mass eigenfunctions in the superposition, which will correspond to the wavefunction being a superposition of flavor eigenstates as well, which in turn will produce the behavior described above. In short, a neutrino created with a given flavor can change its flavor during its propagation. The phase that is responsible of oscillation is often written as:

$2 \pi \Delta m^2\, c^3/(2 h |\vec{p}|)\times t\approx 2.54 (\Delta m^2/eV^2) (L/km) (GeV/E) \,$

where $L$ is the distance between the source and the detector. This explains why for atmospheric neutrinos (where the relevant difference of masses is about $\Delta m^2 =2.5\times 10^{-3} eV^2$ and the typical energies are $E\approx 1\, GeV$ ) oscillations become visible for neutrinos travelling several hundred of km, which means neutrinos that reach the detector from below the horizon. In fact the above term gets different from zero as soon as these assumptions are satisfied.

Origins of neutrino mass

The question of how these masses arise has not been answered conclusively. In the Standard Model of particle physics, fermions ("matter particles") 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)). Neutrinos are special, since as electrically-neutral particles they may have another source of mass, Majorana mass (which cannot work for electrically-charged particles since it would allow particles to turn into anti-particles which would violate conservation of electric charge). Neutrinos are also special because only left-handed neutrinos have been observed.

Physicists like to modify successful theories (like the Standard Model) as little as possible, so the smallest modification to the Standard Model, which only has left-handed neutrinos, is to allow these left-handed neutrinos 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.

The most popular solution currently is the "see-saw" model, 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 (say, 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 with 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 recall that right handed neutrinos can help to explain the origin of matter through a mechanism known as leptogenesis.

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.

Neutrino parameters and their values

$\sin^2(2\Theta_{13}) < 0.2^{}_{}$ (at 90 % Confidence Level (C.L.), $\Delta m^2_{atm} = 2.0\cdot 10^{-3} eV^2$ ) Limit from the Chooz Experiment. This corresponds to $\Theta_{13} < 13^\circ$ .
$\sin^2(2\Theta_{12}) = 0.8^{+0.2}_{-0.2}$ . This corresponds to $\Theta_{12}=\Theta_{sol}=33\pm 3^\circ$ ('sol' stands for solar)
$\sin^2(2\Theta_{23}) = 1^{+0}_{-0.1}$ , corresponding to $\Theta_{23}=\Theta_{atm}=45\pm 7^\circ$ ('atm' for atmospheric)
$\Delta m^2_{21}=\Delta m^2_{sol}= 7.0^{+2}_{-3}\cdot 10^{-5} eV^2$
$\Delta m^2_{31}=\Delta m^2_{atm}= 2.4^{+0.6}_{-0.5}\cdot 10^{-3} eV^2$
$\alpha=\frac{\Delta m^2_{sol}}{\Delta m^2_{atm}}$ . $α < 1$ and survival probabilities (e.g. $P(\bar\nu_e\rightarrow\bar\nu_e)$ ) are often expanded in $α$ .

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

It would be nice, if these values could be corrected/ updated by real experts from the Neutrino community, where necessary.

Here you can find all Neutrino data in one plot.

Neutrino oscillation

Contents

Evidence for neutrino oscillations

The effect of neutrino masses

Origins of neutrino mass

Neutrino parameters and their values

Neutrino experiments

Measuring $Θ 13$

See also

Neutrino oscillation

Contents

Evidence for neutrino oscillations

The effect of neutrino masses

Origins of neutrino mass

Neutrino parameters and their values

Neutrino experiments

Measuring Θ13

See also

Measuring $Θ 13$