Neutrino oscillation

Neutrino oscillation is the observed periodic conversion of neutrinos from one lepton flavor to another and back. This allows a neutrino that z. B. was generated as an electron neutrino, also appear as a muon or tau neutrino and therefore, depending on the type of detector, escape detection .

The neutrino oscillation is explained as an interference effect between different neutrino components that propagate with different phase velocities . The probabilities of encountering this or that neutrino type vary sinusoidally with the path covered by the neutrino; Periods and amplitudes of the variations depend on the neutrino energy and the medium of propagation (see also MSW effect ).

The neutrino oscillation was investigated as a theoretical possibility by Bruno Pontecorvo in 1957 - if neutrinos were not massless, as was assumed at the time. A first experimental indication of neutrino oscillation was the deficit of low-energy solar neutrinos, which was observed with the Homestake experiment from the late 1960s . This was confirmed with the Kamiokande II experiment from 1987, a Cherenkov detector that was also able to determine the direction of origin. As a result, numerous other neutrino experiments were carried out, with higher-energy neutrinos from the sun, from cosmic radiation in the earth's atmosphere, from nuclear reactors and from accelerators, in order to differentiate between three alternative parameter ranges of the model and numerous alternative models, some with still massless neutrinos. It was only with antineutrinos in the KamLAND experiment, which has been running since 2002 , that it was possible to show that the original model is the right one. The possible parameter range of masses and mixing angles in a vacuum could be limited.

The observed upper mass limits are small enough that the standard model of elementary particle physics remains valid for practical reaction rate prediction. However, the mathematical description of the neutrino masses in the standard model is a theoretical problem because more particles or interactions are required than in the original model. The observed neutrino oscillation results in the non-maintenance of the lepton family number .

Solar neutrino deficit

The first experimental evidence of neutrino oscillations was obtained in the field of solar neutrinos. Electron neutrinos are generated in large numbers in nuclear fusion processes inside the sun . In the 1960s, Raymond Davis Jr. began studying solar neutrino flux with a detector in the Homestake mine (chlorine detector). The measured flux of the electron neutrinos, however, only corresponded to less than half of the flux that would be expected based on the luminosity of the sun.

The luminosity of the sun can be measured, from which the expected neutrino flux can be calculated using the measured properties of the atomic nuclei using complex solar models. Some of these models are referred to as the Standard Sun Models (SSM) because the scientific community has agreed on them as reference models after a long discussion . Provided that these SSMs describe the sun correctly, the result of the Homestake experiment could be interpreted as a "disappearance" of the neutrinos.

Today, based on a large number of experiments, the neutrino deficit measured by Davis is explained not by inaccuracies in the solar model, but by the particle properties of the neutrinos: on their way from the center of the sun to the neutrino detectors of our earth, the electron neutrinos "oscillate" into others Types of neutrinos (muon and tau neutrinos).

Davis and Masatoshi Koshiba (detection of cosmic neutrinos from the 1987A supernova in the Kamiokande detector) received the Nobel Prize in Physics for the Homestake experiment in 2002 .

Theoretical basis

Two assumptions are needed:

• need for a different neutrino masses possess
• on the other hand, the associated mass eigenstates of the neutrinos should be twisted compared to the states adapted to the weak interaction. The overlay is called the Pontecorvo-Maki-Nakagawa-Sakata-Mixture - sometimes just referred to as the MNS-Mixture, s. Subsequent chapter - analogous to the CKM mix in the quark sector.

This should be explained for the case of a 2- flavor oscillation of highly relativistic neutrinos , where the interaction states are ( with 3-flavor). ${\displaystyle \nu _{\alpha }\rightarrow \nu _{\beta }}$ ${\displaystyle \nu _{\alpha },\nu _{\beta }}$ ${\displaystyle \nu _{e},\nu _{\mu },\nu _{\tau }}$

The mixture is then characterized by a parameter, the mixing angle : ${\displaystyle \Theta _{m}}$

${\displaystyle \left({\begin{array}{c}\nu _{\alpha }\\\nu _{\beta }\end{array}}\right)=\left({\begin{array}{cc}\;\;\;\cos \Theta _{m}&\sin \Theta _{m}\\-\sin \Theta _{m}&\cos \Theta _{m}\end{array}}\right)\left({\begin{array}{c}\nu _{1}\\\nu _{2}\end{array}}\right),}$

where and are the mass eigen-states. In contrast to the interaction states, these can not be observed due to the small size of the masses or the effects caused by them. ${\displaystyle \nu _{1}}$ ${\displaystyle \nu _{2}}$

If one considers the neutrino mass eigenstates as a plane wave , then the following applies: ${\displaystyle \nu _{j}}$

${\displaystyle \left|\nu _{j}(t)\right\rangle =\left|\nu _{j}(0)\right\rangle e^{-i(Et-p_{j}x)/\hbar }}$.

For highly relativistic neutrinos with the approximation applies for the momentum : ${\displaystyle m_{\nu }c^{2}\ll E_{\nu }}$ ${\displaystyle p}$

${\displaystyle p={\frac {1}{c}}{\sqrt {E^{2}-m^{2}c^{4}}}\approx {\frac {1}{c}}\left(E-{\frac {m^{2}c^{4}}{2E}}\right)}$.

The time may be because of the flight path be expressed . The plane wave is thus described by: ${\displaystyle t}$ ${\displaystyle v\approx c}$ ${\displaystyle x=L}$ ${\displaystyle t={L}/{c}}$

${\displaystyle \left|\nu _{j}(L)\right\rangle =\left|\nu _{j}(0)\right\rangle e^{-i{\frac {m_{j}^{2}c^{4}}{2E}}{\frac {L}{\hbar c}}}}$.

For the temporal development of the interaction states and thus results from the superposition of two slightly different plane waves: ${\displaystyle \nu _{\alpha }}$ ${\displaystyle \nu _{\beta }}$

${\displaystyle {\begin{array}{lcrcr}\left|\nu _{\alpha }(L)\right\rangle &=&\cos \Theta _{m}\left|\nu _{1}(0)\right\rangle e^{-i{\frac {m_{1}^{2}c^{4}}{2E}}{\frac {L}{\hbar c}}}&+&\sin \Theta _{m}\left|\nu _{2}(0)\right\rangle e^{-i{\frac {m_{2}^{2}c^{4}}{2E}}{\frac {L}{\hbar c}}}\\\left|\nu _{\beta }(L)\right\rangle &=&-\sin \Theta _{m}\left|\nu _{1}(0)\right\rangle e^{-i{\frac {m_{1}^{2}c^{4}}{2E}}{\frac {L}{\hbar c}}}&+&\cos \Theta _{m}\left|\nu _{2}(0)\right\rangle e^{-i{\frac {m_{2}^{2}c^{4}}{2E}}{\frac {L}{\hbar c}}}\end{array}}}$.

If the two mass eigenstates after a finite route no longer live the same, so it is possible to find the other condition in an originally generated interaction state posts . Then the following applies to the oscillation probability: ${\displaystyle \langle \nu _{\beta }(0)|\nu _{\alpha }(L)\rangle \neq 0}$

${\displaystyle P(\nu _{\alpha }\rightarrow \nu _{\beta })=\left|\langle \nu _{\beta }(0)|\nu _{\alpha }(L)\rangle \right|^{2}\approxeq \sin ^{2}\left({\frac {\Delta m^{2}c^{4}}{4E}}{\frac {L}{\hbar c}}\right)\cdot \sin ^{2}\left(2\Theta _{m}\right)}$

with the difference of the mass squares of the flavors . ${\displaystyle \Delta m^{2}}$

The MSW effect occurs with neutrino oscillations in matter (only when the density changes) (named after Stanislaw Michejew , Alexei Jurjewitsch Smirnow and Lincoln Wolfenstein ). This causes a resonant amplification of the oscillation for certain electron densities and neutrino mass differences in matter .

In the standard model , neutrinos are massless and only appear as particles with negative chirality (left-handed particles). With the observation of neutrino oscillations, these assumptions are no longer tenable; these oscillations therefore offer a first insight into physics beyond the standard model. The changes to the standard model necessary for neutrinos with mass include e.g. B .:

• the appearance of right-handed neutrinos; they are not subject to electroweak or strong interaction , i.e. H. they are sterile .
• the appearance of Majorana neutrinos; they are their own antiparticles , so that the preservation of the lepton number is no longer guaranteed.

PMNS matrix

( Pontecorvo-Maki-Nakagawa-Sakata matrix , formerly MNS matrix , without Pontecorvo , and also called neutrino mixing matrix )

The solar and atmospheric neutrino experiments have shown that the neutrino oscillations result from a deviation between the flavor and mass eigenstates of the neutrinos. The relationship between these eigenstates is given by

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

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

in which

• ${\displaystyle \left|\nu _{\alpha }\right\rangle }$denotes a neutrino with a certain flavor α. α = e (electron), μ (muon) or τ (tauon).
• ${\displaystyle \left|\nu _{i}\right\rangle }$is a neutrino with a certain mass, indexed with = 1, 2, 3. ${\displaystyle i}$
• ${\displaystyle {}^{*}}$means the complex conjugation (for antineutrinos this must be omitted from the second equation and added to the first equation).

Inside is the PMNS matrix. It is the analogue of the CKM matrix for quarks and the mixing matrix of the electroweak interaction parameterized by the Weinberg angle . If this matrix were the identity matrix , then the flavor eigen-states would be the same as the mass eigen-states. However, the experiments mentioned show that this is not the case. ${\displaystyle U_{\alpha i}}$

If the usual three neutrino theory is consistent, then it has to be a 3 × 3 matrix, with only two different neutrinos (i.e. two flavors) it would be a 2 × 2 matrix, with four neutrinos a 4 × 4 -Matrix. In the case of three flavors it is given by:

${\displaystyle {\begin{aligned}\mathbf {U} &={\begin{bmatrix}U_{e1}&U_{e2}&U_{e3}\\U_{\mu 1}&U_{\mu 2}&U_{\mu 3}\\U_{\tau 1}&U_{\tau 2}&U_{\tau 3}\end{bmatrix}}\\\\&={\begin{bmatrix}1&0&0\\0&c_{23}&s_{23}\\0&-s_{23}&c_{23}\end{bmatrix}}{\begin{bmatrix}c_{13}&0&s_{13}e^{-i\delta }\\0&1&0\\-s_{13}e^{i\delta }&0&c_{13}\end{bmatrix}}{\begin{bmatrix}c_{12}&s_{12}&0\\-s_{12}&c_{12}&0\\0&0&1\end{bmatrix}}{\begin{bmatrix}e^{i\alpha _{1}/2}&0&0\\0&e^{i\alpha _{2}/2}&0\\0&0&1\end{bmatrix}}\\\\&={\begin{bmatrix}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta }\\-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta }&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta }&s_{23}c_{13}\\s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta }&-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta }&c_{23}c_{13}\end{bmatrix}}{\begin{bmatrix}e^{i\alpha _{1}/2}&0&0\\0&e^{i\alpha _{2}/2}&0\\0&0&1\end{bmatrix}}\\\\\end{aligned}}}$

where c _ij  = cos θ _ij and s _ij  = sin θ _ij . The phase factors α ₁ and α ₂ are only different from zero if the neutrinos are so-called Majorana particles (this question is still undecided) - but this is relatively insignificant for the neutrino oscillation. In the case of neutrino-free double beta decay , these factors only affect the rate. The phase factor δ is only different from zero if the neutrino oscillation violates the CP symmetry . This is expected, but has not yet been observed experimentally. If the experiment should show that this 3 × 3 matrix should not be unitary , then sterile neutrinos (English: sterile neutrino ) or other new physics beyond the standard model would be required (the same applies to the CKM matrix ).

The flavor mixture of the mass eigenstates depends on the medium. Current values ​​in a vacuum: The mass differences in the neutrino mass spectrum are given by

${\displaystyle \delta m^{2}:=m_{2}^{2}-m_{1}^{2}=7{,}54\cdot 10^{-5}\,\mathrm {eV} ^{2}>0\ ,\quad \Delta m^{2}:=m_{3}^{2}-{\frac {m_{1}^{2}+m_{2}^{2}}{2}}={\begin{cases}\;\;\;2{,}43\cdot 10^{-3}\,\mathrm {eV} ^{2}&(\mathrm {NH} )\\-2{,}42\cdot 10^{-3}\,\mathrm {eV} ^{2}&(\mathrm {IH} )\end{cases}},}$

where NH describes the normal hierarchy with and IH describes the inverse hierarchy with . The angles are as follows: ${\displaystyle \Delta m^{2}>0}$ ${\displaystyle \Delta m^{2}<0}$

${\displaystyle s_{12}^{2}=0{,}307\,,\;s_{23}^{2}={\begin{cases}0{,}386&(\mathrm {NH} )\\0{,}392&(\mathrm {IH} )\end{cases}}\,,\ s_{13}^{2}={\begin{cases}0{,}0241&(\mathrm {NH} )\\0{,}0244&(\mathrm {IH} )\end{cases}}\,,\ \delta ={\begin{cases}1{,}08\pi &(\mathrm {NH} )\\1{,}09\pi &(\mathrm {IH} )\end{cases}}.}$

In addition, they are . This results in the following MNS matrices: ${\displaystyle \alpha _{i}=0}$

${\displaystyle {\begin{aligned}\mathbf {U} _{\mathrm {NH} }={\begin{pmatrix}0{,}822&0{,}547&-0{,}150+0{,}0381\mathrm {i} \\-0{,}356+0{,}0198\mathrm {i} &0{,}704+0{,}0131\mathrm {i} &0{,}614\\0{,}442+0{,}0248\mathrm {i} &-0{,}452+0{,}0166\mathrm {i} &0{,}774\end{pmatrix}}\\\\\mathbf {U} _{\mathrm {IH} }={\begin{pmatrix}0{,}822&0{,}547&-0{,}150+0{,}0429\mathrm {i} \\-0{,}354+0{,}0224\mathrm {i} &0{,}701+0{,}0149\mathrm {i} &0{,}618\\0{,}444+0{,}0278\mathrm {i} &-0{,}456+0{,}0186\mathrm {i} &0{,}770\end{pmatrix}}.\end{aligned}}}$

Experiments

techniques

• Water Cherenkov experiments: These detectors are suitable for real-time detection of neutrinos of higher energy (above 5 MeV). The Cherenkov radiation from the neutrino's reaction partners is detected with light sensors, so-called photomultipliers . This type includes e.g. For example, the Japanese Super Kamiokande detector with 50,000 tons of water with more than 10,000 photomultipliers and the Canadian SNO detector, which was filled with 1,000 tons of heavy water (D ₂ O). These two detectors were able to detect the higher-energy neutrinos from the sun and could show that neutrino oscillations take place.
• Scintillator experiments: Detectors filled with organic liquid scintillators are suitable for detecting neutrinos at low energies. As with the water Cherenkov detectors, information about the energy of the neutrino and the exact time of the neutrino reaction is obtained from the signal of the scintillation light. Large scintillation detectors are e.g. B. the Borexino detector, a solar neutrino experiment in Italy, or the KamLAND detector, a reactor neutrino experiment in Japan. In reactor experiments, the liquid scintillator is often additionally loaded with a metal such as gadolinium for more efficient neutrino detection. Examples include the Double Chooz (France), RENO (South Korea) and Daya Bay (China) experiments .
• Radiochemical experiments such as the Homestake experiment measure the flow of electron neutrinos over a longer period of time. Such experiments take advantage of the fact that the beta decay can be reversed through neutrino capture. For example, converts ⁷¹ Ga through capture of an electron neutrinos in ⁷¹ Ge of an electron emission to below. As in the former GALLEX experiment in Gran Sasso (later GNO), these individual atoms can then be chemically separated from the detector and detected by the decay.
• Noble gas experiments: Here the neutrino reaction is demonstrated in liquid noble gas at low temperatures. An example of this type of detector is the ICARUS detector, which, like GALLEX or Borexino, was operated in the LNGS underground laboratory in Italy. At ICARUS, argon is used as the noble gas . Such large mass liquid noble gas detectors with argon or xenon as the detector material are also suitable for searching for other weakly interacting particles, such as candidates for dark matter .
• Other approaches have been pursued , for example, in the OPERA (direct detection of the tau neutrino appearance) or the MINOS detector.

Solar neutrino experiments

As described above, the first experimental evidence for neutrino oscillations came from the field of solar neutrino research. After the results of the gallium experiments GALLEX (LNGS, Italy) and SAGE (Baksan, Russia), the solar electron-neutrino deficit could no longer be explained solely by adaptations to the solar model. At the latest with the results of the SNO (Sudbury Neutrino Observatory) experiment (Canada) it was possible to prove that the deficit is caused by the conversion of electron neutrinos into other types of neutrinos. The solar neutrino experiments are particularly sensitive to one of the three mixing angles (s ₁₂ ), as well as to the difference between the closer together mass eigenstates. One therefore speaks of the “solar mixing angle” or the “solar mixing parameters”.

Atmospheric neutrino experiments

Another natural source of neutrinos is our earth's atmosphere. The interaction of cosmic rays with the upper layers of our atmosphere creates showers of particles. a. also neutrinos. By comparing the flow of neutrinos that are generated directly in the atmosphere above the detector and those that are generated on the opposite side of the earth, the Superkamiokande experiment (Japan) was able to show that neutrinos oscillate. Atmospheric neutrino experiments are particularly sensitive to the mixing angle s ₂₃ and to the difference between the mass eigenstates that are further apart. One speaks here of the "atmospheric mixing parameters".

Reactor experiments

In addition to the natural sources of neutrinos, there are also sources made by humans. A strong source of neutrinos are e.g. B. Nuclear reactors. Neutrino oscillations have also been demonstrated in such experiments. The KamLAND experiment (Japan) contributed significantly to the more precise determination of the solar mixing parameters. The third, for a long time completely unknown, neutrino mixing angle s ₁₃ was detected in the years 2011 and 2012 by the reactor neutrino experiments Double Chooz (France), RENO (South Korea) and Daya Bay (China). This mixing angle is of fundamental importance for future experiments looking for CP violation in the leptonic sector.

Accelerator experiments

Neutrino oscillations have also been detected at particle accelerators on three continents. These accelerators generate neutrinos with high energy, which are detected in detectors at a distance of several 100 km from the accelerator. These experiments are well suited for the precision measurement of atmospheric mixing parameters.

A pioneering experiment in this area was the K2K experiment (Japan), in which neutrinos generated by the KEK were detected in the Super-Kamiokande detector. After the generation, some of the electron neutrinos were detected in the KEK's near-range detector and from this it was predicted how many neutrinos with or without oscillation should be measured in Kamioka (today Hida ) 250 km away . Only 70% of the electron neutrino events that were predicted without oscillation occurred there. In addition, a shift in the energy spectrum of the detected neutrinos was found, which is characteristic of neutrino oscillations. Recent experiments in this area include a. MINOS and T2K.

literature

• Bogdan Povh , Klaus Rith , Christoph Scholz, Frank Zetsche: Particles and Cores. 8th edition, Springer, Berlin 2009
• Jennifer A. Thomas: Neutrino oscillations - present status and future plans. World Scientific, Singapore 2008, ISBN 978-981-277-196-4 .

Web links

• The Nobel Prize in Physics 2002
• The Nobel Prize in Physics 2015
• Neutrino Mass, Mixing, and flavor change - Particle Data Group (PDG) (PDF file; 18.3 MB)
• OPERA detects its fifth tau neutrino

Individual evidence

1. ^ S. Eidelman et al .: Particle Data Group - The Review of Particle Physics . In: Physics Letters B . 592, No. 1, 2004. Chapter 15: Neutrino mass, mixing, and flavor change (PDF; 466 kB) . Revised September 2005.
2. ↑ Alexei Jurjewitsch Smirnow : Solar neutrinos: Oscillations or No-oscillations? September 2016, arxiv : 1609.02386 .
3. ↑ Fogli et al .: Global analysis of neutrino masses, mixings and phases: entering the era of leptonic CP violation searches. , 2012, arxiv : 1205.5254v3
4. ↑ MH Ahn, et al .: Measurement of Neutrino Oscillation by the K2K Experiment . In: Phys.Rev.D . 74, No. 072003, 2006. arxiv : hep-ex / 0606032 .