Neutrino oscillation

The first experiment that detected 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 chlorine-based detector.[6] 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.[7]

Solar neutrinos have energies below 20 MeV. 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.[1]

Following the theories that were proposed in the 1970s suggesting unification of electromagnetic, weak, and strong forces, a few experiments on proton decay followed in the 1980s. Large detectors such as IMB, MACRO, and Kamiokande II have 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; the first experimental evidence for atmospheric neutrino oscillations was announced in 1998.[8]

Illustration of neutrino oscillations.

Many experiments have searched for oscillation of electron anti-neutrinos produced in nuclear reactors. No oscillations were found until a detector was installed at a distance 1–2 km. Such oscillations give the value of the parameter θ₁₃. Neutrinos produced in nuclear reactors have energies similar to solar neutrinos, of around a few MeV. The baselines of these experiments have ranged from tens of meters to over 100 km (parameter θ₁₂). Mikaelyan and Sinev proposed to use two identical detectors to cancel systematic uncertainties in reactor experiment to measure the parameter θ₁₃.[9]

In December 2011, the Double Chooz experiment found that θ₁₃ ≠ 0.[10] Then, in 2012, the Daya Bay experiment found that θ₁₃ ≠ 0 with a significance of 5.2 σ;[11] These results have since been confirmed by RENO.[12]

Neutrino beams produced at a particle accelerator offer the greatest control over the neutrinos being studied. Many experiments have taken place that study the same oscillations as in atmospheric neutrino oscillation using neutrinos with a few GeV of energy and several-hundred-km baselines. The MINOS, K2K, and Super-K experiments have all independently observed muon neutrino disappearance over such long baselines.[1]

Data from the LSND experiment appear to be in conflict with the oscillation parameters measured in other experiments. Results from the MiniBooNE appeared in Spring 2007 and contradicted the results from LSND, although they could support the existence of a fourth neutrino type, the sterile neutrino.[1]

In 2010, the INFN and CERN announced the observation of a tau particle in a muon neutrino beam in the OPERA detector located at Gran Sasso, 730 km away from the source in Geneva.[13]

T2K, using a neutrino beam directed through 295 km of earth and the Super-Kamiokande detector, measured a non-zero value for the parameter θ₁₃ in a neutrino beam.[14] NOνA, using the same beam as MINOS with a baseline of 810 km, is sensitive to the same.

The idea of neutrino oscillation was first put forward in 1957 by Bruno Pontecorvo, who proposed that neutrino–antineutrino transitions may occur in analogy with neutral kaon mixing.[2] Although such matter–antimatter oscillation had 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[18] and further elaborated by Pontecorvo in 1967.[3] One year later the solar neutrino deficit was first observed,[19] and that was followed by the famous article by Gribov and Pontecorvo published in 1969 titled "Neutrino astronomy and lepton charge".[20]

The concept of neutrino mixing is a natural outcome of gauge theories with massive neutrinos, and its structure can be characterized in general.[21] In its simplest form it is expressed as a unitary transformation relating the flavor and mass eigenbasis and can be written as

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

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

where

• ${\displaystyle \ \left|\nu _{\alpha }\right\rangle \ }$ is a neutrino with definite flavor α = e (electron), μ (muon) or τ (tauon),
• ${\displaystyle \ \left|\nu _{i}\right\rangle \ }$ is a neutrino with definite mass ${\displaystyle \ m_{i}\ ,}$ ${\displaystyle i=1,2,3,}$
• the asterisk (${\displaystyle ^{*}}$) represents a complex conjugate; for antineutrinos, the complex conjugate should be removed from the first equation and inserted into the second.

${\displaystyle U_{\alpha i}}$ 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[22]

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

where c_ij ≡ cos θ_ij , and s_ij ≡ sin θ_ij . The phase factors 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 has not yet been observed experimentally. If experiment shows this 3×3 matrix to be not unitary, a sterile neutrino or some other new physics is required.

Since ${\displaystyle \left|\,\nu _{j}\,\right\rangle }$ are mass eigenstates, their propagation can be described by plane wave solutions of the form

${\displaystyle \left|\,\nu _{j}(t)\,\right\rangle =e^{-i\,\left(\,E_{j}t\,-\,{\vec {p}}_{j}\cdot {\vec {x}}\,\right)}\left|\,\nu _{j}(0)\,\right\rangle ~,}$

where

• quantities are expressed in natural units ${\displaystyle (\,c=1,\hbar =1\,)~,}$ and ${\displaystyle ~i^{2}\,\equiv \,-1~,}$
• ${\displaystyle E_{j}}$ is the energy of the mass-eigenstate ${\displaystyle j}$,
• ${\displaystyle t}$ is the time from the start of the propagation,
• ${\displaystyle {\vec {p}}_{j}}$ is the three-dimensional momentum,
• ${\displaystyle {\vec {x}}}$ is the current position of the particle relative to its starting position

In the ultrarelativistic limit, ${\displaystyle \left|{\vec {p}}_{j}\right|=p_{j}\gg m_{j}~,}$ we can approximate the energy as

${\displaystyle E_{j}={\sqrt {\,p_{j}^{2}+m_{j}^{2}\;}}\simeq p_{j}+{\frac {m_{j}^{2}}{\,2\,p_{j}\,}}\approx E+{\frac {m_{j}^{2}}{\,2\,E\,}}~,}$

where E is the energy of the wavepacket (particle) to be detected.

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 10⁶ in all cases. Using also t ≈ L, where L is the distance traveled and also dropping the phase factors, the wavefunction becomes

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

Eigenstates with different masses propagate with different frequencies. The heavier ones oscillate faster compared to the lighter ones. Since the mass eigenstates are combinations of flavor eigenstates, this difference in frequencies 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

${\displaystyle P_{\alpha \rightarrow \beta }\,=\,{\Bigl |}\,\left\langle \,\left.\nu _{\beta }(L)\,\right|\,\nu _{\alpha }\,\right\rangle \,{\Bigr |}^{2}\,=\,\left|\,\sum _{j}\,U_{\alpha j}^{*}\,U_{\beta j}\,e^{-i{\frac {m_{j}^{2}\,L}{2E}}}\,\right|^{2}~.}$

This is more conveniently written as

${\displaystyle {\begin{aligned}P_{\alpha \rightarrow \beta }=\delta _{\alpha \beta }&{}-4\,\sum _{j>k}\,\operatorname {\mathcal {R_{e}}} \left\{\,U_{\alpha j}^{*}\,U_{\beta j}\,U_{\alpha k}\,U_{\beta k}^{*}\,\right\}\,\sin ^{2}\left({\frac {\Delta _{jk}m^{2}\,L}{4E}}\right)\\&{}+2\,\sum _{j>k}\,\operatorname {\mathcal {I_{m}}} \left\{\,U_{\alpha j}^{*}\,U_{\beta j}\,U_{\alpha k}\,U_{\beta k}^{*}\,\right\}\,\sin \left({\frac {\Delta _{jk}m^{2}\,L}{2E}}\right)~,\end{aligned}}}$

where ${\displaystyle \Delta _{jk}m^{2}\ \equiv m_{j}^{2}-m_{k}^{2}~.}$

The phase that is responsible for oscillation is often written as (with c and ${\displaystyle \hbar }$ restored)

${\displaystyle {\frac {\Delta _{jk}(mc^{2})^{2}\,L}{4\hbar c\,E}}={\frac {{\rm {GeV}}\,{\rm {fm}}}{4\hbar c}}\times {\frac {\Delta _{jk}m^{2}}{{\rm {eV}}^{2}}}{\frac {L}{\rm {km}}}{\frac {\rm {GeV}}{E}}\approx 1.27\times {\frac {\Delta _{jk}m^{2}}{{\rm {eV}}^{2}}}{\frac {L}{\rm {km}}}{\frac {\rm {GeV}}{E}}~,}$

where 1.27 is unitless. In this form, it is convenient to plug in the oscillation parameters since:

• The mass differences, Δm², are known to be on the order of 10⁻⁴ eV² = (10⁻² 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 CP-violation (δ is zero), then the second sum is zero. Otherwise, the CP asymmetry can be given as

${\displaystyle A_{\mathsf {CP}}^{(\alpha \beta )}=P(\nu _{\alpha }\rightarrow \nu _{\beta })-P({\bar {\nu }}_{\alpha }\rightarrow {\bar {\nu }}_{\beta })=4\,\sum _{j>k}\,\operatorname {\mathcal {I_{m}}} \left\{\,U_{\alpha j}^{*}\,U_{\beta j}\,U_{\alpha k}U_{\beta k}^{*}\,\right\}\,\sin \left({\frac {\Delta _{jk}m^{2}\,L}{2E}}\right)}$

In terms of Jarlskog invariant

${\displaystyle \operatorname {\mathcal {I_{m}}} \left\{\,U_{\alpha j}\,U_{\beta j}^{*}\,U_{\alpha k}^{*}\,U_{\beta k}\,\right\}=J\,\sum _{\gamma ,\ell }\varepsilon _{\alpha \beta \gamma }\,\varepsilon _{jk\ell }~,}$

the CP asymmetry is expressed as

${\displaystyle A_{\mathsf {CP}}^{(\alpha \beta )}=16\,\sin \left({\frac {\Delta _{21}m^{2}\,L}{4E}}\right)\sin \left({\frac {\Delta _{32}m^{2}\,L}{4E}}\right)\sin \left({\frac {\Delta _{31}m^{2}\,L}{4E}}\right)\,J\,\sum _{\gamma }\,\varepsilon _{\alpha \beta \gamma }}$

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 meaningful cases in which only two neutrinos participate significantly. In this case, it is sufficient to consider the mixing matrix

${\displaystyle U={\begin{pmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \end{pmatrix}}.}$

Then the probability of a neutrino changing its flavor is

${\displaystyle P_{\alpha \rightarrow \beta ,\alpha \neq \beta }=\sin ^{2}(2\theta )\,\sin ^{2}\left({\frac {\Delta m^{2}L}{4E}}\right)\quad {\text{ [natural units] .}}}$

Or, using SI units and the convention introduced above

${\displaystyle P_{\alpha \rightarrow \beta ,\alpha \neq \beta }=\sin ^{2}(2\theta )\,\sin ^{2}\left(1.27\,{\frac {\Delta m^{2}L}{E}}\,{\frac {\rm {[eV^{2}]\,[km]}}{\rm {[GeV]}}}\right)~.}$

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 mix (superposition) of ν_μ and ν_τ . 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.

Spring-coupled pendulums

Time evolution of the pendulums

Lower frequency normal mode

Higher frequency normal mode

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.

The description of the system in terms of the two pendulums 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.

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 ${\displaystyle {\tfrac {1}{2}}{\tfrac {mg}{L}}x^{2}}$, where g is the standard gravity, L is the length of the pendulum, m is the mass of the pendulum, and x is the horizontal displacement of the pendulum. As an isolated system the pendulum is a harmonic oscillator with a frequency of ${\displaystyle {\sqrt {g/L\;}}\,}$. The potential energy of a spring is ${\displaystyle {\tfrac {1}{2}}kx^{2}}$ where k is the spring constant and x is the displacement. With a mass attached it oscillates with a period of ${\displaystyle {\sqrt {k/m\;}}\,}$. With two pendulums (labeled a and b) of equal mass but possibly unequal lengths and connected by a spring, the total potential energy is

${\displaystyle V={\frac {m}{2}}\left({\frac {g}{L_{a}}}x_{a}^{2}+{\frac {g}{L_{b}}}x_{b}^{2}+{\frac {k}{m}}(x_{b}-x_{a})^{2}\right).}$

This is a quadratic form in x_a and x_b, which can also be written as a matrix product:

${\displaystyle V={\frac {m}{2}}{\begin{pmatrix}x_{a}&x_{b}\end{pmatrix}}{\begin{pmatrix}{\frac {g}{L_{a}}}+{\frac {k}{m}}&-{\frac {k}{m}}\\-{\frac {k}{m}}&{\frac {g}{L_{b}}}+{\frac {k}{m}}\end{pmatrix}}{\begin{pmatrix}x_{a}\\x_{b}\end{pmatrix}}.}$

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

${\displaystyle {\begin{pmatrix}x_{a}\\x_{b}\end{pmatrix}}={\begin{pmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\end{pmatrix}}}$

then

${\displaystyle V={\frac {m}{2}}{\begin{pmatrix}x_{1}\ x_{2}\end{pmatrix}}{\begin{pmatrix}\lambda _{1}&0\\0&\lambda _{2}\end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\end{pmatrix}}}$

where λ₁ and λ₂ are the eigenvalues of the matrix. The variables x₁ and x₂ describe normal modes which oscillate with frequencies of ${\displaystyle {\sqrt {\lambda _{1}\,}}}$ and ${\displaystyle {\sqrt {\lambda _{2}\,}}}$. When the two pendulums are identical (L_a = L_b), θ is 45°.

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.

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 Δm².

If three neutrinos are considered, the probability for each neutrino to appear is somewhat complex. The graphs below show the probabilities for each flavor, with the plots in the left column showing a long range to display the slow "solar" oscillation, and the plots in the right column zoomed in, to display the fast "atmospheric" oscillation. The parameters used to create these graphs (see below) are consistent with current measurements, but since some parameters are still quite uncertain, some aspects of these plots are only qualitatively correct.[23]

Electron neutrino oscillations, long range. Here and in the following diagrams black means electron neutrino, blue means muon neutrino and red means tau neutrino.[23]	Electron neutrino oscillations, short range[23]
Muon neutrino oscillations, long range[23]	Muon neutrino oscillations, short range[23]
Tau neutrino oscillations, long range[23]	Tau neutrino oscillations, short range[23]

The illustrations were created using the following parameter values:[23]

• sin²(2θ₁₃) = 0.10 (Determines the size of the small wiggles.)
• sin²(2θ₂₃) = 0.97
• sin²(2θ₁₂) = 0.861
• δ = 0 (If the actual value of this phase is large, the probabilities will be somewhat distorted, and will be different for neutrinos and antineutrinos.)
• Normal mass hierarchy: m₁ ≤ m₂ ≤ m₃
• Δm²
  ₁₂ = 0.759×10⁻⁴ eV²
• Δm²
  ₃₂ ≈ Δm²
  ₁₃ = 23.2×10⁻⁴ eV²

The most popular conjectured 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 reciprocal 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. Because the Standard Model has only one fundamental mass scale,[lower-alpha 2] all particle masses[lower-alpha 3] must arise in relation to this scale.

There are other varieties of seesaw[27] and there is currently great interest in the so-called low-scale seesaw schemes, such as the inverse seesaw mechanism.[28]

The addition of right-handed neutrinos has the effect of adding new mass scales, unrelated to the mass scale of the Standard Model, hence the observation of heavy right-handed neutrinos would reveal physics beyond the Standard Model. Right-handed neutrinos would help to explain the origin of matter through a mechanism known as leptogenesis.

There are alternative ways to modify the standard model that are similar to the addition of heavy right-handed 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.