Neutrino oscillations
January 23, 2024I took the opportunity in a previous post to summarize under a couple of dates the history of neutrinos. There is another aspect that I very much like about neutrinos, which makes them so unique.
Let me start by a small reminder. Neutrinos are neutral leptons with an extremely small mass compared to other particles. Due to their inherent properties, their interactions occur exclusively through the weak force.
Before I continue, did I say extremely small mass? Well, that is not exactly what the Standard Model predicts. In fact, its formulation requires neutrinos to be massless particles in order to mathematically work. But if, like me, you look closely at the diagram, neutrinos do not have a mass equal to 0; they have an upper bound. This detail might first seem unsignificant, but it actually is a consequence of the great discovery that neutrinos oscillate.
Neutrino oscillation is the phenomenon in which a neutrino changes its lepton flavour (\( e, \mu, \tau\)). It was first mentioned in Mesonium and anti-mesonium by Bruno Pontecorvo in Mai 1957 (Sov.Phys.JETP 6 (1957) 429) and first mathematically formulated in Remarks on the unified model of elementary particles by Ziro Maki, Masami Nakagawa, and Shoichi Sakata in November 1962 (10.1143/PTP.28.870).
Following Gell-Mann and Pais observation of \( K^0\) oscillation as \( K^0 \leftrightarrow \overline{K}^0\) indicating a superposition of \( K_1^0\) and \( K_2^0\) (10.1103/PhysRev.97.1387), Pontecorvo suggested in the first paper that there exist other mixed neutral particle besides the \( K^0\) which differs from their anti-counterparts and for which the particle\(\leftrightarrow\)antiparticle transitions are not strickly forbidden. From this it follows that, besides the \( K^0\), the only system consisting of presently-known constituents which could be a mixed particle would be a mesonium defined as the bound system (\( \mu^+ e^-\)). He proposed the oscillation \( (\mu^+ e^-)\) \(\leftrightarrow\) \((\nu \bar{\nu^{}})\) \(\leftrightarrow\) \((\mu^- e^+)\), and suggested also the possibility of neutrino\(\leftrightarrow\)antineutrino oscillation.
In the second paper, the assumption behind the mathematical formulation of the neutrino oscillation is that there are two neutrino representations (also called states): true neutrinos (mass states) represented by \(\nu_1\) and \(\nu_2\), and weak neutrinos (flavour states) represented by \(\nu_e\) and \(\nu_\mu\). The reason why the formulation includes only two flavours of neutrinos is because the third flavour wasn't postulated at that time. But since then, the mathematical formulation has been extended to the three known neutrino flavours. A key characteristic of these representations is related to their role in the oscillation process. The flavour states are present at the detection or production of the neutrino, while the mass state occurs during the propagation, as illustrated in the following diagram.
The relationship between these two types of states is expressed using the so-called PMNS matrix. This \( 3 \times3\) unitary matrix usually denoted as \(U\) parameterizes the transformation as the following:
\[ |\nu_\alpha\rangle = \sum_{\alpha} U_{\alpha i} \, |\nu_i\rangle \Longleftrightarrow \begin{pmatrix} \nu_e \\ \nu_\mu \\ \nu_\tau \\ \end{pmatrix} = \underbrace{\begin{pmatrix} U_{e 1} & U_{e 2} & U_{e 3} \\ U_{\mu 1} & U_{\mu 2} & U_{\mu 3} \\ U_{\tau 1} & U_{\tau 2} & U_{\tau 3} \\ \end{pmatrix}}_{PMNS} \begin{pmatrix} \nu_1 \\ \nu_2 \\ \nu_3 \\ \end{pmatrix} \]
In general, there are nine degrees of freedom in any unitary \( 3 \times 3\) matrix. But in the case of the PMNS matrix, five of them are absorbed as lepton field phases, leaving a total of four parameters. It is typically described by three mixing angles (\( \theta_{12}, \theta_{23}, \theta_{13}\)) and a CP-violating phase (\( \delta_{CP} \equiv \delta \) ) that affects oscillation rates based on event order. The matrix can then be expressed as:
\[ \begin{array}{rl} \begin{pmatrix} U_{e 1} & U_{e 2} & U_{e 3} \\ U_{\mu 1} & U_{\mu 2} & U_{\mu 3} \\ U_{\tau 1} & U_{\tau 2} & U_{\tau 3} \\ \end{pmatrix} &= \begin{pmatrix} c_{12} c_{13} & s_{12}c_{13} & s_{13}e^{-i \delta} \\ - s_{12} c_{23} - c_{12} s_{13} s_{23} e^{i \delta} & c_{12} c_{23} - s_{12} s_{13} s_{23} e^{i \delta} & c_{13} s_{23} \\ s_{12} s_{23} - c_{12} s_{13} c_{23} e^{i \delta} & - c_{12} s_{23} - s_{12} s_{13} c_{23} e^{i \delta} & c_{13} c_{23} \\ \end{pmatrix} \\ &= \begin{pmatrix} 1 & 0 & 0 \\ 0 & c_{23} & s_{23} \\ 0 & - s_{23} & c_{23} \\ \end{pmatrix} \begin{pmatrix} c_{13} & 0 & s_{13}e^{- i \delta} \\ 0 & 1 & 0 \\ - s_{13} e^{i \delta} & 0 & c_{13} \\ \end{pmatrix} \begin{pmatrix} c_{12} & s_{12} & 0 \\ - s_{12} & c_{12} & 0 \\ 0 & 0 & 1 \\ \end{pmatrix} \end{array}\]
where \( s_{ij}\) and \( c_{ij}\) are used to denote \( \sin \theta_{ij}\) and \( \cos \theta_{ij}\) respectively. In the case of Majorana neutrinos, two extra complex phases \( \eta_1\) and \( \eta_2\) are needed, but in general we assume \( \eta_1 = \eta_2 = 0\).
So far, the neutrino oscillation had only been theorized, and no clear link to neutrino mass had been established. However, a problem emerged in the mid-60s, compelling scientists to expand these theoretical frameworks to better explain their observations.
The Homestake experiment, conducted in Lead from 1967 to 1994, had one main purpose: collecting and counting neutrinos emitted by nuclear fusion taking place in the Sun. This specific choice of neutrino source is driven by the fact that solar neutrinos are exclusively electron neutrinos. And as the detector was located roughly 1.5 km below the surface, it was naturally shielded from cosmic rays.
The detection method is based upon the inverse beta decay (\( n + \nu_e \to p + e^-\)). A \(390\mathrm{m}^3\) tank filled with \(\mathrm{C}_2\mathrm{Cl}_4\) was used in order to detect the capture reaction \( {}^{37}\mathrm{Cl} + \nu_e \to {}^{37}\mathrm{Ar} + e^- \). The collected \({}^{37}\mathrm{Ar} \) was filling a small gas counter which served as a way to determine how many \( \nu_e \) were captured.
In 1968, Raymond Davis, Don S. Harmer, and Kenneth C. Hoffman reported in Search for Neutrinos from the Sun (10.1103/PhysRevLett.20.1205) a neutrino flux from \({}^8\mathrm{B}\) decay in the sun, denoted as \( \phi ({}^8\mathrm{B}) \), equal or less than \( 2\) \( \times \) \( 10^6 \, \mathrm{cm}^{-2}\mathrm{s}^{-1}\) at the earth, while theory around that time predicted a flux between \( 2.5 \) \( \times \) \( 10^6 \, \mathrm{cm}^{-2}\mathrm{s}^{-1} \) and \(3.6 \) \( \times \) \( 10^7 \, \mathrm{cm}^{-2}\mathrm{s}^{-1}\).
Indeed, in November 1967, John N. Bahcall, Martin Cooper and Pierre Demarque reported a result such that \( 1.7 \) \( \times \) \( 10^7 \, \mathrm{cm}^{-2}\mathrm{s}^{-1} \) \( < \phi ({}^8\mathrm{B}) < \) \( 3.6 \) \( \times \) \( 10^7 \, \mathrm{cm}^{-2}\mathrm{s}^{-1}\) in Dependence of the \(^8\mathrm{B}\) Solar Neutrino Flux on Heavy Element Composition (10.1086/149373). In May 1968, John N. Bahcall and Neta A. Bahcall, Giora Shaviv reported a flux considering several configurations such that \( 2.5 \) \( \times \) \( 10^6 \, \mathrm{cm}^{-2}\mathrm{s}^{-1}\) \( < \phi ({}^8\mathrm{B}) < \) \( 1.35 \) \( \times \) \( 10^7 \, \mathrm{cm}^{-2}\mathrm{s}^{-1}\) in Present Status of the Theoretical Predictions for the \(^{37}\mathrm{Cl}\) Solar-Neutrino Experiment (10.1103/PhysRevLett.20.1209). And in July 1968, John N. Bahcall and Giora Shaviv obtained a flux equal to \( 1.3(1 \pm 0.6) \) \( \times \) \( 10^7 \, \mathrm{cm}^{-2}\mathrm{s}^{-1}\), which is equivalent to \( 5.2 \) \( \times \) \( 10^6 \, \mathrm{cm}^{-2}\mathrm{s}^{-1} \) \( < \phi ({}^8\mathrm{B}) < \) \( 2.08 \) \( \times \) \( 10^7 \, \mathrm{cm}^{-2}\mathrm{s}^{-1}\) in Solar Models and Neutrino Fluxes (10.1086/149641).
This discrepancy, known as the solar neutrino problem, raised numerous questions among physicists at that time. The only reasonable explanation was electron neutrinos were changing their flavour, and were therefore not detected. In Neutrino astronomy and lepton charge (10.1016/0370-2693(69)90525-5), Bruno Pontecorvo and Vladimir Gribov computed in 1969 the neutrinos oscillation probability. Once extended to the three families of neutrinos, the oscillation probability looks like the following:
\[ \begin{array}{rl} P (\nu_\alpha \to \nu_\beta) &= |\langle\nu_\beta | \nu_\alpha (t)\rangle|^2 \\ &= |\sum_{i} \sum_{j} U_{\alpha i}^\dagger U_{\beta j} \langle\nu_j | \nu_i (t)\rangle |^2 \\ &= \delta_{\alpha, \beta} - 4 \sum_{i < j} \mathrm{Re} \big[ U_{\alpha i} U_{\beta i}^\dagger U_{\alpha j}^\dagger U_{\beta j} \big] \sin^2 \frac{\Delta m_{ij}{}^2 L}{4 E} \\ & \phantom{=} + 2 \sum_{i < j} \mathrm{Im} \big[ U_{\alpha i} U_{\beta i}^\dagger U_{\alpha j}^\dagger U_{\beta j} \big] \sin \frac{\Delta m_{ij}{}^2 L}{2 E} \end{array} \]
where \( \mathrm{Re}\) and \( \mathrm{Im}\) correspond to the real and imaginary parts of the subsequent expression, \( L \) the distance traveled by the neutrino, \( E \) the energy of the neutrino, and \( \Delta m_{ij}{}^2\) is the mass difference \( m_i{}^2 - m_j{}^2\).
But wait, are we talking about neutrino mass?
Indeed, for the first time we have a direct relationship between neutrino oscillations and neutrino mass. In short, observing neutrino oscillation implies that the probability is nonzero, indicating a nonzero mass difference. Consequently, this confirms that neutrinos are massive particles.
But what if these observations were wrong due, for example, to a detector problem? One way of solving this problem, or at least testing whether an error had occurred, is to repeat the measurement. As reported in A review of the homestake solar neutrino experiment written by Raymond Davis and published in 1994 (10.1016/0146-6410(94)90004-3), the Homestake experiment measured between 1970 and 1992 an average neutrino flux of \( (1.92 \pm 0.28) \times 10^6 \, \mathrm{cm}^{-2}\mathrm{s}^{-1}\). In the same paper, Kamiokande II, another neutrino experiment based in japan, reported a flux value of \((2.39 \pm 0.35) \times 10^6 \, \mathrm{cm}^{-2}\mathrm{s}^{-1}\). Theoretical frameworks also evolved, and the predicted value was revised to \(5.76 \times 10^6 \, \mathrm{cm}^{-2}\mathrm{s}^{-1}\) (using a non-standard solar model of Sienkiewicz, Bahcall, and Paczynski from 1990, Mixing and the Solar Neutrino Problem (10.1086/168351)). The discrepancy remained, but it was not enough evidence to claim a direct observation of neutrino oscillation.
So far, only solar neutrinos were used, but there are many possible sources of neutrinos, divided into controlled and natural sources. Controlled sources include neutrinos from nuclear reactions. Natural sources include solar neutrinos, but also atmospheric neutrinos. They are produced as follows: a cosmic ray strikes an air nucleus full of neutrons and protons in the atmosphere. This initial collision produces many pions. These pions decay mainly (99.99%) into muons and muon neutrinos, and muons decay almost always (\(\approx\)100%) into electrons and electron and muon neutrinos. Because of these subsequent reactions, muon neutrinos are produced at twice the rate of electron neutrinos.
Another specificity of atmospheric neutrinos is that they can come from anywhere in the Universe. But on Earth, we distinguish two sources, depending on their direction. Upward facing sources are neutrinos going through the Earth before reaching the detector, whereas downward facing sources are neutrinos coming directly on the detector. Traversing the Earth makes the neutrino source very clean, as other particles are absorbed by matter. Using the zenith angle \( \theta\), upward facing neutrinos have a positive \(\cos \theta\), and downward facing neutrinos have a negative \(\cos \theta\).
It is not insignificant to introduce atmospheric neutrinos, as it turns out that this is the source that was targeted by one of the two experiments that were decisive in confirming neutrino oscillations in the 90s.
Super-Kamiokande, an upgraded version of the aformentioned Kamiokande II detector, is a cylindrical neutrino detector located 1,000 meters underground in the Mozumi Mine near Hida, Japan. It is filled with 50000 tons of ultra-pure water and equipped with inward-facing photomultiplier tubes (PMTs) to detect Cherenkov radiation from neutrino interactions, as well as outward-facing PMTs to monitor background radiation. Its primary goal was to detect atmospheric neutrinos by observing the interactions \(\nu_e + N \to e + X\) and \(\nu_\mu + N \to \mu + X\), and to compare their rates to predictions from models with and without neutrino oscillations.
The Sudbury Neutrino Observatory (SNO) was a spherical neutrino detector located 2,100 meters underground in a nickel mine near Sudbury, Canada. It was filled with 1000 tons of heavy water (\(D_2O\)) and equipped with photomultiplier tubes (PMTs) arranged around the sphere to detect Cherenkov radiation from neutrino interactions. SNO's primary goal was to study solar neutrinos by detecting charged current interactions such as \( \nu_e + d \to p + p + e^- \) and neutral current interactions sensitive to all neutrino flavors, allowing comparisons between these two rates.
In both cases, the detection principle is based on the Cherenkov radiation. The underlying principle is the following: when a charged particle with a velocity \(v\) travels faster than the speed of light \(c\) in a medium, also known as radiator, with a reflective index \(n\), it emits light. The angle \( \theta_c\) at which the light is emitted is given by \( \cos \theta = \frac{1}{n \beta}\), where \(\beta = \frac{v}{c}\).
In August 1998, the Super-Kamiokande Collaboration published a paper titled Evidence for Oscillation of Atmospheric Neutrinos (10.1103/PhysRevLett.81.1562) where they expose their comparison of models. It turns out that the data was consistent with the model including two-flavor \( \nu_e \leftrightarrow \nu_\mu \) oscillations, with \(\sin^2 2 \theta > 0.82 \) and \( 5 \times 10^{-4} < \Delta m^2 < 6 \times 10^{-3}\) eV\(^2\) at 90% confidence level. Shortly after, the SNO Collaboration published two paper about their recent measurements, called Measurement of the Rate of \( \nu_e + d \to p + p + e^- \) Interactions Produced by \( ^{8}\mathrm{B} \) Solar Neutrinos at the Sudbury Neutrino Observatory (10.1103/PhysRevLett.87.071301) in July 2001 and Direct Evidence for Neutrino Flavor Transformation from Neutral-Current Interactions in the Sudbury Neutrino Observatory (10.1103/PhysRevLett.89.011301) in June 2002. In these two papers, they confirm that the sun's neutrino flux is not composed solely of electron neutrino (\( \phi_{e}^{\text{paper 1}} \) \(= \) \((1.75 \pm ^{0.15}_{0.14} ) \) \(\times \) \(10^6 \, \mathrm{cm}^{-2}\mathrm{s}^{-1}\) and \(\phi_{e}^{\text{paper 2}} \) \(=\) \( (1.76 \pm 0.10 )\) \(\times\) \( 10^6 \, \mathrm{cm}^{-2}\mathrm{s}^{-1}\)), but also includes a muon and tau component (\( \phi_{\mu, \tau}^{\text{paper 2}} \) \(=\) \( (3.41 \pm ^{0.66}_{0.64} ) \) \(\times\) \( 10^6 \, \mathrm{cm}^{-2}\mathrm{s}^{-1} \)), which they determine with precision, and the total sum of the flux (\( \phi_{tot}^{\text{paper 1}} \) \(=\) \( (5.44 \pm 0.99 ) \) \(\times\) \( 10^6 \, \mathrm{cm}^{-2}\mathrm{s}^{-1} \) and \( \phi_{tot}^{\text{paper 2}} \) \(=\) \( (5.09 \pm ^{0.64}_{0.61} ) \) \(\times\) \( 10^6 \, \mathrm{cm}^{-2}\mathrm{s}^{-1} \)) is in agreement with the theory.
These two cumulated proofs were giving enough evidence of the discovery of neutrino oscillation, and served as a way of confirming what had been long observed. These results were rewarded with a Nobel Prize in Physics, awarded to Takaaki Kajita and Arthur McDonald in 2015.
Of course, this is not the end of the story, as there are still many things to be solved, such as the origin of neutrino mass, the problem of mass hierarchy, the value of certain parameters making up the PMNS matrix, and so on. We'll see what the future brings.