# Atmospheric neutrino anomaly

Nimmy Sarah Alex

Assumption College (Autonomous), Changanacherry 686101, Kerala

Prof. Sanjib Kumar Agarwalla

Associate Professor-G, Institute of Physics, Bhubaneswar 751005, Odisha

## Abstract

Neutrinos are the neutral, fermionic, spin-half, omnipresent elementary particles with negligible mass. They are the second most abundant particle in the Universe after photons and interact only via weak interactions. These particles are produced in various natural and artificial processes like nuclear fusion in Sun, cosmic ray interactions and subsequent hadron decay reactions in atmosphere, radioactive decay reactions in nuclear reactors, and particle accelerators. Neutrinos came into the picture by the successful explanation of beta-decay reaction by Wolfgang Pauli in 1930, followed by its detection by Cowan and Reines in 1956. Neutrinos exist in three different flavors–electron-type, muon-type, and tau-type. Since neutrinos interact feebly, they can bring crucial information about their parent sources when they reach to us with a wide range of energies and traversing different path lengths from various sources. We studied the production and detection mechanisms of neutrinos as a precursor to this project. The project deals with atmospheric neutrinos, produced by the interaction of primary cosmic rays with atmospheric particles and subsequent decay of pions and kaons. The hadron decay reaction in atmosphere notifies that atmospheric neutrinos contain muon-type and electron-type neutrinos in the ratio 2:1 and these neutrinos are detected by various Water Cherenkov Detectors and Tracking Calorimeters like Kamiokande, Super-Kamiokande, Soudan-2, and IMB. These detectors calculated the Double Ratio, the ratio of muon-like events to electron-like events at the detector according to the data to the same ratio got via a computational model of experiment. The Double Ratio which was supposed to be unity, is decreased, almost reduced by a factor of 2, and is regarded as the Atmospheric Neutrino Anomaly. The phenomena of swapping of neutrino flavors during its propagation, called flavor-based Neutrino Oscillations explained Atmospheric Neutrino Anomaly. The sole aim of this project is to study the Atmospheric Neutrino Anomaly and flavor-based Neutrino Oscillation in a wide range of energies and baselines, through νμ→νμ Survival Channel.

Keywords: neutrinos, solar neutrino anomaly, atmospheric neutrino anomaly, neutrino oscillation, survival probability

## Outline

The most generous, massive particle in the universe, the most elusive neutrinos yields an interesting area of research in particle physics. Since the proposal of massless, neutral particle by Wolfgang Pauli in 1930, the neutrinos fascinated mankind by its characteristic properties. That our body emits about 350 million neutrinos a day and each cubic meter of Universe encompasses about 30 million neutrinos from the Big Bang fascinates mankind. We receive about 400 trillion neutrinos from the Sun every second and Earth transmits about 50 billion neutrinos per second. Our body produces 1012 neutrinos every second. These neutrinos never disturb the day-to-day life but make it keep going. Neutrinos carries the information from its source and reach the Earth much earlier than light. The studies on the neutrino can lead us to study the dominance of matter over antimatter and the existence of life on Earth. The studies on neutrino oscillations presented us an idea of mass for neutrinos and hence lifted us to investigations beyond the Standard Model. Researches on solar and atmospheric neutrinos lead to Solar and Atmospheric Neutrino Anomalies. We establish the explanations of these anomalies via mass induced Flavor Based Neutrino Oscillations in vacuum in this project.

## Neutrinos in Picture

Since the discovery of radioactivity by Henry Becquerel, comprehensive studies were made in the field of radioactive disintegrations. The beta decay of a particle was expected to be a two-body decay process:

$A\rightarrow B+\overline e$

From the energy and momentum conservations applied in the center-of-mass frame, we can arrive at

$E_A=E_B+E_{e^-}$

And​

$P_A=P_B+P_{e^-}=0$

where E and P are the energy and momenta of the particles in the reaction. On solving for the energy of ejected electrons, we may obtain:

$E_{e^-}=\frac{M_A^2+M_{e^-}^2-M_B^2}{2M_A}$

Thus the expelled electron is of distinct energy since the values of MA, Me-, and MB are known. Unfortunately, the beta decay experiments demonstrated a continuous energy spectrum for the electrons ejected.

Representation of Beta Decay Spectrum

Beta-decay reaction again violated the conservation of momentum. For the successful fulfillment of momentum conservancy, the newly created lighter nuclei and the ejected electron should recoil at an angle of 1800, but the experiments revealed a recoiling angle other than 1800. This two body beta decay also violates the angular momentum conservation and the spin-statistics theorem.

The proposal of a neutral, massless particle called the NEUTRINO by Wolfgang Pauli in 1930 solved this disparity. From then, the beta decay reaction was interpreted as a three-body decay reaction, where the ejected electron divides its energy with the neutrinos generated.

$A\rightarrow B+e^-+\overline{\nu_e}$
$n\rightarrow p+e^-+\overline{\nu_e}$

Such a reaction now justifies the energy and​ momentum conservation, and spin-statistics.​[1]

## The Very First Discovery

The identification of this little ghost particles is a tedious, onerous process. Since they interact only via weak interactions and the cross-section of the weak interaction is insignificant, we can never identify neutrinos. Hence, what points to the discovery is the interactions and products created because of neutrino interactions.

The general beta decay was represented as

$n\rightarrow p+e^-+\overline{\nu_e}\;\;\;\;\;\;\;\;(\beta^-\;decay)$

A nucleus (Z, N) with Z protons and N neutrons emits an electron and an electron anti-neutrino to form the (Z+1, N-1) nucleus.

$(Z,N)\rightarrow(Z+1,N-1)+e^-+\overline{\nu_e}$

Inverse beta decay reaction is obtained by transferring the electron antineutrino to the reaction side as its antiparticle.

$\nu_e+(Z,N)\rightarrow(Z+1,N-1)+\;e^-\;\;\;\;(inverse\;\beta^-\;decay)$
$(Z,N)\rightarrow(Z-1,N+1)+\;e^++\nu_e\;\;\;\;(\beta^+\;decay)\;$
$\overline{\nu_e}+(Z,N)\rightarrow(Z-1,N+1)+e^+\;\;\;(inverse\;\beta^+\;decay)$
$\overline{\nu_e}+p\rightarrow n+e^+\;\;\;\;​(Cowan\;and\;Reines\;reaction)$

In the mid-1950s, Cowan and Reines set up a nuclear reactor at the Savannah River in South California. This experiment used 400 liters of Cadmium Chloride solution with water and observed the inverse beta decay reaction. Alike with a very large flux of antineutrinos (5 x 1013), only two or three events happened every hour. The antineutrino from a nuclear reactor interacts with the protons in target materials releasing neutron and positron. These positrons annihilate suddenly with an electron and it produces simultaneously two photons.

$e^-+e^+\rightarrow\gamma+\gamma$

The neutron is captured by the cadmium nuclei and it emits another photon after some 15µs of positron annihilation. This series of photon emission within the time frame justifies the interaction of neutrinos and hence is the method to identify them.​​​​[2]

Pictorial Representation of Cowan and Reines Experiment (Google image)

## Objectives of the Research

The research focuses mainly on the properties of neutrinos for which a little awareness of neutrino sources and detectors is requisite. In this project, we will be dealing with the Solar and Atmospheric Neutrino Anomalies. The explanation to these oddities is obtained via mass induced Flavor Based Neutrino Oscillations. In this project, I have learned the Two Flavor Neutrino Oscillation in vacuum via the Survival Channel. Theoretical calculations for neutrino oscillation and the plots obtained for some standard values are presented in this project.

## Scope

Neutrino oscillations immediately points to the mass of neutrinos. Hence this study may lead us to think beyond the Standard Model for Particle Physics, which treats neutrinos as massless particles. These studies can lead to solutions for some open problems of the Universe like the CP violation in the neutrino sector and the mass hierarchy issue. Future studies will help us understand the Matter-Antimatter Asymmetry in the universe and hence eventually lead to the idea of existence of life on the Universe.

## LITERATURE REVIEW

Neutrinos are the subtle, chargeless, particles that interact only via weak interactions and thus make the detection difficult. I make a little yet detailed analysis of neutrino sources and detectors as a precursor. The solar and atmospheric Neutrino Anomalies are studied and a detailed study of its explanation is also carried out.

## Neutrino Sources

About millions and trillions of neutrinos are produced every second from various neutrino sources around the Universe. The neutrino sources are mainly classified in two streams as Natural and Artificial Neutrino Sources. The theoretical studies suggest that neutrinos that are formed at Big Bang were also present in the Universe and these can render useful informations about the sources.

## Natural Neutrino Sources

The natural neutrino sources include stars, supernova, Sun, Big Bang, Gamma Ray Bursts and Cosmic-ray interactions in the upper atmosphere. The neutrinos from these sources have different energy ranges and hence traverse different pathlengths or baselines.

## Relic neutrinos

At the time of Big Bang, all the fundamental particles, the electrons, protons, and positrons were in thermal equilibrium. As the universe expanded, the temperature dropped down and the thermal equilibrium was lost. The particles began to freeze out or decouple creating Z0 bosons and hence neutrinos, in 1 second.

$\gamma\;\gamma\leftrightarrow e^+e^-\leftrightarrow Z^0\leftrightarrow\nu\;\overline\nu$

The temperature significantly reduces and the decoupling of particles takes place, maintaining a thermal equilibrium and creating a cosmic microwave background. It is proposed that an analogous neutrino background is possible, with the neutrinos created instantly at the point of the Big Bang. Such neutrinos were termed Relic Neutrinos. The universe model suggests that the energy of these neutrinos to be about 0.1 meV and hence is challenging to recognize. They are so copious that about 330 neutrinos are present per cm3 and interact only via neutral current (Z0) interactions.

## Solar neutrinos

Sun is an ample source of electron neutrinos and these are the finest source to study the Solar processes. Solar neutrinos are generated by the fusion processes in the Sun via two distinct chains-pp chain and CNO cycle.

pp chain

The proton-proton cycle in the Sun produces about 98.4% of total Solar energy.

$\begin{array}{l}\;p^++p^+\rightarrow{}^2H+e^++\nu_e\end{array}$
${}^2H+p^+\rightarrow{}^3He+\gamma$
${}^{\;3}He+{}^3He\rightarrow{}^4He+2p^+$

In the initial cycle, two protons fuse to form a deuterium and an electron neutrino. The neutrinos generated through this process were called the pp neutrinos and is the most expected path of Solar Energy production.

The p-e-p reaction, where 2 protons fuse in the vicinity of an electron is 230 times less likely than the pp reaction.

$\begin{array}{l}p^++e^-+p^+\rightarrow{}^2H+\nu_e\end{array}$
$\begin{array}{l}{}^2H+p^+\rightarrow{}^3He+\gamma\\\end{array}$
${}^3He+{}^3He\rightarrow{}^4He+2p^+$

Rarely the 3He nucleus produced will combine with 4He nucleus to produce Berrylium-7 which transforms to Li-7 and electron neutrinos, which are formed with very less branching ratio of 12%.

${}^3He+{}^4He\rightarrow{}^7Be+\gamma$
${}^7Be+e^{-\;}\rightarrow{}^7Li+\nu_e$
${}^7Li+p^+\rightarrow{}^4He+{}^4He$

In the pp III chain with a branching ratio of 0.1%, 8B neutrinos are produced by the decay of 8B nucleus by the reaction:

${}^3He+{}^4He\rightarrow{}^7Be+\gamma$
${}^7Be+p^+\rightarrow{}^8B+\gamma\;$
${}^8B\rightarrow{}^8Be+e^++\nu_e$
${}^8Be\rightarrow{}^4He+{}^4He$

These solar neutrinos are of extremely limited energy and cannot be easily detected. Standard Solar Model predicts the energy to be 0.4MeV.

Carbon Nitrogen Oxygen cycle (CNO cycle)

This cycle contributes to 1.6% of Solar Energy and can be represented as :

$\begin{array}{l}{}^{12}C+P\rightarrow{}^{13}N+\gamma\end{array}$
$\begin{array}{l}{}^{13}N\rightarrow{}^{13}C+e^++\nu_e\;\;(E_\nu\;\leq\;1.2MeV)\end{array}$
$\begin{array}{l}{}^{13}C+p\rightarrow{}^{14}N+\gamma\\{}^{14}N+\gamma\rightarrow{}^{15}O+\gamma\end{array}$
${}^{\;15}O\rightarrow{}^{15}N+e^++\nu_e\;\;\;(E_\nu\leq1.73MeV)$
${}^{15}N+p\rightarrow{}^{12}C+{}^4He$

​​These neutrinos are of 1.2-1.5 MeV energies.

Solar neutrinos are usually detected by the Gallium experiments. Chlorine experiments observe only 7Be neutrinos and big water experiments like Super-K and SNO detects 8B neutrinos.

Fluxes of Neutrinos Produced in Various Interactions in the Solar Sector.

## Neutrinos from supernova

Supernovae is an outbreak of a massive star resulting in the discharge of much energy in a week as the Sun radiates in 10 years. Neutrinos are generated in enormous numbers in such outbursts and carry valuable information about the massive/giant star. Supernova explosion occurs either by the core collapse of a massive star or by the thermonuclear explosion of white dwarf stars within a binary system.

The mechanism inside a massive star:

$e^-+{}^ZA\rightarrow{}^{Z-1}A+\nu_e$

These neutrinos have 99% of released energy with an average of 10 MeV, thus making CC interactions feasible for the electron type neutrinos. Supernova Event Watch System (SNEWS) is a collaboration of neutrino experiments that will check for these neutrinos if two detectors see a huge pulse of neutrinos at the same time.

## Astrophysical neutrino sources

Such neutrinos are produced by two processes:

1. Acceleration Process, where protons sped up to high energy (about 1016 GeV) interacting with background photons or protons and subsequent meson showers.

$p+p,\;p+\gamma\rightarrow\pi^0,\pi^\pm,k^\pm\;$

​The mesons decay to neutrinos during flight. Supernova acceleration, Black holes, Quasars, Active Galactic Nuclei, and Gamma-Ray Bursts are some acceleration mechanisms.

2. Decay/Annihilation of exotic phenomena leftover after Big Bang. These are vague, theoretical concepts like cosmic strings, evaporating black holes, magnetic monopoles, quark nuggets, Q-balls, etc...

These neutrinos are of extreme energies and fewer fluxes and hence large observatories or detectors are required to identify them. Water Cherenkov detectors/ICECUBE detectors are the typical astrophysical neutrino detectors.

## Geo-neutrinos

Are produced due to decay of unstable elements like Uranium, Thorium, Pottasium, inside the Earth. The measurements of geoneutrinos can address the question of radioactive Earth and can determine its nature. The first geoneutrinos are identified by the KamLAND experiment, which observed antineutrinos of 1.8 MeV energy.

## Atmospheric neutrinos

The primary cosmic rays are comprised of 95% protons, 5% alpha particles and <1% of heavier nuclei and electrons. The primary particles responsible for the production of atmospheric neutrinos are those with energies lesser than 1012 MeV. The atmosphere on being bombarded with primary cosmic rays produce an avalanche of hadrons. These hadrons will decay on their flight and atmospheric neutrinos are produced. The main energy chain is:

$\mathrm\pi^+\rightarrow\mathrm\mu^++{\mathrm\nu}_{\mathrm\mu}$
$\begin{array}{l}\mathrm\mu^+\rightarrow\overline{{\mathrm\nu}_{\mathrm\mu}}\;+e^++\nu_e\end{array}$
$\pi^-\rightarrow\mu^-+\overline{\nu_\mu}$
$\mu^-\rightarrow e^-+\overline{\nu_e}+\nu_\mu$

The energy of these neutrinos ranges from 1 GeV to 100s of GeVs. The ratio of muon type neutrinos to electron type neutrinos i.e,$\frac{\nu_\mu\;+\;\overline{\nu_\mu}}{\nu_e\;+\;\overline{\nu_e}}$, is 2, theoretically. The atmospheric neutrino detectors are in position on or below the Earth's surface thus varying the flight distances from 15 km to 13000 km. The experiments like Super-K in Japan, Soudan-2, Frejus, etc.. are designed to search for atmospheric neutrinos.​[3]

## Reactor neutrinos

These are usually terrestrial electron-type antineutrinos by the radioactive decay of unstable ions like 238U and 239Pu. The reactors produce antineutrinos isotropically and can aid background estimates by operating in beam on and beam off modes. The neutrinos of energy range 3 MeV to 8 MeV are produced and the detection mechanism is the inverse beta decay reaction.

$\overline{\nu_e}+p\rightarrow e^++n$

​with a threshold of 1.8 MeV. KamLAND detects neutrinos from reactors all over Japan.

## Neutrino beamlines

To carry out experiments with neutrinos, they are produced by mimicking atmospheric cosmic ray interactions in labs. High Energy protons from a proton synchrotron is forced to collide with a target producing beams of kaons and pions, which on their path through long, evacuated decay pipes, produces neutrinos.​[4]

$\begin{array}{l}M^+\rightarrow\mu^++\nu_\mu\;\;\;\;(M\;=\;\pi,K)\\\end{array}$
$\mu^+\rightarrow e^++\nu_e+\overline{\nu_\mu}$

## Methods of Neutrino Detection

As observed, neutrinos generated at various sources differ in their energies and hence span different baselines. Therefore, different neutrino detection techniques are employed and a few are discussed. The neutrinos, since neutral move unhindered and unrecognized. The existence of neutrinos is thus singled out by studying the particles left behind by the interaction of neutrinos on target materials. Therefore, the detectors should have fine tracking capability and should be low mass detectors. The detection technique depends on the energy and type of neutrinos and the interaction (charged current/neutral current (discussed later)) that we are looking for.​[3]

${}_N^AZ+\nu_e\rightarrow\;{}_{(N-1)}^A(Z+1)+e^-$

​This technique detects neutrinos, by tracing the radioactive decay of unstable nuclei formed by neutrino interactions. This technique gives information on the number of electron neutrino events but, all other information regarding the energy, direction, and all other flavors are lost. Solar Neutrino detection mainly uses this method and the principle of inverse beta decay is employed here. The experiments that use this technique are:

Homestake experiment: This uses 615 tonnes of cleaning fluid (perchlorate-ethylene) as the target material. The reaction:

${}^{37}Cl+\nu_e{\;\rightarrow\;^{37}}Ar+e^-$

SAGE, GALLEX, GNO: uses gallium atoms as the target material and have a threshold of 233 keV.

${}^{71}Ga+\nu_e\rightarrow{}^{71}Ge+e^-$

## ​Water Cherenkov technique

The detection of neutrinos is carried out by observing the Cherenkov cone developed as the result of the passage of charged particles through a medium, with a velocity higher than the velocity of light in that medium. Particles are differentiated by the shape of the Cherenkov ring. The success of the experiment relies on transparent medium and neutrino energy (it should be higher than the Cherenkov limit). This method reconstructs the direction of the charged particle, but it fails for neutral particles and those with energy lesser than the Cherenkov limit.

Super-Kamiokande is the Water-Cherenkov detector, that uses 50 kilotonnes cylindrical volume of ultra-pure water, and 11,147 photomultiplier tubes, that detects photons. Super-K usually detects atmospheric neutrinos and employs charged current interactions for detection and fails to observe those generated via neutral current interactions.

Sudbury Neutrino Observatory in Canada employs 1000 tonnes of heavy water and 9700 photomultiplier tubes. SNO detects neutrinos via both charged current interactions and neutral current interactions and is a better detector of Solar Neutrinos.

$\begin{array}{l}\nu_e+d\rightarrow e^-+p+p\;\;\;(1.4\;MeV)\end{array}$
$\begin{array}{l}\nu+e^-\rightarrow\nu+e^-\;\;(elastic\;scattering(CC))\\\end{array}$
$\nu+d\rightarrow\nu+p+n\;(neutral\;current)$

## Scintillation technique

Detects mainly electron-antineutrinos from reactors with very low energies of the range of 2-3 MeV, by observing the scintillation lights produced by the charged particles using photosensors.

KamLAND in Kamioka mine in Japan consists of 1000 tonnes of liquid scintillator in a spherical weather balloon and is surrounded by a non-scintillator fluid for shielding. 1280 photomultiplier tubes are employed and the neutrino signal is observed via the reactions:

$\begin{array}{l}\overline\nu+p\rightarrow n+e^+\\\end{array}$
$n+p\rightarrow d+\gamma$

​KamLAND detects electron antineutrinos from reactors and the Solar neutrinos.

## Tracking and hybrid detectors

The neutrinos with energies of few GeV are tracked depending on their range, magnetic tracking, and shower calorimetry.

NuTeV, a tracking detector uses iron plates as target material and scintillators and drift chambers for tracking. The muons generated enters the magnetic channel and is detected by techniques like fine-grained trackings, muon spectrometers, hadronic and electromagnetic calorimetry, etc.

## Solar Neutrino Anomaly

Arthur Eddington proposed nuclear energy as the source of solar energy in the 1920s. The nuclear fusion process that may happen inside the Sun’s core was a vague idea until 1938 when Hans Bethe developed a reaction mechanism that suggested the CNO cycle for the heavier stars and pp chain for the lighter stars (like Sun). There are 6 major and 2 minor thermonuclear fusion reactions in the solar core. We encapsulate the neutrino production reaction:

$\begin{array}{l}{}^1H+{}^1H\rightarrow{}^2H+e^++\nu_e\;\;pp-I\\{}^7Be+e^-\rightarrow{}^7Li+\nu_e+\gamma\;\;pp-II\\{}^8B\rightarrow{}^8Be+e^++\nu_e\;\;\;pp-III\\{}^{13}N\rightarrow{}^{13}C+e^++\nu_e\;\;\;CN\\{}^{15}O\rightarrow{}^{15}N+e^++\nu_e\;\;\;CN\\{}^{17}F\rightarrow{}^{17}O+e^++\nu_e\;\;\;\;NO\end{array}$

and the two rarely occuring three body decays:

$\begin{array}{l}{}^1H+{}^1H+e^-\rightarrow{}^2H+\nu_e\;\;\;pep\\{}^3He+{}^1H+e^-\rightarrow{}^4He+\nu_e\;\;hep\;\end{array}$

The pp-I neutrinos have relatively fewer energies of the order of 0.26 MeV, whereas pp-III neutrinos are of higher (1-14 MeV) of energy. The pep neutrinos and the neutrinos from electron capture by two different states of 7Be produce monochromatic νe lines in the energy spectrum, whereas all other processes give a continuous spectrum. The studies showed net Solar Neutrino flux to be 70 billion neutrinos per square cm per sec.(refer fig:3).

The results of flux calculations by various detectors are discussed:

Homestake experiment: This deep underground detector that uses Chlorine as the target, measured a neutrino capture rate of 2.56±0.25 SNU, (1 SNU = 10-36 neutrino capture per target atom per second); against the theoretical estimate of 8.1±1.2 SNU. About two-thirds of the neutrinos are missing, and this discrepancy became known as the Solar Neutrino Anomaly. However, the experiment faced a fundamental mistake of having a threshold of 0.8 MeV and hence detects only 7Be and 8B neutrinos.

$\nu_e\;+\;{}^{37}Cl\;\rightarrow\;e^-\;+\;{}^{37}Ar\;\;$

Super-Kamiokande: with the threshold of 5 MeV, Super-K detects those neutrinos produced due to the elastic scattering of neutrino on an electron of water molecule and measured a neutrino capture rate of 0.45±0.02 SNU, against a model prediction of 1.0±0.2 SNU.

$\nu_x\;+\;e^-\;\rightarrow\;\nu_x\;+\;e^-\;\;\;(x\;=\;e,\mu)$

SAGE and GALLEX: The Cl and water experiments were sensitive to relatively rarer pep and 8B neutrinos, whereas the Ga experiments observed the bulk of pp neutrinos. SAGE observed a neutrino capture rate of 70.8±5 SNU against model prediction of 129±9 SNU. Whereas GALLEX observed 77.5±8 SNU, 40% lower than the prediction.

$\nu_e\;+\;{}^{71}Ga\;\rightarrow\;e^-\;+\;{}^{71}Ge$

All these attempts were mainly concerned about the electron neutrinos and it is found that the disparity in all these experiments was energy-dependant. The lower the energy of neutrinos, the lesser is the discrepancy.

Solar Neutrino data analysed by various neutrino detectors. The blue bars represents the experimental data, whereas others represents the theoretical data.

Hence, Solar Neutrino Anomaly was an area of concern and two solutions were drawn:

1. The Standard Solar Model predicted by Bahcall was wrong.

2. Neutrinos produced was wrongly interpreted.

But the various helio-seismological studies show sharp agreement with the Standard Solar Model and hence it is confirmed that some information on neutrinos is missing.

The final soution is obtained by the SNO experiment. In order to understand this breakthrough experiment in Solar Neutrino Problem an idea of the weak interaction pathways is made.​[5]

## Two different types of weak interactions

1.Charged Current/Charge Changing (CC) Interaction is represented to be:

$\nu_e+(Z,N)\rightarrow(Z+1,N-1)+e^-\;$

​In this process, the interaction is mediated by charged bosons (W+,W-). Only the electron neutrinos take part in this process.

2. Neutral Current (NC process) is represented as:

$\nu_x+(Z,N)\rightarrow(Z,N)^\ast+\nu_x$

​The nucleus either gets excited to a higher energy state or disintegrates. This interaction is mediated by Z0 bosons and all neutrino flavors will interact via this process. ​[6]

We find that all the above neutrino detection techniques relied on charged current interactions and hence νμ and ντ neutrinos are left undetected. Also, the threshold energies of various experiments hinders the detection of lower energy neutrinos. Hence there is a demand to measure the total Solar Neutrino flux, regardless of the flavor and SNO was one such experiment, that calculated the total flux of Solar Neutrinos.

## Two-in-one D2O experiment

The SNO experiment used a heavy water target. The deuteron atom D2O requires only 2 MeV to break apart and hence solar neutrinos of any flavor with energy up to 30 MeV can interact via Neutral Current interaction. SNO could detect neutrinos via three distinct interactions.

1. Elastic scattering channel

$\nu+e^-\rightarrow\nu+e^-$

​electron neutrinos interact via both charged current and neutral current interactions, whereas νμ and ντ​ take part only in the neutral current process. Thus total neutrino flux is obtained to be $\varphi_{(T)}=\varphi_{\nu_e}+0.15(\varphi_{\nu_\mu}+\varphi_{\nu_\tau})$ and is measured as 2.39±0.26 x 10-8 cm-2 s-1

2. Charged current channel

$\nu_e+d\rightarrow p+p+e^-$

​This interaction estimates only the νe flux and is found to be 1.76 ± 0.01 x 10-8 cm-2 s-1. This step ensures that the scarcity in ​νe is observed in the SNO experiment also. $\varphi_{(T)}\;=\;\varphi_{\nu_e}$

3. Neutral current interaction

$\nu+d\rightarrow n+p+\nu$

​Since all neutrino flavors interact via this method, this measures the total solar neutrino flux and it was obtained to be 5.09 ± 0.63 x 10-8 cm-2 s-1. $\varphi_{(T)}=\varphi_{\nu_e}+\varphi_{\nu_\mu}+\varphi_{\nu_\tau}$

This total flux was in agreement with the Standard Solar Model thus validating SSM. $\varphi_{(SSM)}=5.05\pm1.01\times10^{-8}\;cm^{-2}\;s^{-1}$

Measurement by the SNO experiment of muon and tau neutrino flux from the Sun as a function of the electron neutrino flux, showing that the total flux is consistent with solar models.

The independent flavor fluxes were calculated using these processes and (νμτ ) flux was obtained to be 3.33±0.63x10-8 cm-2 s-1; which was three times larger than νe flux. This contradicts the fact that solar neutrinos are solely electron neutrinos and hence concludes that this anomaly was the result of some changes in electron neutrinos produced at Sun, on their way to detectors. The occurrence of νμ and ντ fluxes suggests the swapping/oscillation between neutrino flavors, called as Neutrino Oscillations.​[2]

The failure of Cl and Ga experiments is due to the fact that there wouldn't be sufficient energy for νμ and ντ to take part in charged current interactions; A maximum energy of 30 MeV is insufficient to produce corresponding charged leptons (muon with rest mass energy of 105 MeV and tau with rest mass energy of 1777 MeV), thus νμ and ντ escapes the detection process.

Kamiokande and Super-K, even though used electron scattering mechanisms, failed to find νμ and ντ due to less sensitivity of the setups. Thus Neutrino Oscillation was considered the solution for the Solar Neutrino Anomaly.

## Atmospheric Neutrino Anomaly

Around 1900 Cosmic rays were discovered and it was found that primary cosmic rays consists of energetic protons, which fly around all directions in the Universe. When these cosmic rays hit the upper atmosphere, they interact with the nuclei of atoms there and hadrons (pions and kaons) are produced. These pions and kaons have very short lifetimes (of the order of 10-8 sec and 10-11 sec respectively) and decay to muons and neutrinos. Muons developed, again disintegrates in 10-6 sec, producing neutrinos. These neutrinos produced as the result of cosmic ray interactions are referred to as atmospheric neutrinos. The dominant part of the decay chain is represented as:

$\begin{array}{l}\pi^+\rightarrow\mu^++\nu_\mu\\\mu^+\rightarrow e^++\nu_e+\overline{\nu_\mu}\\\pi^-\rightarrow\mu^-+\overline{\nu_\mu}\\\mu^-\rightarrow e^-+\nu_\mu+\overline{\nu_e}\end{array}$
This diagram shows cosmic rays interacting with an air nucleus in the atmosphere, producing atmospheric neutrinos. These neutrinos are typically produced around 15 kilometers above the ground.

The atmospheric neutrinos have energies between 100 MeV to 100 GeV and is detected in underground laboratories through scattering off of nuclei. Since of higher energies, νμ and ντ in atmospheric sector can induce charged current interactions.

The ratio of fluxes of electron type to muon type neutrinos irrespective of their flavors was found to be nearly 2, at lower energies and horizontal trajectories. However, the ratio deviates to slightly higher than 2 at vertical trajectories and higher energies.

$\frac{\varphi_{\nu_\mu}}{\phi_{\nu_e}}\cong2$

​These neutrinos are usually observed by Water Cherenkov detectors (Kamiokande, IMB) and Tracking Calorimetries (Frejus, NUSEX). Most of these detectors gave the results in the form of a double ratio, R which is the ratio of neutrino fluxes by observation (experiment) and by Monte Carlo simulation (theoretical) methods.

$R\;=\frac{({\displaystyle\frac{\nu_\mu}{\nu_e}})\;_{Observed}}{({\displaystyle\frac{\nu_\mu}{\nu_e}})_{MonteCarlo}}$

If the observed composition agrees with the theoretical prediction double ratio cancels out and should be unity, but various experiments proved that the double ratio varied from unity and the observations are shown in the table.

Measurement of double ratio for various atmospheric neutrino experiments
 Experiment Type Of Experiment R Super-Kamiokande Water Cherenkov 0.675±0.085 Soudan-2 Iron Tracking Calorimetry 0.69±0.13 IMB Water Cherenkov 0.54±0.12 Kamiokande Water Cherenkov 0.60±0.07 Frejus Iron Tracking Calorimetry 1.0±0.15
The double-ratio of muon to electron neutrino rates for the atmospheric neutrino measurements that characterized the atmospheric neutrino anomaly. Square data points represent iron tracking detectors, circles represent water Cherenkov detectors

Hence it is understood that fewer neutrino events were observed than predicted, and either less νμ events had occurred or more νe events were created in the atmosphere. The double ratio predicted by Frejus was wrong for some systematic errors.

## More evidence from Super-Kamiokande

Super Kamiokande, the Water Cherenkov detector, disclosed its most significant results in 1998. Super-K interpreted the following results:

1. It addresses the deviation of the double ratio in the Sub-GeV range (visible energy less than 1.33 GeV). The Super-K analyzed atmospheric neutrino data by two different flux models in two distinct observatories-Kamioka and IMB, and it drew an overlap in the results. It measured the double ratio to be 0.61±0.03±0.05. It also identified that double ratio was independent of fiducial position, eliminating entering cosmogenic background, such as neutrons.

2. In the Sub-GeV range, the double ratio is defined as a function of the zenith angle, and no significant deviation is noticed in the results, except for higher baselines. The dependence of the number of events in the zenith angle is washed out in low energy range because of poor angular correlation between the neutrino and the outgoing lepton. For higher energy events, an intriguing dependence of double ratio on the zenith angle is obtained.

3. In the Sub-GeV range, about half of the muon neutrinos are found to be missing over the full range of zenith angles. Super-K examines only the fully contained events for the analysis since the neutrinos are considered in the lower energy ranges. As the energy of neutrinos is reduced, it is found that more of the muons neutrinos are lost from the data.

Zenith angle distributions of νµ and νe-initiated atmospheric neutrino events detected by Super-Kamiokande. The left column shows the νe (“e-like”) events, where as the right column depicts νµ events. The top row show low energy events where the neutrino energy was less than 1 GeV, whilst the bottom row shows events where the neutrino energy was greater than 1 GeV. The red line shows what should be expected from standard cosmic ray models and the black points show what Super-Kamiokande actually measured.

4. In the multi-GeV range, fully contained as well as the partially contained events are analyzed. The results show the source of distortion, and a deficit of muon like events, can be directly seen with zenith angle distribution. The muon neutrino data is satisfied for the zenith angle of (0-90)0, for those neutrinos coming downwards​, but deviates and almost reduces by half to those neutrinos coming upwards through Earth, with zenith angle (90-180)0. Thus, atmospheric neutrino anomaly was mainly associated with the muon neutrinos. With a sample of roughly 400 muon like events, an up-down asymmetry can be drawn. It is found that high energy muon neutrinos were detected at half the expected rate if they traveled long distances. This implies the disappearance of muon neutrinos with maximal mixing.​​​[7]

The zenith angle distribution for multi-GeV FC and partially-contained events from the first 414 days of Super-Kamiokande data. The markers represent the events counted by Super-Kamiokande, the boxes represent the expected event counts.

5. The Sub-GeV and Multi-GeV electron like events agreed with expectation suggesting that electron neutrinos were not measurably affected by neutrino oscillation at these energies.

The Sub-GeV double ratio as well as the Multi-GeV up-down asymmetry suggests a clear preference for neutrino oscillations.

6. It is found that the number of electron and muon events were gradually reduced as the analysis moves from Sub-GeV to Multi-Gev range. This is due to the decrease in cosmic ray flux towards higher energies. $N(E)dE\sim E^{-x}dE$, where x = 2.7 below 1015 eV energy.​[7]

The all-particle spectrum of cosmic rays. The arrows and values between parentheses indicate the integrated flux above the corresponding energies.

Super-K analyzed the results not only by double ratio and up-down assymetry but subdivided it in single ring Sub-GeV, single ring Multi-GeV and partially contained event samples and performed a combined fit over 70 bins to the two flavor hypothesis. Both νμ→ντ and νμ→νewere considered; with the later statistically favored by several standard deviations.

In νμ→ντ oscillation, the experimental signature is the disappearance of muon interactions, because the most oscillated tau neutrinos have energy below the threshold of 3.5 GeV; and those that are over threshold generally do not produce the clean single event required for identification of lepton flavor.​[7]

## The Opera experiment

Oscillation Project with Emulsion tracking Apparatus was the first experiment to detect tau neutrinos. The Super Proton Synchrotron at CERN fires pulses to carbon target, simulating reactions as in the atmosphere, thereby producing neutrinos. The apparatus consists of 2 opera supermodules, with 150000 bricks arranged parallelly interleaved with plastic scintillator counters. Each supermodules also contain a magnetic spectrometer that identifies the charge and momentum of penetrating particles. Tau particles produced by the interaction of tau neutrinos were observed by the scintillator counters in May 2010, for the first time. Thus this experiment affirms the assumption made by the Super-K, validating νμ→ντ oscillation as solution to Atmospheric Neutrino Anomaly. The experiment has observed about 5 tau particle to date.

## Neutrino Mixing

Neutrinos are found to exist in three different flavors - electron neutrinos (νe) muon neutrinos (νμ) and tau neutrinos (ντ). Neutrinos are also classified in terms of their masses - neutrino[1] (ν1), neutrino[2] (ν2) and neutrino[3] (ν3), with masses m1, m2, and m3 respectively. The mass eigen states of these neutrinos are not identical to the flavor eigenstates, and they are mixed. i.e, every time we create a particular flavor state, say νe, it will be in one of these mass states, satisfying the conservation laws. Since the value of these masses are unknown, it is assumed that at a weak interaction vertex, a particular flavor state will be a linearly superposed state of all the three mass states, and vice versa.

Mixing of flavor states and mass states
$\vert\nu_\alpha>\;=\;\sum_{k=1,2,3}U_{\alpha k}\;\left|\nu_k>\right.$

where, |να> : flavor state; |νk> : mass eigen states

${\left|\nu\right.}_\alpha>=U_{\alpha1}\;\left|\nu_1>\right.+U_{\alpha2}\;\left|\nu_2>+U_{\alpha3}\;\left|\nu_3>\right.\right.$

Neutrino Oscillation is the result of the quantum superposition of its mass states. Whenever a neutrino gets produced at the source, it is in a definite flavor (say νeμτ), the mass state is undefined. Once it starts propagating it will propagate in definite mass states and hence go out of phase as it traverses with different velocities. The neutrino mass states interact at the detector and the quantum superposition of these mass states may result in different flavor states​[3]​.

A method to understand Neutrino Oscillation is drawing the analogy with the propagation of a plane-polarized light wave. A light wave propagating in z-direction will have circular polarization either towards the right or left. Thus a plane wave can be resolved as a superposition of left and right circularly polarized waves. On propagating through an optical medium, the circular components travel with different velocities. When the light emerges out from the medium, these circular components with different phases superpose to form the plane-polarized wave, but the plane of polarization will be slightly tilted or rotated from the initial direction. For a neutrino, the mass states are analogous to the circular components and the flavor states can be compared to the plane-polarized wave. A particular neutrino flavor generated at the source may oscillate into a different flavor due to the phase difference generated by the mass eigenstates (due to differences in masses of these states) during propagation.​​​[2]

## Methodology

This section deals with the mathematics and methodology used in studying the neutrino oscillation phenomenon. In this project, I've studied the two flavor neutrino oscillation, which can be further extended to the three flavor sector to draw the complete details of neutrino oscillations.

The mathematics of two flavor neutrino oscillations

## Mixing matrix

The two flavor states, say |να> and |νβ>, with < να | νβ > = δαβ are connected to the mass states or the eigen states of the hamiltonian, |ν1> and |ν2>, with < ν1 | ν2 > = δ12, by a unitary matrix U as

$\begin{pmatrix}\vert\nu_\alpha>\\\vert\nu_\beta>\end{pmatrix}\;=\;\begin{pmatrix}U_{\alpha1}&U_{\alpha2}\\U_{\beta1}&U_{\beta2}\end{pmatrix}\;\begin{pmatrix}\vert\nu_1>\\\vert\nu_2>\end{pmatrix}$

​​For an nxn matrix, there are n2 real parameters, among which $\frac{1}{2}n\left(n-1\right)$ are mixing angles and $\frac{1}{2}n\left(n+1\right)$are phases. And for Dirac fermions (2n-1) phases are eliminated through redefinition of field and only $\frac12(n-1)(n-2)$physical phases are left. So for parameterization of 2x2 unitary matrix, single mixing angle is required and hence the mixing matrix is obtained as [8]

$U\;=\;\begin{pmatrix}U_{\alpha1}&U_{\alpha2}\\U_{\beta1}&U_{\beta2}\end{pmatrix}\;=\;\begin{pmatrix}\cos\theta&\sin\theta\\-\sin\theta&\cos\theta\end{pmatrix}$

The discussion on how flavor mixing affects propagation of neutrinos in vacuum is proceeded with 2 flavor neutrino mixing in the atmospheric sector.

## Probability for two flavor neutrino oscillation

​The initial flavor state at a space-time point (0,0) is a linear combination of the two mass states:

$\vert\nu_{\alpha(0,0)}>\;=\;U_{\alpha k}\;\vert\nu_{k(0,0)}>$

This flavor state propagates from source to detector in its mass states and these mass states evolve in space-time as a plane wave. The Time-Dependent Schrὃdinger Equation (TDSE) for the neutrino state at a space-time point (x,t) is obtained as:​

$i\frac\partial{\partial t}\vert\nu_{k(x,t)}>=E\vert\nu_{k(x,t)}>=-\frac1{2m_k}\frac{\partial^2}{\partial x^2}\vert\nu_{k(x,t)}>\;\;:for\;k=1,2$

The solution, |νk(x,t)> will be:

$\vert\nu_{k(x,t)}>\;=\;e^{-i(E_kt-P_kx)}\;\vert\nu_{k(0,0)}>\;=\;e^{-i\phi_k}\;\vert\nu_{k(0,0)}>$

where Pk​ is the four momentum of neutrino mass state and x is the four space vector.

At some point the flavor state |να> evolves as:

$\vert\nu_{\alpha(x,t)}>\;=\;\underset{k=1,2}{\sum U_{\alpha k}}\;\vert\nu_{k(x,t)}>\;=\underset{k=1,2}{\sum U_{\alpha k}}\;e^{-i\phi_k}\;\vert\nu_{k(0,0)}>$

​On inversion of inital mixing matrix, we have:

$\vert\nu_{k(0,0)}>\;=\;\sum_\beta\;U_{\beta k}^\ast\vert\nu_{\beta(0,0)}>\;$

Therefore,

$\vert\nu_{\alpha(x,t)}>\;=\;\sum_{k=1,2}U_{\beta k}^\ast e^{-i\varphi_k}U_{\alpha k}\;\vert\nu_{\beta(0,0)}>$

Similarly, by using the bra-notation, we can obtain

$<\nu_{\beta(x,t)}\vert\;=\;\sum_{k=1,2}U_{\alpha k}e^{i\varphi_k}U_{\beta k}^\ast\;<\nu_{\alpha(0,0)}\vert$

The transition amplitude i.e, amplitude for detecting a neutrino of flavor β at the detector when flavor α is produced at the source is given by:

$\begin{array}{l}\begin{array}{l}A_{(\nu_{\alpha(0,0)}\rightarrow\nu_{\beta(x,t)})}\;=\;<\nu_{\beta(x,t)}\vert\nu_{\alpha(0,0)}>\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\;{\textstyle\sum_\alpha}{\textstyle\sum_k}\;U_{\alpha k}\;e^{i\phi_k}U_{\beta k}^\ast\;<\nu_{\alpha(0,0)}\vert\nu_{\alpha(0,0)}>\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\;{\textstyle\sum_k}U_{\alpha k}\;e^{i\phi_k}U_{\beta k}^\ast\;\;\\(since\;flavor\;states\;are\;orthogonal;<\nu_{\alpha(0,0)}\vert\nu_{\beta(0,0)}>\;=\;\delta_{\alpha\beta})\end{array}\\\end{array}$

The oscillation probability is the coherent sum:

$\begin{array}{l}\begin{array}{l}P_{(\nu_{\alpha(0,0)}\rightarrow\nu_{\beta(x,t)})}=\vert{\textstyle\sum_k}\;U_{\alpha k}\;e^{i\phi_k}U_{\beta k}^\ast\vert^2\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;={\textstyle\sum_j}\;U_{\alpha j}^\ast\;e^{-i\phi_j}U_{\beta j}\;{\textstyle\sum_k}U_{\alpha k}\;e^{i\phi_k}U_{\beta k}^\ast\end{array}\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;={\textstyle\sum_{j,k}}\;U_{\alpha j}^\ast U_{\beta j}U_{\alpha k}U_{\beta k}^\ast e^{-i(\phi_j-\phi_k)}\end{array}$

Using the mixing matrix the two flavor neutrino oscillation probability is calculated:

$k=1,\;j=1,\;P\;=\vert U_{\alpha1}\vert^2\vert U_{\beta1}\vert^2=\;\sin^2\theta\cos^2\theta$
$k=1,j=2,\;P=U_{\alpha1}U_{\alpha2}^\ast U_{\beta2}U_{\beta1}^\ast\;e^{-i(\phi_2-\phi_1)}=-\sin^2\theta\cos^2\theta\;e^{-i(\phi_2-\phi_1)}$
$k=2,j=1,\;P=U_{\alpha2}U_{\alpha1}^\ast U_{\beta1}U_{\beta2}^\ast\;e^{-i(\phi_1-\phi_2)}=-\sin^2\theta\cos^2\theta\;e^{-i(\phi_1-\phi_2)}$
$k=2,j=2,\;P=\vert U_{\alpha2}\vert^2\vert U_{\beta2}\vert^2=\sin^2\theta\cos^2\theta$

Hence,

$\begin{array}{l}P_{(\nu_{\alpha(0,0)}\rightarrow\nu_{\beta(x,t)})}\\=\;\vert U_{\alpha1}\vert^2\vert U_{\beta1}\vert^2+\vert U_{\alpha2}\vert^2\vert U_{\beta2}\vert^2+U_{\alpha1}U_{\alpha2}^\ast U_{\beta2}U_{\beta1}^\ast\;e^{-i(\phi_2-\phi_1)}+U_{\alpha1}^\ast U_{\alpha2}U_{\beta1}U_{\beta2}^\ast\;e^{i(\phi_2-\phi_1)}\\=\;2\sin^2\theta\cos^2\theta-\sin^2\theta\cos^2\theta\;(e^{i(\phi_2-\phi_1)}+e^{-i(\phi_2-\phi_1)})\\=\;2\sin^2\theta\cos^2\theta(1-\cos(\phi_2-\phi_1))\\=\;2\sin^2\theta\cos^2\theta\times2\sin^2(\frac{\phi_2-\phi_1}2\;)\\=\;\sin^2(2\theta)\sin^2(\;\frac{\phi_2-\phi_1}2\;)\end{array}$

The phase term (φ21)/2 can be calculated.​

We have

$\phi_i\;=\;E_it-P_ix$

​This implies,

$\phi_2-\phi_1\;=\;(E_2-E_1)t-(P_2-P_1)x$

Since neutrinos have relativistic properties, we can assume that t=x=L, and hence

$E_i^2=m_i^2+p_i^2$
$\Rightarrow\;\;p=\sqrt{E_i^2-m_i^2}=E_i\sqrt{1-\frac{m_i^2}{E_i^2}}=E_i(1-\frac{m_i^2}{2E_i^2})$
$\Rightarrow\phi_2-\phi_1=(E_2-E_1)L-\left[E_2(1-\frac{m_2^2}{2E_2^2})-E_1(1-\frac{m_1^2}{2E_1^2})\right]L$
$\Rightarrow\phi_2-\phi_1=\;\left(\frac{m_2^2}{2E_2}-\frac{m_1^2}{2E_1}\right)$

​Since we modeled mass eigenstates as plane waves, we can assume that mass eigenstates are created with either the same momentum or the same energy. Even if the analysis is made by assuming mass states as wave packets, we will get similar results; without assuming equal momentum for the mass states. Hence we obtain,

$\phi_2-\phi_1=\left(\frac{m_2^2}{2E_2}-\frac{m_1^2}{2E_1}\right)L=\frac{\triangle m_{21}^2}{2E}$

​ where E1=E2=E, and m22-m12= ∆m212

The probability equation turns out to be

$P_{\left(\nu_\alpha\rightarrow\nu_\beta\right)}=\sin^2\left(2\theta\right)\cdot\sin^2\left(\frac{\triangle m_{21}^2L}{4E}\right),in\;natural\;units$

If we measure L in units of kilometers and E in units of GeV and pay attention to all the ħ and c we will be left out with,

$\begin{array}{l}\frac{\triangle m_{21}^2\;L}{4\;E}\;in\;natural\;units\;=\;\frac{\triangle m_{21}^2(eV^2)\;\times\;L(Km)\;x\;10^{-3}}{4.E(GeV)\;\times\;0.197\;\times\;10^{-12}(eV)\;\times\;10^9(Km)}\\\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\;1.267\frac{\triangle m_{21}^2(eV^2)\;\times\;\;L\;(Km)}{E(GeV)}\end{array}$

Thus the probability equation is obtained as :

$P_{(\nu_{\alpha(0,0)}\rightarrow\nu_{\beta(x,t)})}\;\;=\;1.267\frac{\triangle m_{21}^2(eV^2)\;\times\;L\;(Km)}{E(GeV)}$

This is the complete Two Flavor Neutrino Oscillation probability or the Transition Probability.

The Survival Probability, the probability of detecting the same neutrino flavor as observed at the source can be obtained by subtracting transition probability from unity.

$\begin{array}{l}P_{(\nu_{\alpha(0,0)}\rightarrow\nu_{\alpha(x,t)})}\;\;=1-P_{(\nu_{\alpha(0,0)}\rightarrow\nu_{\beta(x,t)})}\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=1-1.267\frac{\triangle m_{21}^2(eV^2)\;\times\;L\;(Km)}{E(GeV)}\end{array}$

## Discussion on Probability Equation

$P_{(\nu_\alpha\;\rightarrow\;\nu_\beta)}=\;\sin^2(2\theta)\times\sin^2(1.267\times\triangle m^2\times L/E)$

The equation has to be analyzed:

1. The angle (θ): The mixing angle defines how neutrino flavor states differ from its mass states. At θ = 00, the oscillations will be minimum and hence no swapping from να to νβ will occur. All the να produced at the source will be detected as να itself at the detector. The oscillations are maximal at θ = π/4; all the να produced will be detected as νβ at the detector.

2. The mass squared difference (Δm2): Neutrino Oscillations occur due to the phase change produced by the mass eigenstates during the flight. The phase change is the result of differences in velocities arising due to the difference in mass eigenstates of these neutrinos. Hence it is clear that neutrinos should have some mass to exhibit neutrino oscillations. For neutrino oscillations to happen the neutrinos should have non-zero masses and it should be unequal.

3. L/E ratio: The only factor experimentalists can control. L is the baseline distance, the distance from source to detector and E is the energy of neutrinos the detector scans. The oscillation probability will change as L and E change. This factor can be manipulated to analyze the mass of neutrinos. If a particular value of Δm2 is known, L and E are altered to improve the sensitivity of the experiment, such that

$\begin{array}{l}\;\;\;1.267\triangle m^2\frac LE=\frac\pi2\\\end{array}$
$\Rightarrow\;\frac LE=\frac\pi{2.534\;\triangle m^2}$

The argument of sinefunction takes maximum value at each (2n+1)π/2 angle. The oscillation becomes maximum for

$1.267\triangle m^2\frac LE=(2n+1)\frac\pi2,\;where\;n=0,1,2,...$
The oscillation probability as a function of the baseline, L, for a given set of parameters: ∆m32 2= 3 × 10−3eV 2, sin2(2θ) = 0.8 and Eν = 1GeV (Figure taken from Prof. Mark Thomson’s Particle Physics lecture notes).

## ​ RESULTS AND DISCUSSIONS

In this section I present the results obtained to study Neutrino Oscillations. The transition probability and survival probability for various combinations of the parameters are obtained using C++ programs and GNU plots. I have studied the two flavor neutrino oscillations in the atmospheric sector, where νμ is oscillated to ντ, the mixing angle involved is θ23 and the mass squared difference is Δm322.

Survival probability curve for two flavor neutrino oscillation with θ23=450 and Δm322=0.003 eV2, for different energies

The probability of transition and survival is a sinusoidal function that oscillates from 0 to 1. The maximum amplitude of neutrino oscillation is decided by the ${\mathrm{sin}}^{2}\left(2\theta \right)$function and the oscillation length is decided by the $\sin^2(1.267\times\triangle m^2\times L/E)$ term in the probability equation. If ${\mathrm{sin}}^{2}\left(2\theta \right)$attains the value 1, oscillation is referred to as maximal and a 0 points no oscillation.

## The Plots

The Transition Probabilities in the atmospheric sector using νμ→ντ channel is obtained for a particular baseline (295 Km, 810 Km, 1300 Km and 3000 Km) by varying Δm322 values (2.4x10-3 eV2, 2.5x10-3 eV2, 2.6x10-3 eV2), for constant θ23=450..

Plot 1

Transition probability curve for 295 km baseline and θ23 = 450 for different values of Δm322

​By calculation of oscillation maxima, it can be found that the first oscillation maxima is obtained at 0.59 GeV, and second oscillation maxima at 0.198 GeV. The oscillation maxima is higher for higher mass squared difference and hence oscillation will depend on the mass states of two flavors.

Plot 2

Transition Probability Curve for 295 Km and Δm322 = 2.5x10-3 eV2 for different values of θ23

It is found that oscillation is maximum for θ23 = 450 and is reduced as θ23 is reduced. i.e, at θ23= 400 or 500, all of the νμ is not converted to ντ.

Plot 3

Transition Probability Curve for 810 Km baseline and θ23 = 450 for different values of Δm322

Altering the baseline, alters the probability and behaviour of oscillation changes. Now the oscillation maxima occurs at an energy of 1.63 GeV and second oscillation maxima occurs at 0.544 GeV. i.e, as the baseline length increased, peak of transition probability is also varied.

Plot 4

Transition probability curve for 810 Km and Δm322 = 2.5x10-3 eV2 for different values of θ23

In this plot also, one can observe that the oscillation maxima is determined by the mixing angle, whereas the frequency of oscillation is a characteristics of the L/E ratio.

Now, I can provide the Survival Probabilities in the atmospheric sector using νμ→νμ channel is obtained for a particular baseline (295 Km, 810 Km, 1300 Km and 3000 Km) by varying Δm322 values (2.4x10-3 eV2, 2.5x10-3 eV2, 2.6x10-3 eV2). It can be inferred that transition and survival probabilities will adds up upto unity.

Plot 5

Survival probability curve for 295 Km baseline and θ23 = 450 by varying Δm322

Plot 6

Survival probability curve for 295 Km and Δm322 = 2.5x10-3 eV2 for different values of θ23

Plot 7

Survival Probability Curve for 810 Km baseline and θ23 = 450 by varying Δm322

Plot 8

Survival Probability Curve for 810 Km and Δm322 = 2.5x10-3 eV2 for different values of θ23

I can present a few more plots to establish the atmospheric neutrino oscillation phenomena as a characteristic of L/E ratio and the mixing angle and the mass of neutrinos.

Plot 9

Transition probability curve with 1300 km baseline and Δm322 = 2.5x10-3 eV2 for different values of θ23

Plot 10

Survival probability curve for 1300 Km and θ23 = 450 for different values of Δm322

Plot 11

Transition probability curve for 3000 Km and Δm32 = 2.5x10-3 eV2 for different values of θ23

Plot 12

Survival Probability Curve for 3000 Km and Δm322=2.5x10-3 eV2 for different values of θ23

Plots 9 and 10, or plots 11 and 12 together show that probability of oscillation by Survival Channel and Disappearance Channel will always sum up to unity. The oscillation maxima in Transition Channel compliments the oscillation minima in Survival Channel and vice-versa.

## Explanation of atmospheric neutrino anomaly

We can summarize the results obtained for the atmospheric neutrino sector:

1. The peak of oscillation is the function of the mixing angle. It is found that oscillation is maximal for θ=450, and is slightly reduced for other mixing angles. This implies that muon neutrinos generated at the sources may not get fully oscillated to electron neutrinos, and it depends on the mixing angle.

2. The frequency of oscillation is controlled by several factors like Δm2, L and E.The variations in any of these factors will surely alter the plots. i.e, the probability to find a particular flavor of the neutrino is not always constant or identical, but changes as a function of parameters.

3. The Survival probability and the Transition probability will always sum up to unity.

4. The explanation of anomaly using the disappearance channel also suggests a mechanism of oscillation to sterile neutrino flavor, rather than to tau neutrino. However, we rule out this concept because sterile neutrinos are suppressed at higher energies because of the matter effects involved while neutrino propagation.

Since this project deals with mass induced flavor based neutrino oscillations in the vacuum, matter effects on neutrino oscillation is the very next, new arena to be studied.

The neutrino oscillation demands some mass to the neutrinos and through the probability equation, we are introduced to the mass squared differences. Super-K analysis released some results on m2 values, towards lower energy ranges, against the Kamiokande results that gave some higher values for m2. The lower m2 values favored by nature hinders the experiments like OPERA and MINOS to identify tau neutrinos. Later experiments like MACRO and Soudan released their neutrino oscillation studies and the compilation of these results gives the preferable values of mixing angle in the atmospheric sector to be sin2(2θ23)=0.92 and the m322=2.5×10-3 eV2, with 90% confidence level.​[7]​​[9]

The 90% confidence level regions for six atmospheric neutrino samples individually fit for a two-flavor atmospheric neutrino oscillation. The small bold region is the interval with all subsamples simultaneously fit.

## 40 Routes

As a part of studying neutrino sources and detectors, the propagation of neutrinos from different sources to detectors is analyzed and the distance the neutrinos have to traverse is obtained. A C++ program is written to obtain the distance the neutrino will travel (with the distance through crust, mantle, and core specified) between two different locations if their geographical positions are given (latitude and longitude). I have also studied the energy for first and second oscillation maxima for these base lengths. The results are also presented herewith. 40 different routes are analyzed and are given below. The notion considered is {North[+], South[-]} and {East[+],West[-]}.

The neutrino sources considered are:

• FermiLAB, Chicago (41.83194,-88.25722)
• CERN, Geneva (46.233567,6.055202)
• RAL, UK (51.57333,-1.31472)
• J-Parc, Japan (36.46170,140.59937)

The neutrino detectors are listed below:

• Homestake mine, SD, USA (44.3692,-103.7574)
• India-Based Neutrino Obsrvatory, India (9.956938,77.283062)
• Sudbury Neutrino Observatory LAB, SNOLAB, Canada (46.471857,-81.186755)
• Laboratori Nazionali del Gran Sasso, LNGS (42.419928,13.51721)
• Pyhasalmi Mine, Finland (63.66107, 26.03799)
• Canfranc Underground Laboratory, CUL, Spain (42.7491825,-0.518578)
• Kamioka Japan (36.42572,137.301306)
• China Jinping Underground Laboratory, CJUL, China(28.15323,101.7114)
• Modane Underground Laboratory (LSM) Frejus, France (45.17855,6.689021)
• Soudan, Minessota, USA (47.823333,-92.23722)

For the calculation, the radius of Earth was considered to be 6378 km, and the width of the crust as 78 km, that of mantle is 2900 km, and the radius of core as 3400 km.

Baseline and Energy Calculation for neutrinos from FermiLAB to various neutrino detectors.
 Neutrino Source Neutrino Detector Shortest Distance(km) Distance Traversed through Energy (GeV) Crust(km) Mantle(km) Core(km) 1st Oscillation Maxima 2nd Oscillation Maxima Fermi LAB Homestake mine 1286.87 1286.87 0 0 2.5950 0.8650 INO, India 11392.7 197.639 7546.26 3648.84 22.9733 7.6578 SNOLAB 764.343 764.343 0 0 1.5413 0.5138 Soudan 2, Minessota 736.685 736.685 0 0 1.4855 0.4950 LNGS 7342.05 310.588 7031.46 0 14.8052 4.9351 Pyhasalmi 6617.48 346.372 6271.11 0 13.34414 4.4480 CUL, Spain 6540.02 350.705 6189.32 0 13.1879 4.3960 Kamioka, Japan 9124.59 247.997 8876.59 0 18.3997 6.1332 CJUL, China 10410.8 216.659 10194.1 0 20.9933 6.9978 Frejus, France 6823.2 335.379 6487.82 0 13.7589 4.5863
Shortest distance calculation for neutrinios from CERN.
 Neutrino Source Neutrino Detector Shortest Distance(km) Distance Traversed through Energy (GeV) Crust(km) Mantle(km) Core(km) 1st Oscillation Maxima 2nd Oscillation Maxima CERN Homestake mine 7342.1 310.586 7031.51 0 14.8053 4.9351 INO, India 7305.78 312.198 6993.58 0 14.7321 4.9107 SNOLAB 6073.99 379.33 5694.65 0 12.2482 4.0827 Soudan 2, Minessota 6577.58 348.59 6228.99 0 13.2637 4.4212 LNGS 731.056 731.056 0 0 1.4742 0.4914 Pyhasalmi 2288.87 1408.7 880.172 0 4.6034 1.5345 CUL, Spain 649.636 649.636 0 0 1.3099 0.4366 Kamioka, Japan 8736.47 259.344 8447.13 0 17.6170 5.8724 CJUL, China 7650.73 297.538 7353.19 0 15.4277 5.1426 Frejus, France 127.456 127.456 0 0 0.2571 0.0857
Calculations for neutrinos from RAL
 Neutrino Source Neutrino Detector Shortest Distance(km) Distance Traversed through Energy (GeV) Crust(km) Mantle(km) Core(km) 1st Oscillation Maxima 2nd Oscillation Maxima Rutherford Appleton Laboratory (RAL) Homestake mine 6069.61 379.625 5689.99 0 12.2393 4.0797 INO, India 7778.34 292.464 7485.87 0 15.6850 5.2283 SNOLAB 5387.28 431.622 4955.66 0 10.8634 3.6211 Soudan 2, Minessota 5888.12 392.147 5495.97 0 11.8734 3.9578 LNGS 1510.49 1510.49 0 0 3.0459 1.0153 Pyhasalmi 2076.61 2076.61 0 0 4.1875 1.3958 CUL, Spain 983.275 983.275 0 0 1.9827 0.6609 Kamioka, Japan 8604.65 263.441 8341.2 0 17.3512 5.7837 CJUL, China 7831.84 290.389 7541.45 0 15.7929 5.2643 Frejus, France 923.886 923.886 0 0 1.8630 0.6210

Baseline and pathlength through Crust, Mantle and Core of Earth for neutrinos from J-PARC.
 Neutrino Source Neutrino Detector Shortest Distance(km) Distance Traversed through Energy (GeV) Crust(km) Mantle(km) Core(km) 1st Oscillation Maxima 2nd Oscillation Maxima J-PARC Homestake mine 8232.33 275.759 7956.57 0 16.6005 5.5335 INO, India 6638.55 345.212 6293.34 0 13.3866 4.4622 SNOLAB 8938.44 253.311 8685.13 0 18.0243 6.0081 Soudan 2, Minessota 8488.43 267.165 8221.26 0 17.1169 5.7056 LNGS 8821.64 256.765 8564.87 0 17.7888 5.9296 Pyhasalmi 7079.1 322.664 6756.44 0 14.2750 4.7583 CUL, Spain 9271.78 243.953 9027.82 0 18.6965 6.2322 Kamioka, Japan 295.147 295.147 0 0 0.5952 0.1984 CJUL, China 3694.02 663.913 3030.11 0 7.4489 2.4830 Frejus, France 8892.42 254.66 8637.76 0 17.9315 5.9772

## CONCLUSION

It all began with the beta decay spectrum. The identification of these ghost particles gave not only a solution to the beta decay spectrum but also opened a new branch in the High Energy Physics. This project begins with an explanation of the beta decay reaction with neutrinos. Subsequently, the elusive particle is studied. The neutrino sources and detectors are studied which lead us to the Solar and Atmospheric Neutrino Anomalies. These anomalies were explained successfully by introducing the concepts of Neutrino Oscillations. In this project, the fundamental studies in Neutrino Oscillation is made using two flavors. The plots for the transition and survival probabilities are obtained and analyzed. This method now is extended to the three flavor sector and the matter effects can also be included to study the complete neutrino oscillations. Further studies may lead us to the CP violation and the mass hierarchy issues, some of the open problems in high energy physics. In vast studies on neutrino oscillations can guide one to the wide arena of matter-antimatter asymmetry and the reason for the existence of life.

## ACKNOWLEDGEMENTS

This project had led a little fellow down in one corner of the Earth to a big world of research. I would like to thank the 'Science Academy' for providing me an opportunity to experience the vast areas of research for the past few months. Also, I will thank 'Institute Of Physics Bhubaneswar, Odisha' for providing the requirements for the project titled 'Atmospheric Neutrino Anomaly'.

I hereby express my sincere and thankful gratitude to Prof. Sanjib Kumar Agarwalla, my project guide, to have led me to this fascinating arena of Neutrino Physics. He had provided me the maximum possible research materials and spent his valuable time discussing and sharing the ideas from the basics. As a man of simplicity and care, he pampered us through the whole project.

I should express my love and gratitude to Dr.Jesly Jacob, my mentor, who consistently encouraged me to be a part of this fellowship program. I thank Ms.Soumya and Mr. Vinayan for their sequential care and help during the project period.

I will also mention my deep love towards my fellow groupmates MD Ful Hossain SK and Sujit Sahoo who taught and helped me to understand complex ideas and to my little sister Nandana Rajeev for her subsequent and timely help.

#### References

• Roy. Eighty years of neutrino physics. [arXiv:0809.1767[hep-ph]]

• Rajasekaran (2016). The story of the neutrino. [arXiv:1606.08715v1[physics.pop-ph]].

• Gouvea (2004). TASI Lectures on Neutrino Physics.[arXiv:hep-ph/0411274v1]

• Griffiths (2004). Introduction to Elementary Particles Wiley-VCH, Weinheim.

• Kajita et al. (2016). Establishing Atmospheric Neutrino Oscillations with Super-K. Science Direct, 908, 14-29.

• Kayser (2008). Neutrino Oscillation Phenomenology. [arXiv:0804.1121v3[hep-ph]]

• Kajita (2016). Discovery of Atmospheric Neutrino Oscillations. Reviews of Modern Physics,88.

#### Source

• Fig 3: https://neutrino-history.in2p3.fr/solar-neutrinos/
• Fig 4: http://www.sns.ias.edu/~jnb/SNviewgraphs/snviewgraphs.html
• Fig 5: PDG2014http://antares.in2p3.fr/News/news_nobel2015.html
• Fig 6: Takaaki Kajita in the Proceedings of the Japan Academy, Series B, Physical and Biological Sciences (10.2183/pjab.86.303)https://iopscience.iop.org/article/10.1088/1367-2630/6/1/194
• Table 1: https://warwick.ac.uk/fac/sci/physics/staff/academic/boyd/warwick_week/neutrino_physics/lec_oscillations.pdf
• Fig 7: https://www.sciencedirect.com/science/article/pii/S0550321316300554
• Fig 8: https://www.researchgate.net/publication/2008250_Neutrino_Oscillations_Masses_and_Mixing
• Fig 9: https://www.sciencedirect.com/science/article/pii/S0550321316300554
• Fig 10: S.Swordy http://physik.uibk.ac.at/hephy/lectures/seminar/2002ws/article2/
• Fig 11: http://www-sk.icrr.u-tokyo.ac.jp/sk/sk/neutrino-e.html
• Fig 12: https://warwick.ac.uk/fac/sci/physics/staff/academic/boyd/warwick_week/neutrino_physics/lec_oscillations.pdf
• Fig 26: https://www.sciencedirect.com/science/article/pii/S0550321316300554
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