Review Article | Open Access
A. Upadhyay, M. Batra, "Phenomenology of Neutrino Mixing in Vacuum and Matter", International Scholarly Research Notices, vol. 2013, Article ID 206516, 15 pages, 2013. https://doi.org/10.1155/2013/206516
Phenomenology of Neutrino Mixing in Vacuum and Matter
The current status and some perspectives of the phenomenology of massive neutrinos is reviewed. We start with the phenomenology of neutrino oscillations in vacuum and in matter. We summarize the results of neutrino experiments using solar, atmospheric. The fundamental theory of flavor changing neutrinos that has confirmed the neutrino oscillations and the various parameters affecting these oscillations have been discussed in detail. Specifically we will take the solar and atmospheric neutrino case. The oscillation plots will be discussed in detail, based on their behavior in vacuum and matter. Both normal and inverted mass hierarchy hypotheses are tested and both are consistent with observation. Finally the sensitivity of theta 13 over these probability oscillations has been analyzed and commented.
Neutrinos are the most elusive in nature. They interact much less through coupling with , and, boson. Several past and currently going neutrino experiments have not only resolved the mystery of solar and atmospheric neutrinos but also proved the massive nature of neutrinos. The flavor oscillation phenomenon leads to nonzero mass in the theories beyond a standard model in contradiction to the well-established standard model. The history of neutrino oscillation traces back to Gribov and Pontecorvo  but the credit for providing the evidence of neutrino oscillation goes to Cleveland et al.  for observing deficit in ’s neutrino flux about 1/3rd as given by the standard solar model  through the chlorine Homestake experiment . Several independent experiments like GALLEX  and SAGE  agreed with Cleveland et al.  by observing the neutrino flux more than half as expected from the solar model predictions. The mystery of solar neutrinos finally got resolved with the experiment Sudbury Neutrino Observatory  which proved that the deficit was due to flavor transitions. The experiment was different in the sense that it used a heavy water detector to detect 8B neutrinos via charged current interaction with electron only and neutral current interaction with total neutrino flux. Atmospheric neutrino anomaly is related to the electron-to-muon-neutrino ratio which reduces to half of the expected value and a possible explanation is the change of flavor from . Solar and atmospheric neutrino oscillations, neutrino masses, and mixing parameters have been proved by various detectors with different sensitivity and accuracy. The recent journey of detectors in neutrino physics starts from Super-Kamiokande , KamLAND , T2 K , MINOS,  and finally end with detectors like SOUDAN  and OPERA . The KamioKande  measured solar neutrinos to be half as per the SM whereas two experiments Gallex and Sage [4, 5] measured 56–60% of neutrino capture rate as predicted by the standard model. Then Super-Kamiokande I, II, and III [14–16] provided evidence for nonzero mass and also produced observation consistent with -neutrinos changing into -neutrinos. The latest searches at these experiments are more concerned with oscillation parameters, CP violation, and mass hierarchies. KAMLAND experiment was also able to investigate geographically produced antineutrinos [15, 17] and the best fitted values of eV2 and for as calculated using KAMLAND experiment. K2K [16, 18] was another long baseline experiment to study oscillation from to in the atmospheric region and confirmed the deficit of muon as observed in Super-Kamiokande. MINOS  at the Fermi National Laboratory studied muon oscillations produced from pion and kaon decay in the energy range of 1–10 GeV and focused primarily on the measurement of with the precision better than 10%.
The most recent parameter of our interest is because it has opened some of the most fundamental questions like CP violation. The nonzero value of may change the old picture of neutrino oscillation completely. Our main focus is the study of the flavor oscillations in the solar and atmospheric neutrinos. The detailed theory is presented with the aim that oscillation shows variations in vacuum and matter for the solar and atmospheric cases. The parameters affecting the neutrino oscillations are all included in our studies. Few experiments are analyzed in detail and their significance is mentioned. Special attention is given to the probability oscillation curve and its sensitivity to the recent bound on .
Neutrino oscillations are periodic transitions between different flavor neutrinos in neutrino beams. The theory of neutrino oscillations has been studied by various pioneer scientists via different approaches including classical treatment by moving them to quantum mechanical treatment of wave packets  and then finally with quantum field theoretic treatment . In this paper we present a detailed formalism for neutrino using the classical treatment only. The dependence of states on the time is given by the Schrodinger equation: where is the total Hamiltonian in flavor space and the general solution for (1) is where is the state at the initial time .
Flavor neutrino states and mass eigen states can be related through leptonic mixing matrix as At time , Application of the Hamiltonian operator will give the following: where The left-handed neutrino at the time is written as Similarly, Neutrino energies are different; hence the flavor oscillation in neutrinos gives the amplitude of the transition during the time and can be written as Analogously, the amplitude of the transition during the time is given by Probability for the transition of neutrinos and antineutrinos will be From (11), we can find possible relations between probabilities: A mixed neutrino state is characterized by their momentum with and mass with .
And Hence the energy difference between two such states will be where .
As , therefore Here the assumption is made that neutrinos are having different energy in different mass eigenstates. Let us suppose the denotes the difference between production and detection time for the ultrarelativistic neutrinos and is the distance between source and the detector: The unitary condition suggests that Hence Analogously, for the case of antineutrino, The transition probability is given as Finally for any complex and , , Similarly, for the antineutrino oscillation, probability becomes
3. Two-Flavor Oscillation Probability in Vacuum
For two-flavor neutrino oscillations, the mixing matrix can be defined in terms of some rotating angle in 2 dimensions where is actually PMNS matrix; therefore The weak eigenstates are rotated by an angle with respect to the mass eigenstates and to allow mixing between and . After some time where we have used For neutrinos to be relativistic, the substitution and can be made: Here represents the mixing between two mass eigen states, is length of source from the detector also known as baseline, is the energy of neutrinos produced from the source. The functional dependence on is called a spectral dependence where . The mass squared difference values and the mixing angle are the significant quantities to be measured. Physicists just probe the different mass eigenvalues and predict the mixing at which it occurs. Mixing angle dependence of the transition probability is expressed by . If we change from to , the mixing angle dependence remains as such which confirms with degeneracy of oscillation the probability for and . Two possibilities here correspond to two physically different mixings for two mass eigen states: if , the electron neutrino is composed more of , and if , then muon neutrino is composed more of . Moreover, transition to a different flavor is not possible if which led us to a survival probability. Direct information about the mixing angle can be obtained from the average neutrino oscillation probability .
4. Three-Flavor Probability Oscillations in Vacuum
In case of three-flavor neutrino oscillations, the standard parameterization of mixing matrix can be achieved by using three vectors, and performing the Euler rotations introduces three mixing angles and one complex phase factor. For three flavor and three mass eigen states, it can be written as Consider the unitary 3 × 3 mixing matrix for the Dirac neutrinos and introduce the standard parameters (three mixing angles and one phase) which characterize it: The first Euler rotation performed at the angle around the vector produces new orthogonal and normalized vectors as Here and ; Second rotation at the angle around vector the introduces the CP phase : In the matrix , where .
Similarly rotation around vector at the angle is as follows The phase is responsible for the effects of the CP violation which can take values from 0 to . The mixing angles are parameters for three-neutrino oscillation in vacuum; all real parts of the quadratic products of elements of the mixing matrix entering in the three-neutrino oscillation probabilities are given as . The individual probability expression for three neutrino flavors changing into others can be achieved by solving for individual matrix elements. It is shown below that the probability for neutrino oscillation depend on three mixing angle and two-mass-squared difference; Similarly other expressions can also be derived shown in Table 1.
From the table, it is clear that out of three types of mass mixing, , , and only two are independent: Thus neutrino oscillations are only sensitive to the mass squared difference in spite of the actual mass. The solar experiments have inferred the sign of from the MSW effect but the sign of mass squared difference in atmospheric is not known and the condition observed at experiments, , predicts that both kinds of neutrino mixing can occur; one is normal hierarchy (NH) having two light states and one heavier , and the other is inverted hierarchy (IH), is lightest state which assume masses in order . The two hierarchies are shown in Figure 1.
5. Atmospheric Neutrino Oscillation Probability
Atmospheric neutrinos are created by interactions of primary cosmic rays with nuclei in the atmosphere. The neutrinos generate upwardly-going and horizontal muons through decays. The phenomena of atmospheric neutrino oscillations arise from the deficit in upwardly and downward going muon neutrino. Let us calculate the probability for an electron neutrino changing into muon neutrino, as we already discussed that there are three possible cases for electron neutrino changing into muon neutrino, that is, from one mass eigenstate to another: For atmospheric neutrinos, The survival probability of electron neutrinos is In case of vacuum, Similarly we can calculate the probability for muon neutrinos changing into tau neutrinos: Now we will calculate the survival probability of muon neutrinos; that is, These are six probability terms from where we can find the probability for one neutrino changing into another depending upon their flavor.
6. Solar Neutrino Oscillation Probability
The difference in the number of solar neutrinos predicted from solar models and the number of neutrinos flowing through earth led to a solar neutrino problem; this created the solar neutrinos as the target for researchers as it can provide a more elaborated picture of stellar evolution and energy resources. The various experiments are focused to measure the solar neutrino flux. Solar neutrino events can also be analyzed by a Monte Carlo simulation study of uncertainties which made use of fluxes from 100 standard solar models . Solar neutrino experimental data constrains that mass squared difference is only taken where other mass differences are neglected: Similarly,
7. Neutrino Oscillations in Matter
Since neutrinos are weakly interacting, they might interact with matter either through charged-current (CC) or neutral-current (NC) interactions . For the charged current interactions, only the electrons participate via exchange. Mikheev and Smirnov  noticed a resonance behavior for specific oscillation and matter density parameters. Therefore the probabilities for neutrino oscillation differ from their vacuum counterparts. Neutral-current flavor interactions can occur for any type of neutrino flavor. Moreover, a neutral-current interaction leads to the addition of an extra term in Hamiltonian for flavor oscillation; such a term produces a shift in the eigenvalues, but charged-current interactions give the contribution not only in the form of a change in eigen states but also adds to Hamiltonian an energy proportional to , where = Fermi coupling constant and = number of electrons per unit volume.
Neutrinos are produced in flavor eigen states, , created by the interaction of weak gauge bosons with the charged leptons between the source, production point of the neutrinos, and the detector. The state of neutrinos with momentum satisfies the equation When is a free Hamiltonian, the state can be expanded over the total system of states of flavor neutrinos , with momentum , Here , , , and is the amplitude of probability to find in state which is described by .