My friend, Lin Tang, has been my source of encouragement, support, and happiness throughout this work. Finally, this work is especially dedicated to my parents, who sacrificed so much to provide the best opportunities for my brother and me.
Histories of Neutrino
Brief Thermal History of Neutrino
After the neutrino becomes non-relativistic, gravity begins to affect the distribution of relic trinos. With the current cosmic microwave background temperature T0 = 2.73 K, n0ν = 56 cm−3 is therefore the current number density of relic trinos per taste.
Brief History of Neutrino Studies
The free-flowing neutrino becomes non-relativistic when the temperature of the universe falls below the neutrino mass. Twenty-six years after the neutrino was named, the neutrino was first discovered in a nuclear reactor experiment by Cowan and Reines [11].
Current Knowledge of Neutrino
Due to the complexity of the transmission probability, P(E0, z), there is no general analytical solution for the observable spectrum F(E0). In the study of the viability of the (2+2) scheme, the normal (2+2) scheme will be used and the conclusion will be the same for the reverse (2+2) scheme.
NEUTRINO MASS AND THE Z-BURST MODEL
The Original Z-Burst Model
Still, the secondary particles in the Z-Burst process, which are caused by EHEνCR above the GZK cut-off energy, are one of the models to offer plausible explanations of this phenomenon [ 59 ]. Recent studies comparing different proposals to measure the neutrino mass have recommended the Z-Burst model as one of the most promising ways to detect the CνB and measure the absolute neutrino mass in the near future [ 60 , 61 ].
Improve The Z-Burst Model With New Cosmology Model
This is still a good approximation, although the neutrino mass is not as large as expected then. 1.2, the heaviest neutrino will have a mass about 10−1 eV, which is still considerably larger than the momentum, about 0.7 meV, of the relic trinos.
Neutrino Cosmic Ray Spectroscopy
- Extremely High-Energy Neutrino Cosmic Ray Emission Spectrum 11
Together, these EHνCR source distribution models should include possibilities for decreasing Z-Burst absorption. But the Z-Burst absorption dip maximum is usually located near the low energy end.
Overview
It will therefore have no significant contribution to the Z-Burst absorption drop in the relative observable spectrum of the neutrino cosmic ray. This absolute mass is essential for detecting the Z-Burst absorption dip in future experiments.
NUMERICAL CALCULATION AND RESULTS
Overview of the Method and the Study
- The Numerical Integration Method
- The Parameters of Z-Burst Model
A total of seven parameters are involved in the calculation of the relative observable spectrum of neutrino cosmic rays. The different contributions of these parameters to the final results of the relative observable spectrum of cosmic ray neutrino will be investigated in the next section.
The Non-Contributing Parameters
- The Curvature Energy Fraction Ω k
- The Mass Energy Fraction Ω M
- The Hubble Constant h
- Coefficient of the Power-Law Spectrum α
The coefficient of the power law spectrumα plays a more complicated role for the Z-Burst absorption dip in the relative observable spectrum of neutrino cosmic rays. In fact, the maximum difference of the relative observable spectrum of cosmic ray neutrino between α = 1 and α = 3 is about 3%.
The Contributing Parameters
- The Mass of Neutrino
- Relic Neutrino Number Density n 0
- Source Distribution of Extremely High-Energy Neutrino Cosmic Rays 27
The exact position is also determined by the source distribution of cosmic ray neutrinos. The source distribution of extremely high-energy neutrino cosmic rays plays a key role in determining the absorption dip of a Z-burst.
Discussion and Conclusion
- Neutrino Flavors
- Z-Burst Absorption Signatures
- Experimental Aspects
- Conclusion
In this chapter, the sum rule (2+2) of the neutrino model with all small mixing angles ² and the effect of ground matter will be calculated using a numerical method. 27 is an obvious relaxation of the sum rule when small mixing angles² and the matter effect are included in the neutrino oscillation. In this work, the roles of the three small mixing angles ² and the effect of earth substance in the sum rule are studied.
2+2 NEUTRINO OSCILLATION AND SUM RULE
Basics of Neutrino Oscillation With Two Neutrino Oscillation
The three phases in a general 2×2 unitary matrix were absorbed in redefinition of the ν fields. When the neutrino beam travels in vacuum, the evolution of the beam, in the flavor basis, can be described as. Since the mass eigenstates |νji are the eigenstates of the free space Hamiltonian H, so hνj|H|νki=Ejδj,k,.
Neutrino Oscillation Experiments
- Solar Neutrino Oscillation
- Atmospheric Neutrino Oscillation
- Man-made Neutrino Oscillation Experiments
Some solar neutrino experiments can detect neutrinos with energies above 0.2 MeV. The LSND and KARMEN experiments are designed to detect the appearance of electron neutrinos in a muon neutrino beam. The CHOOZ and KamLAND experiments are designed to detect the disappearance of electron antineutrinos, while the controversial LSND [27].
Four-Neutrino Oscillation’s Parameters
- Four-Neutrino Mass Spectra
The effects of the CP-violating phases are essentially to change the allowed ranges of the three small mixing angles ²µµ, ²µe and ²ee from [0, π/2] to [−π/2, π/2]. So, in this work, the CP-violating phases will be replaced by letting the values of the three small mixing angles have both positive and negative signs. Three small mixing angles: The allowable ranges for the three small mixing angles are correlated.
Matter Effect in Neutrino Oscillation
The effects that matter has on neutrino oscillations are called the MSW matter effect after the three dominant players in the theoretical formulation of resonance. When all small mixing angles² are turned on, the effect of MSW matter is very important in solar and atmospheric neutrino oscillations due to solar matter and terrestrial matter, respectively. This contribution of the ground matter effect is studied numerically in the next section and analytically in Appendix A.
Sum Rule and Product Rule of (2+2) Neutrino Model
This is because the potential matrix induced by matter is diagonal to the basis of the flavor. From this equation, the oscillation amplitude for solar neutrinos arriving at the earth at. It will be shown through numerical calculation in the next chapter that small mixing angles ² and the effect of soil matter can change the unitary result quite significantly.
Formalism of (2+2) Neutrino Oscillation in Matter
27, the relative importance of terrestrial matter effects for atmospheric neutrino oscillation with small mixing angles² is evident. Relaxing the sum rule raises a question about the assumptions used in fitting the global data. Also, it is believed that the inclusion of small mixing angles, ²µe and ²µ, in the global.
NUMERICAL CALCULATION AND RESULTS OF (2+2) SUM RULE
Numerical Calculation Procedure
- Solar Neutrino Calculation
- Atmospheric Neutrino Calculation
- The Exclusion Regions From Experiments
For each energy value, the procedure described in the previous section will be performed to obtain the probabilities. All probabilities for energy points will be averaged to obtain the overall probabilities of the specified energy bin. 60 also describes the situation where the neutrino beam travels only through the Earth's mantle.
Calculation Results
- Zero th Order Sum Rule
- Sum Rule With Small Mixing Angles and The Earth Matter Effect 63
- Sum Rule for Anti-neutrinos
- Product Rule of (2+2) Neutrino Oscillation
In this section, the Sum Rule will be studied with all the small mixing angles non-zero and the earth material effect turned on. To answer this important question, the Sum Rule is calculated with only one of the small mixing angle ²s turned on and with the earth matter effect turned on. The ²µe's significant effect on the Sum Rule indicates that at least the small mixing angles.
Conclusions
The last of the small mixing angles²ee does not contribute much to the deviation from the Sum rule. In the current study, only one of the small mixing angles ²µe must be non-zero. Instead, we show that resonances are not crucial for the relaxation of the sum rule for the (2+2) neutrino model.
WHAT HAVE WE LEARNED
Viability of Sterile Neutrino and (2+2) Sum Rule
The four-neutrino models and associated sterile neutrinos required by the LSND data have been questioned due to recent poor fit with global data and due to a contradiction between data and the sum rule of the (2+2) model. In short, the arguments against the (2+2) model are weakened by the significant relaxation of the sum rule due to the inclusion of the small mixing angles² and the earth matter effect. The MiniBooNE experiment [31], which is already collecting data, will provide the final judgment on the validity of the LSND claim.
Detection of Relic Neutrino and the Neutrino Absolute Mass
Even assuming the 10−1 scale neutrino mass, detection of the Z-Burst dip before the year of 2013 remains difficult. If a much lighter neutrino mass is nature's choice, direct detection of the relic neutrino and measurement of the neutrino absolute mass by the Z-Burst absorption dip will remain just a beautiful theoretical idea without evidence. Even with this lighter neutrino mass, detection of the Z-Burst absorption dip is quite possible this decade, or during the next decade in the worst case scenario.
Future Work on the Small Mixing Angles and the Z-Burst Model
Either the high-energy edge of the Z-Burst absorption dip or the maximum of the absorption dip can be used to derive the absolute mass of the heaviest neutrino. As for measuring the dip in Z-burst absorption in the relative observable spectrum of cosmic ray neutrinos, patience is needed while we await the construction of the proposed detectors and the collection of data. Meanwhile, further neutrino oscillation experiments will provide better information on the neutrino mass spectrum, a useful input for Z-Burst absorption drop calculations.
The Mixing Matrix and The Hamiltonian with Earth Matter
It will greatly simplify the analytical procedure to find the eigenstates of the Hamiltonian with matter if the mass square part of the Hamiltonian, Eqn. The new basis, in which the square mass part of the Hamiltonian is in relative block-diagonal form, is called the proper basis. In the proper basis, the squared mass part of the Hamiltonian is relatively block diagonal.
Possible Resonances
Atmospheric Resonance
The motivation for moving to this correct basis is to transform the matter-effect Hamiltonian into this relatively simple structure, in which all significant terms are within the relative block diagonals and the off-diagonal elements are relatively smaller than the diagonal elements. In this form the following discussions of the resonances and the perturbation calculations can be performed much more easily.
Solar Resonance
Resonances Are Not Essential
If the other two small mixing angles are turned on, there may be more resonances, which may improve the relaxation of the sum rule of the (2+2) neutrino model.
Approximations
- Approximate Solution for 2 × 2 Matrix
- Atmospheric Block
- Solar Block
- Overall Transformation Matrix
- Final State From Initial | ν µ i Through Earth
These typical values will help to identify the small elements in the Hamiltonian with matter, thus making it easy to perform the perturbation calculation for the analytical study. We have verified with the numerical calculations that W can be discarded completely without losing much accuracy for the sterile neutrino oscillation probability Pµ→s, but not for the muon neutrino oscillation probability Pµ→µ. are the eigenvalues and corresponding unnormalized eigenvectors. So the final state of the neutrino after traveling through the earth is with constant nuclear density.
Average Over Energy and Zenith Angle
For the average over the zenith angle θz, or equivalently, the length through the earth in Eq.
Thus, the significant relaxation with respect to the zero-order sum rule is actually due to the effect of small angles on the muon-neutrino probability Pµ→µ, i.e., the muon-neutrino probability on non-muon neutrinos, Pµ→6µ. Another note worth noting is the apparently good agreement between the approximate result and the numerical result at small θτ sin Fig. So the almost perfect match with the numerical result around θτ with ~0.17 must be pure luck.
Muon Neutrino Oscillation Probability
This is exactly the atmospheric ratio for the zero-order sum rule without matter effect, discussed in section 4.5. The atmospheric ratio Ratm will therefore be slightly larger than the value of the zero-order sum rule without matter, as in Eq.A-54. This is precisely why the zeroth order sum rule with matter, plotted in Figure 23, is slightly above the zero order sum rule without matter.
Conclusion
- The Total Neutrino Mass versus The Heaviest Neutrino
- The Observable Spectrum of Neutrino Cosmic Ray
- The Relative Observable Spectrum of Neutrino Cosmic Ray
- The Standard ROS of Neutrino Cosmic Ray
- The Contribution of Ω k to Transmission Probability T
- The Contribution of Ω M in Transmission Probability T
- The Contribution of Ω M to the ROS of Neutrino Cosmic Ray
- The Contribution of h to Transmission Probability T
- The Contribution of h to the ROS of Neutrino Cosmic Ray
- The Contribution of α to the ROS of Neutrino Cosmic Ray
- The ROS of Neutrino Cosmic Ray for neutrino mass m ν = 0.2 eV
- The ROS of Neutrino Cosmic Rays Composed of Equal Flux of Neutrinos with
- The ROS of Neutrino Cosmic Rays Composed of Equal Flux of Neutrino with
- The ROS of Neutrino Cosmic Ray for Different Relic Neutrino Number Density
- The ROS of Neutrino Cosmic Rays from Gaussian and Step-function Sources 29
- The ROS of Neutrino Cosmic Rays from Gaussian Distributions with σ =
- The Current Upper Limit of Neutrino Cosmic Ray Flux
- The Improvement of The Future Experiments on Neutrino Cosmic Ray Flux . 36
- The Neutrino Beam Travel Through the Earth
- Classification of Super-Kamiokande Neutrino Events
- Zero th Order Sum Rule With Matter Effect
- The Sum Rule for 0.5 - 1.5 GeV
- The Sum Rule for 1.5 - 30 GeV
- The Sum Rule for 30 - 500 GeV
- The Sum Rule for 50 - 150 GeV
- Small Mixing Angles at 0.5 - 1.5 GeV
- Small Mixing Angles at 1.5 - 30 GeV
- Small Mixing Angles at 30 - 500 GeV
- Small Mixing Angles at 50 - 150 GeV
- The Sum Rule of Anti-neutrinos of 1.5 - 30 GeV
- The Product Rules of the (2+2) Model
- Parameter Values Yielding The Atmospheric Resonance, for E = 100 GeV
- The Solar Resonance
- The Approximate Analytical P µ → s Compared with the Exact Numerical Result 95
37] Karlsruhe Tritium Neutrino Experiment (KATRIN), http://www-ik.fzk.de/tritium/. Simkovic, Majorana neutrino masses, neutrinoless double beta decay and nuclear matrix elements, hep-ph. Lawrence et al., Haverah Park Collaboration, J. 62] Extreme Universe Space Observatory, http://www.euso-mission.org/. 63] Orbital Wide Angle Light Collectors, http://owl.gsfc.nasa.gov/. 65] Pierre Auger Observatory, http://www.auger.org/. Particle Data Group, http://pdg.lbl.gov), Phys. Gorham et al., An experimental limit on the cosmic diffuse flux of ultrahigh energy neutrinos, Astrof.
