Phenomenology of Neutrino Mixing in Vacuum and Matter

Abstract

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.

1. Introduction

Neutrinos are the most elusive in nature. They interact much less through coupling with 𝑊⁺,  𝑊⁻  and,  𝑍⁰ 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 [1] but the credit for providing the evidence of neutrino oscillation goes to Cleveland et al. [2] for observing deficit in 𝜈_𝑒’s neutrino flux about 1/3rd as given by the standard solar model [3] through the chlorine Homestake experiment [2]. Several independent experiments like GALLEX [4] and SAGE [5] agreed with Cleveland et al. [2] 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 [6] 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 𝜈_𝜇⇔𝜈_𝜏,𝜈_𝜇⇔𝜈_𝑒. 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 [7], KamLAND [8], T2 K [9], MINOS, [10] and finally end with detectors like SOUDAN [11] and OPERA [12]. The KamioKande [13] 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 𝜇-neutrinos changing into 𝜏-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 Δ𝑚²₂₁=7.58+0.14(stat)^+0.15_−0.15(syst)×10⁻⁵ eV² and tan𝜃₁₂=0.56+^0.10_−0.07(stat)^+0.10_−0.06(syst) for tan𝜃₁₂<1 as calculated using KAMLAND experiment. K2K [16, 18] was another long baseline experiment to study oscillation from 𝜈_𝜇 to 𝜈_𝑒 in the atmospheric region and confirmed the deficit of muon as observed in Super-Kamiokande. MINOS [17] 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 Δ𝑚²₂₃ with the precision better than 10%.

The most recent parameter of our interest is 𝜃₁₃ because it has opened some of the most fundamental questions like CP violation. The nonzero value of 𝜃₁₃ 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 𝜃₁₃.

2. Theory

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 [18] and then finally with quantum field theoretic treatment [19]. 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: 𝑖𝜕|Ψ(𝑡)⟩|Ψ(𝑡,𝜕𝑡=𝐻)⟩ (1) where 𝐻 is the total Hamiltonian in flavor space and the general solution for (1) is |Ψ(𝑡)⟩=𝑒^{−𝑖𝐻𝑡}|Ψ(0)⟩, (2) where |Ψ(0)⟩ is the state at the initial time (𝑡=0).

Flavor neutrino states and mass eigen states can be related through leptonic mixing matrix 𝑈 as 𝜈_𝑙=³_𝑖=0𝑈^∗_𝑙𝑖𝜈_𝑖. (3) At time (𝑡=0), |Ψ(0)⟩=𝜈_𝑖. (4) Application of the Hamiltonian operator will give the following: 𝐻𝜈_𝑖=𝐸_𝑖𝜈_𝑖, (5) where 𝐸_𝑖=𝑝²_𝑖+𝑚²_𝑖. (6) The left-handed neutrino at the time 𝑡≥0 is written as 𝜈_𝑙𝑡=𝑒^{−𝑖𝐻𝑡}𝜈_𝑙=³_𝑖=1𝑒^{−𝑖𝐸𝑖𝑡}𝑈^∗_𝑙𝑖𝜈_𝑖. (7) Similarly, 𝜈_𝑙𝑡=𝑒^{−𝑖𝐻𝑡}𝜈_𝑙=³_𝑖=1𝑒^{−𝑖𝐸𝑖𝑡}𝑈_𝑙𝑖𝜈_𝑖. (8) Neutrino energies 𝐸_𝑖(𝑖=1,2,3) are different; hence the flavor oscillation in neutrinos gives the amplitude of the transition 𝜈_𝑙→𝜈_𝑙′ during the time 𝑡 and can be written as 𝐴𝜈_𝑙⟶𝜈_𝑙′=³_𝑖=1𝑈_𝑖𝑙′𝑒^{−𝑖𝐸𝑖𝑡}𝑈^∗_𝑙𝑖. (9) Analogously, the amplitude of the transition 𝜈_𝑙→𝜈_𝑙′ during the time 𝑡 is given by 𝐴𝜈_𝑙⟶𝜈_𝑙′=³_𝑖=1𝑈^∗_𝑖𝑙′𝑒^{−𝑖𝐸𝑖𝑡}𝑈_𝑙𝑖. (10) Probability for the transition of neutrinos and antineutrinos will be 𝑃𝜈_𝑙⟶𝜈_𝑙′=³_𝑖=1𝑈_𝑖𝑙′𝑒^{−𝑖𝐸𝑖𝑡}𝑈^∗_𝑙𝑖²,𝑃𝜈_𝑙⟶𝜈_𝑙′=³_𝑖=1𝑈^∗_𝑖𝑙′𝑒^{−𝑖𝐸𝑖𝑡}𝑈_𝑙𝑖². (11) From (11), we can find possible relations between probabilities: _𝑙𝑃𝜈_𝑙⟶𝜈_𝑙′=1,_𝑙′𝑃𝜈_𝑙⟶𝜈_𝑙′=1,_𝑙′𝑃𝜈_𝑙⟶𝜈_𝑙′=1,_𝑙𝑃𝜈_𝑙⟶𝜈_𝑙′=1. (12) A mixed neutrino state is characterized by their momentum 𝑝 with 𝑝_𝑖=𝑝 and mass 𝑚_𝑖 with 𝑚²_𝑖/𝑝²≪1.

And 𝐸_𝑖𝑚≅𝑝+^2𝑖.2𝑝 (13) Hence the energy difference between two such states will be 𝐸_𝑖−𝐸_𝑗=Δ𝑚²_𝑗𝑖,2𝑝 (14) where Δ𝑚²_𝑗𝑖=𝑚²_𝑖−𝑚²_𝑗.

As 𝐸≈𝑝, therefore 𝐸_𝑖−𝐸_𝑗=Δ𝑚²_𝑗𝑖.2𝐸 (15) Here the assumption is made that neutrinos are having different energy in different mass eigenstates. Let us suppose the 𝑡 denotes the difference between production and detection time for the ultrarelativistic neutrinos and 𝐿 is the distance between source and the detector: 𝐸𝑡≅𝐿,_𝑖−𝐸_𝑗𝑡=Δ𝑚²_𝑗𝑖𝑃𝜈2𝐸𝐿,_𝑙⟶𝜈_𝑙′=³_𝑖=1𝑈_𝑙′𝑖𝑒^{−𝑖(Δ𝑚2𝑗𝑖𝐿/2𝐸)}𝑈^∗_𝑙𝑖². (16) The unitary condition suggests that _𝑖𝑈_𝑙′𝑖𝑈^∗_𝑙𝑖=𝛿_𝑙𝑙′. (17) Hence 𝑃𝜈_𝑙⟶𝜈_𝑙′=𝛿_𝑙𝑙′+_𝑖≠𝑗𝑈_𝑖𝑙′(𝑒^{−𝑖(Δ𝑚2𝑗𝑖𝐿/2𝐸)}−1)𝑈^∗_𝑙𝑖². (18) Analogously, for the case of antineutrino, 𝑃𝜈_𝑙⟶𝜈_𝑙′=𝛿_𝑙𝑙′+_𝑖≠𝑗𝑈^∗_𝑖𝑙′𝑒^{−𝑖(Δ𝑚2𝑗𝑖𝐿/2𝐸)}−1𝑈_𝑙𝑖². (19) The transition probability 𝜈_𝑙→𝜈_𝑙′ is given as 𝑃𝜈_𝑙⟶𝜈_𝑙′=_𝑖,𝑘𝑈_𝑙′𝑖𝑈^∗_𝑙′𝑘𝑈^∗_𝑙𝑖𝑈_𝑙𝑘𝑒^{−𝑖(Δ𝑚2𝑗𝑖/2𝐸)𝐿}=_𝑖𝑈_𝑙′𝑖²𝑈_𝑙𝑖²+2Re_𝑖>𝑘𝑈_𝑙′𝑖𝑈^∗_𝑙′𝑘𝑈^∗_𝑙𝑖𝑈_𝑙𝑘𝑒^{−𝑖(Δ𝑚2𝑗𝑖/2𝐸)𝐿},_𝑖𝑈_𝑙′𝑖²𝑈_𝑙𝑖²=𝛿_𝑙𝑙′−2Re_𝑖>𝑘𝑈_𝑙′𝑖𝑈^∗_𝑙𝑘𝑈^∗_𝑙𝑖𝑈_𝑙𝑘=𝛿_𝑙𝑙′−2Re_𝑖>𝑘𝑈_𝑙′𝑖𝑈^∗_𝑙′𝑘𝑈^∗_𝑙𝑖𝑈_𝑙𝑘+2Re_𝑖>𝑘𝑈_𝑙′𝑖𝑈^∗_𝑙′𝑘𝑈^∗_𝑙𝑖𝑈_𝑙𝑘𝑒^{−𝑖(Δ𝑚2𝑗𝑖/2𝐸)𝐿}=𝛿_𝑙𝑙′−2Re_𝑖>𝑘𝑈_𝑙′𝑖𝑈^∗_𝑙′𝑘𝑈^∗_𝑙𝑖𝑈_𝑙𝑘×1−𝑒^{−𝑖(Δ𝑚2𝑘𝑖𝐿/2𝐸)}. (20) Finally for any complex 𝑎 and 𝑏, Re(𝑎𝑏)=Re(𝑎)Re(𝑏)−Im(𝑎)Im(𝑏), 𝑃𝜈_𝑙⟶𝜈_𝑙′=𝛿_𝑙𝑙′−2Re_𝑖>𝑘𝑈_𝑙′𝑖𝑈^∗_𝑙′𝑖𝑈^∗_𝑙𝑖𝑈_𝑙𝑘×1−cosΔ𝑚²_𝑘𝑖𝐿2𝐸+2_𝑖>𝑘Im𝑈_𝑙′𝑖𝑈^∗_𝑙′𝑘𝑈^∗_𝑙𝑖𝑈_𝑙𝑘sinΔ𝑚²_𝑘𝑖𝐿.2𝐸 (21) Similarly, for the antineutrino oscillation, probability becomes 𝑃𝜈_𝑙⟶𝜈_𝑙′=𝛿_𝑙𝑙′−2Re_𝑖>𝑘𝑈_𝑙′𝑖𝑈^∗_𝑙′𝑖𝑈^∗_𝑙𝑖𝑈_𝑙𝑘×1−cosΔ𝑚²_𝑘𝑖𝐿2𝐸−2_𝑖>𝑘Im𝑈_𝑙′𝑖𝑈^∗_𝑙′𝑘𝑈^∗_𝑙𝑖𝑈_𝑙𝑘sinΔ𝑚²_𝑘𝑖𝐿.2𝐸 (22)

3. Two-Flavor Oscillation Probability in Vacuum

For two-flavor neutrino oscillations, the mixing matrix can be defined in terms of some rotating angle 𝜃 in 2 dimensions where 𝑈 is actually PMNS matrix; therefore 𝜈_𝑒𝜈_𝜇=cos𝜃sin𝜃𝜈−sin𝜃cos𝜃𝜈¹₂,𝜈_𝑒(𝑡=0)=𝜈_𝑒=cos𝜃𝜈₁+sin𝜃𝜈₂,𝜈_𝜇(𝑡=0)=𝜈_𝜇=−sin𝜃𝜈₁+cos𝜃𝜈₂. (23) The weak eigenstates are rotated by an angle 𝜃 with respect to the mass eigenstates |𝜈₁⟩ and |𝜈₂⟩ to allow mixing between 𝜈_𝜇 and 𝜈_𝑒. After some time 𝑡𝜈_𝜇(𝑡=𝑡)=𝜈_𝜇(𝑡)=−sin𝜃𝜈₁𝑒^{(−𝑖𝐸1𝑡/ℎ)}+cos𝜃𝜈₂𝑒^{−𝑖𝐸2𝑡/ℎ}=−sin𝜃𝜈₁𝑒^{−𝑖(𝑝+(𝑚21/2𝑝))𝑡/ℎ}+cos𝜃𝜈₂𝑒^{−𝑖(𝑝+(𝑚22/2𝑝))𝑡/ℎ}, (24) where we have used 𝐸₁=𝑝²+𝑚²₁^1/2,𝐸₂=𝑝²+𝑚²₂^1/2,𝑃𝜈_𝑒⟶𝜈_𝜇=𝜈_𝜇∣𝜈_𝑒(𝑡)²,𝜈_𝜇=cos𝜃𝜈₁+sin𝜃𝜈₂,𝑃𝜈_𝑒⟶𝜈_𝜇=𝜈_𝜇∣𝜈_𝑒(𝑡)²=𝑒^{𝑖𝑧−𝑖𝑧}sin²𝜃cos²𝜃1−𝑒^{(𝑖Δ𝑚2/2𝑝)𝑥}×1−𝑒^{(−𝑖Δ𝑚2/2𝑝)𝑥}. (25) For neutrinos to be relativistic, the substitution 𝑝=𝐸_𝑣 and 𝑥=𝐿 can be made: 𝑃𝜈_𝑒⟶𝜈_𝜇=sin²𝜃cos²𝜃1−𝑒^{(𝑖Δ𝑚2/2𝐸𝑣)𝐿}×1−𝑒^{(−𝑖Δ𝑚2/2𝐸𝑣)𝐿}.𝑃𝜈_𝑒⟶𝜈_𝜇(𝐿,𝐸)=sin²2𝜃sin²1.27Δ𝑚²𝐿𝐸_𝑣. (26) Here 𝜃 represents the mixing between two mass eigen states, 𝐿 is length of source from the detector also known as baseline, 𝐸 is the energy of neutrinos produced from the source. The functional dependence on 𝐿/𝐸 is called a spectral dependence where 𝐹(𝐿/𝐸)=𝐿/𝐸. 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 (sin2𝜃)². If we change from 𝜃 to 𝜋/2−𝜃, the mixing angle dependence remains as such which confirms with degeneracy of oscillation the probability for 𝜃 and 𝜋/2−𝜃. Two possibilities here correspond to two physically different mixings for two mass eigen states: if 𝜃<𝜋/4, the electron neutrino is composed more of 𝜈₁, and if 𝜃>𝜋/4, then muon neutrino is composed more of 𝜈₂. Moreover, transition to a different flavor is not possible if Δ𝑚²𝐿/2𝐸≪1 which led us to a survival probability. Direct information about the mixing angle can be obtained from the average neutrino oscillation probability ⟨𝑃(𝜈_𝛼→𝜈_𝛽)⟩=(1/2)(sin2𝜃)².

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 𝜈₁𝜈₂𝜈₃𝜈=𝑈𝜈^𝑒_𝜇𝜈_𝜏. (27) 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: |𝑖),⟨⟩⟩(𝑖=1,2,3𝑖∣𝑘=𝛿_𝑖𝑘. (28) The first Euler rotation performed at the angle 𝜃₁₂ around the vector |3⟩ produces new orthogonal and normalized vectors as |1⟩⁽¹⁾=𝑐₁₂|1⟩+𝑠₁₂|2⟩,|2⟩⁽¹⁾=−𝑠₁₂|1⟩+𝑐₁₂|2⟩,|3⟩⁽¹⁾=|3⟩. (29) Here 𝑐₁₂=cos𝜃₁₂ and 𝑠₁₂=sin𝜃₁₂; |𝜈⟩⁽¹⁾=𝑈⁽¹⁾|𝜈⟩|𝜈⟩suchthat₍₁₎=1⁽¹⁾2⁽¹⁾3⁽¹⁾,|𝜈⟩=|1⟩|2⟩|3⟩,𝑈⁽¹⁾=𝑐₁₂𝑠₁₂0−𝑠₁₂𝑐₁₂0.001 (30) Second rotation at the angle 𝜃₁₃ around vector the |2⟩⁽¹⁾ introduces the CP phase 𝛿: |1⟩⁽²⁾=𝑐₁₃|1⟩⁽¹⁾∣+𝑠₁₃𝑒^−𝑖𝛿|3⟩⁽¹⁾,|2⟩⁽²⁾=|2⟩⁽¹⁾,|3⟩⁽²⁾=−𝑠₁₃𝑒^−𝑖𝛿|1⟩⁽¹⁾+𝑐₁₃|3⟩⁽¹⁾. (31) In the matrix |𝑣⟩⁽²⁾=𝑈⁽²⁾|𝑣⟩⁽¹⁾,  where  𝑈⁽²⁾=^{𝑐130𝑠13𝑒−𝑖𝛿010}_{−𝑠13𝑒−𝑖𝛿0𝑐13}.

Similarly rotation around vector |1⟩⁽²⁾ at the angle 𝜃₂ is as follows |1⟩^mix=|1⟩⁽²⁾,|2⟩^mix=𝑐₂₃|2⟩⁽²⁾+𝑠₂₃|3⟩⁽²⁾,|3⟩^mix=−𝑠₂₃|2⟩⁽²⁾+𝑐₂₃|3⟩⁽²⁾,𝑣^mix=𝑈⁽³⁾|𝑣⟩⁽²⁾,𝑈⁽³⁾=1000𝑐₂₃𝑠₂₃0−𝑠₂₃𝑐₂₃,𝑣^mix=𝑈|𝑣⟩,where𝑈=𝑈⁽³⁾𝑈⁽²⁾𝑈⁽¹⁾,𝑈=1000𝑐₂₃𝑠₂₃0−𝑠₂₃𝑐₂₃𝑐₁₃0𝑠₁₃𝑒^−𝑖𝛿𝑠010₁₃𝑒^−𝑖𝛿0𝑐₁₃𝑐₁₂𝑠₁₂0−𝑠₁₂𝑐₁₂0,𝑐001𝑈=¹³𝑐₁₂𝑐₁₃𝑠₁₂𝑠₁₃𝑒^−𝑖𝛿−𝑐₂₃𝑠₁₂−𝑐₁₂𝑠₁₃𝑠₂₃𝑒^𝑖𝛿𝑐₁₂𝑐₂₃−𝑠₁₂𝑠₁₃𝑠₂₃𝑒^𝑖𝛿𝑐₁₃𝑠₂₃𝑠₂₃𝑠₁₂−𝑠₁₃𝑐₁₂𝑐₂₃𝑒^𝑖𝛿𝑐₁₂𝑠₂₃−𝑠₁₃𝑐₂₃𝑠₁₂𝑒^𝑖𝛿𝑐₁₃𝑐₂₃. (32) The phase 𝛿 is responsible for the effects of the CP violation which can take values from 0 to 2𝜋. 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 Re(𝑈_𝑙′𝑖𝑈^∗_𝑙′𝑘𝑈^∗_𝑙𝑖𝑈_𝑙𝑘). 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; Re𝑈_𝑙′𝑖𝑈^∗_𝑙′𝑘𝑈^∗_𝑙𝑖𝑈_𝑙𝑘=𝑈₂₂𝑈^∗₂₁𝑈^∗₁₂𝑈₁₁,for𝑙^′=𝑐=2,𝑙=1,𝑖=2,𝑘=1₂₃𝑐₁₂−𝑠₁₃𝑠₂₃𝑠₁₂𝑒^𝑖𝛿−𝑐₂₃𝑠₁₂−𝑠₁₃𝑠₂₃𝑐₁₂𝑒^−𝑖𝛿×𝑐₁₃𝑠₁₂𝑐₁₃𝑐₁₂1=−4𝑐²₁₃sin2𝜃₁₂×sin2𝜃₁₂𝑐²₂₃−𝑠²₂₃𝑠²₁₃+cos2𝜃₁₂sin2𝜃₂₃𝑠₁₃cos𝛿₁₃,Re𝑈_𝑙′𝑖𝑈^∗_𝑙′𝑘𝑈^∗_𝑙𝑖𝑈_𝑙𝑘=𝑈₂₃𝑈^∗₂₂𝑈^∗₁₃𝑈₁₂=𝑐₁₃𝑠₂₃𝑐₂₃𝑐₁₂−𝑠₁₃𝑠₂₃𝑠₁₂𝑒^−𝑖𝛿𝑠₁₃𝑒^𝑖𝛿𝑐₁₃𝑠₁₂=−𝑐²₁₃𝑠₁₃𝑠₁₂𝑠₂₃𝑠₁₃𝑠₂₃𝑠₁₂−𝑐₂₃𝑐₁₂cos𝛿₁₃,Re𝑈_𝑙′𝑖𝑈^∗_𝑙′𝑘𝑈^∗_𝑙𝑖𝑈_𝑙𝑘=𝑈₂₃𝑈^∗₂₁𝑈^∗₁₃𝑈₁₁=𝑐₁₃𝑠₂₃−𝑐₂₃𝑠₁₂−𝑠₁₃𝑠₂₃𝑐₁₂𝑒^−𝑖𝛿𝑠₁₃𝑒^𝑖𝛿𝑐₁₃𝑐₁₂=−𝑐²₁₃𝑠₁₃𝑐₁₂𝑠₂₃𝑐₂₃𝑠₁₂𝑒^𝑖𝛿+𝑠₁₃𝑠₂₃𝑐₁₂. (33) Similarly other expressions can also be derived shown in Table 1.

Table 1 

From the table, it is clear that out of three types of mass mixing, Δ𝑚²₂₁,  Δ𝑚²₃₁,  and  Δ𝑚²₃₂ only two are independent: Δ𝑚²₃₂+Δ𝑚²₂₁−Δ𝑚²₃₁=0. (34) 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 Δ𝑚² from the MSW effect but the sign of mass squared difference in atmospheric is not known and the condition observed at experiments, Δ𝑚²_sol≪Δ𝑚²_atmos, predicts that both kinds of neutrino mixing can occur; one is normal hierarchy (NH) having two light states and one heavier (𝑚₁≪𝑚₂<𝑚₃)_, and the other is inverted hierarchy (IH), 𝑚₃ is lightest state which assume masses in order (𝑚₃≪𝑚₁<𝑚₂) [19]. The two hierarchies are shown in Figure 1.

Figure 1 

Normal and inverted hierarchy.

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: 𝑃𝜈_𝑒⟶𝜈_𝜇=𝛿₁₂−1−44𝑐²₁₃sin2𝜃₁₂×sin2𝜃₁₂𝑐²₂₃−𝑠²₂₃𝑠²₁₃+cos2𝜃₁₂sin2𝜃₂₃𝑠₁₃cos𝛿₁₃sinΔ𝑚²₂₁𝐿2𝐸+𝛿₃₂−4−𝑐²₁₃𝑠₁₃𝑠₁₂𝑠₂₃𝑠₁₃𝑠₂₃𝑠₁₂−𝑐₂₃𝑐₁₂cos𝛿₁₃×sinΔ𝑚²₃₂𝐿4𝐸+𝛿₃₁−4−𝑐²₁₃𝑠₁₃𝑐₁₂𝑠₂₃𝑐₂₃𝑠₁₂𝑒^𝑖𝛿+𝑠₁₃𝑠₂₃𝑐₁₂×sinΔ𝑚²₃₁𝐿.4𝐸 (35) For atmospheric neutrinos, Δ𝑚²₃₂≅Δ𝑚²₃₁≅Δ𝑚²_atm,Δ𝑚²₂₁≅Δ𝑚²_sol𝑃𝜈≅0,_𝑒⟶𝜈_𝜇=−4−𝑠²₁₂𝑠²₂₃𝑠²₁₃𝑐²₁₃+𝑠₁₂𝑠₂₃𝑠₁₃𝑐₂₃𝑐₁₂𝑐²₁₃cos𝛿₁₃−𝑠²₁₃𝑐²₁₃−𝑠₁₂𝑠₂₃𝑠₁₃𝑐₂₃𝑐₁₂𝑐²₁₃cos𝛿₁₃×sinΔ𝑚²_atm𝐿4𝐸=sin²2𝜃₁₃sin²𝜃₂₃sinΔ𝑚²_atm𝐿,𝑃𝜈4𝐸_𝑒⟶𝜈_𝜏=sin²2𝜃₁₃cos²𝜃₂₃sinΔ𝑚²_atm𝐿.4𝐸 (36) The survival probability of electron neutrinos is 𝑃𝜈_𝑒⟶𝜈_𝑒=1−𝑃𝜈_𝑒⟶𝜈_𝜇+𝑃𝜈_𝑒⟶𝜈_𝜏=1−sin²2𝜃₁₃sinΔ𝑚²_atm𝐿.4𝐸 (37) In case of vacuum, 𝑃𝜈_𝑒⟶𝜈_𝜇=𝑃𝜈_𝜇⟶𝜈_𝑒,𝑃𝜈_𝜇⟶𝜈_𝑒=sin²2𝜃₁₃sin²𝜃₂₃sinΔ𝑚²_atm𝐿.4𝐸 (38) Similarly we can calculate the probability for muon neutrinos changing into tau neutrinos: 𝑃𝜈_𝜇⟶𝜈_𝜏=sin²2𝜃₂₃cos⁴𝜃₁₃sinΔ𝑚²_atm𝐿.4𝐸 (39) Now we will calculate the survival probability of muon neutrinos; that is, 𝑃𝜈_𝜇⟶𝜈_𝜇=1−𝑃𝜈_𝜇⟶𝜈_𝑒+𝑃𝜈_𝜇⟶𝜈_𝜏=1−sin²2𝜃₁₃sin²𝜃₂₃+sin²2𝜃₂₃cos⁴𝜃₁₃sinΔ𝑚²_atm𝐿.4𝐸 (40) 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 [20]. Solar neutrino experimental data constrains that mass squared difference Δ𝑚²₂₁ is only taken where other mass differences are neglected: 𝑃𝜈_𝑒⟶𝜈_𝜇=𝛿₁₂−1−44𝑐²₁₃sin2𝜃₁₂×sin2𝜃₁₂𝑐²₂₃−𝑠²₂₃𝑠²₁₃+cos2𝜃₁₂sin2𝜃₂₃𝑠₁₃cos𝛿₁₃×sinΔ𝑚²₂₁𝐿,𝑃𝜈2𝐸_𝑒⟶𝜈_𝜇−1=−44𝑐²₁₃sin2𝜃₁₂×sin2𝜃₁₂𝑐²₂₃−𝑠²₂₃𝑠²₁₃+cos2𝜃₁₂sin2𝜃₂₃𝑠₁₃cos𝛿₁₃×sinΔ𝑚²₂₁𝐿=2𝐸sin²2𝜃₁₂cos²𝜃₁₃×cos²𝜃₂₃−sin²𝜃₂₃sin²𝜃₁₃+14sin4𝜃₁₂sin2𝜃₁₃cos𝜃₁₃sinΔ𝑚²₂₁𝐿.2𝐸 (41) Similarly, 𝑃𝜈_𝑒⟶𝜈_𝜏=𝛿₁₂1−44𝑐²₁₃sin2𝜃₁₂×sin2𝜃₁₂𝑐²₂₃𝑠²₁₃−𝑠²₂₃+cos2𝜃₁₂×sin2𝜃₂₃𝑠₁₃cos𝛿₁₃×sinΔ𝑚²₂₁𝐿=4𝐸−sin²2𝜃₁₂cos²𝜃₁₃×cos²𝜃₂₃sin²𝜃₁₃−sin²𝜃₂₃−14sin4𝜃₁₂sin2𝜃₁₃sin2𝜃₂₃cos𝜃₁₃×sinΔ𝑚²₂₁𝐿,𝑃𝜈2𝐸_𝑒⟶𝜈_𝑒=1−sin²2𝜃₁₂cos⁴𝜃₁₃sinΔ𝑚²₂₁𝐿.2𝐸 (42)

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 [21]. For the charged current interactions, only the electrons participate via 𝑊^± exchange. Mikheev and Smirnov [22] 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 𝑉=√2𝐺_𝐹𝑁_𝑒, where 𝐺_𝐹 = Fermi coupling constant and 𝑁_𝑒 = number of electrons per unit volume.

Neutrinos are produced in flavor eigen states, |𝜐_𝛼⟩, (𝛼=𝑒,𝜇,𝜏) 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 |𝜓(𝑡)⟩ of neutrinos with momentum 𝑝 satisfies the equation 𝑖𝜕𝜕𝑡=𝐻₀𝜓(𝑡). (43) When 𝐻₀ is a free Hamiltonian, the state |𝜓(𝑡)⟩ can be expanded over the total system of states of flavor neutrinos 𝜈_𝑙, with momentum 𝑝, 𝜓(𝑡)=𝑎_𝑙(𝑡)𝜈_𝑙. (44) Here 𝜈_𝑙=_𝑖𝑈^∗_𝑙𝑖𝜈_𝑖, (45)𝐻₀|𝜈_𝑖⟩=𝐸_𝑖|𝜈_𝑖⟩,  𝐻₀|𝜈_𝑖⟩=𝐸_𝑖|𝜈_𝑖⟩,  𝐸_𝑖=𝑝²_𝑖+𝑚²_𝑖≃𝑝+𝑚²_𝑖/2𝐸, and 𝑎_𝑙(𝑡)=⟨𝜈_𝑙∣𝜓(𝑡)⟩ is the amplitude of probability to find 𝜈_𝑙 in state which is described by |𝜓(𝑡)⟩.

Therefore, 𝑖𝜕𝑎^′_𝑙(𝑡)𝜕𝑡=_𝑙𝜈^′_𝑙∣𝐻₀∣𝑎_𝑙(𝑡), (46) where 𝜈^′_𝑙∣𝜈_𝑙=𝑈_𝑙′𝑙,𝜈_𝑙∣𝜈_𝑙′=𝑈^∗_𝑙′𝑙. (47) Taking into account this relation, for the free Hamiltonian in the flavor representation we have the following expression: 𝜈_𝑙′∣𝐻₀∣𝜈_𝑙=_𝑙𝑈_𝑙′𝑖𝐸_𝑖𝑈^∗_𝑙𝑖∼𝑝+_𝑙𝑈_𝑙′𝑖𝑚²_𝑖𝑈2𝐸^∗𝑙𝑖. (48) Therefore neutrino evolution equation in the flavor representation is as follows: 𝑖𝜕𝑎(𝑡)𝑚𝜕𝑡=𝑈²𝑈2𝐸^†𝑎(𝑡). (49) Let us introduce the function 𝑎^′(𝑡)=𝑈^†𝑎(𝑡). (50) We find that the function 𝑎^′(𝑡) satisfies the following equation, multiplying by 𝑈^† on both sides, then 𝑖𝜕𝑎𝜕𝑡^′(𝑡)=𝑚²𝑎2𝐸^′(𝑡). (51) It is obvious that the solution of the equation has the form 𝑎^′(𝑡)=𝑒^{−𝑖(𝑚2/2𝐸)(𝑡−𝑡0)}𝑎^′(𝑡₀), where, 𝑎^′(𝑡₀) is the wave function at the initial time 𝑡₀.

As the neutrinos propagate in matter, electron neutrino plays a very special role due to coherent forward scattering of neutrinos from electrons so that it leads to an additional contribution in oscillation probability. The forward scattering of electrons of matter with scattering of electrons of matter with neutrinos is called charge current interaction. Therefore the probabilities for neutrino oscillation differ from their vacuum counterparts. Earth matter effects for the long baseline have been nicely studied earlier for both hierarchies [23]. Charged current interaction can give contribution only to the process of the elastic scattering of 𝜈_𝑒 on electrons. Moreover, neutral current interactions occur for all flavors, leading to the addition of an extra term in Hamiltonian for the flavor oscillation probability. For the low energy, an effective Hamiltonian of the neutrino interaction was obtained from the diagonal matrix element ⟨𝑝mat|𝐻^CC_𝐼|𝑝mat⟩, where 𝐻^CC_𝐼=(𝐺_𝐹/√2)2𝜈_𝑒𝐿(𝑥)𝛾_𝛼𝜈_𝑒𝐿𝑒(𝑥)𝛾^𝛼(1−𝛾₅)𝑒(𝑥) and the vector |𝑝mat⟩=|𝑝⟩|mat⟩. Now substituting the Hamiltonian we obtain ⟨𝑝mat|(𝐺_𝐹/√2)2𝜈_𝑒𝐿(𝑥)𝛾_𝛼𝜈_𝑒𝐿𝑒(𝑥)𝛾^𝛼(1−𝛾₅)𝑒(𝑥)|𝑝mat⟩=⟨𝑝|⟨mat|(𝐺_𝐹/√2)2𝜈_𝑒𝐿(𝑥)𝛾_𝛼𝜈_𝑒𝐿𝑒(𝑥)𝛾^𝛼(1−𝛾₅)𝑒(𝑥)|𝑝⟩|mat⟩.

But ⟨𝑝|⟨mat|(𝐺_𝐹/√2)2𝜈_𝑒𝐿(𝑥)𝛾_𝛼𝜈_𝑒𝐿𝑒(𝑥)𝛾^𝛼𝛾₅𝑒(𝑥)|𝑝⟩|mat⟩=0 as for unpolarized matter ⟨mat|𝑒(𝑥)𝛾^𝛼𝛾₅𝑒(𝑥)|mat⟩=0.

Also ⟨mat|𝑒(𝑥)𝛾^𝛼𝑒(𝑥)|𝑝⟩|mat⟩=⟨mat|𝑒(𝑥)𝑒(𝑥)|𝑝⟩|mat⟩𝛿_𝛼0=𝑛_𝑒(𝑥)𝛿_𝛼0, where 𝑛_𝑒(𝑥) is the number density at the point 𝑥. Also, ⟨𝑝|𝜈_𝑒𝐿(𝑥)𝛾_𝛼𝜈_𝑒𝐿|𝑝⟩=1.

Substituting all these values, 𝐻^mat_𝐼(𝑡)=√2𝐺_𝐹𝑛_𝑒(𝑡)𝛽. Using 𝛽_{𝜈𝑒,𝜈𝑒}=1, all other elements of matrix 𝛽 are equal to zero.

Let us now consider the NC interaction. Induced by the 𝑍⁰ exchange, the Hamiltonian of NC interactions of neutrinos with electrons and nucleons has the form 𝐻^CC_𝐼(𝑥)𝐺=2^𝐹√2_{𝑙=𝑒,𝜇,𝜏}𝜈_𝑙𝐿(𝑥)𝛾^𝛼𝜈_𝑙𝐿(𝑥)𝑗^NC_𝛼(𝑥), (52) where 𝑗^NC_𝛼(𝑥) is the sum of electron and nucleon (quark) neutral current. For the vector part of the effective hadron neutral current, 𝜈^NC_𝛼^(𝑁)(𝑥)=12𝑁(𝑥)𝛾_𝛼𝜏₃𝑁(𝑥)−2sin²𝜃_𝑤𝑝(𝑥)𝛾_𝛼𝑝(𝑥),𝑝𝑛as𝑁=,𝜏₃=,100−1 (53) where 𝜃_𝑤 is the weak angle. Similarly, for the effective part of electron current, 𝜈^NC_𝛼^(𝑛)(𝑥)=−12𝑛(𝑥)𝛾_𝛼𝑛(𝑥),𝜈^NC_𝛼^(𝑒)(𝑥)=−12+2sin²𝜃_𝑤𝑒(𝑥)𝛾_𝛼𝑒(𝑥). (54) For the corresponding matter matrix elements, we have mat∣𝜈^NC_𝛼^(𝑒)(𝑥)=−1∣mat2+2sin²𝜃_𝑤𝑛_𝑒(𝑥)𝛿_𝛼0,mat∣𝜈^NC_𝛼^(𝑝)(𝑥)=1∣mat2−2sin²𝜃_𝑤𝜌_𝑝(𝑥)𝛿_𝛼0,mat∣𝜈^NC_𝛼^(𝑛)(𝑥)1∣mat=−2𝜌_𝑛(𝑥)𝛿_𝛼0. (55) For the neutral matter, 𝑛_𝑒(𝑥)=𝜌_𝑝(𝑥) we conclude that the contributions of electron and proton NC to the effective Hamiltonian cancel each other. Thus mat∣𝑗^NC_𝛼(𝑥)1∣mat=−2𝜌_𝑛(𝑥)𝛿_𝛼0. (56) By taking into account the effective charged current interaction, only 𝜈_𝑒−𝑒 CC interaction gives a contribution to the effective Hamiltonian. Thus the evolution equation of neutrino has the form 𝑖𝜕𝑎(𝑡)=𝑈𝑚𝜕𝑡²𝑈2𝐸^†+√2𝐺_𝐹𝑛_𝑒(𝑡)𝛽𝑎(𝑡). (57) Similarly, for the antineutrinos, effective Hamiltonian differs in sign from the neutrino-electron interactions; thus 𝐻^mat_𝐼(𝑥)=−√2𝐺_𝐹𝑛_𝑒(𝑡)𝛽. (58) Here we are more concerned with the matter effects of a constant density. The total Hamiltonian of neutrino in matter 𝐻=𝑈(𝑚²/2𝐸)𝑈^†+√2𝐺_𝐹𝑛_𝑒𝛽, where 𝑈=^{cos𝜃sin𝜃}_{−sin𝜃cos𝜃} then the total effective Hamiltonian is 𝐻=(1/2)Tr𝐻+𝐻^𝑚. Here (1/2)Tr𝐻=((𝑚²₁+𝑚²₂)/4𝐸)+(1/2)√2𝐺_𝐹𝑛_𝑒 and 𝐻^𝑚 is the traceless part of Hamiltonian: 𝐻^𝑚=14𝐸−Δ𝑚²cos2𝜃+𝐴Δ𝑚²sin2𝜃Δ𝑚²sin2𝜃Δ𝑚²,√cos2𝜃−𝐴where𝐴=22𝐺_𝐹𝑛_𝑒𝐸𝐻^𝑚=𝑈^𝑚𝐸^𝑚𝑈^𝑚†,𝑈^𝑚=cos𝜃^𝑚sin𝜃^𝑚−sin𝜃^𝑚cos𝜃^𝑚,𝐸^𝑚=𝐸^𝑚₁00𝐸^𝑚₂, (59) where 𝐸^𝑚_1,2=±(1/4𝐸)(Δ𝑚²cos2𝜃−𝐴)²+(Δ𝑚²sin2𝜃)².

From equation, we find that the mixing angle 𝜃^𝑚  is given as: cos2𝜃^𝑚=Δ𝑚²cos2𝜃−𝐴Δ𝑚²cos2𝜃−𝐴²+Δ𝑚²sin2𝜃²,sin2𝜃^𝑚=Δ𝑚²sin2𝜃Δ𝑚²cos2𝜃−𝐴²+Δ𝑚²sin2𝜃². (60) Three expressions for atmospheric neutrinos for the normal hierarchy are given as follows: 𝑃_{𝜇𝜏matter}=cos𝜃^𝑚₁₃²∗sin2𝜃₂₃²∗sin1.27∗Δ𝑚₃₁+𝐴+Δ𝑚^𝑚₃₁∗𝐿𝐸²+sin𝜃^𝑚₁₃²∗sin2𝜃₂₃²∗sin1.27∗𝐴+Δ𝑚₃₁−Δ𝑚^𝑚₃₁∗𝐿𝐸²−cos2𝜃₂₃²∗sin2𝜃^𝑚₁₃²∗sin𝜃₂₃²∗sin1.27∗Δ𝑚^𝑚₃₁∗𝐿𝐸²,𝑃_{𝜇𝜇matter}=1−cos𝜃^𝑚₁₃²∗sin2𝜃₂₃²∗sin1.27∗𝐴+Δ𝑚₃₁−Δ𝑚^𝑚₃₁∗𝐿𝐸²,𝑃_{𝜇𝑒matter}=sin2𝜃^𝑚₁₃²∗sin𝜃₂₃²∗sin1.27∗Δ𝑚^𝑚₃₁∗𝐿𝐸². (61) And for the inverted hierarchy, 𝑃_{𝜇𝜏matter}=cos𝜃^𝑚₁₃²∗sin2𝜃₂₃²∗sin1.27∗𝐴−Δ𝑚₃₁+Δ𝑚^𝑚₃₁∗𝐿𝐸²+sin𝜃^𝑚₁₃²∗sin2𝜃₂₃²∗sin1.27∗𝐴+Δ𝑚₃₁−Δ𝑚^𝑚₃₁∗𝐿𝐸²−cos2𝜃₂₃²∗sin2𝜃^𝑚₁₃²∗sin𝜃₂₃²∗sin1.27∗Δ𝑚^𝑚₃₁∗𝐿𝐸²,𝑃_{𝜇𝜇matter}=1−cos𝜃^𝑚₁₃²∗sin2𝜃₂₃²∗sin1.27∗𝐴−Δ𝑚₃₁−Δ𝑚^𝑚₃₁∗𝐿𝐸²,𝑃_{𝜇𝑒matter}=sin2𝜃^𝑚₁₃²∗sin𝜃₂₃²∗sin1.27∗Δ𝑚^𝑚₃₁∗𝐿𝐸². (62)

8. Oscillation Parameters and Masses

Neutrino oscillation is new physics beyond the standard model and requires the addition of a new field and new parameters to the standard model. Although atmospheric and solar neutrinos are having different origin, yet they exhibit the same phenomenon of oscillations among their flavors. The parameters involved in three-flavor oscillations can be estimated from the neutrinos coming from the sun and atmospheric and nuclear reactors and accelerators. Solar experiments detect neutrinos generated in the core of the sun due to thermonuclear reactions and must have the energy of the order of 0.2–15 MeV, whereas atmospheric experiments detect the neutrino produced in a cascade initiated by Cosmic rays collisions with nuclei in the Earth’s atmosphere and has a source detector at a distance of several tons of meters with a range 𝐿/𝐸<1 m/MeV. Oscillation parameters historically fall into four categories in mixing angle-mass splitting parameter space: vacuum oscillations (VAC), “LOW”, small mixing angle (SMA), and large mixing angle (LMA). Similarly based on the distance from source to detector, detectors can be SBL, LBL, VSBL, and VLBL. A global analysis of different experiments can provide best fit values of different parameters. The two large mixing angles 𝜃₁₂ and 𝜃₂₃ have been found to be of the order of ~34° and ~45°, respectively, but the third mixing angle 𝜃₁₃ is suppressed by different experiments. Recent searches for 𝜃₁₃, however, oppose suppression of this mixing angle. Moreover sign of Δ𝑚²₃₁ is still unknown; therefore two types of neutrino spectrum are possible, one is normal hierarchy (𝑚₁≪𝑚₂<𝑚₃) whereas the other is inverted hierarchy which assume masses in order (𝑚₃≪𝑚₁<𝑚₂). Three mixing angles and the mass differences from global analysis of data [24] are given below: Δ𝑚²₂₁=7.59±0.20^+0.61_−0.69×10⁻⁵eV²,Δ𝑚²₃₁=+2.46±0.12(±0.37)×10⁻³eV²=−2.36±0.11±0.37eV²,𝜃₂₃=42.8^+4.7_−2.9^+10.7_−7.3,𝜃₁₂=34.4±1.0^+3.2_−2.9,𝜃₁₃=5.6^+3.0_−2.7. (63)

The parameter affecting the neutrino oscillation in matter is given by 𝐴=2√2𝐺_𝐹𝑛_𝑒𝐸 where 𝐺_𝐹 is Fermi constant, 𝑛_𝑒 is the electron number density, and 𝐸 is the energy of neutrino beam. The parameter 𝐴 is almost constant for the long baseline experiment with high precision as the beam does not penetrate deeply into the earth without affecting the different layers of the earth, but in sun, density decreases exponentially from center to core of earth which further leads to resonance enhancement of oscillations known as the MSW effect [22]. We here restrict ourselves to the matter effects with constant density for atmospheric neutrinos only. The graph for probability versus energy at three different lengths in vacuum as well as in matter (for normal and inverted hierarchy) for 𝜈_𝜇 to 𝜈_𝑒, 𝜈_𝜇 to 𝜈_𝜏, and 𝜈_𝜇 to 𝜈_𝜇 shows that the presence of matter affects the neutrino oscillation probability the most in case of the 𝜈_𝜇 to 𝜈_𝑒 oscillations. This may be due to the reason that the charged current interaction with the electrons of the medium directly affects the oscillation probability in case of electrons only. This proves the fact that the matter effects are flavor dependent. Moreover, oscillations become more frequent at low energy and the increase in length results in increase of the energy range which is more sensitive to oscillations but affects to a very lesser extent of magnitude of transition probability. Oscillation probability is the more of magnitude in case of 𝜇 to 𝜏 rather than an electron which concludes that a mass eigen state is more composed of 𝜇 and 𝜏 rather than an electron for atmospheric neutrinos especially. The smallness of 𝜈_𝑒  part of total mass eigen state is measured by the mixing angle 𝜃₁₃. For solar neutrino oscillation probability, reactor-based neutrino experiments and detectors are designed as different experiments are designed in order to be sensitive to different values of Δ𝑚², by choosing the appropriate value of 𝐿/𝐸, the neutrino beam consisting of 𝜈_𝑒 when coming out of the source and having energy only ~1 MeV that is, in order to satisfy Δ𝑚²𝐿/2𝐸∼1 and 𝐿 can be of order such that 𝐿/𝐸<=1 km/MeV. Solar neutrino experiments have sensitivity for Δ𝑚² of very small value. SNO has been extremely successful so far to prove neutrino flavor change because it was designed in such a way to detect the changed flavor of solar neutrinos, that is, the flavor in which 𝜈_𝑒 changed. After analyzing the results from flux ratios, it was found that day-night asymmetry has more sensitivity to the value of Δ𝑚²₁₂. The results favored large mixing angle solutions. The CHOOZ [25] experiment put forward that the electron antineutrino survival probability and its oscillation probability from electron to muon and tau are excluded for Δ𝑚²≥8×10⁻⁴ eV² and at maximum mixing sin²2𝜃≥0.17. The chain of experiment continued with the KamLAND [8] experiment which provided evidence of antineutrino disappearance probability for the very first time with a long baseline length of the order of ~175km as shown in Figure 6. All reactor-based neutrinos consider only the vacuum part of the oscillation probability as matter effects are negligible at low energy. Transitions from electron to 𝜇’s and 𝜏’s are shown in Figures 7 and 8. Two plots here show that the maximum oscillations are from 𝑒 to 𝜇 for solar neutrinos.

9. Summary and Conclusion

In summary, two flavor oscillations are discussed here for the probability measurements and moreover generalized to three flavor despite in vacuum and matter. In the era of neutrino physics, vast amount of the information for oscillation parameters we have in hand, still the situation is not clear yet. Neutrino oscillations helped to solve the greatest mystery in solar world but actual mass of the neutrinos is still unknown. The recent searches for neutrinos are more focused on the estimation of parameter 𝜃₁₃ [25] and the speed of neutrinos. Some more pieces of information on the mass hierarchy and CP violating phase can help us to get more detailed information on oscillation phenomenology. Future experiments are more concerned to find the accurate and precise measurement of 𝜃₁₃. The accurate determination of 𝜃₁₃ can provide the more elaborated information on CP violating phase 𝛿 and mass hierarchy. It is proposed that a sign of Δ𝑚²₃₁ can also be well determined. Reactor experiments are unique in the sense that they can give a value of 𝜃₁₃ up to a higher precision value. That’s why more recent experiments like Double CHOOZ, T2K, Nova, Daya Bay, and MINOS are designed in such a way that they are expected to measure this angle. Nuclear reactor experiments are more efficient in determination of 𝜃₁₃ as for the case of long baseline experiments sensitivity is reduced due to effect of transition parameters and matter effects. Nuclear reactors are very intense sources of 𝜈_𝑒’s coming from the 𝛽-decay of the neutron-rich fission fragments. If relaxation 𝜃₁₃ is avoided, that antineutrino disappearance probability becomes 𝑃𝜈_𝑒⟶𝜈_𝑒=1−sin²_2𝜃13sin²Δ𝑚²₃₁𝐿2𝐸−cos4_𝜃13sin²_𝜃12sin²Δ𝑚²₃₁𝐿.2𝐸 (64) Thus sensitivity of 𝜃₁₃ can be seen more at small energy and short baseline rather than long baseline as shown in Figure 9. The present information about this angle can be determined through global analysis of data. Gonzalez-Garcia et al. predicted the following limits on 𝜃₁₃ from various experiments [26]: sin²_𝜃13<()()()0.0600.089)(solar+KAMLAND0.0270.058)(CHOOZ+atm+K2K+MINOS0.0350.056)(Globaldata. (65) But what would happen if this angle is large? Earlier experiments assumed 𝜃₁₃ as zero because of some discrete symmetries present in the mixing parameters. Now the question arises how these symmetries are changed if 𝜃₁₃ is large or the new parameter ranges are compatible with the PMNS matrix 𝑈 and whether the more resolved values of two mass differences is expected or not? What are the changes expected in the neutrino beam energy to measure more accurately the mixing parameters?

Appendix

Oscillation Plots for Vacuum and Matter

For more details, see Figures 2, 3, 4, 5, 6, 7, 8, and 9 and Table 1.

Figure 2 

Oscillations and survival probabilities.

Figure 3 

Figure 4 

Figure 5 

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(b)

(c)

Figure 6 

(a)

(b)

(c)

Figure 7 

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Figure 8 

Figure 9