# Phenomenology of Neutrino Mixing in Vacuum and Matter

Authors: A. Upadhyay; M. Batra
Journal: ISRN High Energy Physics
 (2013)

---

## 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.


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## Body

## 1. Introduction

Neutrinos are the most elusive in nature. They interact much less through coupling withW+,  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 νe’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 νμ⇔ντ,νμ⇔νe. 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 Δm212=7.58+0.14(stat)(syst)×10-5-0.15+0.15 eV2 and tanθ12=0.56+(stat)-0.070.10(syst)-0.06+0.10 for tanθ12<1 as calculated using KAMLAND experiment. K2K [16, 18] was another long baseline experiment to study oscillation from νμ to νe 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 Δm232 with the precision better than 10%.The most recent parameter of our interest isθ13 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:
(1)i∂|Ψ(t)〉∂t=H|Ψ(t)〉,
where H is the total Hamiltonian in flavor space and the general solution for (1) is
(2)|Ψ(t)〉=e-iHt|Ψ(0)〉,
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 matrixU as
(3)|νl〉=∑i=03Uli*|νi〉.
At time (t=0),
(4)|Ψ(0)〉=|νi〉.
Application of the Hamiltonian operator will give the following:
(5)H|νi〉=Ei|νi〉,
where
(6)Ei=pi2+mi2.
The left-handed neutrino at the time t≥0 is written as
(7)|νl〉t=e-iHt|νl〉=∑i=13e-iEitUli*|νi〉.
Similarly,
(8)|νl-〉t=e-iHt|νl-〉=∑i=13e-iEitUli|νi-〉.
Neutrino energies Ei(i=1,2,3) are different; hence the flavor oscillation in neutrinos gives the amplitude of the transition νl→νl′ during the time t and can be written as
(9)A(νl⟶νl′)=∑i=13Uil′e-iEitUli*.
Analogously, the amplitude of the transition νl-→νl′- during the time t is given by
(10)A(νl-⟶νl′-)=∑i=13Uil′*e-iEitUli.
Probability for the transition of neutrinos and antineutrinos will be
(11)P(νl⟶νl′)=|∑i=13Uil′e-iEitUli*|2,P(νl-⟶νl′-)=|∑i=13Uil′*e-iEitUli|2.
From (11), we can find possible relations between probabilities:
(12)∑lP(νl⟶νl′)=1,∑l′P(νl-⟶νl′-)=1,∑l′P(νl⟶νl′)=1,∑lP(νl-⟶νl′-)=1.
A mixed neutrino state is characterized by their momentum p with pi=p and mass mi with mi2/p2≪1.And(13)Ei≅p+mi22p.
Hence the energy difference between two such states will be
(14)Ei-Ej=Δmji22p,
where Δmji2=mi2-mj2.AsE≈p, therefore
(15)Ei-Ej=Δmji22E.
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:
(16)t≅L,(Ei-Ej)t=Δmji22EL,P(νl⟶νl′)=|∑i=13Ul′ie-i(Δmji2L/2E)Uli*|2.
The unitary condition suggests that
(17)∑iUl′iUli*=δll′.
Hence
(18)P(νl⟶νl′)=|δll′+∑i≠jUil′(e-i(Δmji2L/2E)-1)Uli*|2.
Analogously, for the case of antineutrino,
(19)P(νl-⟶νl’-)=|δll′+∑i≠jUil′*(e-i(Δmji2L/2E)-1)Uli|2.
The transition probability νl→νl′ is given as
(20)P(νl⟶νl′)=∑i,kUl′iUl′k*Uli*Ulke-i(Δmji2/2E)L=∑i|Ul′i|2|Uli|2+2Re∑i>k(Ul′iUl′k*Uli*Ulke-i(Δmji2/2E)L),∑i|Ul’i|2|Uli|2=δll′-2Re∑i>k(Ul′iUlk*Uli*Ulk)=δll′-2Re∑i>k(Ul′iUl′k*Uli*Ulk)+2Re∑i>k(Ul′iUl′k*Uli*Ulke-i(Δmji2/2E)L)=δll′-2Re∑i>k(Ul′iUl′k*Uli*Ulk)×(1-e-i(Δmki2L/2E)).
Finally for any complex a and b, Re(ab)=Re(a)Re(b)-Im(a)Im(b),
(21)P(νl⟶νl’)=δll’-2Re∑i>k(Ul′iUl′i*Uli*Ulk)×(1-cosΔmki2L2E)+2∑i>kIm(Ul′iUl′k*Uli*Ulk)sinΔmki2L2E.
Similarly, for the antineutrino oscillation, probability becomes
(22)P(νl¯⟶νl′¯)=δll’-2Re∑i>k(Ul′iUl′i*Uli*Ulk)×(1-cosΔmki2L2E)-2∑i>kIm(Ul′iUl′k*Uli*Ulk)sinΔmki2L2E.


## 3. Two-Flavor Oscillation Probability in Vacuum

For two-flavor neutrino oscillations, the mixing matrix can be defined in terms of some rotating angleθ in 2 dimensions where U is actually PMNS matrix; therefore
(23)(νeνμ)=(cosθsinθ-sinθcosθ)(ν1ν2),|νe(t=0)〉=|νe〉=cosθ|ν1〉+sinθ|ν2〉,|νμ(t=0)〉=|νμ〉=-sinθ|ν1〉+cosθ|ν2〉.
The weak eigenstates are rotated by an angle θ with respect to the mass eigenstates |ν1〉 and |ν2〉 to allow mixing between νμ and νe. After some time t(24)|νμ(t=t)〉=|νμ(t)〉=-sinθ|ν1〉e(-iE1t/h)+cosθ|ν2〉e-iE2t/h=-sinθ|ν1〉e-i(p+(m12/2p))t/h+cosθ|ν2〉e-i(p+(m22/2p))t/h,
where we have used
(25)E1=(p2+m12)1/2,E2=(p2+m22)1/2,P(νe⟶νμ)=|〈νμ∣νe(t)〉|2,〈νμ|=cosθ〈ν1|+sinθ〈ν2|,P(νe⟶νμ)=|〈νμ∣νe(t)〉|2=eiz-izsin2θcos2θ(1-e(iΔm2/2p)x)×(1-e(-iΔm2/2p)x).
For neutrinos to be relativistic, the substitution p=Ev and x=L can be made:
(26)P(νe⟶νμ)=sin2θcos2θ(1-e(iΔm2/2Ev)L)×(1-e(-iΔm2/2Ev)L).P(νe⟶νμ)(L,E)=sin22θsin2(1.27Δm2LEv).
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(να→νβ)〉=(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(27)(ν1ν2ν3)=U(νeνμντ).
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:
(28)|i〉(i=1,2,3),〈i∣k〉=δik.
The first Euler rotation performed at the angle θ12 around the vector |3〉 produces new orthogonal and normalized vectors as
(29)|1〉(1)=c12|1〉+s12|2〉,|2〉(1)=-s12|1〉+c12|2〉,|3〉(1)=|3〉.
Here c12=cosθ12 and s12=sinθ12;
(30)|ν〉(1)=U(1)|ν〉suchthat|ν〉(1)=(|1(1)〉|2(1)〉|3(1)〉),|ν〉=(|1〉|2〉|3〉),U(1)=(c12s120-s12c120001).
Second rotation at the angle θ13 around vector the |2〉(1) introduces the CP phase δ:
(31)|1〉(2)=c13|1〉(1)∣+s13e-iδ|3〉(1),|2〉(2)=|2〉(1),|3〉(2)=-s13e-iδ|1〉(1)+c13|3〉(1).
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
(32)|1〉mix=|1〉(2),|2〉mix=c23|2〉(2)+s23|3〉(2),|3〉mix=-s23|2〉(2)+c23|3〉(2),|vmix〉=U(3)|v〉(2),U(3)=(1000c23s230-s23c23),|vmix〉=U|v〉,whereU=U(3)U(2)U(1),U=(1000c23s230-s23c23)(c130s13e-iδ010s13e-iδ0c13)(c12s120-s12c120001),U=(c13c12c13s12s13e-iδ-c23s12-c12s13s23eiδc12c23-s12s13s23eiδc13s23s23s12-s13c12c23eiδc12s23-s13c23s12eiδc13c23).
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(Ul′iUl′k*Uli*Ulk). 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;
(33)Re(Ul′iUl′k*Uli*Ulk)=(U22U21*U12*U11),forl′=2,l=1,i=2,k=1=(c23c12-s13s23s12eiδ)(-c23s12-s13s23c12e-iδ)×(c13s12)(c13c12)=-14c132sin2θ12×[sin2θ12(c232-s232s132)+cos2θ12sin2θ23s13cosδ13],Re(Ul′iUl′k*Uli*Ulk)=(U23U22*U13*U12)=(c13s23)(c23c12-s13s23s12e-iδ)(s13eiδ)(c13s12)=-c132s13s12s23(s13s23s12-c23c12cosδ13),Re(Ul′iUl′k*Uli*Ulk)=(U23U21*U13*U11)=(c13s23)(-c23s12-s13s23c12e-iδ)(s13eiδ)(c13c12)=-c132s13c12s23(c23s12eiδ+s13s23c12).
Similarly other expressions can also be derived shown in Table 1.Table 1
Probability
i
,
k
Re(Ul′iUl′k*Uli*Ulk)
P
(
ν
e
→
ν
μ
)
i
=
2
,
k
=
1
(
-
1
/
4
)
c
13
2
sin
2
θ
12
[
sin
2
θ
12
(
c
23
2
-
s
23
2
s
13
2
)
+
cos
2
θ
12
sin
2
θ
23
s
13
cos
δ
13]
i
=
3
,
k
=
2
c
13
2
s
13
s
12
s
23
(
s
13
s
23
s
12
-
c
23
c
12
cos
δ
13
)
i
=
3
,
k
=
1
c
13
2
s
13
c
12
s
23
(
c
23
s
12
e
i
δ
+
s
13
s
23
c
12
)
P
(
ν
e
→
ν
τ
)
i
=
2
,
k
=
1
(
1
/
4
)
c
13
2
sin
2
θ
12
[
sin
2
θ
12
(
c
23
2
s
13
2
-
s
23
2
)
+
cos
2
θ
12
sin
2
θ
23
s
13
cos
δ
13]
i
=
3
,
k
=
2
-
c
13
2
s
13
s
12
c
23
(
s
13
c
23
s
12
+
s
23
c
12
cos
δ
13
)
i
=
3
,
k
=
1
-
c
13
2
s
13
c
12
c
23
(
s
13
c
23
c
12
-
s
23
s
12
cos
δ
13
)
P
(
ν
e
→
ν
e
)
i
=
2
,
k
=
1
(
1
/
4
)
c
13
4
sin
2
2
θ
12
i
=
3
,
k
=
2
(
1
/
4
)
s
12
2
sin
2
2
θ
13
i
=
3
,
k
=
1
1
/
4
c
12
2
sin
2
2
θ
13
P
(
ν
μ
→
ν
τ
)
i
=
2
,
k
=
1
(
1
/
16
)
sin
2
2
θ
12
sin
2
2
θ
23
(
1
+
s
13
2
)
2
-
(
1
/
4
)
(
sin
2
2
θ
12
+
sin
2
2
θ
23
)
s
13
2
-
(
1
/
16
)
sin
4
θ
12
sin
4
θ
23
(
1
+
s
13
2
)
s
13
cos
δ
13
+
(
1
/
4
)
sin
2
2
θ
12
sin
2
2
θ
23
s
13
2
cos
2
δ
13
i
=
3
,
k
=
2
(
-
1
/
4
)
sin
2
θ
23
c
13
2
[
sin
2
θ
23
(
c
12
2
-
s
12
2
s
13
2
)
+
sin
2
θ
12
cos
2
θ
23
s
13
cos
δ
13
]
i
=
3
,
k
=
1
(
1
/
4
)
sin
2
θ
23
c
13
2
[
sin
2
θ
23
(
c
12
2
s
13
2
-
s
12
2
)
+
sin
2
θ
12
cos
2
θ
23
s
13
cos
δ
13
]
P
(
ν
μ
→
ν
μ
)
i
=
2
,
k
=
1
(
1
/
4
)
sin
2
2
θ
12
(
c
23
4
+
s
23
4
s
13
2
)
+
(
1
/
4
)
(
1
-
(
1
/
2
)
sin
2
2
θ
12
)
sin
2
2
θ
23
s
13
2
+
(
1
/
4
)
sin
4
θ
12
sin
2
θ
23
(
c
23
2
-
s
23
2
s
13
2
)
s
13
cos
δ
13
-
(
1
/
4
)
sin
2
2
θ
12
sin
2
2
θ
23
s
13
2
cos
2
δ
13
i
=
3
,
k
=
2
s
23
2
c
13
2
(
c
12
2
c
23
2
+
s
12
2
s
23
2
s
13
2
-
(
1
/
2
)
sin
2
θ
12
sin
2
θ
23
s
13
cos
δ
13
)
i
=
3
,
k
=
1
s
23
2
c
13
2
(
s
12
2
c
23
2
+
c
12
2
s
23
2
s
13
2
+
(
1
/
2
)
sin
2
θ
12
sin
2
θ
23
s
13
cos
δ
13
)
P
(
ν
τ
→
ν
τ
)
i
=
2
,
k
=
1
(
1
/
4
)
sin
2
2
θ
12
(
s
23
4
+
c
23
4
s
13
2
)
+
(
1
/
4
)
(
1
-
(
1
/
2
)
sin
2
2
θ
12
)
sin
2
2
θ
23
s
13
2
+
(
1
/
4
)
sin
4
θ
12
sin
2
θ
23
(
s
23
2
-
c
23
2
s
13
2
)
s
13
cos
δ
13
-
(
1
/
4
)
sin
2
2
θ
12
sin
2
2
θ
23
s
13
2
cos
2
δ
13
i
=
3
,
k
=
2
c
23
2
c
13
2
(
c
12
2
s
23
2
+
s
12
2
c
23
2
s
13
2
+
(
1
/
2
)
sin
2
θ
12
sin
2
θ
23
s
13
cos
δ
13
)
i
=
3
,
k
=
1
c
23
2
c
13
2
(
s
12
2
s
23
2
+
c
12
2
c
23
2
s
13
2
-
(
1
/
2
)
sin
2
θ
12
sin
2
θ
23
s
13
cos
δ
13
)From the table, it is clear that out of three types of mass mixing,Δm212,  Δm312,  and  Δm322 only two are independent:
(34)Δm322+Δm212-Δm312=0.
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, Δmsol2≪Δmatmos2, 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:(35)P(νe⟶νμ)=δ12-4[-14c132sin2θ12×[sin2θ12(c232-s232s132)+cos2θ12sin2θ23s13cosδ13]14]sinΔm212L2E+δ32-4[-c132s13s12s23(s13s23s12-c23c12cosδ13)]×sinΔm322L4E+δ31-4[-c132s13c12s23(c23s12eiδ+s13s23c12)]×sinΔm312L4E.
For atmospheric neutrinos,
(36)Δm322≅Δm312≅Δmatm2,Δm212≅Δmsol2≅0,P(νe⟶νμ)=-4(-s122s232s132c132+s12s23s13c23c12c132cosδ13-s132c132-s12s23s13c23c12c132cosδ13)×sinΔmatm2L4E=sin2(2θ13)sin2(θ23)sinΔmatm2L4E,P(νe⟶ντ)=sin2(2θ13)cos2(θ23)sinΔmatm2L4E.
The survival probability of electron neutrinos is
(37)P(νe⟶νe)=1-[P(νe⟶νμ)+P(νe⟶ντ)]=1-sin2(2θ13)sinΔmatm2L4E.
In case of vacuum,
(38)P(νe⟶νμ)=P(νμ⟶νe),P(νμ⟶νe)=sin2(2θ13)sin2(θ23)sinΔmatm2L4E.
Similarly we can calculate the probability for muon neutrinos changing into tau neutrinos:
(39)P(νμ⟶ντ)=sin2(2θ23)cos4(θ13)sinΔmatm2L4E.
Now we will calculate the survival probability of muon neutrinos; that is,
(40)P(νμ⟶νμ)=1-[P(νμ⟶νe)+P(νμ⟶ντ)]=1-sin2(2θ13)sin2(θ23)+sin2(2θ23)cos4(θ13)sinΔmatm2L4E.
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 Δm212 is only taken where other mass differences are neglected:
(41)P(νe⟶νμ)=δ12-4[-14c132sin2θ12×[sin2θ12(c232-s232s132)+cos2θ12sin2θ23s13cosδ13]-14c132sin2θ12]×sinΔm212L2E,P(νe⟶νμ)=-4[-14c132sin2θ12×[sin2θ12(c232-s232s132)+cos2θ12sin2θ23s13cosδ13]-14c132sin2θ12]×sinΔm212L2E=[14(sin4θ12sin2θ13cosθ13)sin22θ12cos2(θ13)×(cos2θ23-sin2θ23sin2θ13)+14(sin4θ12sin2θ13cosθ13)]sinΔm212L2E.
Similarly,
(42)P(νe⟶ντ)=δ12-4[∑14c132sin2θ12×{∑sin2θ12(c232s132-s232)+cos2θ12×sin2θ23s13cosδ13∑}]×sinΔm212L4E=[∑-sin22θ12cos2θ13×(cos2θ23sin2θ13-sin2θ23)-14sin4θ12sin2θ13sin2θ23cosθ13∑]×sinΔm212L2E,P(νe⟶νe)=1-[sin22θ12cos4θ13]sinΔm212L2E.


## 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=2GFNe, where GF = Fermi coupling constant and Ne = number of electrons per unit volume.Neutrinos are produced in flavor eigen states,|υα〉, (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
(43)i∂∂t=H0|ψ(t)〉.
When H0 is a free Hamiltonian, the state |ψ(t)〉 can be expanded over the total system of states of flavor neutrinos νl, with momentum p,
(44)|ψ(t)〉=∑al(t)|νl〉.
Here
(45)|νl〉=∑iUli*|νi〉,H0|νi〉=Ei|νi〉,  H0|νi〉=Ei|νi〉,  Ei=pi2+mi2≃p+mi2/2E, and al(t)=〈νl∣ψ(t)〉 is the amplitude of probability to find νl in state which is described by |ψ(t)〉.Therefore,(46)i∂al′(t)∂t=∑l〈νl′∣H0∣al(t)〉,
where
(47)〈νl′∣νl〉=Ul′l,〈νl∣νl′〉=Ul′l*.
Taking into account this relation, for the free Hamiltonian in the flavor representation we have the following expression:
(48)〈νl′∣H0∣νl〉=∑lUl′iEiUli*~p+∑lUl′imi22EUli*.
Therefore neutrino evolution equation in the flavor representation is as follows:
(49)i∂a(t)∂t=Um22EU†a(t).
Let us introduce the function
(50)a′(t)=U†a(t).
We find that the function a′(t) satisfies the following equation, multiplying by U† on both sides, then
(51)i∂∂ta′(t)=m22Ea′(t).
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 νe 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|HICC|pmat〉, where HICC=(GF/2)2νeL-(x)γανeLe-(x)γα(1-γ5)e(x) and the vector |pmat〉=|p〉|mat〉. Now substituting the Hamiltonian we obtain 〈pmat|(GF/2)2νeL-(x)γανeLe-(x)γα(1-γ5)e(x)|pmat〉=〈p|〈mat|(GF/2)2νeL-(x)γανeLe-(x)γα(1-γ5)e(x)|p〉|mat〉.But〈p|〈mat|(GF/2)2νeL-(x)γανeLe-(x)γαγ5e(x)|p〉|mat〉=0 as for unpolarized matter 〈mat|e-(x)γαγ5e(x)|mat〉=0.Also〈mat|e-(x)γαe(x)|p〉|mat〉=〈mat|e-(x)e(x)|p〉|mat〉δα0=ne(x)δα0, where ne(x) is the number density at the point x. Also, 〈p|νeL-(x)γανeL|p〉=1.Substituting all these values,HImat(t)=2GFne(t)β. Using βνe,νe=1, all other elements of matrix β are equal to zero.Let us now consider the NC interaction. Induced by theZ0 exchange, the Hamiltonian of NC interactions of neutrinos with electrons and nucleons has the form
(52)HICC(x)=2GF2∑l=e,μ,τν¯lL(x)γανlL(x)jαNC(x),
where jαNC(x) is the sum of electron and nucleon (quark) neutral current. For the vector part of the effective hadron neutral current,
(53)ναNC(N)(x)=12N¯(x)γατ3N(x)-2sin2θwp¯(x)γαp(x),asN=(pn),τ3=(100-1),
where θw is the weak angle. Similarly, for the effective part of electron current,
(54)ναNC(n)(x)=(-12)n¯(x)γαn(x),ναNC(e)(x)=(-12+2sin2θw)e¯(x)γαe(x).
For the corresponding matter matrix elements, we have
(55)〈mat∣ναNC(e)(x)∣mat〉=(-12+2sin2θw)ne(x)δα0,〈mat∣ναNC(p)(x)∣mat〉=(12-2sin2θw)ρp(x)δα0,〈mat∣ναNC(n)(x)∣mat〉=-12ρn(x)δα0.
For the neutral matter, ne(x)=ρp(x) we conclude that the contributions of electron and proton NC to the effective Hamiltonian cancel each other. Thus
(56)〈mat∣jαNC(x)∣mat〉=-12ρn(x)δα0.
By taking into account the effective charged current interaction, only νe-e CC interaction gives a contribution to the effective Hamiltonian. Thus the evolution equation of neutrino has the form
(57)i∂a(t)∂t=(Um22EU†+2GFne(t)β)a(t).
Similarly, for the antineutrinos, effective Hamiltonian differs in sign from the neutrino-electron interactions; thus
(58)H-Imat(x)=-2GFne(t)β.
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†+2GFneβ, where U=(cosθsinθ-sinθcosθ) then the total effective Hamiltonian is H=(1/2)TrH+Hm. Here (1/2)TrH=((m12+m22)/4E)+(1/2)2GFne and Hm is the traceless part of Hamiltonian:
(59)Hm=14E(-Δm2cos2θ+AΔm2sin2θΔm2sin2θΔm2cos2θ-A),whereA=22GFneEHm=UmEmUm†,Um=(cosθmsinθm-sinθmcosθm),Em=(E1m00E2m),
where E1,2m=±(1/4E)(Δm2cos2θ-A)2+(Δm2sin2θ)2.From equation, we find that the mixing angleθm  is given as:
(60)cos2θm=Δm2cos2θ-A(Δm2cos2θ-A)2+(Δm2sin2θ)2,sin2θm=Δm2sin2θ(Δm2cos2θ-A)2+(Δm2sin2θ)2.
Three expressions for atmospheric neutrinos for the normal hierarchy are given as follows:
(61)Pμτmatter=(cos[θ13m])2*(sin[2θ23])2*(sin[(1.27*(Δm31+A+Δm31m)*L)E])2+(sin[θ13m])2*(sin[2θ23])2*(sin[(1.27*(A+Δm31-Δm31m)*L)E])2-(cos[2θ23])2*(sin[2θ13m])2*(sin[θ23])2*(sin[(1.27*Δm31m*L)E])2,Pμμmatter=(1-cos[θ13m])2*(sin[2θ23])2*(sin[(1.27*(A+Δm31-Δm31m)*L)E])2,Pμematter=(sin[2θ13m])2*(sin[θ23])2*(sin[(1.27*Δm31m*L)E])2.
And for the inverted hierarchy,
(62)Pμτmatter=(cos[θ13m])2*(sin[2θ23])2*(sin[(1.27*(A-Δm31+Δm31m)*L)E])2+(sin[θ13m])2*(sin[2θ23])2*(sin[(1.27*(A+Δm31-Δm31m)*L)E])2-(cos[2θ23])2*(sin[2θ13m])2*(sin[θ23])2*(sin[(1.27*Δm31m*L)E])2,Pμμmatter=(1-cos[θ13m])2*(sin[2θ23])2*(sin[(1.27*(A-Δm31-Δm31m)*L)E])2,Pμematter=(sin[2θ13m])2*(sin[θ23])2*(sin[(1.27*Δm31m*L)E])2.


## 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 rangeL/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 Δm312 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:
(63)Δm212=7.59±0.20-0.69+0.61×10-5eV2,Δm312=+2.46±0.12(±0.37)×10-3eV2=-2.36±0.11±0.37eV2,θ23=42.8-2.9+4.7(-7.3+10.7),θ12=34.4±1.0(-2.9+3.2),θ13=5.6-2.7+3.0.The parameter affecting the neutrino oscillation in matter is given byA=22GFneE where GF is Fermi constant, ne 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 νμ to νe, νμ to ντ, and νμ to νμ shows that the presence of matter affects the neutrino oscillation probability the most in case of the νμ to νe 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 νe  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 νe 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 νe changed. After analyzing the results from flux ratios, it was found that day-night asymmetry has more sensitivity to the value of Δm122. 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 parameterθ13 [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 Δm312 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
(64)P(ν¯e⟶ν¯e)=1-sin2θ132sin2Δm312L2E-cos4θ13sinθ122sin2Δm312L2E.
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]:
(65)sinθ132<{0.060(0.089)(solar+KAMLAND)0.027(0.058)(CHOOZ+atm+K2K+MINOS)0.035(0.056)(Globaldata)}.
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?



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