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Home , Mixing angle, Neutrino oscillation

OF

SPANDAN MONDAL UNDERGRADUATE PROGRAMME, INDIAN INSTITUTE OF SCIENCE, BANGALORE

Project Instructor: PROF. SOUROV ROY

INDIAN ASSOCIATION FOR THE CULTIVATION OF SCIENCE, KOLKATA

Contents

1. Introduction ...... 1 2. and : The Basics ...... 1 2.1. Elementary Particles ...... 1 2.2. (ℏ = 푐 = 1) ...... 2 2.3. Dirac’s bra-ket Notation in QM ...... 3 Ket Space ...... 3 Bra Space ...... 3 Operators ...... 3 The Associative Axiom ...... 4 Matrix Representation...... 4 2.4. Rotations in Quantum Mechanics ...... 5 2.5. The Klein-Gordon Equation ...... 6 2.6. The Dirac Equation ...... 6 2.7. Bilinear covariants and Weak Interactions ...... 8 3. Neutrino Flavours, Mass Eigenstates and Mixing...... 8 3.1. Introduction ...... 8 3.2. Leptonic Mixing ...... 9 4. in Vacuum ...... 10 4.1. Detecting a flavour change ...... 10 4.2. Finding the oscillation probability ...... 10 4.3. Analysis and Discussions ...... 14 5. Neutrino Oscillation in Matter ...... 16 5.1. Factors affecting in Matter ...... 16 5.2. Finding the Vacuum Hamiltonian ...... 17 5.3. Finding the Hamiltonian in Matter ...... 18 5.4. Finding the Oscillation Probability in Matter ...... 19 5.5. Discussions ...... 19 6. Properties of Neutrinos ...... 21 6.1. The Mass-squared Splittings ...... 21 6.2. The Flavour and Mass States and Mixing ...... 21 7. Conclusions ...... 22 8. References ...... 23

[i]

1. Introduction

In recent years, there have been several breakthroughs in neutrino physics and it has emerged as one of the most active fields of research. The of particle physics describes neutrinos as massless, chargeless elementary particles that come in three different flavours [1]. However, recent experiments indicate that neutrinos not only have mass, but also have multiple mass eigenstates that are not identical to the flavour states, thereby indicating mixing. As an evidence of mixing, neutrinos have been observed to change from one flavour to another during their propagation – a phenomenon called neutrino oscillation.

Neutrinos can be produced from four different sources, and accordingly they are termed as solar, atmospheric, reactor and accelerator neutrinos. In the past few decades, several experiments have confirmed the event of flavour change in each of these neutrinos. Besides, the values of the parameters affecting the probabilities of neutrino oscillation have been experimentally determined in most of the cases.

In this project, we have studied the reasons and derived the probabilities of neutrino flavour change, both in vacuum and in matter. We have also studied the parameters affecting this probability. We have discussed the special case of two-neutrino . Lastly, we have discussed some basic properties of neutrinos that are reflected in the previous derivations and highlighted a few relevant open problems.

To begin with, we have also studied the relevant topics in introductory High Physics and Quantum Mechanics to familiarize with the notations, units and methodologies that would be required for the subsequent project work.

2. Particle Physics and Quantum Mechanics: The Basics

2.1. Elementary Particles

During the 5th century B.C., Greek philosophers Leucippus and Democritus considered that if one keeps on cutting matter into smaller and smaller bits one will eventually end up with a fundamental bit which cannot be further subdivided. They called this smallest bit an atom. This is probably man’s first attempt at finding out what the world is made of.

It goes without saying that 25 centuries worth of research has considerably refined the concept of atom, and today we have a finer understanding of what the world is made of. We know today that there is no single fundamental particle, but a number of them. Our current understanding of elementary particles can be summarized in Table 2.1 [1].

Table 2.1: Elementary Particles Name Spin Charge Q Number B Number L 1 1 2 up (u), charm (c), top (t) 0 + 2 3 3 1 1 1 down (d), strange (s), bottom (b) 0 − 2 3 3 1 (푒−), (휇−), (휏−) 0 1 -1 2 1 휈 , 휈 , 휈 (neutrino) 0 1 0 푒 휇 휏 2 훾 (photon) 1 0 0 0 Guage 푊±, 푍 (weak bosons) 1 0 0 ±1, 0 푔푖 () 1 0 0 0

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Spin is in the units of ℏ. Charges are defined such that an electron has -1 units of charge. Each and lepton has a corresponding antiparticle with its B, L and Q having reversed signs.

All the particles which undergo strong interactions are called hadrons. and neutrinos do not undergo such interactions and are called leptons. While the electron is charged and can therefore undergo electromagnetic interactions, the neutrino participates exclusively in weak interactions.

Both quarks and leptons can be represented in terms of generation, wherein each generation forms a doublet. The representation for leptons has been shown in Table 2.2.

Table 2.2: Generations of Leptons First Generation Second Generation Third Generation 휈 휈휇 휈 푄 = 0 ( 푒 ) ( ) ( 휏 ) 푒− 휇− 휏− 푄 = −1

2.2. Natural Units (ℏ = 푐 = 1)

The two fundamental constants, viz., the Planck’s constant, ℎ, and the velocity of light in vacuum, 푐, appear repeatedly in expressions in relativistic quantum mechanics and high energy physics. It is convenient to use a system of units in which the use of these two constants can be avoided, thereby hugely simplifying such expressions.

This can be achieved by choosing units ℏ = 푐 = 1, where ℏ is the reduced Planck’s constant. ℎ ℏ = = 1.055 × 10−34 J sec 2휋 푐 = 2.998 × 108 m sec−1

In high energy physics, quantities are measured in units of GeV, which is convenient as rest mass-energy of a proton is of the order of 1 GeV.

Keeping in mind the dimensions of ℏ (ML2/T) and 푐 (L/T), it is possible to represent other quantities in terms of ℏ, 푐 and mass, 푚. For example, energy can be expressed as 퐸 = 푚푐2, and time can be expressed as 푡 = ℏ⁄푚푐2.

To find the conversion formulae, we might proceed as follows:

푚 kg ≡ 푚푐2 Energy units

1 kg ≡ 1 × (2.998 × 108)2 J (2.998 × 108)2 J = J 1.6 × 10−19 eV = 5.618 × 10−35 eV = 5.618 × 10−26 GeV. Table 2.3 shows a few other useful conversion factors.

Table 2.3: Conversion factors for MKS to Natural units Quantity Conversion factor Actual dimension GeV Mass 1 kg = 5.62 × 10−26 GeV 푐2 ℏ푐 Length 1 m = 5.07 × 1015 GeV−1 GeV ℏ Time 1 sec = 1.52 × 1024 GeV−1 GeV

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2.3. Dirac’s bra-ket Notation in QM

Ket Space In quantum mechanics, it is postulated that a physical state of a system is completely specified by a function, Ψ(푟, 푡), called the wave function. As per the notation developed by Dirac in 1939, the physical state of a quantum mechanical system can be equivalently represented by a state vector in a complex [2]. Such a vector is called a ket and is denoted by |훼⟩.

We shall state a few properties of kets:

1. |훼⟩ + |훽⟩ = |훾⟩ ( 1 )

2. 푐|훼⟩ = |훼⟩푐 where c is any . If 푐 = 0, the resulting ket is a null ket. ( 2 )

3. 퐴 ∙ (|훼⟩) = 퐴|훼⟩ where 퐴 is an corresponding to an observable. ( 3 )

4. If 퐴|훼′⟩ = 훼′|훼′⟩ then |훼′⟩ is called an eigenket of operator 퐴. ( 4 )

Bra Space Now, we postulate that corresponding to every ket |훼⟩ there exists a bra, denoted by ⟨훼| in a space called the bra space. Thus there is a one-to-one correspondence between ket and bra spaces, 퐷퐶 |훼⟩ ↔ ⟨훼| ( 5 ) where DC stands for dual correspondence.

퐷퐶 It is postulated that 푐|훼⟩ ↔ 푐∗⟨훼|. ( 6 )

We define an inner product between a bra and a ket as, (⟨훽|) ∙ (|훼⟩) = ⟨훽|훼⟩ such that, 1. ⟨훽|훼⟩ = ⟨훼|훽⟩∗ ( 7 ) 2. ⟨훼|훼⟩ ≥ 0 ( 8 )

Two kets |훼⟩ and |훽⟩ are called orthogonal if ⟨훼|훽⟩ = 0. ( 9 )

Operators

An operator acts on a ket from the left and the resulting product is another ket. 퐴 ∙ (|훼⟩) = 퐴|훼⟩

An operator acts on a bra from the right and the resulting product is another bra. (⟨훼|) ∙ 퐴 = ⟨훼|퐴

In general, the ket 퐴|훼⟩ and the bra |훼⟩퐴 are not dual to eachother. But, 퐷퐶 퐴|훼⟩ ↔ |훼⟩퐴† ( 10 ) where the operator 퐴† is the Hermitian adjoint of 퐴. 퐴 is called Hermitian if, 퐴 = 퐴†. ( 11 )

We define the outer product of |훽⟩ and ⟨훼| as (|훽⟩) ∙ (⟨훼|) = |훽⟩⟨훼| which, unlike the inner operator, is not a simple number but is regarded as an operator.

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The Associative Axiom

By the associative axiom of multiplication [2], (|훽⟩⟨훼|) ∙ |훾⟩ = |훽⟩ ∙ (⟨훼|훾⟩) = |훽⟩⟨훼|훾⟩. ( 12 )

By another illustration of this axiom, ⟨훽| ∙ (퐴|훼⟩) = (⟨훽|퐴) ∙ |훼⟩.

Because both sides of this equation are same, we use a more compact expression, ⟨훽|퐴|훼⟩. It can be shown that [2] ∗ ⟨훽|퐴|훼⟩ = ⟨훼|퐴†|훽⟩ ( 13 )

Matrix Representation

Hermitian operators (defined in 11) are of particular interest in quantum mechanics as they often turn out to be operators representing some physical observables. It can be proved that eigenvalues of a Hermitian operator 퐴 are real and that the eigenkets of 퐴 corresponding to the different eigenvalues are orthogonal [2].

Thus, if 훼′ be an eigenvalue of 퐴, 훼′ = 훼′∗ ( 14 ) and, ⟨훼′′|훼′⟩ = 0, 훼′′ ≠ 훼′ ( 15 )

Now, for an arbitrary ket |훼⟩ in the ket space spanned by eigenkets of 퐴, we can expand |훼⟩ as,

|훼⟩ = ∑ 푐훼′|훼′⟩ 훼′ ( 16 ) Multiplying both sides by ⟨α′′|, we get ′′ ′′ ′ ⟨α |α⟩ = ∑ 푐훼′⟨α |α ⟩ 훼′ Using (15), ′′ 푐훼′′ = ⟨훼 |훼⟩ ( 17 ) So, |훼⟩ = ∑ |훼′⟩⟨훼′|훼⟩ 훼′ ( 18 ) Since |훼⟩ is arbitrary, we must have, ∑ |훼′⟩⟨α′| = 1. 훼′ ( 19 ) We might compare equation (18) to the expansion of a vector 퐕 in Euclidean space:

퐕 = ∑ 퐞̂퐢(퐞̂퐢 ∙ 퐕) 푖 Clearly, |훼′⟩⟨α′|, when operates on |훼⟩, selects that portion of ket |훼⟩ that is parallel to |훼′⟩. So |훼′⟩⟨α′| is known as the projection operator along the base ket |훼′⟩.

We can proceed similarly and using (19) twice, we can write an arbitrary operator 푋 as, 푋 = ∑ ∑ |훼′′⟩⟨훼′′|푋|훼′⟩⟨α′| 훼′′ 훼′

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( 20 ) If 푁 be the dimensionality of the ket space, we have a total of 푁2 terms. We may arrange them in an 푁 × 푁 matrix as,

⟨훼(1)|푋|훼(1)⟩ ⟨훼(1)|푋|훼(2)⟩ … 푋 = (⟨훼(2)|푋|훼(1)⟩ ⟨훼(2)|푋|훼(2)⟩ …) ( 21 ) ⋮ ⋮ ⋱ Similarly,

⟨훼(1)|푋†|훼(1)⟩ ⟨훼(1)|푋†|훼(2)⟩ … † 푋 = (⟨훼(2)|푋†|훼(1)⟩ ⟨훼(2)|푋†|훼(2)⟩ …) ⋮ ⋮ ⋱ Using (13), it is clear that the matrix representing 푋† is the complex conjugate transposed matrix of 푋.

2.4. Rotations in Quantum Mechanics

We recall from classical mechanics that rotations in three dimensions can be represented by real, orthogonal 3 × 3 matrices. The new and old components of a vector are related via the roation matrix R as follows:

푉푥′ 푉푥 (푉푦′) = (푅) (푉푦) ( 22 )

푉푧′ 푉푧

For example, a finite rotation of 휙 about the z-axis is achieved by using an operator 푅푧(휙) on a vector, where 푅푧(휙) is given by cos 휙 − sin 휙 0

푅푧(휙) = (sin 휙 cos 휙 0) ( 23 ) 0 0 1 For an infinitesimal rotation, using Taylor’s expansion and ignoring terms of order 휀3 and higher, 휀2 1 − −휀 0 2 푅 (휀) = ( 휀2 ) ( 24 ) 푧 휀 1 − 0 2 0 0 1 In quantum mechanics, we can proceed by analogy to state that there exists an operator 풟(푅) in the appropriate ket space corresponding to R such that, |훼⟩푅 = 풟(푅)|훼⟩ ( 25 ) where |훼⟩ and |훼⟩푅 are the original and rotated systems respectively. It can be shown that the appropriate infinitesimal operator can be written as [1]

푈휀 = 1 − 푖퐽휀 ( 26 ) where 퐽, the generator of rotations can be defined as a component of the angular . (In classical mechanics we know that angular momentum is the generator of rotations.)

A rotation through finite angle 휃 may be thought to be built from 푛 successive small rotations. 푛 푛 푖퐽휃 −푖퐽휃 푈(휃) = (푈휀) = (1 − ) → 푒 ( 27 ) 푛 푛→∞

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2.5. The Klein-Gordon Equation

We recall that the Schrödinger equation for a free particle can be obtained starting from the classical, nonrelativistic energy-momentum relation, 푝2 퐸 = . ( 28 ) 2푚 Substituting 휕 퐸 → 푖ℏ , 퐩 = −푖ℏ훁 ( 29 ) 휕푡 into (28), and shifting to natural units, we get the Schrödinger equation, 휕휓 1 푖 + ∇2휓 ( 30 ) 휕푡 2푚 where 휓(퐱, 푡) is a complex wavefunction.

Similarly, we start off with the relativistic energy-momentum relation (in natural units),

퐸2 = 퐩2 + 푚2. ( 31 )

Then we use the relations (29) in (31) and obtain the Klein-Gordon equation, 휕2휙 − + ∇2휙 = 푚2휙 ( 32 ) 휕푡2 We may use the D’Alembertian operator, 2 휕2 ≡ − ∇2 ( 33 ) 휕푡2 in (32) to get a more compact form of the Klein-Gordon equation, 2 ( + 푚2) 휙 = 0. ( 34 )

Taking the free particle solution 휙 = 푁푒푖퐩∙퐱−푖퐸푡 ( 35 ) and using it in (34) we get, −퐸2휙 + 푝2휙 + 푚2휙 = 0 ( 36 ) or, we get the energy eigenvalues of the Klein-Gordon equation as 1/2 퐸 = ±(퐩ퟐ + 푚2) ( 37 ) which gives negative values of E in addition to positive ones.

Dirac explained these negative solutions postulating that all negative energy states are occupied by 퐸 < 0 electrons. Whenever an electron is excited from 퐸 < 0 to 퐸 > 0 state, the corresponding absence of electron, or presence of a “hole” in the vacuum sea is explained as the presence of an antiparticle (positron, +e) of energy +퐸 [1]. This explains pair production of an electron and positron.

Stückelberg and Feynman suggested that a negative energy solution describes a particle that propagates backward in time or, equivalently, a positive energy antiparticle propagating forward in time. Thus an emission of a positron with energy+퐸 is the same as the absorption of an electron of energy – 퐸 [1]. Mathematically, it is easy to see 푒−푖(−퐸)(−푡) = 푒−푖퐸푡.

2.6. The Dirac Equation

∂ As we can see from (34), the Klein-Gordon equation is quadratic in and ∇. However, Dirac wanted to write the ∂t ∂ relativistic Schrödinger equation in a form that is linear in (implying linear in ∇ as well, since the equation must ∂t be covariant). Thus, we can begin with the general form

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퐻휓 = (훂 ∙ 퐏 + 훽푚)휓 ( 38 ) where 훼푖(푖 = 1, 2, 3) and 훽 are constants. However, we must satisfy (31) simultaneously. In other words, 퐻2휓 = (퐏ퟐ + 푚2)휓 ( 39 ) and (38) must together represent the Dirac Equation.

Starting from (38), we write (in Einstein summation convention) 2 퐻 휓 = (훼푖푃푖 + 훽푚)(푎푗푃푗 + 훽푚)휓 2 2 2 2 = (훼푖 푃푖 + (훼푖훼푗 + 훼푗훼푖)푃푖푃푗 + (훼푖훽 + 훽훼푖)푃푖푚 + 훽 푚 )휓 Comparing with (39), 2 2  훼푖 = 훽 = 0 ( 40 )  훼푖훼푗 + 훼푗훼푖 = 훼푖훽 + 훽훼푖 = 0. Hence 훼푖’s and 훽 anticommute with one another. ( 41 )

By (41) 훼푖’s and 훽 cannot be numbers and must be represented by matrices. The lowest dimensionality matrices satisfying these are 4 × 4. They are not unique. One representation, known as Dirac-Pauli representation is as follows [1] 0 훔 퐼 0 훂 = ( ) , 훽 = ( ) 훔 0 0 −퐼 where 훔 are the Pauli matrices [1]: 0 1 0 −푖 1 0 휎 = ( ), 휎 = ( ), 휎 = ( ) 1 1 0 2 푖 0 3 0 −1 Making necessary operator substitutions given by (29), we can rewrite (38) as 휕휓 푖 = −푖훂 ∙ 훁휓 + 푚휓 ( 42 ) 휕푡 with 훼푖’s and 훽 satisfying (40) and (41). Multiplying (42) from the left by 훽, we get ∂ψ 푖훽 = −푖훽훂 ∙ 훁ψ + 훽푚휓 ∂t which can be rewritten as 휇 (푖훾 휕휇 − 푚)휓 = 0 ( 43 ) ∂ where 훾휇 ≡ (훽, 훽훂) and 휕 = ( , 훁) (in four vector notation). 훾휇’s are known as the Dirac 훾 matrices. 휇 ∂t Since 훾0 = 훽, † 훾0 = 훾0 ( 44 ) (훾0)2 = 퐼 ( 45 )

But, for 훾푘, 푘 = 1, 2, 3, † † 훾푘 = (훽훼푘) = 훼푘훽 = −훾푘 ( 46 )

We take the Hermitian conjugate of the Dirac equation 휕휓 휕휓 푖훾0 + 푖훾푘 − 푚휓 = 0 휕푡 휕푥푘 where 푘 = 1, 2, 3, and using (44) and (46) we get, 휕휓† 휕휓† −푖 훾0 − 푖 (−훾푘) − 푚휓† = 0 ( 47 ) 휕푡 휕푥푘 Multiplying (47) by 훾0 from the right, 휕휓† 휕휓† −푖 훾0훾0 + 푖 훾푘훾0 − 푚휓†훾0 = 0 휕푡 휕푥푘 Or, 휕휓† 휕휓† −푖 훾0훾0 − 푖 훾0훾푘 − 푚휓†훾0 = 0 휕푡 휕푥푘 Or,

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휕휓̅ 휕휓̅ −푖 훾0 − 푖 훾푘 − 푚휓̅ = 0 휕푡 휕푥푘 where 휓̅ = 휓†훾0. Or, 휇 푖휕휇휓̅훾 + 푚휓̅ = 0 ( 48 )

Multiplying (43) by 휓̅ from the left and (48) by 휓 from the right and adding, 휇 휇 휇 휓̅훾 휕휇휓 + (휕휇휓̅)훾 휓 = 휕휇(휓̅훾 휓) = 0 ( 49 ) which gives the continuity equation.

Thus the four-vector current density of, say, an electron of charge –e can be given by [1] 퐽휇 = −푒휓̅훾휇휓 ( 50 )

2.7. Bilinear covariants and Weak Interactions

We can further generalize current densities by using products of 훾-matrices instead of just one as in (50). We can list a number of bilinear quantities of the form

(ψ̅)(4 × 4)(ψ). For notational simplicity, we define

훾5 ≡ 푖훾0훾1훾2훾3. ( 51 )

Table 2.4: Bilinear Covariants Scalar 휓̅휓 Vector ψ̅γμψ Tensor 휓̅σ휇휈휓 Axial Vector ψ̅γ5γμψ Pseudoscalar ψ̅γ5ψ

Experiments indicate that leptons participate in weak interactions in a special combination of two of the bilinear covariants. For electron and an electron-flavoured neutrino [1], 1 Jμ = − ψ̅ γμ(1 − γ5)ψ 2 푒 휈 which takes a form of 푉 − 퐴 (Vector ‘minus’ Axial Vector, (γμ − γμγ5)).

3. Neutrino Flavours, Mass Eigenstates and Mixing

3.1. Introduction

As discussed earlier, neutrinos come in three flavours, electron neutrinos (휈푒), muon neutrinos (휈μ), and tau neutrinos (휈τ). Each flavour is, of course, associated with a corresponding antiparticle called an antineutrino. The Standard Model states that neutrinos are massless and chargeless, and only undergo weak interactions.

However, it has been observed that neutrinos can change their flavours during their travel. That is, a neutrino which was generated with a certain flavour might end up having a different flavour after travelling some distance. Such flavour changes require, as we shall show in subsequent sections, that neutrino flavours have

8 different masses with significant mixing. This, in turn, implies that neutrinos are not massless and, therefore, can participate in gravitational as well.

3.2. Leptonic Mixing

To begin with, we shall assume that neutrinos have masses. Thus there is a spectrum of neutrino mass eigenstates, 휈푖, 푖 = 1, 2, …, each with mass 푚푖. We can, however, distinguish between neutrinos in terms of their flavours as well. We define three flavour states, 휈훼, 훼 = 푒, 휇, 휏, as observed experimentally. Mixing may be described with the observation that each of the three flavours of neutrinos can be expressed as a superposition of mass eigenstates. To understand this experimentally, consider the leptonic decay + 푊 → 휈푖 + ℓ̅̅훼̅ ( 52 ) where ℓ훼 is a charged lepton of flavour 훼. Mixing implies that every time the above decay produces a particular ℓ̅̅훼̅, the accompanying neutrino mass eigenstate is not the same 휈푖, but can be any 휈푖 even if the lepton has a fixed flavour. Thus, each 푣훼 is actually a superposition of several eigenstates 휈푖’s, of which, of course, only one state can be discerned during a single observation.

Thus, we can write a flavour state | 휈훼⟩ in the form [3]:

| 휈훼⟩ = ∑ 푈훼푖 | 휈푖⟩ 푖 ( 53 ) The 푈훼푖’s may be written in a matrix form, called leptonic mixing matrix. A typical leptonic mixing matrix assuming 푖 = 1, 2, 3 and 훼 = 푒, 휇, 휏 would look like 푈푒1 푈푒2 푈푒3 푈 = (푈휇1 푈휇2 푈휇3) 푈휏1 푈휏2 푈휏3

The Standard Model states 푈 is unitary. Thus, 푈푈† = 푈†푈 = 퐼 ( 54 )

Inverting (53), we can express each mass eigenstate as a superposition of flavours ∗ | 휈푖⟩ = ∑ 푈훼푖 | 휈훼⟩ 훼 ( 55 ) The corresponding mixing matrix is, of course, 푈†.

Assuming there are 3 mass eigenstates, the Lagrangian can be expressed in mass eigenstate (basis) terms as: ℒ = 휈̅푚휈 푚1 0 0 휈1 = (휈1̅ 휈2̅ 휈3̅ ) ( 0 푚2 0 ) (휈2) ⏟ 0 0 푚 3 휈3 푀퐷 ( 56 ) ℒ can also be expressed in flavour basis, as

푀11 푀12 푀13 휈훼 ℒ = (휈훼̅ 휈휇̅ 휈휏̅ ) (푀21 푀22 푀23) (휈휇 ) ⏟푀 31 푀 32 푀 33 휈휏 푀 ( 57 )

Clearly, 푀 would be diagonal if there was no mixing. Using (53) and (55) in (57), and assuming 푀 to be symmetric, it follows † 푀퐷 = 푈 푀푈 ( 58 )

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2 From (53) and (55) it is clear that the flavour-훼 fraction in 휈푖, and, equivalently, mass-푖 fraction in 휈훼 is |푈훼푖| . This is, therefore, the probability that the neutrino will have mass 푚푖 when a ℓ̅̅훼̅ is observed in the decay (52).

4. Neutrino Oscillation in Vacuum

4.1. Detecting a flavour change

The term ‘neutrino oscillation’ refers to the phenomenon of neutrino flavour change, for reasons stated later on. However, as discussed earlier, neutrinos participate in weak interactions only (and very weakly in gravity, owing to their extremely small masses), which makes it difficult to detect them. But the charged lepton produced alongside a neutrino can be easily detected and its flavour can be identified. Thus we can determine the flavour of the produced neutrino, say 훼.

Similarly, at the end of a path of length 퐿, the neutrino reacts with the detector and produces a charged lepton. Determining the flavour of the charged lepton at the detector, we can find the final flavour of the neutrino, say 훽.

If 훼 ≠ 훽, then the neutrino has changed its flavour in its journey. This neutrino flavour change, 휈훼 → 휈훽 is a quantum mechanical effect, and we can try and find out the probability of such a change, 푃(휈훼 → 휈훽).

4.2. Finding the oscillation probability

Since each 휈훼 is a superposition of 휈푖’s, we have to individually add the contribution from each travelling 휈푖 while finding 푃(휈훼 → 휈훽).

We shall first find the amplitude, denoted by Amp(휈훼 → 휈훽). Each 휈푖’s contribution will depend on three factors [3]:

 The amplitude for 휈푖 when a ℓ̅̅훼̅ is produced at the source. As explained earlier, this is given by 푈훼푖.  The amplitude for 휈푖 to propagate from source to detector. Let this be Prop(휈푖). ̅̅̅ ∗  The amplitude for 휈푖 when a ℓ훽 is detected at the detector. This is given by 푈훽푖.

Thus, the amplitude of flavour change 휈훼 → 휈훽,

∗ Amp(휈훼 → 휈훽) = ∑ 푈훼푖Prop(휈푖)푈훽푖 푖 ( 59 ) We shall now find the value of Prop(휈푖). In the rest frame of the neutrino, its state vector as a function of time 휏, follows the simple Schrödinger equation, 휕 푖 |휈 (휏)⟩ = 푚 |휈 (휏)⟩ 휕휏 푖 푖 푖 whose solution is given by −푚푖휏 |휈푖(휏)⟩ = 푒 |휈푖(0)⟩.

The amplitude of 휈푖 travelling for time 휏0 is given by −푚푖휏0 ⟨휈푖(0)|휈푖(휏0)⟩ = 푒

Thus if 휏푖 be the time taken by the 휈푖 neutrino to travel from its source to the detector in its rest frame (proper time 휏푖), then,

10

−푚푖휏푖 Prop푅푒푠푡(휈푖) = ⟨휈푖(0)|휈푖(휏푖)⟩ = 푒 ( 60 )

However, we need Prop(휈푖) in the lab frame. So we shall need to use a Lorentz transform to find the corresponding expression in the lab frame. The lab frame variables are [3]:  Distance between source and detector, 퐿.  Laboratory-frame time, 푡.

 Energy of mass eigenstate 휈푖, 퐸푖.  Momentum of mass eigenstate 휈푖, 푝푖. By Lorentz invariance, 푚푖휏푖 = 퐸푖푡 − 푝푖퐿 ( 61 )

To eliminate 푝푖, we use the relation 푚2 푝 = √퐸2 − 푚2 ≅ 퐸 − 푖 ( 62 ) 푖 푖 2퐸 2 2 where we have used the fact that rest energy of neutrinos are extremely tiny, 푚푖 ≪ 퐸 . Using (62) in (61), 푚2 푚 휏 ≅ 퐸푡 − 퐸퐿 + 푖 퐿 푖 푖 2퐸 푚2 = 퐸(푡 − 퐿) + 푖 퐿. 2퐸 ( 63 ) Here we have used the same 퐸 for different mass eigenstates. It can be justified as follows. Suppose two different components, 휈푖 and 휈푗, have different 퐸푖 and 퐸푗. By the time they reach the detector, they have phases of 푒−푖퐸푖푡 and 푒−푖퐸푗푡 respectively, where 푡 is travel time in lab frame. Thus the detector detects an interference caused by two components with a phase difference of 푒−푖(퐸푖−퐸푗)푡, which vanishes for an average over time for 푖 ≠ 푗 [3]. Thus only components with same energy are detected.

The term 퐸(푡 − 퐿) is, therefore, common to every interfering mass eigenstate. Thus, considering only the 푖- dependent part 푚2 −푖 푖 퐿 푃푟표푝(휈푖) = 푒 2퐸 . ( 64 )

Thus, (59) can be finally written as 푚2 −i 푖 퐿 2퐸 ∗ Amp(휈훼 → 휈훽) = ∑ 푈훼푖e 푈훽푖. 푖 ( 65 ) Next, we shall find 푃(휈훼 → 휈훽) from (65) [3]. 2 푃(휈훼 → 휈훽) = |Amp(휈훼 → 휈훽)| ∗ 2 푚2 푚 −i 푖 퐿 −i 푗 퐿 2퐸 ∗ 2퐸 ∗ = (∑ 푈훼푖e 푈훽푖) (∑ 푈훼푗e 푈훽푗) 푖 푗 퐿 푖 (푚2−푚2) ∗ ∗ 2퐸 푗 푖 = ∑ ∑ 푈훼푖푈훽푖푈훼푗푈훽푗푒 푖 푗 퐿 푖 ∆푚2 ∗ ∗ ∗ ∗ 2퐸 푗푖 = ∑ 푈훼푖푈훽푖푈훼푖푈훽푖 + ∑ 푈훼푖푈훽푖푈훼푗푈훽푗푒 푖 푖≠푗 ( 66 ) where 2 2 2 ∆푚푗푖 = (푚푗 − 푚푖 ) ( 67 )

We derive the following identity: 푒푖퐴 = cos 퐴 + 푖 sin 퐴

11

퐴 = 1 − 2 sin2 + 푖 sin 퐴 2 ( 68 ) 퐿 푖 ∆푚2 We shall write (66) as a sum of four sums by expanding 푒 2퐸 푗푖 using (68), as

푃 푃1 푃2 3 ⏞ ⏞ ⏞ 퐿 푃(휈 → 휈 ) = ∑ 푈∗ 푈 푈 푈∗ + ∑ 푈∗ 푈 푈 푈∗ − 2 ∑ 푈∗ 푈 푈 푈∗ sin2 (∆푚2 ) 훼 훽 훼푖 훽푖 훼푖 훽푖 훼푖 훽푖 훼푗 훽푗 훼푖 훽푖 훼푗 훽푗 푗푖 4퐸 푖 푖≠푗 푖≠푗 퐿 + 푖 ∑ 푈∗ 푈 푈 푈∗ sin (∆푚2 ). 훼푖 훽푖 훼푗 훽푗 푗푖 2퐸 ⏟푖≠ 푗 푃4 ( 69 ) Next, we shall evaluate each part of the expression individually. 퐿 푃 = ∑ 푈∗ 푈 푈 푈∗ sin2 (∆푚2 ) 3 훼푖 훽푖 훼푗 훽푗 푗푖 4퐸 푖≠푗 퐿 퐿 = ∑ 푈∗ 푈 푈 푈∗ sin2 (∆푚2 ) + ∑ 푈∗ 푈 푈 푈∗ sin2 (∆푚2 ) 훼푖 훽푖 훼푗 훽푗 푖푗 4퐸 훼푖 훽푖 훼푗 훽푗 푗푖 4퐸 푖>푗 푖<푗 퐿 퐿 = ∑ 푈∗ 푈 푈 푈∗ sin2 (∆푚2 ) + ∑ 푈∗ 푈 푈 푈∗ sin2 (∆푚2 ) 훼푖 훽푖 훼푗 훽푗 푖푗 4퐸 훼푗 훽푗 훼푖 훽푖 푖푗 4퐸 푖>푗 푖>푗 퐿 = ∑ sin2 (∆푚2 ) (푈∗ 푈 푈 푈∗ + 푈∗ 푈 푈 푈∗ ) 푖푗 4퐸 훼푖 훽푖 훼푗 훽푗 훼푗 훽푗 훼푖 훽푖 푖>푗 퐿 = ∑ sin2 (∆푚2 ) (푈∗ 푈 푈 푈∗ + 푈 푈∗ 푈∗ 푈 ) 푖푗 4퐸 훼푖 훽푖 훼푗 훽푗 훼푖 훽푖 훼푗 훽푗 푖>푗 퐿 ∗ = ∑ sin2 (∆푚2 ) [푈∗ 푈 푈 푈∗ + (푈∗ 푈 푈 푈∗ ) ] 푖푗 4퐸 훼푖 훽푖 훼푗 훽푗 훼푖 훽푖 훼푗 훽푗 푖>푗 퐿 = 2 ∑ ℜ(푈∗ 푈 푈 푈∗ ) sin2 (∆푚2 ) 훼푖 훽푖 훼푗 훽푗 푖푗 4퐸 푖>푗 ( 70 ) where ℜ(푍) denotes the real part of a complex number 푍.

2 2 2 Note, in the second step we have replaced the first ∆푚푗푖 with ∆푚푖푗 keeping the sign same as sin is an even function.

We shall proceed similarly for 푃4: 퐿 푃 = ∑ 푈∗ 푈 푈 푈∗ sin (∆푚2 ) 4 훼푖 훽푖 훼푗 훽푗 푗푖 2퐸 푖≠푗 퐿 퐿 = − ∑ 푈∗ 푈 푈 푈∗ sin (∆푚2 ) + ∑ 푈∗ 푈 푈 푈∗ sin (∆푚2 ) 훼푖 훽푖 훼푗 훽푗 푖푗 2퐸 훼푖 훽푖 훼푗 훽푗 푗푖 2퐸 푖>푗 푖<푗 퐿 퐿 = − ∑ 푈∗ 푈 푈 푈∗ sin (∆푚2 ) + ∑ 푈∗ 푈 푈 푈∗ sin (∆푚2 ) 훼푖 훽푖 훼푗 훽푗 푖푗 2퐸 훼푗 훽푗 훼푖 훽푖 푖푗 2퐸 푖>푗 푖>푗 퐿 = ∑ sin (∆푚2 ) (푈∗ 푈 푈 푈∗ − 푈∗ 푈 푈 푈∗ ) 푖푗 2퐸 훼푗 훽푗 훼푖 훽푖 훼푖 훽푖 훼푗 훽푗 푖>푗 퐿 ∗ = ∑ sin (∆푚2 ) [(푈∗ 푈 푈 푈∗ ) − 푈∗ 푈 푈 푈∗ ] 푖푗 2퐸 훼푖 훽푖 훼푗 훽푗 훼푖 훽푖 훼푗 훽푗 푖>푗 퐿 = ∑ sin (∆푚2 ) [−2푖ℑ(푈∗ 푈 푈 푈∗ )] 푖푗 2퐸 훼푖 훽푖 훼푗 훽푗 푖>푗

12

퐿 = −2푖 ∑ ℑ(푈∗ 푈 푈 푈∗ )sin (∆푚2 ). 훼푖 훽푖 훼푗 훽푗 푖푗 2퐸 푖>푗

( 71 ) where ℑ(푍) denotes the imaginary part of a complex number 푍. In the second step we have replaced the first 2 2 ∆푚푗푖 with ∆푚푖푗 and changed the sign of that term as sin is an odd function.

We shall evaluate the two terms 푃1 and 푃2 jointly.

∗ ∗ ∗ ∗ 푃1 + 푃2 = ∑ 푈훼푖푈훽푖푈훼푖푈훽푖 + ∑ 푈훼푖푈훽푖푈훼푗푈훽푗 푖 푖≠푗 ∗ ∗ = ∑ ∑ 푈훼푖푈훽푖푈훼푗푈훽푗 푖 푗 ∗ ∗ = ∑(푈훼푖푈훽푖) ∑(푈훼푗푈훽푗) 푖 푗 2 ∗ = |∑ 푈훼푖푈훽푖| . 푖 ( 72 ) To evaluate this, we shall use the unitary property of 푈 (54). Assuming three mass eigenstates, 푈 and 푈† can be written as

∗ ∗ ∗ 푈푒1 푈푒2 푈푒3 푈푒1 푈휇1 푈휏1 † ∗ ∗ ∗ 푈 = (푈휇1 푈휇2 푈휇3), 푈 = (푈푒2 푈휇2 푈휏2) ( 73 ) ∗ ∗ ∗ 푈휏1 푈휏2 푈휏3 푈푒3 푈휇3 푈휏3

∗ ∗ ∗ ∑ 푈푒푖푈푒푖 ∑ 푈푒푖푈휇푖 ∑ 푈푒푖푈휏푖 푖 푖 푖 1 0 0 † ∑ 푈 푈∗ ∑ 푈 푈∗ ∑ 푈 푈∗ ∴ 푈푈 = 휇푖 푒푖 휇푖 휇푖 휇푖 휏푖 = (0 1 0) 푖 푖 푖 0 0 1 ∗ ∗ ∗ ∑ 푈휏푖푈푒푖 ∑ 푈휏푖푈휇푖 ∑ 푈휏푖푈휏푖 ( 푖 푖 푖 ) ( 74 ) Or, in other words, 1 if 훼 = 훽 ∑ 푈 푈∗ = { 훼푖 훽푖 0 if 훼 ≠ 훽 푖 = 훿훼훽 ( 75 ) where 훿 is the Kronecker delta function.

Thus, using the result (75) in (72), we get 푃1 + 푃2 = 훿훼훽. ( 76 )

Now, putting (71), (72) and (76) in (69), we get 퐿 푃(휈 → 휈 ) = 훿 − 2 ∙ 2 ∑ ℜ(푈∗ 푈 푈 푈∗ ) sin2 (∆푚2 ) 훼 훽 훼훽 훼푖 훽푖 훼푗 훽푗 푖푗 4퐸 푖>푗 퐿 + 푖 [−2푖 ∑ ℑ(푈∗ 푈 푈 푈∗ )sin (∆푚2 )] 훼푖 훽푖 훼푗 훽푗 푖푗 2퐸 푖>푗 퐿 퐿 = 훿 − 4 ∑ ℜ(푈∗ 푈 푈 푈∗ ) sin2 (∆푚2 ) + 2 ∑ ℑ(푈∗ 푈 푈 푈∗ )sin (∆푚2 ) 훼훽 훼푖 훽푖 훼푗 훽푗 푖푗 4퐸 훼푖 훽푖 훼푗 훽푗 푖푗 2퐸 푖>푗 푖>푗 ( 77 )

13

4.3. Analysis and Discussions

1. To extend (77) to antineutrinos, we assume that CPT invariance holds true. Under this assumption, the process 휈̅̅훼̅ → 휈̅̅훽̅ is the CPT-mirror image of 휈훽 → 휈훼

푃(휈̅̅훼̅ → 휈̅̅훽̅) = 푃(휈훽 → 휈훼).

∗ ∗ ∗ ∗ Interchanging 훼 and 훽 in 푈훼푖푈훽푖푈훼푗푈훽푗 gives 푈훽푖푈훼푖푈훽푗푈훼푗, which is nothing but the complex conjugate of ∗ ∗ 푈훼푖푈훽푖푈훼푗푈훽푗. Thus in the expression of 푃(휈훽 → 휈훼), we simply need to reverse the sign of the term ∗ ∗ containing ℑ(푈훼푖푈훽푖푈훼푗푈훽푗) in (77).

Thus,

푃(휈̅̅훼̅ → 휈̅̅훽̅) = 푃(휈훽 → 휈훼) 퐿 퐿 = 훿 − 4 ∑ ℜ(푈∗ 푈 푈 푈∗ ) sin2 (∆푚2 ) − 2 ∑ ℑ(푈∗ 푈 푈 푈∗ )sin (∆푚2 ) 훼훽 훼푖 훽푖 훼푗 훽푗 푖푗 4퐸 훼푖 훽푖 훼푗 훽푗 푖푗 2퐸 푖>푗 푖>푗 ( 78 )

2. As the probability of neutrino flavour change is a sum of sinusoidal and sine-squared functions, it necessarily 퐿 oscillates with the value of . Hence the term “Neutrino Oscillation” is used. 퐸

2 3. If all neutrinos were massless, then the mass squared difference, ∆푚푖푗 = 0. Which reduces (77) and (78) to

푃(휈̅̅훼̅ → 휈̅̅훽̅) = 푃(휈훼 → 휈훽) = 훿훼훽 which gives a zero probability of the event 휈훼 → 휈훽 for 훼 ≠ 훽. This means the fact that neutrinos have been observed to undergo flavour changes in vacuum implies that neutrinos are not massless and, additionally, mass eigenstates are not degenerate.

4. Since the entire calculation has been done for the case of neutrinos travelling in vacuum, it is clear that the phenomenon of flavour change does not arise from interactions with matter, but arises from the time evolution of a neutrino itself.

∗ 5. If there was no leptonic mixing, all off-diagonal terms in 푈훼푖 would be zeroes. Thus at least one among 푈훼푖 ∗ ∗ and 푈훼푗 in 푈훼푖푈훽푖푈훼푗푈훽푗 is a zero for 푖 > 푗. Which again reduces (77) and (78) to

푃(휈̅̅훼̅ → 휈̅̅훽̅) = 푃(휈훼 → 휈훽) = 훿훼훽 This means the fact that neutrinos have been observed to undergo flavour changes in vacuum implies leptonic mixing.

퐿 6. We shall include the ℏ and 푐 terms in ∆푚2 to get a quantitative estimate. This can be done by dimensional 푖푗 4퐸 퐿 퐿 analysis. Since ∆푚2 is an argument for a trigonometric function, it must be dimensionless. ∆푚2 has a 푖푗 4퐸 푖푗 4퐸 푀2퐿 푀푠2 퐿 푐3 푀푠2 퐿3푠 dimension of [ ] = [ ]. From observation, ∆푚2 has a dimension [ ] [ ] = [1]. So, 푀퐿2 퐿 푖푗 4퐸 ℏ 퐿 푠3푀퐿2 푠2 퐿 푐3 ∆푚2 is the correct expression in S.I. 푖푗 4퐸 ℏ To be able to write the numerical values of mass in eV, length in km and energy in GeV, we convert to expression from SI, to suitable units using conversion factors in Table 1.3: 3 2 퐿 2 퐿푚푒푡푒푟 푐 ∆푚푖푗 | = ∆푚푘푔 4퐸 푁푎푡푢푟푎푙 4퐸퐽표푢푙푒 ℏ 3 8 3 2 10 퐿푘푚 1 (2.998 × 10 ) = ∆푚푒푉 35 2 × −10 × −34 (5.61 × 10 ) 4퐸퐺푒푉 1.6 × 10 1.055 × 10

14

2 퐿푘푚 = 1.27∆푚푒푉 퐸퐺푒푉 퐿(km) = 1.27∆푚2 (eV2) . 푖푗 퐸(GeV) ( 79 ) 2 2 2 2 where, for example, ∆푚푖푗(eV ) and ∆푚푒푉 are both used to mean ∆푚푖푗 expressed in electron volts. Experimentally, the sinusoidal terms are discernable as long as their arguments are of the order of unity or 2 −5 5 larger. Thus, if we need a ∆푚푖푗 sensitivity of, say, 10 with 퐸 = 1 GeV, we shall need a path of 퐿 = 10 km.

2 7. Equation (77) and (78) contain the term ∆푚푖푗, but do not contain the mass of each mass eigenstate explicitly. Hence, although we can find out the squared-mass splitting from neutrino oscillation experiments, we cannot find out the mass of each eigenstate.

2 2 8. Suppose only two mass eigenstates 휈1 and 휈2were significant, with ∆푚21 = ∆푚 . Correspondingly, the two flavour states are 휈푒 and 휈휇. We expect a corresponding 2 × 2 mixing matrix 푈 to exist, which must be unitary. A 2 × 2 has 1 rotation angle and 3 phase factors. It can be shown, that phase factors have no effect on oscillation and hence can be omitted [3]. Thus, the only possible unitary matrix with one angle parameter is: 푈푒1 푈푒2 cos 휃 sin 휃 푈 = ( ) = ( ) 푈휇1 푈휇2 − sin 휃 cos 휃 ( 80 ) and consequently, 푈∗ 푈∗ † 푒1 휇1 cos 휃 − sin 휃 푈 = ( ∗ ∗ ) = ( ) 푈푒2 푈휇2 sin 휃 cos 휃

Using this in (77), ∗ ∗ 4푈훼2푈훽2푈훼1푈훽1 = −4 sin 휃 cos 휃 cos 휃 sin 휃 = − sin2 2휃. Thus, from (77), for 훼 ≠ 훽, 퐿 퐿 푃(휈 → 휈 ) = 훿 − (− sin2 2휃) sin2 (∆푚2 ) + 2 × 0 × sin (∆푚2 ) 훼 훽 훼훽 푖푗 4퐸 푖푗 2퐸 퐿 = sin2 2휃 sin2 (∆푚2 ) 푖푗 4퐸 ( 81 )

Note, the same equation applies for 푃(휈̅̅훼̅ → 휈̅̅훽̅) since the last term vanishes for a real 푈 matrix.

)

훽

휈

→

훼

휈

( 푃

2 −3 2 Figure 4.1: Probability vs. Energy plot with L = 1000 km, ∆푚푖푗 = 2.4 × 10 푒푉 at 휃 = 45°.

15

From (81) we plot 푃(휈훼 → 휈훽) vs. Energy 퐸 for the two-neutrino case. For a definite 퐿, the probability will vary with 퐸 somewhat as in Fig. 4.1. Hence a proper choice for the range of 퐸 ensures proper sensitivity (Smaller values of 퐸 will cause very rapid fluctuations, while larger values will be monotonous.)

5. Neutrino Oscillation in Matter

5.1. Factors affecting Neutrinos in Matter

In the previous section, we calculated the oscillation probability of neutrinos travelling in vacuum. But that is usually not enough as neutrino experiments often involve detecting neutrinos that originate at a source on the ’s surface, travel through the bulk material of the earth and get detected at a detector placed on the surface several kilometres away. In such cases, we need to consider matter effects as well.

The presence of matter in a neutrinos path may affect its oscillation probability in two ways:

 Interaction with matter may cause a flavour change in a neutrino. But the Standard Model predicts that all neutrino-matter interactions are flavour conserving [3]. Hence we do not consider this possibility.

 Neutrinos can undergo forward scattering while interacting with ambient particles, which will give rise to an extra interaction . This can happen in two ways: 1. A neutrino 휈훼 can exchange a 푊 with the corresponding charged lepton ℓ훼 only. The only charged lepton that is found in the bulk earth material in significant numbers is the electron. Hence this effect will be observed in a 휈푒 neutrino (and its antiparticle 휈̅푒) only. The corresponding interaction potential will obviously be proportional to the number density of electrons, 푁푒, in the bulk matter. According to the Standard Model, this potential will also be proportional to the Fermi coupling constant 퐺퐹 [1, 3]. The Standard Model gives,

+√2퐺퐹푁푒 for 휈푒 푉푊 = { −√2퐺퐹푁푒 for 휈̅푒 ( 82 ) 2. According to the Standard Model, a neutrino of any flavour can exchange a 푍 boson with an ambient electron, proton or neutron. Also, the 푍-couplings to electron and proton are equal and opposite. Thus, since the bulk earth matter is electrically neutral, the electron and proton contributions via 푍-boson exchange will cancel out. Finally, a flavour-independent potential 푉푍 will stand out, and will be proportional to the number of neutrons per unit volume, 푁푛. The Standard Model gives [3], √2 − 퐺 푁 for 휈 2 퐹 푛 푉푍 = √2 + 퐺 푁 for 휈̅ { 2 퐹 푛 ( 83 ) To begin with, let’s consider the Schrödinger equation in the lab-frame for a neutrino traveling through matter: 휕 푖 |휈(푡)⟩ = ℋ|휈(푡)⟩ ( 84 ) 휕푡 where |휈(푡)⟩ is the multi-component neutrino state vector with one component for each of the neutrino flavours [3]. For two neutrino flavours 푒 and 휇, for example, 푓푒(푡) |휈(푡)⟩ = ( ) ( 85 ) 푓휇(푡) where 푓훼(푡) is the amplitude of the neutrino being a 휈훼 at time 푡. Therefore, ℋ is a 2 × 2 matrix in 휈푒-휈휇 space.

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5.2. Finding the Vacuum Hamiltonian

We will first work with two-neutrino case in vacuum and find out the components of ℋVac, which is the Hamiltonian corresponding to the vacuum case. To find out (훼, 훽) component of the ℋVac, we shall use (21). Starting from (21) and applying (53), the 휈훼-휈훽 component of ℋVac comes out to be:

⟨휈훼|ℋVac|휈훽⟩ = ⟨∑푖 푈훼푖 휈푖|ℋVac| ∑푗 푈훽푗 휈푗⟩

= ∑⟨푈훼푖휈푖|ℋVac|푈훽푖휈푖⟩ 푖 (all terms with ⟨휈푖|휈푗⟩ with 푖 ≠ 푗 vanish due to orthogonality.) ∗ = ∑ 푈훼푖푈훽푖퐸푖⟨휈푖|휈푖⟩ 푖 ∗ 2 2 = ∑ 푈훼푖푈훽푖√푝 + 푚푖 푖 ( 86 ) where we have used the fact that 퐸푖 is the energy of mass eigenstate 휈푖.

Using (86) and the matrix in (80) we evaluate each of the four terms in ℋVac. We denote the terms as ℋαα, 푚2 ℋ , ℋ , and ℋ respectively. Also, we use the highly relativistic approximation√푝2 + 푚2 = (푝 + 푖 ). αβ βα ββ 푖 2푝

ℋαα : 푚2 푚2 ℋ = cos2 휃 (푝 + 1) + sin2 휃 (푝 + 2) αα 2푝 2푝 푚2 푚2 = (cos2 휃 + sin2 휃)푝 + cos2 휃 1 + (1 − cos2 휃) 2 2푝 2푝 푚2 푚2 푚2 = 푝 + cos2 휃 1 + 2 − cos2 휃 2 2푝 2푝 2푝 cos2 휃 푚2 = 푝 − ∆푚2 + 2 2푝 2푝 푚2 2 cos2 휃 − 1 ∆푚2 = 푝 + 2 − ∆푚2 − ; 2푝 4푝 4푝 2 cos2 휃 − 1 2푚2 − ∆푚2 = − ∆푚2 + 푝 + 2 4푝 4푝 ∆푚2 푚2 + 푚2 = − cos 2휃 + 푝 + 1 2. 4푝 4푝

Proceeding similarly for ℋββ: 푚2 푚2 ℋ = sin2 휃 (푝 + 1) + cos2 휃 (푝 + 2) ββ 2푝 2푝 ∆푚2 푚2 + 푚2 = cos 2휃 + 푝 + 1 2. 4푝 4푝 ℋαβ, ℋβα: 푚2 푚2 ℋ = ℋ = − cos 휃 sin 휃 (푝 + 1) + sin 휃 cos 휃 (푝 + 2) αβ βα 2푝 2푝 푚2 푚2 = (− sin 휃 cos 휃 + sin 휃 cos 휃)푝 − cos 휃 sin 휃 1 + sin 휃 cos 휃 2 2푝 2푝 푚2 푚2 = sin 휃 cos 휃 ( 2 − 1) 2푝 2푝 ∆푚2 = sin 2휃 . 4푝

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Thus, 2 2 2 ∆푚 − cos 2휃 sin 2휃 푚1 + 푚2 1 0 ℋ = [ ] + (푝 + ) [ ] Vac 4푝 sin 2휃 cos 2휃 4푝 0 1 ( 87 ) As only the relative phases of the interfering contributions matter, and consequently only the relative energies matter, we can freely subtract a multiple of the from the expression of ℋVac [3]. This will not affect the differences between the eigenvalues of ℋ.

Also, for a highly relativistic neutrino, we can approximate 푝 ≅ 퐸. Thus (87) becomes ∆푚2 − 푐표푠 2휃 푠푖푛 2휃 ℋ = [ ] ( 88 ) 푉푎푐 4퐸 푠푖푛 2휃 푐표푠 2휃

5.3. Finding the Hamiltonian in Matter

Now we shall construct the corresponding Hamiltonian ℋ푀 for propagation in matter. As discussed earlier, we expect contributions from two more factors, the interaction potentials 푉푊 and 푉푍 [3]. 1 0 1 0 ℋ = ℋ + 푉 [ ] + 푉 [ ]. ( 89 ) 푀 푉푎푐 푊 0 0 푍 0 1  Since the 푉푊 term affects only 휈푒’s, only the upper left term, corresponding to 휈푒-휈푒 element, is non- vanishing.

 Since the 푉푍 term affects all flavours equally, a diagonal identity matrix is required. We rewrite (89) as, 1 0 1 0 1 0 ℋ = ℋ + 푉 [ ] + 푉 [ ] + 푉 [ ]. 푀 Vac 푊 0 −1 푊 0 1 푍 0 1 As discussed earlier, we can conveniently add or subtract any multiples of the identity matrix from the expression of the Hamiltonian without affecting the relative energy eigenvalues. Thus, the equation reduces to 1 0 ℋ = ℋ + 푉 [ ]. ( 90 ) 푀 푉푎푐 푊 0 −1 Using (82) and (88), (푉푊/2) 2 − (cos 2휃 − ) sin 2휃 ∆푚 (∆푚2/4퐸) ℋ = 푀 4퐸 (푉 /2) sin 2휃 (cos 2휃 − 푊 ) [ (∆푚2/4퐸) ] ∆푚2 −(cos 2휃 − 푥) sin 2휃 = [ ]. 4퐸 sin 2휃 (cos 2휃 − 푥) ( 91 ) where,

(푉푊/2) 2√2퐺퐹푁푒퐸 푥 = = (∆푚2/4퐸) ∆푚2 ( 92 ) For simplicity, we wish to write (91) in the form of (88). It can be done if we find an 푋 such that,

cos 2휃푀 = (cos 2휃 − 푥)푋 sin 2휃푀 = (sin 2휃푀)푋 ∆푚2 ∆푚2 = 푀 푋 Solving these equations, we get 푠푖푛2 2휃 푠푖푛2 2휃 = ( 93 ) 푀 푠푖푛2 2휃+(푐표푠 2휃−푥)2

2 2 2 2 ∆푚푀 = ∆푚 √푠푖푛 2휃 + (푐표푠 2휃 − 푥) ( 94 )

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Thus, 2 ∆푚푀 − 푐표푠 2휃푀 푠푖푛 2휃푀 ℋ푀 = [ ]. ( 95 ) 4퐸 푠푖푛 2휃푀 푐표푠 2휃푀

2 2 So, the Hamiltonian in matter is identical to that in vacuum if ∆푚 and 휃 are replaced, respectively, by ∆푚푀 and 2 휃푀 using (93) and (94). Thus the effective ∆푚 and 휃 are different in matter from that in vacuum.

5.4. Finding the Oscillation Probability in Matter

From (53) and (80), we get, for the two-neutrino case (note, 휃 has to be replaced with 휃푀 in case of matter) |휈푒⟩ = |휈1⟩ cos 휃푀 + |휈2⟩ sin 휃푀 |휈휇⟩ = −|휈1⟩ sin 휃푀 + |휈2⟩ cos 휃푀 ( 96 ) The eigenvalues of ℋM from (95) are ∆푚2 ∆푚2 휆 = + 푀 , 휆 = − 푀 1 4퐸 2 4퐸 ( 97 ) Thus solving (84) with |휈(휏 = 0)⟩ = |휈푒⟩ and using (96) and (97), ∆푚2 ∆푚2 −푖 푀푡 +푖 푀푡 |휈(푡)⟩ = |휈1⟩푒 4퐸 푐표푠 휃푀 + |휈2⟩푒 4퐸 푠푖푛 휃푀 ( 98 )

The probability that |휈(푡)⟩ is detected as |휈휇⟩ at time 푡 is 2 푃푀(휈푒 → 휈휇) = |⟨휈휇|휈(푡)⟩| 2 ∆m2 ∆m2 −푖 M푡 +푖 M푡 = |− sin 휃푀 푒 4E cos 휃푀 + cos 휃푀 푒 4E sin 휃푀|

2 ∆m2 ∆m2 +푖 M푡 −푖 M푡 = |sin 휃푀 cos 휃푀 (푒 4E − 푒 4E )|

2 ∆m2 = |sin 휃 cos 휃 (2푖 sin M 푡)| 푀 푀 4E ∆m2 = sin2 2휃 sin2 ( M 푡) 푀 4E 퐿 = sin2 2휃 sin2 (∆m2 ). 푀 M 4E ( 99 ) In the last step we have replaced 푡 with 퐿 as we have considered the highly relativistic case.

2 2 As we can get back the case of vacuum oscillations just by replacing ∆푚푀 and 휃푀 with ∆푚 and 휃 respectively, we get from (99), 퐿 푃(휈 → 휈 ) = sin2 2휃 sin2 (∆m2 ) 푒 휇 4E which is exactly (81), as expected.

5.5. Discussions

Clearly, matter effects are significantly determined by the quantity 푥. We would like to find out the extent to which 푥 could be significant. We shall calculate the value of 푥 for an atmospheric neutrino that travels through the earth’s mantle on its way to a detector.

We know, the squared-mass splitting for atmospheric neutrinos is about ∆푚2 = 2.4 × 10−3 eV2 [4, 5]. The value of Fermi coupling constant ≈ 1.67 × 10−23 eV−2. Density of mantle ≅ 3000 kg/m3.

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3000 kg Number of (protons + neutrons) in 1 m3 = = 1.80 × 1030. 1.67×10−27 kg 1.80×1030 So number density of electrons, 푁 = number density of protons = m−3 = 8.98 × 1029 m−3. 푒 2 (Assuming number of protons and neutrons are almost equal in mantle elements.)

From Table 1.3, we find, 1 m ≡ 5.07 × 1015 GeV−1. So, 1 m−3 = 7.67 × 10−48 GeV3. 29 −3 −18 3 9 3 So 푁푒 = 8.98 × 10 m = 7.19 × 10 GeV = 6.89 × 10 eV . Putting this value in (92), 2√2 × 1.17 × 10−23 eV−2 × 6.89 × 109 eV3 |푥| = 퐸 2.4 × 10−3 eV2 = 9.50 × 10−11퐸 eV−1 퐸 ≈ . 10.53 GeV ( 100 )

The value of 푥 is proportional to the neutrino energy 퐸. Thus, for a neutrino with high energy, say 20 GeV, the ∆m2 푠푖푛2 2θ matter effect is large. To demonstrate the dependence, we plot M vs. 퐸 and M vs. 퐸 in Fig 5.1 based on ∆푚2 푠푖푛2 2θ equations(93) and (94).

2 ∆푚 2 푀 푠푖푛 2θM

∆푚2 푠푖푛2 2θ

∆푚2 푠푖푛2 2휃 Figure 5.1: Plots for 푀 vs. 퐸 and 푀 vs. 퐸 at 휃 = 30°. ∆푚2 푠푖푛2 2휃 In special cases, matter effects can be dramatically large. Consider the case where 휃 is very small and 푥 ≅ cos 2휃. sin2 2휃 From (93) we can see that even though sin2 2휃 ≪ 1, sin2 2θ ≅ = 1, its maximum value. Thus, the M sin2 2휃 presence of matter in the path of a neutrino may cause a dramatic amplification of the in such special cases [3]. This phenomenon, known as the Mikheyev–Smirnov–Wolfenstein (MSW) resonance effect, was believed to occur in the case of neutrinos within solar matter. However, later experiments showed that the vacuum mixing angle for solar neutrinos is larger (휃 ≅ 34°).

As we noted earlier, 푥 will have an opposite sign if an antineutrino is used instead of a neutrino. Correspondingly, 2 as equations (93) and (94) suggest, the values of ∆푚푀 and 휃푀 will be different for neutrinos and antineutrinos. This can be a method to differentiate between neutrinos and their antiparticles, but as we shall see in the next 2 section, the relative hierarchies of the values of 푚푖 ’s and consequently the sign of the parameter 푥 is also not known.

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6. Properties of Neutrinos

6.1. The Mass-squared Splittings

In the discussion till now, we had assumed in some places that there are a total of 3 mass eigenstates for notational simplicity. But we do not know the total number of mass eigenstates for sure. However, the fact that there are at least three, can be easily understood from studying solar and atmospheric neutrinos.

2 −5 2  Solar neutrinos have a squared-mass splitting of ∆푚sol ≅ 8.0 × 10 eV [6]. We attribute the 2 2 2 −5 2 corresponding splitting to 휈1 and 휈2. Thus, ∆푚21 ≡ 푚2 − 푚1 ≅ 8.0 × 10 eV . 2 −3 2  Atmospheric neutrinos have a squared-mass splitting of ∆푚atm ≅ 2.4 × 10 eV [4, 5]. We must have one more mass eigenstate 휈3 to accommodate for this. So we attribute the atmospheric splitting to 휈2 and 휈3. 2 2 2 −3 2 Thus, ∆푚32 ≡ 푚3 − 푚2 ≅ 2.4 × 10 eV .

2 2 2 In the above representation, we have assumed 푚3 ≫ 푚2 > 푚1. However, as (91) indicates, the value of 푥, the parameter that determines matter effects in neutrino oscillations, reverses sign for a particular mass pair if the corresponding hierarchy is reversed. That is, if we do not assume a definite hierarchy, we cannot tell apart a neutrino and an antineutrino, and vice versa.

퐿 As discussed earlier, the ∆푚2 sensitivity of an experiment can be regulated by regulating the value of . Thus to 퐸 detect solar neutrinos, we might have a setup with 퐸 ≈ 1 GeV and 퐿 ≈ 105 km. Whereas, for atmospheric neutrinos, a setup with 퐸 ≈ 1 GeV and 퐿 ≈ 103 km would be ideal.

What if there are more than three mass eigenstates? If such mass eigenstates exist, their linear combinations will give rise to one or more flavours of neutrinos each of which does not form an isospin doublet with a lepton. Thus they cannot couple to 푊 or 푍 bosons unlike the three observed neutrino flavours. This implies, they cannot participate in weak interactions. Such neutrinos are called sterile neutrinos [7].

6.2. The Flavour and Mass States and Mixing

We know, each of the flavour states is a superposition of mass eigenstates, where the fraction of 휈푖 in 휈훼 is given 2 2 by |푈훼푖| , or equivalently, fraction of 휈훼 in 휈푖 is |푈훼푖| .

2 2 2 Experimentally, it turns out that |푈푒3| is very small [8]. A quantity 휃13 defined as |푈푒3| = sin 휃13 is the corresponding mixing angle. Experiments show that this mixing angle has a value of ~9°, indicating very little mixing [9]. Thus, 휈3 is essentially superposition of only 휈휇 and 휈휏. This 휈휇-휈휏 mixing angle is, however, large (~45°), indicating maximal mixing.

2 1 2 2 1 For 휈 , |푈 |2 ≈ |푈 | ≈ |푈 |2 ≈ . For 휈 , |푈 |2 ≈ . |푈 | ≈ |푈 |2 ≈ . [3] 2 푒2 휇2 휏2 3 1 푒1 3 휇2 휏2 6 Table 6.1 summarizes the above experimental observations.

Table 6.1: Table indicating mixing probabilities in neutrinos

휈푒 휈휇 휈휏

2 1 1 휈 |푈 |2 ≪ 1 |푈 | ≈ |푈 |2 ≈ 3 푒3 휇3 2 휏3 2

1 2 1 1 휈 |푈 |2 ≈ |푈 | ≈ |푈 |2 ≈ 2 푒2 3 휇2 3 휏2 3

2 2 1 1 휈 |푈 |2 ≈ |푈 | ≈ |푈 |2 ≈ 1 푒1 3 휇1 6 휏1 6

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Recalling that the matrix 푈 is a 3 × 3 unitary matrix, we should be able to rewrite 푈 in terms of (at least) 3 rotation angle and 1 phase factor. It is useful to represent 푈 as [3],

푖훿 1 0 0 푐13 0 푠13푒 푐12 푠12 0 푈 = [0 푐23 푠23] × [ 0 1 0 ] × [−푠12 푐12 0]. 푖훿 ⏟0 − 푠 23 푐 23 ⏟− 푠 13 푒 0 푐 33 ⏟ 0 0 1 Atmospheric Cross Solar

Note, the four independent factors are 휃12, 휃23, 휃13 and 훿, and 푐12 represents cos 휃12, and 푠12 ≡ sin 휃12.

Approximating atmospheric oscillation as a two neutrino 휈휇-휈휏 problem, 휃23 ≈ 휃atm. This is a valid approximation as 휈푒 is almost inexistent in 휈3, which participates in atmospheric oscillation. Turns out, this 휃atm is approximately 45°, implying maximal mixing [4, 5].

Similarly we can approximate oscillation as a two neutrino problem, 휃12 ≈ 휃sol. This is a valid approximation as 휈푒 is essentially either 휈1 or 휈2 and solar neutrinos are born as 휈푒 only. Experimentally, 휃sol ≈ 33.9° indicating large, though not maximal, mixing [6].

푖훿 The cross-mixing matrix contains a CP violating term 훿. However, since it is multiplied with 푠13푒 , its effect is negligible as long as 휃13 is zero or very small. As discussed earlier, 휃13 ≈ 9° [8, 9], which is small, but cannot be ignored.

7. Conclusions

As we have seen, assuming the existence of non-degenerate mass eigenstates of neutrinos gives us a probability- based model that generously accommodates for the experimentally observed phenomenon of neutrino oscillation. The existence of the phenomenon itself refutes the assumption that neutrinos are massless, and makes us look beyond the Standard Model. Numerous experiments in the recent decades have not only helped us reconfirm the phenomenon, but also have helped us determine the related parameters with better and better accuracies.

However, neutrino physics is still an active area of research with many more open questions and unsolved problems. We are in the process of building yet more sensitive detectors to make more precise measurements and observations.

One such open question, for example, is whether more than three mass eigenstates exist. As discussed, that would imply the presence of sterile neutrinos. The LSND experiment data suggests presence of another mass- squared splitting in the range of 0.2~10 eV2 [7]. To acccommodate for this splitting we shall definitely require one more mass state that is separated by a large magnitude from the other three.

We would also like to know the explicit values of the masses of the mass eigenstates. As discussed earlier, neutrino oscillation experiments can only give the relative squared-splittings of these values, from which we can, at best, find a range of values for the masses of the mass eigenstates.

Is there any difference between a neutrino and its antineutrino? We would like to know whether neutrinos are Majorana particles (particle identical to its antiparticle) or Dirac particles (particles and antiparticles are distinct). For this we might want to find out whether a neutrinoless is possible. A 훽-decay refers to the − − process 푛 → 푝 + 푒 + 휈̅푒. A neutrinoless double beta decay would refer to the process, 2푛 → 2푝 + 2푒 . To explain this, we must introduce two virtual 휈̅푒’s for each of the beta decay processes. But since the 휈̅푒 term doesn’t appear in the final product, each of them must have annihilated the other, which would imply that at least one flavour of the neutrino is a Majorana particle.

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Thus the phenomenon of neutrino oscillation and the discovery of neutrino mass has brought along with them several unsolved problems in particle physics, and we would obviously want to find ways to solve these.

8. References

[1] F. Halzen and A. D. Matrin, “Quarks and Leptons: An Introductory Course in Modern Particle Physics”.

[2] J. J. Sakurai, “Modern Quantum Mechanics”.

[3] B. Kayser, “Neutrino Physics,” eConf C040802 (2004) L004, arXiv:hep-ph/0506165.

[4] Super-Kamiokande Collaboration, “A Measurement of atmospheric neutrino oscillation parameters by SUPER-KAMIOKANDE,” Phys.Rev. D71 (2005) 112005, arXiv:hep-ex/0501064.

[5] K2K Collaboration, “Measurement of Neutrino Oscillation by the ,” Phys.Rev. D74 (2006) 072003, arXiv:hep-ex/0606032.

[6] KamLAND Collaboration, “Measurement of neutrino oscillation with KamLAND: Evidence of spectral distortion,” Phys.Rev.Lett. 94 (2005) 081801, arXiv:hep-ex/0406035.

[7] LSND Collaboration, “Evidence for neutrino oscillations from the observation of anti-neutrino(electron) appearance in a anti-neutrino(muon) beam,” Phys.Rev. D64 (2001) 112007, arXiv:hep-ex/0104049.

[8] T2K Collaboration, “Indication of Electron Neutrino Appearance from an Accelerator-produced Off-axis Beam,” Phys.Rev.Lett. 107 (2011) 041801, arXiv:1106.2822 [hep-ex].

[9] Daya Bay Collaboration, “Improved Measurement of Electron Antineutrino Disappearance at Daya Bay,” Chin.Phys. C37 (2013) 011001, arXiv:1210.6327 [hep-ex].

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