thesis

Neutrino Oscillations

Rohit R.

S. R. 11-01-00-10-91-11-1-09106

A thesis presented for the degree of Bachelor of Science

(Research)

Undergraduate ProgrammeIndian Institute of Science

Bengaluru - 560012India

20 April, 2015

Abstract


Rohit R.

Although the field of neutrino oscillations has been experimentally active for manyyears now, the sterile neutrino has eluded detection. We attempt to come up with a

proposal to use matter effects on neutrino oscillations to enhance sterile neutrinooscillations and hence detect the sterile neutrino.

1

Acknowledgements

First and foremost, I would like to thank my mentors Prof. Sudhir Vempati and Prof. UmaShankar for their invaluable help and guidance. I would also like to thank all my friends for theirsupport and scintillating discussions. This endeavour would definitely have not been possiblewithout my parents support and faith in me. I would also like to thank IISc for giving me theopportunity to work in this field. A special thanks to the Kishore Vaigyank Protsahana Yojanafor financial support.

2

Contents

1 Introduction 5

2 Neutrino Oscillations 62.1 2-flavour oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 n-flavour Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.1 Decomposition of the n-neutrino problem . . . . . . . . . . . . . . . . . . 102.3 Oscillations in the presence of matter . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.1 Uniform Matter Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3.2 Varying matter density: The Adiabatic approximation . . . . . . . . . . . 162.3.3 Non-adiabatic effects and the Adiabaticity Parameter . . . . . . . . . . . 172.3.4 The big picture: Looking at the Solar Neutrino problem . . . . . . . . . . 19

3 Experiments in Neutrino Physics: Measuring the Oscillation Parameters 213.1 Detection of Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Solar neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.1 The Solar Neutrino Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2.2 Measuring the Solar Neutrino Flux . . . . . . . . . . . . . . . . . . . . . . 233.2.3 The Solution of the Solar Neutrino Problem . . . . . . . . . . . . . . . . . 27

3.3 Experiments involving Atmospheric Neutrinos . . . . . . . . . . . . . . . . . . . . 283.3.1 Cosmic Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.3.2 Production of Atmospheric Neutrinos . . . . . . . . . . . . . . . . . . . . 293.3.3 Characteristics of the atmospheric neutrino flux[10] . . . . . . . . . . . . . 293.3.4 Zenith angle dependence of the atmospheric neutrino flux . . . . . . . . . 303.3.5 Measuring the Atmospheric Neutrino Flux . . . . . . . . . . . . . . . . . . 30

3.4 Accelerator experiments to measure 13 . . . . . . . . . . . . . . . . . . . . . . . 33

4 Sterile Neutrino Oscillations 364.1 Derivation of matter term for s oscillations . . . . . . . . . . . . . . . . . . 36

4.1.1 Certain Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2 Propagation in the Earths crust . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2.1 Baseline at Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.3 Outlook and Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

A The vacuum Hamiltonian in a more illuminating form 41

3

CONTENTS

B Some Mathematica demos 42B.1 3-flavour Oscillations in Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . 42B.2 Solar Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

B.2.1 Survival Probability of e . . . . . . . . . . . . . . . . . . . . . . . . . . . 44B.2.2 Density profile of the Sun and sin2 . . . . . . . . . . . . . . . . . . . . . 45B.2.3 The Adiabaticity parameter . . . . . . . . . . . . . . . . . . . . . . . . . . 45B.2.4 The Exclusion Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4

Chapter 1

Introduction

Neutrinos are very light leptons with masses of the order of meVs. Since they are neutralparticles, they do not take part in electromagnetic interactions. They come in three flavours:e, , and being produced/annihilated alongside the complementary lepton in weak processes.Neutrinos produced alongside electrons are of e flavour; and similarly for muons and taons.Since neutrinos are affected only by the weak and gravitational interactions (which are extremelyweak), they have very high mean free paths even in dense matter and hence may be used toprobe and understand the internal structure of otherwise inaccessible bodies like stars and otherastrophysical structures.

Whether or not neutrinos are massless was an open question until the recent discovery ofthe phenomenon of neutrino oscillations, which conclusively establish the existence of massiveneutrinos. This phenomenon is the main topic of discussion of this thesis. This article isorganised as follows. In Chapter II, we attempt to review the theoretical framework behindthe phenomenon of neutrino oscillations. In Chapter III, we shall look at the experimentaladvances in the field and the various estimations of the parameters associated with oscillations.In Chapter IV, we shall take a brief look at a simple model incorporating sterile neutrinos andconsider a stand alone accelerator-based experiment to look for sterile neutrinos.

5

Chapter 2


Neutrino Oscillations refers to the phenomenon whereby the flavour of a neutrino - earlierbelieved to be a constant of motion, varies with time without any external influence. Thechief reasons for this phenomenon are the non-trivial relation between the flavour and massneutrino eigenstates (meaning that states with a definite mass do not have a definite flavourand vice versa), and the non-zero mass difference between neutrino mass eigenstates. Hence,this phenomenon could be used to calculate the mass differences of neutrino states and also toestablish the existence of the massive neutrino.

We would like to note that the derivation outlined here is a highly simplified version whichdoes not take into account various conceptual issues[1, 2, 3, 4, 5, 6, 7, 8]. However, approachesthat do rectify these issues (field-theoretical, wave-packet based) are more elaborate and givethe same results.

2.1 2-flavour oscillations

We shall first look at the simplified case of oscillations between two flavours because, as shallbe seen later, the general case may be decoupled into many copies of these 2-flavour oscillations.Now, the left-handed components of the flavour states are unitary linear combinations of theleft-handed components of the mass eigenstates. Denoting the two flavour states by , (flavour states shall stand for left-handed states from here on) and the mass states by 1, 2,(

(t)(t)

)= U

(1(t)2(t)

)(2.1)

where U is the unitary mixing matrix. Now the mass (Hamiltonian) eigenstates evolve underSchrodinger dynamics as follows:

d

dt

(1(t)2(t)

)=

(E1 00 E2

).

(1(t)2(t)

)(2.2)

i.e.,

(1(t)2(t)

)=

(eE1t 00 eE2t

)(1(0)2(0)

)(2.3)

Call

(t) :=

(eE1t 00 eE2t

)(2.4)

6

CHAPTER 2. NEUTRINO OSCILLATIONS

Now substituting into equation (1),

U ((t)(t)

)= (t)U

((0)(0)

)(2.5)

Left multiply by U and use unitarity of U to obtain the relation between flavour states at thepoint of production (t = 0) and point of detection (at t).(

(t)(t)

)= U(t)U

((0)(0)

)(2.6)

Strictly speaking, the indices and may only stand for the active flavour states sinceneutrinos are produced and detected via weak processes. Sterile neutrinos, if they exist, mustbe detected indirectly via the disappearance of active flavour neutrinos.

It is clear that the amplitudes for oscillation are given by the off-diagonal elements ofU(t)U . Hence there will be no oscillations if this matrix is diagonal. One way this couldhappen is if U is the identity matrix (trivial mixing).

Now, the most general unitary matrix in two dimensions is of the form,

U =

(cos esinesin cos

)(2.7)

But since Dirac fermions can absorb phases, some of the phase factors are physicallyirrelevant and the mixing matrix maybe subject to the transformation

Ui eiUie (2.8)without changing the physics. Especially in the 2-dimensional case, it is easy to see that the onlyphase factor present maybe eliminated in this manner and the general mixing matrix becomes:

U =

(cos sinsin cos

)(2.9)

UU =(

cos sinsin cos

)(eE1t 00 eE2t

)(cos sinsin cos

)=

(eE1tcos2 + eE2tsin2 (eE1t eE2t)sincos(eE1t eE2t)sincos eE1tsin2 + eE2tcos2

)Hence,

P( ) = sin22 sin2 (E1 E2)t2

(2.10)

Applying the ultra relativistic approximation, with p1 p2, v c and E1 E2

Ej pj +m2j2Ej

E := E1 E2 m21 m222E

=m2

2E

And the final result becomes(in natural units):

P( ) = sin22 sin2 m2L

4E= sin22 sin2

L

Losc(2.11)

7


where L := ct is the distance travelled and Losc =4E

m2is defined as the oscillation length. This

is the celebrated neutrino oscillation formula.Note that m2 = 0 P( ) = 0 and hence the mass differences being non-zero is

the second reason for this effect. This dependence of m2 on an experimentally measurablequantity (P( )) allows the measurement of m2 and this establishes the existence ofthe massive neutrino.

In practical cases, the beam of neutrinos tend to have a range of energies as opposed to themono energetic beam we have considered. In such cases, the probability is averaged over thedistribution (say (E)) and this average (denoted P) is the observed quantity.

P =

0P(E)(E)dE (2.12)

Figure 2.1 compares the oscillation probabilities for a mono energetic beam and a beamwith energies normally distributed about some value. Note that for very high energies, theprobability approaches 0.5 as the sine squared dependence on energy averages out to give half.

Hence, it is to be noted that oscillations may be observed only at certain length scales

Figure 2.1: Observed oscillation probabilities in theoretical and realistic sce-narios plotted against 1piL/Losc. The lighter curve is the monoenergetic caseand the bold line shows the averaged out version over a Gaussian energyspectrum of mean E and Std. deviation E/10 (Fig taken from [10]).

(depending upon the mass difference and energy). If oscillation length is given by Losc, it isclear from the figure that if L Losc, oscillation probability is negligible; while if L Losc, itis a constant. Hence, each experiment on neutrinos is said to be sensitive only upto a certainm2 (defined as that for which L/Losc ' 1). Table 2.1 lists several classes of experiments andthe m2 they are sensitive to.

Experiments studying neutrino oscillation provide bounds on transition probabilities in par-ticular channels. Now these may be used to mark out regions in parameter space that areallowed by the measured bounds. To understand this, consider an effective two flavour oscil-lation channel with a mixing of driven by mass difference m2. Say the upper and lowerbounds on the transition probability are given by A and B respectively.

B sin22. sin2 m2L

4E A (2.13)

8


Experiment LmE

MeVm2

eV2

Reactor SBL 102 1 102

Reactor LBL 103 1 103

Accelerator SBL 103 103 1

Accelerator LBL 106 103 103

Atmospheric 107 103 104

Solar 1011 1 1011

Table 2.1: The table lists various classes of experiments and their m2 sensitivities (taken from[10]).

Now for a given baseline and energy, this relation marks out regions in the sin22 m2plane. (Note that any one-one function of onto the required range may be used instead ofsin22. Indeed, various functions such as tan2 and sin22/cos2 are used.) Such plots areknown as exclusion plots and are used to compute the values of the relevant parameters. Figure2.2 shows exclusion plots from various accelerator experiments.

Figure 2.2: Figure shows exclusion plots (90% CL) from various acceleratorexperiments (taken from [10]).

2.2 n-flavour Oscillations

The derivation of the oscillation formula in the case of n massive neutrinos runs on similarlines, and the amplitudes are given by the matrix:

A = U(t)U (2.14)

9


A( ) =nk=1

UkeEktUk (2.15)

Using unitarityn

k=1 UkUk = and simplifying using the ultra-relativistic approximation,

P( ) = +

nk=2

UkUk

[exp

(m

2k1L

2E

) 1]

2

(2.16)

Now for anti-neutrinos, the mixing matrix is simply the complex conjugate of the matrix Uand hence it is easily seen that

P = P (2.17)This is a consequence of CPT invariance that is a feature of any local, unitary quantum fieldtheory. It is also clear from the above equation that the survival probabilities for neutrinos andanti-neutrinos of the same flavour are identical. But the transition probabilities are in generaldifferent unless the mixing matrix happens to be real. In this case, there is CP invarianceand hence the only physical phase in the PMNS matrix, which is the mixing matrix associatedwith the three active flavour neutrinos in the Standard Model (see Section 2.2.1), is known asthe CP-violating phase. It is to be noted that for the same reason, CP violation cannot bedetermined through 2-flavour oscillations (since the mixing matrix can be made real as shownin Eqn. 2.9).

2.2.1 Decomposition of the n-neutrino problem

To show how n-flavour oscillations can reduce to effective two flavour oscillations, considerthe case where where one state is much more massive than others, i.e., mn >> mk k =1, 2, . . . , n 1. Define M2 := m2n1, and m2 := m2k1. Now choose an appropriate lengthscale such that: M2L

2E& 1 (2.18)

which naturally means that|m2|L

2E 1. Using this in the n-neutrino oscillation formula meansthat all phases with m2 are small and hence their exponentials are close to one. This allowsus to neglect all the terms in the summation except the nth term and hence the oscillationformula becomes:

P(n)( ) + UnUn [exp(M2L2E

) 1]2 (2.19)

where the superscript n denotes that the P is calculated keeping in mind that is is an n-neutrino system. Expanding the absolute value,

P(n)( ) 2 |Un|2( |Un|2

)(1 cosM

2L

2E

)(2.20)

Now let us assume that we start out with a neutrino beam of flavour . The survival andtransition probabilities of are respectively given by:

P(n)( ) 1 2 |Un|2(

1 |Un|2)(

1 cosM2L

2E

)(2.21)

10


P(n)( ) 2 |Un|2 |Un|2(

1 cosM2L

2E

)(2.22)

Observe that this is starting to look like a 2-flavour system. Motivated by this, define:

sin22eff := 4 |Un|2 (1 |Un|2) (2.23)

:= 6=

Un1 |Un|2

(2.24)

It is easy to see that sin22eff is always less that unity and that the new flavour is normalised.

Now the survival and transition probabilities look like:

P(n)( ) 1 12

sin22eff

(1 cosM

2L

2E

)(2.25)

P(n)( ) = | ||2

=

6= Un1 |Un|2 |2

6= Un1 |Un|2 UnUn

(1 eM

2L2E

)2

= 12sin22eff

(1 cosM2L2E

)(2.26)

where unitarity of the mixing matrix U was used to obtain the last line.Hence, under the one-dominant m2 condition and at the right length scales, we have shown

that the n-neutrino problem reduces to a 2-flavour oscillation between with effectivemixing angle eff. Note that may be any flavour (chosen according to the physical situation)and is then accordingly defined.

Now according to present data[9],

|m221| = 7.5 105 eV 2 (2.27)|m231| = 2.5 103 eV 2 (2.28)

|m221||m231|

= 0.03 (2.29)

Hence, the physical 3-neutrino system, at small length scales, decomposes into an effective twoneutrino system driven by m231. This is seen in the oscillation of atmospheric neutrinos. To seethis a little more clearly, consider the parametrisation of the 3-neutrino mixing matrix knownas the Pontercovo-Maki-Nagakawa-Sakata (or PMNS) matrix. Ue1 Ue2 Ue3U1 U2 U3

U1 U2 U3

(2.30)

=

1 0 00 cos23 sin230 sin23 cos23

cos13 0 sin13 eCP0 1 0sin13 eCP 0 cos13

cos12 sin12 0sin12 cos12 00 0 1

(2.31)

11


=

c12 c13 s12 c13 s13 eCPs12 c23 c12 s23 s13 eCP c12 c23 s12 s23 s13 eCP s23 c13s12 s23 c12 c23 s13 eCP c12 s23 s12 c23 s13 eCP c23 c13

(2.32)

where the s are the various mixing angles, CP the only physical (and hence CP-violating asdiscussed earlier) phase. (The obvious redactions have been made for sines and cosines in thelast line.) Now, and the effective mixing angle may be calculated taking e as the startingflavour:

sin22eff = 4 |Ue3|2 (1 |Ue3|2)= 4

sin13 eCP 2 (1 |sin13 eCP |2)= sin2213

(2.33)

=

6=eU3

1|Ue3|2

=sin23cos13

1 sin213 +

cos23cos131 sin213

= sin23 + cos23

(2.34)

Hence, short-range atmospheric neutrino oscillations are e oscillations driven bym213 with effective mixing angle 13 at length scales: (using a typical energy scale 1 GeV foratmospheric neutrinos) L & 200 km.

Solar neutrino oscillations, on the other hand, are a manifestation of the energy averagingdiscussed earlier. To see this, consider the survival probability of some flavour as computedfrom Eqn. 2.15 with the one-dominant m2 approximation as before, but now at length scalessuch that m2L

2E' 1 (2.35)

which means that|M2|L

2E 1. So,

P(n)( ) =

|U1|2 + (|U2|2 + + |U(n1)|2 X

)em2L

4E + |Un|2eM2L

4E

2

(2.36)

=

|U1|2 +X em2L4E 2 + 2|Un|2(|U1|2cosM

2L

2E+X cos

(M2 m2) L

2E

)+ |Un|4

(2.37)

Now, cos M2L

2E is a highly oscillating function of energy (in the limit chosen) and cancels outto zero when averaged over an energy distribution. Hence the middle term completely cancelsout (remember |M2| |m2|), and we expand the rest of the terms to get:

P(n)( ) =(1 |Un|2

)2(1 1

2sin22eff

(1 cosm

2L

2E

))+ |Un|4 (2.38)

with

sin2eff :=X

1 |Un|2 (2.39)

12


It is easy to see that this reduces to a 2-flavour oscillation formula in the limit |Un|2 0.Coming back to 3-flavour mixing ( = e), we observe that:

P(3)(e e) =(1 |Ue3|2

)2(1 1

2sin22eff

(1 cosm

212L

2E

))+ |Ue3|4 (2.40)

with

sin2eff =|Ue2|2

1 |Ue3|2 = sin212 (2.41)

And with[9]|Un|2 = |Ue3|2 = sin213 0.01 (2.42)

this reduces to the 2-flavour solar neutrino oscillations.Hence, the 3-flavour problem maybe decomposed into two 2-flavour oscillations (atmospheric

and solar) occurring at two different length scales and hence the parameters of the PMNS matrixassociated with these oscillations have been suitably dubbed:

m2 := m221 (2.43)

:= 12 (2.44)

m2atm := m231 (2.45)

atm := 23 (2.46)

Note that the atmospheric mixing angle is 23 instead of 13. This is because the atmosphericneutrino oscillations are dominantly (recall we started with e), and in the limitcos213 0, sin22eff = sin2223 for these oscillations.

2.3 Oscillations in the presence of matter

Neutrino oscillations were shown to be modified significantly due to their interaction withmatter. The dominant process that contributes towards this effect, as shown by Wolfenstein[12]and by Mikheyev and Smirnov[13] is coherent forward scattering. In this section, we shall studyneutrino oscillations in matter by considering the case of the solar neutrino problem1.

2.3.1 Uniform Matter Density

We shall first consider the rather simple case of oscillations in uniform matter. The case ofinterest shall involve two flavours - e and (taken here to be electron and muon neutrinos butmay be generalised as described later). The dominant process that contributes to this effect iscoherent forward scattering as shown by the Feynmann diagrams in Fig. 2.3.

Note that in the regimes of interest (namely the Sun and the Earth), matter is mostlycomposed of protons, electrons and neutrons. Even in the Suns core, temperatures are nothigh enough to generate heavier leptons via processes such as pair production. Hence, processesinvolving these leptons are not considered. The low energy charged current (CC) Lagrangian isgiven by:

LCC = 4GF2

[eL(p1)eL(p2)][eL(p3)eL(p4)] (2.47)

1The outline of this section shall closely follow [11]

13


W+

e(p4)

e(p2)

e(p3)

e(p1)

Z

f(p2)

x(p4)

f(p1)

x(p3)

Figure 2.3: These Feynmann diagrams show the dominant (coherent forwardscattering) processes in the interaction between neutrinos and matter. fstands for protons, neutrons and electrons, and x stands for any activeflavour neutrino.

Now, coherent forward scattering implies that p1 = p4 = pe and p2 = p3 =p. Using this, aFierz rearrangement gives:

LCC = 4GF2

[eL(p)eL(p)][eL(pe)eL(pe)] (2.48)

Averaging over the momentum of electrons , and applying the non-relativistic reduction formulae

e5e spin = 0 (2.49)

e~e velocity = 0 (2.50)e0e = Ne (2.51)

gives the effective CC Lagrangian

LeffCC =

2GFNeeL0eL (2.52)

Repeating the same procedure for the neutral current (NC) processes:

LNC = 4GF2

[f(p1)

(I3L

(1 5

2

)Qsin2W

)f(p2)

][L(p3)L(p4)] (2.53)

Averaging over the background fields f, the contributions from the electrons and protons cancelout because they have opposite values of I3L and charge.

LeffNC =12GFNn (eL0eL + L0L) (2.54)

Both the CC and NC interaction terms add to the Hamiltonian in the form a potentialenergy and the effective Hamiltonian is hence given by:

Heff = Hvac +[

2GF(Ne 12Nn

)0

0 12GFNn

](2.55)

where

Hvac = E + m21 +m

22

4E+M2

2E(2.56)

14


with

M2 =

2

[ cos2 sin2sin2 cos2

](2.57)

is the vacuum Hamiltonian in the flavour basis (see Appendix A). Casting this into a moresuggestive form,

Heff = E + m21 +m

22

4E 1

2GFnn +

A

4E+

1

4E

[ (cos2 A) sin2sin2 cos2 A

](2.58)

withA = 2

2GFneE (2.59)

gives the modified eigenvalues (the tilde indicating the respective values in matter).

E = E +m21 +m

22

4E 1

2GFnn +

A

4E 1

4E

(cos2 A)2 + 2sin22 (2.60)

and the modified mixing angle

sin22 =2sin22

2sin22 + (cos2 A)2 (2.61)

It is easily seen that nn = 0 and ne = 0 give back the vacuum energy eigenvalues andmixing angles. Note that the sine-squared of the mixing angle is of the form of a resonancecurve with peak at A = 2

2GFneE = cos2 and width = sin2. Hence, mixing is

strongly suppressed for high matter densities and tends to the vacuum angle for low densities,with a sharp peak at the resonance density :

nR :=cos2

2

2GFE(2.62)

Note that the oscillation length is also enhanced during resonance and is given by:

L(R)osc =4E

m2 sin2(2.63)

This shows that even oscillations with a small mixing angle may be dramatically enhanced inthe presence of matter. This is an idea that will be exploited later on.

Note that although we have limited ourselves to electron and muon neutrinos, e may standfor any flavour with a CC interaction with the medium in question and may stand for anyflavour with only NC interactions with the medium. Indeed for the solar neutrino problem, weshall assign e to electron neutrinos and to a linear combination of muon- and taon-neutrinos(Since neither flavour has CC interactions with the solar medium, nor does their combination).It is also clear that if both flavours merely interact with matter via NC interactions, the matterterm merely adds a phase to both mass eigenstates and the oscillations proceed practically asin vacuum.

For antiparticles, the same analysis follows through except for the fact all number densitiesreverse their sign (since the expectation of the number operator for fermions must now bereplaced with that of anti-fermions which have the same value but opposite sign). Hence,depending on the sign of m2, matter can amplify only one of or oscillationsbut not both. This means that even for 2-flavour oscillations, P( ) 6= P( ).Matter oscillations as in the case we have considered are neither CP nor CPT invariant becausethe medium is composed of matter and not anti-matter.

15


2.3.2 Varying matter density: The Adiabatic approximation

In the event that matter density varies along the path of propagation, we go back to theSchrodinger equation in natural units (Eqn. 2.2).

d

dx(f) =

M2

2E(f) (2.64)

(upto scalar matrices, which only contribute an overall phase) with

M2 =1

2

[ cos2 + 2A sin2sin2 cos2

](2.65)

and f denoting the flavour basis (let p denote the mass basis). Now using (f) = U(p)

dU

dx(p) + U

d

dx(p) =

1

2EM2U (p) (2.66)

ddx(p) =

(1

2EU M2U U dU

dx

)(p) (2.67)

=

(1

2E

[m1

2 00 m2

2

] [

cos sinsin cos

] [ sin coscos sin

]d

dx

)(p) (2.68)

ddx(p) =

[m212E

ddx

ddxm222E

](p) (2.69)

where Eqn. 2.68 is correct only upto the addition of scalar matrices.Now, we assume that propagation is slow enough that state, at each point of propagation

collapses to a Hamiltonian (at that point) eigenstate. Hence, states propagate continuously asHamiltonian eigenstates. This is known as the adiabatic approximation.

Calculating the survival probability of e in this limit,

P(ad)ee(x) = |e(x)|e(0)|2 (2.70)inserting complete mass bases , ,

=

,

e(x)|(x)(x)|(0)(0)|e(0)2

(2.71)

=

,

e(x)|(x)e x0 dx

E(x)(0)|e(0)2

(2.72)

=e dxE1cos0cos + e dxE2sin0sin2 (2.73)

=1

2

(1 + cos20cos2 + sin20sin2cos

[ x0dx(E2 E1)

])(2.74)

The subscript 0 denotes that the parameter value is calculated at the point of origin of theneutrino. It is to be noted that the matrix element

| = e x0 dx

E(x) (2.75)

16


because we have made the adiabatic approximation. The sine term vanishes on averaging overbeam energy[11] provided m2 1010eV2 and we get

P(ad)ee(x) =1

2

(1 + cos20cos2

)(2.76)

We close this section by observing that adiabatic effects alone can cause a dramatic reductionin the survival probability[11]. Consider an electron neutrino produced deep in the Sun where ' pi2 and hence e ' 2. This beam propagates adiabatically and reaches the surface of theSun where = and 2 = ecos + sin. The survival in this case is given by:

P(ad)ee = sin2 (2.77)

This is a drastic reduction if happens to be small.

2.3.3 Non-adiabatic effects and the Adiabaticity Parameter

Now, we try to quantify the notion of adiabatcity so as the identify the regimes wherethis approximation may be applied. Examining equation (2.69), the adiabaticity condition isequivalent to saying that the off-diagonal terms of the Hamiltonian are much smaller than thediagonal terms (non-uniformity of medium comes into play via the off-diagonal terms). Butsince any scalar matrix may be freely subtracted from the Hamiltonian without changing thephysics, it only makes sense to talk of the difference of the diagonal terms rather than theirmagnitudes. Hence, the adiabaticity condition translates to the inequality:ddx

m21 m22

2E. (2.78)

Substituting for from Eqn. 2.61,ddx = 2GFE. sin2(cos2 A)2 + 2sin22 .

dnedx (2.79)

And hence the adiabaticity condition translates todnedx

[(cos2 A)2 + 2sin22] 32

2

2GFE2sin2(2.80)

or(x) 1 (2.81)

with

(x) :=(/E)2

2

2GF.sin22

sin32.

1dnedx

(2.82)(x) depends on position via the density gradient

dnedx

. Note that if matter density is veryhigh so that pi/2, then sin2 0 and (x) . On the other end, if matter densityapproaches 0 so that , (x) is once again large (assuming a small mixing angle). In Fig.2.4, (x) is plotted against x for the solar problem to demonstrate that non-adiabatic effectsare not important in the Sun except near resonance points.

To study the behaviour of neutrinos at the resonance regions, define X to be the probability

17


Figure 2.4: The figure shows a plot of (x) against distance from the solarcore at various values of /E. The solid lines represent (x) and the dashedlines sin22. The vacuum mixing angle is taken as 0.1.

of discrete, non-adiabatic jumps between Hamiltonian eigenstates when the neutrino beampasses through a resonance region. Now since non-adiabaticity is important only near resonancepoints, the expression for X shall depend only on the value of at resonance (say R). Thismeans that if R 1, there is full adiabaticity and discrete jumps do not come into play at all.One of the ways this happens is if E . Note that this means that high energy neutrinosare less prone to non-adiabatic effects.

Now an expression for X may be derived using the Landau-Zener method as described in[11].

X = exp (FR) (2.83)with F = pi4 assuming a linear density variation near resonance.

Although this shall suffice for out purposes, it is a semi classical expression which gives onlythe leading term in the limit of large values of the exponent. Also, the Suns density variationis more exponential than linear[14]. The exact solution, obtained for an exponentially varyingmatter density[15] is of the form:

X =exp(R F ) exp

(R F

sin2

)1 exp

(R F

sin2

) (2.84)where F is a quantity that depends on the variation of ne near resonance.

Considering a beam of electron neutrinos originating in the Suns core and travelling towardsits surface, the survival probability (accounting for resonance jumps) is given by:

Pee = (1X)P(ad)ee +XP(ad)e= (1X)2

(1 + cos20 cos2

)+ X2

(1 cos20 cos2

)= 12

(1 + (1 2X)cos20 cos2

) (2.85)18


Now, if the beam is produced somewhere outside the core so that it passes the resonance regiontwice on its way out,

Pee =((1X)2 +X2)P(ad)ee + 2X(1X)P(ad)e

= 12

(1 + (1 2X)2cos20 cos2

) (2.86)Hence, the general expression for survival probability is given by:

P(n)ee =1

2

(1 + (1 2X)ncos20 cos2

)(2.87)

where the superscript n stands for the number of jumps and can take values 1 or 2.

2.3.4 The big picture: Looking at the Solar Neutrino problem

Armed with the formalism developed so far, we shall take a preliminary look at the solarneutrino problem (see Section 3.2). The following simplifications shall be made:

1. All neutrinos are produced in the core of the Sun (defined as the region with r . 0.07R)so that N0 = 98.8 Navo (where Navo := 6.023 1023/cc).

2. ddx ln ne

R

= 10R1 irrespective of where resonance takes place inside the Sun. This is

justified because the density profile is very well approximated by ne 200 Navoe10rR

everywhere in the Sun except in the inner 15%[11]. So, this is not too bad an assumptionunless resonance occurs deep within the Sun.

The density profile of the Sun as in the model considered is an exponentially decreasingprofile and hence looking at the expression for mixing angle in matter, it is clear that (x)drops down continuously and monotonically from 0 to the vacuum mixing angle, which is lessthat pi4 (assuming that ne(x) decreases as well). Now, since resonance is assumed to not occurin the inner 15% of the Sun and production occurs in the inner 7% of the Sun, it is clear thatneutrinos pass resonance only once, if at all. Hence, at the point of production, if 0 pi4 , itmust pass pi4 and undergo resonance once and exactly once. So the condition for resonance isgiven by:

Ecos2 < 2

2GFN0 = 1.5 1011 (2.88)

Non-adiabatic effects, as maybe seen from the expression for Pee are unimportant as longas X 0.5. Let us say that they are unimportant if X 0.05. Now this occurs for (usingthe simplified expression from Eqn. 2.83) R < 3.8. Using the given model for density, thistranslates to:

E.sin22

cos2< 1014eV (2.89)

using r = 7 1010 cm. Fig. 2.5 shows a plot of survival probability against E for variousvalues of vacuum mixing angle. The dotted line marks the onset of non-adiabatic effects, whichbecome important to the left. At the right end, E is too large for either the non-adiabaticity orthe resonance condition to be satisfied. Hence, the survival probability reverts back to the av-erage value in vacuum. As E decreases, there is a sudden plunge due to the onset of resonance.The range of E over which resonance takes place is determined by the width of the resonance.Clearly, the width increases with increasing as expected.

19


Figure 2.5: The survival probability is plotted as a function of E . Non-adiabatic effects become important to the left of the vertical dashed line.In the bottom-most figure, the dashed line has been computed with F =pi4

(1 tan2) (recall Eqn. 2.83).

But the probability rises from the resonance basin as non-adiabatic effects become more andmore important. The E for which these effects become significant decreases with increasing .For = 0.01, non-adiabaticity effects take over even before the onset of resonance and hencethe basin is never reached.

We shall also look at the exclusion plots for at constant energies. There are three branches

Figure 2.6: Exclusion plot for the problem discussed. The dashed lines standfor 0.19 < Pee < 0.35 at E=5 MeV.The dotted lines stand for 0.18 < Pee 1330 MeV ),where (Evis was defined as the energy of the electron that would produced the observed amountof Cherenkov radiation.

The Super-K experiment has thresholds of around 5 MeV [18] and measures the 8B flux.A summary of their results is given in Table 3.4.

The SNO Experiment

The SNO (Sudbury Neutrino Observatory) experiment in Canada used a heavy waterCherenkov detector that used both CC and NC reactions (Eqn.3.6, 3.7) to detect neutrinos.Located 6800 ft underground, it used 1000 tonnes of heavy water contained in a 12 meter di-ameter acrylic vessel and the radiated Chrenkov light was picked up by 9600 photomultipliertubes mounted on a support structure.[16]

e + d e + p + p [CC] (3.6)

x + d x + p + n [NC] (3.7)

25

CHAPTER 3. EXPERIMENTS IN NEUTRINO PHYSICS: MEASURING THEOSCILLATION PARAMETERS

Experiment Reaction 8B Flux (106 cm2s1)

Kamiokande[33] e 2.80 0.19 0.33Super-K[34] e 2.32 0.04 0.05SNO (Ph. III) [17] CC 1.67+0.050.04

+0.070.08

NC 5.54+0.330.31+0.360.34

Borexino[35] e 2.4 0.4 0.1SSM (BPS08[GS][26]) - 5.94(1 0.11)SSM (SHP11[GS][37]) - 5.58(1 0.14)

Table 3.4: The table summarises results by the discussed real-time observation experiments andcompares them with predictions of the SSM (taken from [9]).

Part of Measured Flux BPS08[GS][26] SHP11[GS][37]

spectrum (cm2s1) (cm2s1) (cm2s1)8B (2.4 0.4 0.1) 106 5.94(1 0.11) 106 5.58(1 0.14) 106

7Be (3.10 0.15) 109 5.07(1 0.06) 109 5.00(1 0.07) 106pep (1.0 0.2) 108 1.41(1 0.011) 108 1.44(1 0.012) 109pp (6.6 0.7) 1010 5.98(1 0.006) 1010 5.97(1 0.006) 1010

Table 3.5: The table summarises the measurements by the Borexino experiment (taken from[9, 71, 72]).

Note that the CC reaction is only senstitive to e while the NC reaction is undergone by allactive neutrino flavours. Hence, SNO had the great advantage of being able to measure boththe electron- and the total (active) neutrino flux simultaneously and hence demonstrate thatthe missing electron-neutrinos do indeed manifest as other flavour neutrinos.

The CC and NC events were distinguished on the basis that while the momentum of theelectron produced in the NC reaction had a strong forward peak (with respect to the Earth-Sunaxis), the same for the CC reaction had an approximate angular distribution of 1 13cos.

The SNO experiment was run in three different phases with different mechanisms to de-tect the neutrons from the NC reaction. In the first phase, the neutrons were captured ondeuterium nucleii in D2O. In the second phase, 2 tons of NaCl were added and the neutronswere captured on the 35Cl to raise detection efficiency. The third phase relied on an arrayof proportional counters (the Neutral Current Detection or NCD array) deployed in the heavywater and neutrons were detected via the reaction[17]:

3He + n 3H + p (3.8)The SNO experiment could accurately determine the shape of the 8B spectrum owing to

the fact that the energy of the electrons produced via the CC reactions had a strong correlationwith that of the original neutrinos. A combined analysis of the data from all phases of SNOcould be found in [17]. The SNO detector is currently being upgraded for the planned SNO+

26


experiment[16].

The Borexino experiment

The Borexino experiment also uses a liquid scintillation detector to study solar neutrinos.It came online in 2007, and was highly sensitive to all solar neutrino components, particularlythose below 2 MeV. Borexino recently succeeded in directly measuring the pp neutrino flux[71]- something that had not been possible over the past 30 years due to the inability to suppressthe high background noise at this energy range.

Borexino was also the first experiment to directly observe 7Be neutrinos[72].Borexino has also measured the pep and 8B part of the solar neutrino spectrum (see Table

3.5 for a summary of results and comparison with SSM predictions), and set an upper limit forthe CNO flux, assuming the LMA solution (see Section 3.2.3).

3.2.3 The Solution of the Solar Neutrino Problem

Until recently, the solution to the solar neutrino problem had only been narrowed down toone of three regions in parameter space[22][10] (90% CL):

The small mixing angle (SMA) MSW solution:

4 106eV2 . m2 . 1.2 105eV2

3 103 . sin22 . 1.1 102

The large mixing angle (LMA) MSW solution:

8 106eV2 . m2 . 3.0 105eV2

0.42 . sin22 . 0.74

The vacuum oscillation solution:

6 1011eV2 . m2 . 1.1 1010eV2

0.70 . sin22 . 1

In 2001, the SNO experiment and Super-K[38] provided evidence for flavour conversion insolar neutrino flux[39]. The NC flux measured by SNO showed a good agreement with the8B flux of the solar model[24]. The results were analysed and found to be consistent with theLMA solution (with MSW effects) to the solar neutrino problem[13, 12]. However, SNO alonecould not rule out other possibilities with sufficient significance, and this job was completed bKamLAND.

KamLAND

KamLAND is a long-baseline experiment with a liquid scintillation detector. It observedanti-neutrinos from reactors and was sensitive down to m2 105. Hence, KamLAND shouldnot observe a substantial e disappearance should SMA (mixing angle is too small) or vacuumoscillations (oscillations happen over a much bigger range) happened to be the right solution.

27


KamLAND reported its first results[40] in December 2002, with a clear observation of edisappearance.

R :=Nobs NBGNNoOsc

= 0.611 0.085 0.041 (3.9)with Nobs being the oberved number of events, NBG the background, and NNoOsc the expectednumber of events in case of no oscillations. A combined analysis with other data showed thatLMA was the unique solution to the solar neutrino problem at more than 5 CL [41, 42, 43,44, 45]. Later on, KamLAND also observed the periodic variation of e survival probability,

Figure 3.2: The e survival probability as observed by KamLAND, as a L0/Ewhere L0 = 180 km. The histograms show the expected distributions froma 2 and 3-flavour analysis using the best-fit values of the parameters (takenfrom [9], originally from [46]).

that is expected with oscillations, for the first time (see Fig. 3.2). Including recent data on 13measurements, a 3-flavour oscillation analysis of the solar neutrino and KamLAND gives thecurrent values of the solar oscillation parameters.

m221 = (7.53 0.18) 105eV2 (3.10)sin213 = 0.023 0.002 (3.11)tan212 = 0.436

+0.0290.025 (3.12)

3.3 Experiments involving Atmospheric Neutrinos

3.3.1 Cosmic Rays

Cosmic rays are highly energetic beams of particles that consist of about 90% protons, 9%alpha particles and 1% heavier nucleii. The absolute flux of cosmic rays is of the order of 1000

28


m2s1sr1 for energies of a few GeV.Above a few GeVs the energy spectrum for protons falls off, to a good approximation, as

E2.7[57]. The fluxes are less well known at higher energies. It is to eliminate the uncertainityin cosmic ray fluxes that experiments rely on flux ratios rather than the absolute values. Also,the H to He ratio is not a constant with energy. At around 100 MeV, this ratio is less than 5.It rises to about 10 at energies around 1 GeV and then to about 30 at around 100 GeV[57, 58].

Cosmic rays fluxes are modulated via two processes: the solar wind and the geomagneticcut-off. The solar wind weakens the galactic cosmic ray flux at lower energies. The magnitude ofthis effect on a particle is determined by its momentum to charge ratio (known as rigidity)-theeffect being weaker at higher rigidities. By around 2 GeV, the solar wind influence is nearlynegligible[10].

The Earths magnetic field influences the charged particles in the cosmic ray flux to producewhat are known as geomagnetic effects. To present a somewhat simplified picture, consider apositively charged particle in a uniform magnetic field. Now, this particle moves in a helicalpath around the field lines and in the direction of the field line (or against, depending on itsinitial momentum). This is mechanism behind geomagnetic effects.

It is easy to see that the trajectory of the particle depends on the ratio of the momentumof the particle to the charge (known as the rigidity). Now, particles with a high rigiditymove in a helix with a large radius and hence appear to be uninfluenced by the field. Thisis (roughly speaking) how the high energy part of the cosmic ray spectrum remains untouchedby geomagnetic effects. Meanwhile, particles with a low enough rigidity never make it to theEarths surface (some of them follow the field lines to the Earths magnetic poles) - hencecreating a cut-off rigidity for cosmic ray particles.

The cosmic rays that do make it past the barriers discussed above react with atmosphericnucleii to form neutrinos that constitute the so-called atmospheric neutrino flux.

3.3.2 Production of Atmospheric Neutrinos

The production of atmospheric neutrinos proceeds essentially in three steps. First, theprimary cosmic rays react with nuceii in the atmosphere to give pions and kaons. The pionsthen decay into muons which then give muon neutrinos (branching ratios given in square braces).[9]

pi + () [99.99%] e + () + e(e) [100.0%]

The kaons either directly give muons or decay into pions. [9]

K + () [63.6%]K pi + pi0 [20.7%]K pi + pi + pi [5.59%]K pi0 + e + e(e) [5.07%]

3.3.3 Characteristics of the atmospheric neutrino flux[10]

1. The muon and electron neutrino fluxes follow the power laws [] E3 and [e] E3.5for energies 1 . EGeV . 10 ([e] denoting the flux of e).

2. The largest contributions to the neutrino flux at energy E come from cosmic rays of energy10E.

29


3. From the production mechanisms, it is easy to see that[+][e+e]

' 2. But for E & 1 GeV ,the muons start reaching the Earths surface before decay and hence stop contributing.Therefore, once again looking at the production mechanisms, [e + e] falls and the ratio[+][e+e]

rises. Around 100 GeV , muons cease to be the dominant source of e and the KL

and K+ reactions take over.

4. [e][e] > 1 because protons heavily dominate the primary cosmic ray flux and hence during

hadronic interactions, the production of pi+ is favoured over pi. So [pi+]

[pi] '[+][] ' [e][e] > 1.

5. For the same reason, while' 1 for E . 1 GeV (since muon decay supplies the

corresponding antiparticles), this ratio increases beyond a few GeV when the muons startreaching the Earth before decay.

6. The geomagnetic effects are negligible for E & 5 GeV .

3.3.4 Zenith angle dependence of the atmospheric neutrino flux

There are two main effects (besides oscillations) that lead to atmospheric neutrino flux beinga function of angle.

The cosmic ray reactions are dependant of the gradient of the density of air. As zenithangle approaches pi2 , the density variation becomes less steep and hence pion and kaondecay is enhance as compared to the flux along the vertical. However, it is easy to seethat the angular dependence induced by this mechanism is up-down symmetric, ie, thedependence is on |cos|. The geomagnetic effect causes a zenith as well as azimuthal angle dependence for the low

energy part of the spectrum. This effect is up-down asymmetric but does not effect thehigh energy neutrinos.

Neutrino oscillations also create a zenith angle dependence in the atmospheric neutrinoflux. This is because of the variation in the path length traversed by neutrinos approach-ing the detector on the Earths surface at various angles. Up-going neutrinos have to crossthe entire diameter of the Earth (L 13000 km), while down-going neutrinos merely crosspart of the atmosphere (L 20 km). This effect is clearly not up-down symmetric (depen-dence on cos) and can be decoupled from geomagnetic effects by using the high energycomponent of the spectrum. This zenith angle dependence was exploited to obtain therelevant mass-squared difference and mixing angle for atmospheric neutrino oscillations.

3.3.5 Measuring the Atmospheric Neutrino Flux

Super Kamiokande

The Super-Kamiokande provided evidence of disappearance of atmospheric muon neutrinosby measuring a []:[e] ratio of less than 2 for the valid energy range (see Section 3.3.3)[47]. Thedata, consistent with oscillations, also drew rival interpretations such as the neutrinodecay[48] and quantum decoherence[49]. However, observation of a dip in the L/E distribution ofthe data[20] (see Fig. 3.4), which exhibited a sinusoidal behaviour characteristic of oscillations,ruled out these theories: neutrino decay being disfavoured at 3.4 and decoherence even morestrongly.

30


Figure 3.3: The figure shows the zenith angle distribution for atmosphericneutrinos as measured by Super-K. The dotted histograms show the Monte-Carlo predictions for the no-oscillation hypothesis and the solid histogramsshow the best-fit expectations for oscillations (taken from [9]).

Figure 3.4: The figure shows the results of the analysis of SK-I data. Thesolid line shows the best-fit with 2-flavour oscillations. The dashedand dotted lines show the predictions by the neutrino decay and quantumdecoherence hypotheses (taken from [9], originally from [20]

31


The final step was the demonstration of appearance of tau neutrinos. Now because ofthe high threshold of the corresponding reaction, (the event rate is extremely small) and theshort lifetime of the tau lepton, Super-K cannot detect taon neutrinos as easily as it does theothers. Hence techniques such as neural network analysis were employed in carefully selectingthe occasional tau events from the large collection of data [50][51] and appearance wasestablished at the 3.8 level in 2013.

Although the first measurements of m2atm and atm were made from atmospheric neutrinos,the uncertainties involved prevented accurate measurement of these parameters. Acceleratorbased experiments had more merits on this front since the variables involved like the length ofthe baseline and the energy spectrum of the neutrinos were known more precisely. These shallbe discussed next.

The K2K experiment

The KEK-to-Kamioka (K2K) experiment was a long-baseline experiment with a baseline of250 km. It used a muon neutrino beam from a 12 GeV proton synchrotron (KEK-PS) with anaverage beam energy of about 1.3 GeV. The energy profile of the source was determined by anear detector located 300 m downstream. Super-K served as the far detector.

The K2K experiment started taking data in 1999 and finished its observations in 2004. Thenumber of FC events observed at Super-K was 112 as opposed to the no-oscillation predictionof 158.1+9.28.6. The measured energy spectrum also showed the necessary distortion required byneutrino oscillations. The probability that these results are due to a statistical fluctuation wereestimated to be 0.0015% or 4.3 [52].

The MINOS Experiment

MINOS uses neutrinos produced at the NuMI (Neutrinos at the Main Injector) facility using120 GeV protons from the Fermilab Main Injector. It has a baseline of 735 km and most of itsdata employed a neutrino beam with an energy spread of 1-5 GeV (peaked at 3 GeV).

MINOS data (Fig. 3.5) came down in favour of neutrino oscillations, disfavouring decayand decoherence at 7 and 9 respectively[53]. The result (at 86% CL) disfavoured maximalmixing. Recently, MINOS has also published[54] a combined analysis of disappearance andnue appearance using both accelerator and atmospheric neutrino data. The results now put theparameter values at |m2atm| = (2.28 2.46) 103eV2 (68% CL) and sin2atm = 0.35 0.65(90% CL) for normal hierarchy, and |m2atm| = (2.322.53)103eV2 (68% CL) and sin2atm =0.34 0.67 (90% CL) for inverted hierarchy.

The T2K Experiment

The T2K experiment is the first off-axis long-baseline experiment with a baseline of 295 kmfrom J-PARC in Tokai, Japan to Super-K. It uses a narrow-band beam with a peak energyof 0.6 GeV directed 2.5o off-axis to Super-K.

T2K started its first run in 2010. Recent measurements[55] report 58 1-ring -like eventsagainst an expectation of 205 17 for no oscillations. T2K results[56] currently put down therelevant parameter values to be |m2atm| = (2.51 0.10) 103eV2 (68% CL) and sin2atm =0.514+0.0550.056 (68% CL) for normal hierarchy, and |m2atm| = (2.48 0.10) 103eV2 (68% CL)and sin2atm = 0.511 0.055 (68% CL) for inverted hierarchy. Note that T2K results areconsistent with maximal mixing.

32


Figure 3.5: The top part of the figure shows the energy spectrum of the fullyreconstructed CC events from the MINOS far detector. The bottom panelshows a plot of R (see Eqn. 3.9) against reconstructed neutrino energy. Thebest-fit to oscillations, decay and decoherence is also shown (taken from [9]).

OPERA

The OPERA experiment has been been designed to directly detect tau neutrinos via CCinteractions. It uses an emulsion technique which is able to identify the short-lived leptonson an event-by-event basis. It has a baseline of 730 km length with the source at CERN andthe detector at Gran Sasso in Italy. The average beam energy is 17 GeV.

The detector is a combination of the Emulsion Cloud Chamber and magnetised spec-trometer. and uses neutrinos produced by high-energy protons from the CERN-SPS. OPERAcollected data during 2008 and 2012, and analysis has not yet been completed[9].

3.4 Accelerator experiments to measure 13

13 measurements were carried out by long-baseline experiments (Daya Bay, RENO, DoubleCHOOZ) using e oscillations which were driven by m

2atm. The accelerator driven experiments

discussed earlier (MINOS, T2K, etc) studying oscillations are also sensitive to 13. The results

33


Figure 3.6: The figure shows the 90% CL allowed regions in the atmosphericneutrino parameter space as estimated by T2K (2013[55], 2011[68]) Super-K[69] and MINOS[70] (taken from [9]).

from these experiments shall be discussed next.

CHOOZ and Double CHOOZ

CHOOZ[59] was a long baseline experiment located at the CHOOZ nuclear power stationin France. With an average value of L/E 300 eV2 (L 1 km and E 3 MeV), it studiedthe disappearance of reactor e driven by m

2atm and 13 (recall Section 2.2.1). CHOOZ used

a liquid scintillation detector and neutrinos were detected via the reaction :

e + p e+ + n (3.13)

CHOOZ initially found no evidence for e oscillations[59]. However, Double CHOOZ hasrecently reported a non-zero value for 13, measuring sin

2213 = 0.109 0.030 0.025[60].

Daya Bay and RENO

The Daya Bay experiment uses the Daya Bay nuclear power complex in China as source, anda gadolinium-doped liquid scintillation detector to observe reactor e. RENO studies reactores from the Yonggwang nuclear power plant in Korea with two identical liquid scintillationdetectors located 294 m and 1383 m away from the center of the reactor array.

Daya Bay observed a non-zero deficit in the e flux, with the ratio of observed to expectedevents being R = 0.944 0.007 0.003, and a 3-flavour oscillation analysis yielded sin2213 =0.089 0.010 0.005 . This result excluded the no-oscillation hypothesis at 7.7[61]. Lateron, the bounds were improved to sin2213 = 0.090

+0.0080.009[62]. RENO initially measured R =

0.920 0.009 0.014, and sin2213 = 0.113 0.013 0.019 from a rate-only analysis, hence

34


Figure 3.7: The figure shows the regions in parameter space al-lowed/excluded from various neutrino experiments (taken from [74]).

excluding the no-oscillation hypothesis at the 4.9 . Later on, an improved data analysis[64]gave sin2213 = 0.113 0.013 0.019.

T2K and MINOS

In 2011, T2K, while studying e oscillations reported a non-zero 13 value with a signifi-cance of 2.5[65]. Recently, this was improved to 7.3 [66] and sin2213 = 0.140

+0.0380.032

(0.170+0.0450.037

)was reported for normal (inverted) hierarchies, assuming CP = 0, sin

223 = 0.5 and |m231| =2.4 103 eV2. Comparison with Daya Bays results[62] immediately leads to 6= 0 orsin223 6= 0.5.

MINOS has also recently reported its estimates for 13. The best-fit value at 90 % CL being2 sin223 sin

2213 = 0.051+0.0380.030

(0.093+0.0540.049

)[70] for normal (inverted) hierarchy.

35

Chapter 4

Sterile Neutrino Oscillations

Having taken a brief glance at the recent developments in the field and the current statusof the oscillation parameters, we now attempt to come up with a proposal to look for sterileneutrinos with the aid of the MSW effect.

4.1 Derivation of matter term for s oscillationsThe fact that sterile neutrinos have eluded detection in the experiments discussed so far

places strong constraints on the mixing between active and sterile states[19]. Hence, we shallrecall that even small mixing angles maybe drastically enhanced by means of the MSW effect(recall Section 2.3.1) and use this to derive the baseline for a stand-alone accelerator-drivenexperiment that shall look for sterile neutrinos. The derivation in this section shall closelyfollow that in Section 2.3.1.

Consider an effective two neutrino system with and s, with a small mixing angle anddriven by a m2 = 0.01eV2, propagating through a uniform matter density. Assuming thatthe dominant process is once again coherent forward scattering, the Lagrangian is given by:(assuming that the thermal energy of the particles is much less than the mass of Z)

L = 4GF2

{f(p1)

(I3L

(1 5

2

)Qsin2W

)f(p2)

}{L(p3)L(p4)}

where f stands for the proton, electron and neutron. The conditions for coherent forwardscattering are given by:

p1 = p2 = p

p3 = p4 = p

The effective Lagrangian for the process is obtained by integrating the Lagrangian over p,assuming a thermal distribution for the matter particles and using the formulae:

f5f spin = 0 [unpolarised medium]fjf velocity = 0 [stationary medium]f0f = nf [number operator]

Leff =

2GF

f

nf

(If3L 2sin2WQf

) [L0L]

36

CHAPTER 4. STERILE NEUTRINO OSCILLATIONS

Now, the contribution due to the protons and electrons cancel out because they have oppositevalues of I3L and charge. The only term left is from the neutron density nn.

Leff = 12GFnn L0L

This contributes in the form of a potential for (with no modification for s) and the modifiedHamiltonian becomes:

H = E + m21 +m

22

4E+

4E

[ cos2 sin2sin2 cos2

]

Hvac

+

[ 1

2GFnn 0

0 0

]

H = 14E

[ cos2 + 2A sin2sin2 cos2

]+ I2

where I2 is the identity matrix (addition of the scalar matrix merely adds an overall phase anddoesnt affect the physics), and A is given by:

A =

2GFnnE

This gives the effective mixing angle and mass difference in vacuum (tilde indicating that theparameter value is in the presence of matter effects.)

sin22 =2sin22

(cos2 A)2 + 2sin22

=

(cos2 A)2 + 2sin22

4.1.1 Certain Issues

The usual calculation of resonance neutron density yields (assuming mixing angle to be 0.01rad):

n(R)n = cos22GFE =

0.01eV 21.00021.17105GeV 21GeV = 8.017 1025 cc1

We recall the discussion in Section 2.3.1 (regarding only one of x y and y x beingamplified by matter effects) and conclude that this may be fixed and that a resonance may beachieved by using antiparticles.

4.2 Propagation in the Earths crust

Firstly, we calculate the density of neutrons in the Earths crust. The contribution of theith constituent element toward this is given by:

n(i)n =PiMi

YiNA

where is the average density of the Earths crust (3.2 gcc1), NA is the Avogadro number, Piis the percentage composition of the element (by weight), Mi its molar mass, and Yi its neutronnumber. The total neutron density is then obtained by summing these.

37


Element % by wt. Atomic Atomic Mass Neutron N. Density

Number (g/mol) Number (1023/cc)

O 47.2 8 16.00 8.00 4.49

Si 28.8 14 28.00 14 2.67

Al 7.96 13 26.98 13.98 0.81

Fe 4.32 26 55.84 29.84 0.82

Ca 3.85 20 40.08 20.08 0.35

Na 2.36 11 22.96 11.96 0.28

K 2.14 19 39.10 20.10 0.26

Mg 2.20 12 24.30 12.30 0.15

Total 9.60

Table 4.1: The table shows the computation of the neutron density of continental crust. El-ements below 1% by weight not considered since the oscillation length estimated is importantonly to the order of magnitude. Data regarding chemical compositions of Earths crust takenfrom [73].

Using this to calculate A,

A = 2GFEnn 2(1.17 105GeV 2)(1GeV ) 9.521023cc1

(51000eV 1/cm)3

= 1.19 104eV 2

This gives a modifies mass difference:

=

(cos2 A)2 + 2sin22=

(0.01 eV 2 1.000 + 1.19 104 eV 2)2 + (0.01 eV 2 0.020)2

= 0.988 102 eV 2

And the baseline for the oscillation:

L ' 4E 4 1 GeV

0.01 eV 2= 80 km

4.2.1 Baseline at Resonance

L(R) ' 4E

=4E

sin2 4 1 GeV

0.01 eV 2 0.02 = 2 1013eV 1 4000 km

However, attaining resonance requires a neutron density about 100 times that of the Earthscrust.

38


An alternative would be to increase the energy instead, so that the resonance neutrondensity falls. But since a beam of 100 GeV energy does not seem pragmatic, we attempt tooptimise-tweaking beam energy so that a sizeable amplification is obtained without having toattain resonance. The catch is that increasing energy increases the baseline length. As is

Figure 4.1: Plot of sin22 against Energy(in GeV) for propagation throughthe Earth (generated in Mathematica)

Figure 4.2: Plot of oscillation length (in km) against Energy (in GeV) forpropagation through the Earth (generated in Mathematica)

evident from the figure, this leaves a very narrow window of high energies around 84 GeV toobtain a reasonable degree of amplification. This is owing to the fact that the width of thispeak is proportional to sin22, which in this case is a very small quantity.

The increase in energy also sends up the oscillation length. The resonance baselines atvarious values of m2, E and are given in Tables 4.2.

Although our attempt via the MSW effect did not immediately yield a practical set ofparameters, the idea of using matter effects to enhance oscillations is by no means exhausted.For instance, the parametric resonance effect[67], which is different from MSW, is believed toplay a role in oscillations in the presence of step-function like matter densities like the Earth(the core, mantle and crust densities may be approximated by a step function). This hints atfuture directions in which this work could be continued.

4.3 Outlook and Summary

The field of neutrino physics is at present a very active and developing one. There arenew experiments (like SNO+, OPERA and many others) being planned that promise exciting

39


aaaaaaaaaaa

m2/E

(eV2/GeV)

(rad) 101 102 103 104

103 3980 39500.0 395000 3950000.0102 398.0 3950.0 39500.0 395000.0101 39.8 395.0 3950.0 39500.0100 3.98 39.5 395.0 3950.0

101 0.398 3.95 39.5 395.0

102 0.0398 0.395 3.95 39.5

103 0.00398 0.0395 0.395 3.95

Table 4.2: The resonance baselines (km, rounded off to 3 s.f.) for various values of m2/E(eV2/GeV) and mixing angle (rad) are tabulated at E=1 GeV

results in the years to come. Although the oscillation parameters have been measured, attemptsare still on to improve the accuracy of estimates. In addition, there are several open problemslike determining the sign of the atmospheric mass-squared difference, the magnitude of the CP-violation phase (these two are closely linked) and the question of whether neutrinos are Diracor Majorana particles. There is also the question of the existance of the sterile neutrino, (whichwe attempted to tackle via the MSW effect) and the number of sterile neutrinos. Tackling thesequestions will be the aim of the community in the immediate future.

40

Appendix A

The vacuum Hamiltonian in a moreilluminating form

In this Appendix, we shall take a look at the vacuum Hamiltonian for 2-flavour oscillationsand rewrite it in the form used in Eqn. 2.56. We start from Eqn.2.2,

d

dt

(1(t)2(t)

)=

(E1 00 E2

).

(1(t)2(t)

)(A.1)

and shift to the flavour basis (the mixing matrix is now a constant and can be taken to the leftof the differential operator).

.U d

dt

((t)(t)

)=

(E1 00 E2

)U ((t)(t)

)(A.2)

Hvac = U(E1 00 E2

)U (A.3)

Substituting for the mixing matrix (Eqn. 2.9),

Hvac =(

cos sinsin cos

)(E1 00 E2

)(cos sinsin cos

)(A.4)

Using the ultra-relativistic approximation and simplifying,

Hvac = 12

[(E1 + E2)

(1 00 1

) m

221

2E

( cos2 sin2sin2 cos2

)](A.5)

where m221 := m22m21. Suppressing the identity, we write (employing the notation in Section

2.3.1)

Hvac = E + m21 +m

22

4E+

m2214E

( cos2 sin2sin2 cos2

)(A.6)

which is the required form.

41

Appendix B

Some Mathematica demos

In this appendix, we shall present the code for some Mathematica demos that shall illustratethe concepts discussed about neutrino oscillations.

B.1 3-flavour Oscillations in Vacuum

(* This notebook demonstrates 3 flavour neutrino oscillations in \

vacuum *)

ClearAll["Global*"]

p = 3.1415;

e = 2.71828182845904523536028747135266249775;

X[\[Theta]23_] := {{1, 0, 0}, {0, Cos[\[Theta]23],

Sin[\[Theta]23]}, {0, -Sin[\[Theta]23], Cos[\[Theta]23]}}

Y[\[Theta]13_, \[Delta]_] := {{Cos[\[Theta]13], 0,

Sin[\[Theta]13]*Power[e, -I*\[Delta]]}, {0, 1,

0}, {-Sin[\[Theta]13]*Power[e, I*\[Delta]], 0, Cos[\[Theta]13]}}

Z[\[Theta]12_] := {{Cos[\[Theta]12], Sin[\[Theta]12],

0}, {-Sin[\[Theta]12], Cos[\[Theta]12], 0}, {0, 0, 1}}

U[\[Theta]23_, \[Theta]13_, \[Theta]12_, \[Delta]_] :=

X[\[Theta]23].Y[\[Theta]13, \[Delta]].Z[\[Theta]12]

V[\[Theta]23_, \[Theta]13_, \[Theta]12_, \[Delta]_] :=

Inverse[U[\[Theta]23, \[Theta]13, \[Theta]12, \[Delta]]]

(* Amplitude matrix *)

Q[\[Alpha]_, \[Beta]_,

L_, \[Theta]23_, \[Theta]13_, \[Theta]12_, \[Delta]_, m21_,

m31_, \[Epsilon]_] :=

U[\[Theta]23, \[Theta]13, \[Theta]12, \[Delta]][[\[Alpha],

1]]\[Conjugate]*

U[\[Theta]23, \[Theta]13, \[Theta]12, \[Delta]][[\[Beta], 1]]*

Power[e, -I*2.53*0.*Power[10, L - \[Epsilon]]] +

42

APPENDIX B. SOME MATHEMATICA DEMOS


2]]\[Conjugate]*


Power[e, -I*2.53*Power[10, m21 + L - \[Epsilon]]] +


3]]\[Conjugate]*


Power[e, -I*2.53*Power[10, m31 + L - \[Epsilon]]];

(* Energy in GeV *)

\[Epsilon]0 = 2;

(* Plot of survival and transition probability for electron neutrinos \

with distance (log scale in km) for vaious values of oscillation \

parameters *)

Manipulate[

Plot[{Abs[

Q[1, 1, L, \[Theta]23, \[Theta]13, \[Theta]12, \[Delta], m21,

m31, 2]]^2,

Abs[Q[1, 2, L, \[Theta]23, \[Theta]13, \[Theta]12, \[Delta], m21,

m31, 2]]^2,


m31, 2]]^2, (Abs[


m31, 2]]^2 +


m31, 2]]^2 +


m31, 2]]^2)}, {L, 0,

7}], {m21, -7, -1}, {m31, -7, -1}, {\[Theta]23,

0, \[Pi]/2}, {\[Theta]13, 0, \[Pi]/2}, {\[Theta]12,

0, \[Pi]/2}, {\[Delta], 0, 2 \[Pi]}]

(* Plot of survival and transition probability for electron neutrinos \

into mixtures of \[Mu] and \[Tau] *)

Manipulate[Plot[{


m31, 2]]^2,

Abs[Cos[\[Theta]]*


m31, 2] +

Sin[\[Theta]]*


m31, 2]]^2,

Abs[-Sin[\[Theta]]*


m31, 2] +

Cos[\[Theta]]*

43



m31, 2]]^2, (Abs[


m31, 2]]^2 +

Abs[Cos[\[Theta]]*


m31, 2] +

Sin[\[Theta]]*


m31, 2]]^2 +

Abs[-Sin[\[Theta]]*


m31, 2] +

Cos[\[Theta]]*


m31, 2]]^2)

}, {L, 0, 3*10^5}], {m21, -7, -1}, {m31, -7, -1}, {\[Theta]23,

0, \[Pi]/2}, {\[Theta]13, 0, \[Pi]/2}, {\[Theta]12,

0, \[Pi]/2}, {\[Delta], 0, 2 \[Pi]}, {\[Theta], 0, \[Pi]/2}]

B.2 Solar Neutrinos

In this section, we shall present code that pertains to the toy version of the solar neutrinoproblem we considered in Section 2.3.4.

B.2.1 Survival Probability of e

ClearAll["Global*"];

(* Survival Probability *)

GF = 1.17*10^(-5 - 18); (* eV^-2 *)

n0 = 98.8 * 6.023*10^(23) / 51000^3 ; (* eV^3 *)

R0 = 7*10^(10) *51000 ; (* eV^-1 *)

X[x_, \[Theta]_] :=

Exp[-\[Pi]/4*x*Sin[2 \[Theta]]^2/Cos[2 \[Theta]]*R0/10];

C0[x_, \[Theta]_] := (x*Cos[2 \[Theta]] - 2*1.414*GF*n0 )/

Sqrt[(x*Cos[2 \[Theta]] - 2*1.414*GF*n0 )^2 + (x*

Sin[2 \[Theta]])^2];

P[x_, \[Theta]_] :=

0.5*(1 + (1 - 2 X[x, \[Theta]]) C0[x, \[Theta]]*Cos[2 \[Theta]])

Manipulate[

Plot[P[Power[10, x], \[Theta]], {x, (-15), (-9)},

44


PlotRange -> {0, 1}], {\[Theta], 0.0001, \[Pi]/4 - 0.0001}]

B.2.2 Density profile of the Sun and sin2

n[z_] := n0/98.8*48.8* Exp[-11.1*z^2 / (z + 0.2)]

S[\[Theta]_, x_, z_] :=

x^2*Sin[2 \[Theta]]^2/(x^2*

Sin[2 \[Theta]]^2 + ( x*Cos[2 \[Theta]] - 2*1.414*GF*n[z] )^2)

Plot[n[z]/n0, {z, 0, 1}, PlotRange -> {0, 1}]

Manipulate[

Plot[S[0.1, Power[10, x], z], {z, 0, 1},

PlotRange -> {0, 1}], {x, -16, 2}]

B.2.3 The Adiabaticity parameter

K[z_] := 1/R0 * 48.8/98.8 *

n0 * ( -2*z*11.1 / (z + 0.2) + z^2 *11.1 / (z + 0.2)^2 )*

Exp[-11.1*z^2 / (z + 0.2)];

\[Gamma][x_, z_, \[Theta]_] :=

x^2/(2*1.414*GF)*

Sin[2 \[Theta]]^2/(Sqrt[S[\[Theta], x, z]]^(3)) /Abs[K[z]]

Plot[K[z], {z, 0, 1}]

Plot[\[Gamma][10^(-14), z, 0.1], {z, 0, 1}, PlotRange -> {0, 30}]

B.2.4 The Exclusion Plot

f[\[Zeta]_] := 0.5*ArcCos[( -\[Zeta] + Sqrt[\[Zeta]^2 + 4] )/2 ];

PP[x_, \[Zeta]_] :=

0.5*(1 + (1 - 2 X[x, f[\[Zeta]]]) C0[x, f[\[Zeta]]]*Cos[2*f[\[Zeta]]])

Manipulate[

Plot[PP[Power[10, x], Power[10, \[Zeta]]], {x, -18, -9},

PlotRange -> {0, 1}], {\[Zeta], -4, 7}]

ContourPlot[

PP[Power[10, x]/(5*10^6), Power[10, \[Zeta]]], {\[Zeta], -4,

1}, {x, -8, -3}, Contours -> {0.19, 0.35}]

45

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thesis

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