My M.Sc thesis ..

8/16/2019 My M.Sc thesis ..

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Tezpur University

A F inal Semester project presentedfor the degree of M aster of Science

Connecting Neutrino Mass and Sterile NeutrinoDark Matter in Inverse and Type II Seesaw

Submitted by Mallika P. Shivam

PHY14002

Supervised byDr. Mrinal Kumar DasDepartment of Physics

May 20, 2016


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D epartment of Physics

Tezpur University

C ertificate

This is to certify that the project work "Connecting Neutrino Mass andSterile Neutrino Dark Matter in Inverse and Type II Seesaw" is a bonaderecord of work done by Mallika P Shivam, Roll: PHY14002 under myguidance, submitted to the Department of Physics, Tezpur University, inpartial fulllment of the requirements for the award of the degree in Masterof Science programme in Physics.

(Dr. Mrinal Kumar Das)

Associate Professor

Dept. of Physics

Tezpur University

Date :-

P lace :-

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Acknowledgement

At the very onset, I would like to express my gratitude and thanks to Dr. Mrinal KumarDas, Associate Professor, Department of Physics, Tezpur University for his supervision,encouragement and guidance throughout the semester and also for his support in completingthe project. I would also like to thank my institution, my co-guide Ananya Mukherjee andHappy Borgohain for their help in every step and last but not the least; I thank my projectpartners, Pragyan Phukan and Papori Seal for their support and enthusiasm.

(Mallika Priyadarshini Shivam)

Date :-

Place :-

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Abstract

We present a TeV scale Seesaw Model for exploring the sterile neutrino dark matter andneutrino phenomenology in the light of latest neutrino data. Using a special kind of Diracneutrino mass matrix and Majorana neutrino mass matrix, we xed the sterile neutrino massmatrix in Tribimaximal form (TBM). We then use the Type II seesaw as a perturbationto generate non- vanishing reactor mixing angle θ 13 without much disturbing the otherneutrino oscillation parameters. Then we have studied the variation of neutrino parameterswith Type II perturbation strength for di ff erent values of sterile neutrino Yukawa coupling.

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Contents

1 Introduction To Neutrino And Sterile Neutrino 81.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2 A Brief History of Neutrino . . . . . . . . . . . . . . . . . . . . . . . . 81.3 Current Status of Neutrino . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4 Neutrino Oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.5 Neutrino Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.5.1 Seesaw Mechanism(Type I, Type II And Inverse) . . . . . . . . . 131.6 Sterile Neutrino and Dark Matter . . . . . . . . . . . . . . . . . . . . 16

2 Methodology 17

3 Results, Analysis and Conclusion 213.1 Results and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4 Reference 32

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List of Figures

3.1 Type II perturbation versus sin2θ 13 . . . . . . . . . . . . . . . . . . . . . 263.2 Type II perturbation versus sin2θ 13 . . . . . . . . . . . . . . . . . . . . . 263.3 Variation of sin 2θ 23 with sin 2θ 13 . . . . . . . . . . . . . . . . . . . . . . 273.4 Variation of sin 2θ 23 with sin 2θ 13 . . . . . . . . . . . . . . . . . . . . . . 27

3.5 Variation of sin2θ 12 with sin

2θ 13 . . . . . . . . . . . . . . . . . . . . . . 28

3.6 Variation of sin 2θ 12 with sin 2θ 13 . . . . . . . . . . . . . . . . . . . . . . 283.7 Variation of ∆ m223 with sin

2θ 13 . . . . . . . . . . . . . . . . . . . . . . . 293.8 Variation of ∆ m231 with sin



2θ 13 . . . . . . . . . . . . . . . . . . . . . . . 30

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List of Tables

3.1 Neutrino Oscillation Data for Normal Hierarchy . . . . . . . . . . . . . . 223.2 Neutrino Oscillation Data for Inverted Hierarchy . . . . . . . . . . . . . 223.3 Value of x,y,z for t = 0.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.4 Value of x,y,z for normal hierarchy . . . . . . . . . . . . . . . . . . . . . 24

3.5 value of x,y,z for inverted hierarchy . . . . . . . . . . . . . . . . . . . . 243.6 Summary of Results obtained from graphs . . . . . . . . . . . . . . . . . 31

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Chapter 1

Introduction To Neutrino And SterileNeutrino

1.1. Introduction

Aneutrino is a lepton (an elementary particle with half-integer spin) that interactsonly via the weak subatomic force and by gravity thereby making it very di ffi cultto detect. The mass of the neutrino is tiny compared to other subatomic particles.

Neutrinos are the only identied candidate for dark matter , specically warm dark matter.Neutrinos come in three avors: electron neutrinos (νe), muon neutrinos ( ν µ), and tauneutrinos ( νΓ ). Each avor is also associated with an antiparticle, called anti neutrino ,which also has no electric charge and half-integer spin. They are the least understood andthe most elusive elementary particle known to exist. Not only it passes easily through

matter undetected and unimpeded but also changes its avor on the way. Contrary to themost successful theory of Particle Physics .ie. Standard Model, later convincing evidencethat neutrinos have oscillation among its avor and mass eigen states was repeatedlyreported.Their number far exceeds the count of all the atoms of the entire universe. Althoughthey hardly interact at all, they helped forge the elements of the early universe, they tell ushow the sun shines and they may even cause the titanic explosion of a dying star or may bethe reason behind the mysterious dark matter or why we live in a universe lled with matter.

1.2. A Brief H istory o f Neutrino

• 1931 - A hypothetical particle is predicted by the theorist Wolfgang Pauli. Pauli basedhis prediction on the fact that energy and momentum did not appear to be conserved incertain radioactive decays. Pauli suggested that this missing energy might be carriedoff , unseen, by a neutral particle which was escaping detection.

• 1934 - Enrico Fermi develops a comprehensive theory of radioactive decays, includingPauli’s hypothetical particle, which Fermi coins the neutrino (Italian: "little neutralone"). With inclusion of the neutrino, Fermi’s theory accurately explains many

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experimentally observed results.

• 1959 - Discovery of a particle tting the expected characteristics of the neutrino isannounced by Clyde Cowan and Fred Reines. This neutrino is later determined to bethe partner of the electron.

• 1962 - Experiments at Brookhaven National Laboratory and CERN, the EuropeanLaboratory for Nuclear Physics make a surprising discovery: neutrinos producedin association with muons do not behave the same as those produced in associationwith electrons. They have, in fact, discovered a second type of neutrino (the muonneutrino).

• 1968 - The rst experiment to detect (electron) neutrinos produced by the Sun’sburning (using a liquid Chlorine target deep underground) reports that less than half the expected neutrinos are observed. This is the origin of the long-standing "solarneutrino problem." The possibility that the missing electron neutrinos may havetransformed into another type (undetectable to this experiment) is soon suggested,

but unreliability of the solar model on which the expected neutrino rates are based isinitially considered a more likely explanation

• 1978 - The tau particle is discovered at SLAC, the Stanford Linear Accelerator Center.It is soon recognized to be a heavier version of the electron and muon, and its decayexhibits the same apparent imbalance of energy and momentum that led Pauli topredict the existence of the neutrino in 1931. The existence of a third neutrinoassociated with the tau is hence inferred, although this neutrino has yet to be directlyobserved.

• 1985 - The IMB experiment, a large water detector searching for proton decay butwhich also detects neutrinos, notices that fewer muon-neutrino interactions thanexpected are observed. The anomaly is at rst believed to be an artifact of detectorinefficiencies.

• 1985 - A Russian team reports measurement, for the rst time, of a non-zero neutrinomass. The mass is extremely small (10,000 times less than the mass of the electron),but subsequent attempts to independently reproduce the measurement do not succeed.

• 1988 - Kamiokande, another water detector looking for proton decay , able to distin-guish muon neutrino interactions from those of electron neutrino, reports that theyobserve only about 60% of the expected number of muon-neutrino interactions.

• 1989 - Experiments at CERN’s Large Electron-Positron (LEP) accelerator determinethat no additional neutrinos beyond the three already known can exist.

• 1989 - Kamiokande becomes the second experiment to detect neutrinos from the Sun,and conrms the long-standing anomaly by nding only about 1 / 3 the expected rate.

• 1994 - Kamiokande nds a decit of high-energy muon-neutrino interactions. Muon-neutrinos travelling the greatest distances from the point of production to the detectorexhibit the greatest depletion.

• 1994 - The Kamiokande and IMB groups collaborate to test the ability of water detec-tors to distinguish muon- and electron-neutrino interactions, using a test beam at the

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KEK accelerator laboratory. The results conrm the validity of earlier measurements.The two groups will go on to form the nucleus of the Super-Kamiokande project.

• 1996 - The Super-Kamiokande detector begins operation.

• 1997 - The Soudan-II experiment becomes the rst iron detector to observe thedisappearance of muon neutrinos. The rate of disappearance agrees with that observedby Kamiokande and IMB.

• 1997 - Super-Kamiokande reports a decit of cosmic-ray muon neutrinos and solarelectron neutrinos, at rates agreeing with measurements by earlier experiments.

• 1998 - The Super-Kamiokande collaboration announces evidence of non-zero neutrinomass at the Neutrino ’98 conference.

• 2000- The DONUT Collaboration working at Fermilab announces observation of tau particles produced by tau neutrinos, making the rst direct observation of the tauneutrino.

• 2000 -SuperK announces that the oscillating partner to the muon neutrino is not asterile neutrino, but the tau neutrino.

• 2001 and 2002– SNO announces observation of neutral currents from solar neutrinos,along with charged currents and elastic scatters, providing convincing evidence thatneutrino oscillations are the cause of the solar neutrino problem.

• 2002– Masatoshi Koshiba and Raymond Davis win Nobel Prize for measuring solarneutrinos(as well as supernova neutrinos).

• 2002– KamLAND begins operations in January and in November announces detectionof a decit of electron anti-neutrinos from reactors at a mean distance of 175 km inJapan. The results combined with all the earlier solar neutrino results establish thecorrect parameters for the solar neutrino decit.

• 2004– SuperKamiokande and KamLAND present evidence for neutrino disappear-ance and reappearance, eliminating non-oscillations models.

• 2005– KamLAND announces rst detection of neutrino ux from the earth and makesrst measurements of radiogenic heat from earth.

1.3. C urrent Status of Neutrino

In the last two decades experiments have established the existence of neutrino oscillationsand most of the related parameters have by now been measured with reasonable accuracy.At present neutrino physics is a most vital domain of particle physics and cosmology. The

Nobel Prize in Physics 2015 was awarded jointly to Takaaki Kajita and Arthur B. McDonald"for the discovery of neutrino oscillations, which shows that neutrinos have mass". Thisdiscovery has changed our understanding of the innermost workings of matter and showedthat the Standard Model cannot be the complete theory of the fundamental constituents of the universe. Now the experiments continue and intense activity is underway worldwide inorder to capture neutrinos and examine their properties. Current experimental constraintson the neutrino mass spectrum and the mixing parameters, including the recent observation

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of non- zero mixing angle θ 13 by reactor neutrino disappearance experiments has initiatedthe precision era of lepton avor physics. Therefore it is timely to identify strategies todetermine the remaining parameters of the three active neutrinos, such as the neutrino masshierarchy and hints for one or more sterile neutrinos and their phenomenology need tobe assessed. In the astrophysics domain, the IceCube discovery of neutrino events at thehighest neutrino energies yet measured (

∼PeV) may have initiated neutrino astronomy as a

new discipline for research.

1.4. Neutrino O scillation

Neutrinos are created or detected with a well- dened avor( electron or muon or tau).Howeverin a phenomenon called as neutrino Flavour Oscillation,neutrinos are able to oscillate be-tween the three available avours while they propagate through space. Specically, thisoccurs because the neutrino avor eigen states are not the same as the neutrino mass eigen

states. This allows for the neutrino that was produced as an electron neutrino at a givenlocation to have a calculable probability to be detected as either a muon or tau neutrino afterit has travelled to di ff erent location. This quantum mechanical e ff ect was rst hinted by thediscrepancy between the number of electron neutrinos detected from the sun core failing tomatch the expected numbers called as the Solar Neutrino Problem. In the Standard Modelthe existence of avor oscillations implies the nonzero di ff erences between the neutrinomasses because the amount of mixing between neutrino avours at a given time depends onthe diff erences in their squared –masses.Neutrino Oscillation is a quantum mechanical phenomenon predicted by Bruno Pontecorvowhereby a neutrino created with a specic avor can later be measured with a di ff ererentavor. The probability of measuring a particular avor for a neutrino varies periodicallyas it propagates .Neutrino Oscillation is of theoretical importance since observation of thephenomenon implies that the neutrino has a non zero mass, which is not part of the original

Standard Model of Particle Physics. The avor eigen states of neutrino isνeν µντ

and the

mass eigen states isν1ν2ν3

. The two states are related by a 3 ×3 mixing matrix :

νe

ν µντ

= ( U PMNS )ν1

ν2ν3

(1.1)

And,

U PMNS =U e1 U e2 U e3U ν1 U ν2 U ν3U τ 1 U τ 2 U τ3

(1.2)

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Taking two generations,

νeν µ

= cos θ sinθ

−sin θ cos θ ν1ν2

(1.3)

At t= 0, we have an electron neutrino and muon neutrino which are both mixtures of ν1 andν2.

νe(t = 0) = co sθν1 + sin θ ν2 (1.4)

ν µ(t = 0) = cosθν2 −sin θ ν1 (1.5)At t = t

νe (t ) = cosθ ν1e−iE 1 t + sin θν2e−iE 2 t (1.6)

ν µ (t ) = co sθ ν2e−iE 2 t −sinθν1e−iE 1 t (1.7)

Taking approximations

E 1 = m21 + p21 p + m21 / 2 p, (1.8) E 2 = m22 + p22 p + m22 / 2 p (1.9)

Where we consider the momentum of the neutrino to be large enough so that p1 = p2 = p

Now, from the above equations it can be shown that:

νe (t ) = νe (0) [cos 2θν1e−iE 1 t + sin2θν2e−iE 2 t ] + ν µ (0) [e−iE 1 t + e−iE 2 t ]cos θ sin θ (1.10)

Therefore,the probability of having νe → ν µ oscillation in time t is

P (νe → ν µ) = [ sin2θ sin ( E 1 − E 2

2ht )]

2(1.11)

And, P( ν µ

→ ν µ ) = 1 - P( νe

→ ν µ )

Where,

E 2 − E 1 ≈m22 −m21

2 pc3 ≈

m22 −m212 E

c4 (1.12)

Thus neutrino oscillation implies that there must be neutrino masses because the probabilityof oscillation depends on the di ff erence of their squared masses.

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1.5. Neutrino M ass

In the standard model , neutrinos have exactly zero mass. This is a consequence of thestandard model containing only left handed neutrinos .With no suitable right handed partner,

it is impossible to add a renormalizable mass term to the standard model. Measurmentsindicated that neutrinos spontaneously change avours implying neutrinos do have a mass.

1.5.1. Seesaw Mechanism(Type I, Type II And Inverse)

Type I Seesaw

One approach to add masses to the neutrinos, the so called Seesaw Mechanism is to addright-handed neutrinos and have these couple to left handed neutrinos with a Dirac massterm. Since neutrinos have non zero electric charge, Majorana terms are also possible andthe Majorana mass of the RH neutrino is much larger than SM symmetry breaking scale.

Once we consider Right Handed neutrinos by hand, we have a number of neutrino massterms –

1. Dirac Mass Term—

L Dmass = m D ¯ν Rν L + h.c = 1

2(m D ¯ν Rν L + m DνC R ν̄

C L ) + h.c... (1.13)

2.Majorana Mass term—

L Lmass =

12 m Lν

C Lν L + h.c... (1.14)

L Rmass = 12

m RνC Rν R + h.c (1.15)

Now we write the total mass lagrangian in the form of a mass matrix

L = L Dmass + L Lmass + L Rmass = νC L ν R 0 m DmT D m R ν LνC R (1.16)

After diagonalizing the matrix the following mass eigen states are obtainedm2 ≈ m R ≈1014 GeV.

m1 ≈m2 Dm R

m Dm−1 R mT D = 102 ×102

1014 ≈ 0.1 eV

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Thus we see that light neutrinos of sub ev scale are naturally generated due to the largescale suppression of the other heavy scale RH Majorana neutrino, hence this is named asSeesaw Mechanism.

Type II Seesaw

Several BSM framewoks have been proposed to explain the tiny neutrino mass and thepattern of neutrino mixing. Tiny neutrino mass can be explained by seesaw mechanismswhich broadly fall into three classes: Type I , Type II and Inverse ,whereas the pattern of neutrino mixing can be understood by incorporating avor symmetries. The SM is extendedby three right handed singlet neutrinos and one higgs triplet such that both Type I andType II Seesaw can contribute to neutrino mass. Type I seesaw is assumed to give rise to asymmetric neutrino mass matrix with θ 13 = 0 whereas Type II Seesaw acts as a perturbationwhich breaks the symmetry resulting in non zero θ 13 .Type I seesaw is the simplest possiblerealization, and is implemented by the inclusion of three additional right handed neutrinos(ν i R, i= 1,2,3 ) as SU (2) L singlets with zero U (1)Y charges. On the other hand ,in TypeII Seesaw , the Standard Model is extended by the inclusion of an additional SU (2) L ∆triplet scalar eld having U (1)Y charge twice that of lepton doublets with its 2 ×2 matrixrepresentation as

∆ = ∆ +

√ 2 ∆ ++∆ 0 −

∆ +

√ 2

Thus the gauge invariant lagrangian for Type I and Type II seesaw mechanism is givenbelow

L = ( D µφ)+ ( D µφ) + T r ( D µ∆ )+ ( D µ∆ ) − Llept Y −V (φ, ∆ ) (1.17)

With Vacuum expectation value of the SM Higgs φ0 = ν√ 2 , the trilinear mass term gener-ates an induced VEV for the Higgs triplet as ∆ 0 = ν ∆√ 2 , thus resulting in 6 ×6 neutrinomass matrix after electroweak symmetry breaking—

M ν = m LL m LRmT LR M RR

where, m LR = the Dirac Neutrino mass , m LL= Majorana mass for the light active neutri-nos and M RR is the bare mass term for the heavy sterile majorana neutrinos.

With the mass hierarchy M RR m LR m LL, the seesaw formula for light neutrino mass isgiven bymν = m LL = m I LL + m

II LL

where, the formula for Type I Seesaw is given by

m I LL = −m LR M −1 RRmT LR

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Whereas the Type II Seesaw mechanism contribution to light neutrino mass is given by

m II LL = f ν ν ∆

where, ν ∆ ≡ ∆ 0 = µφ∆ ν2

M 2∆In the low scale Type II mechanism operative at TeV scale, one can consider a very smallvalue of trilinear mass parameter to be µφ∆ 10−8 GeV.The tiny trilinear mass µφ∆ parameter controls the neutrino overall mass scale.

Inverse Seesaw Model

In spite of explaining the smallness of neutrino mass, such Type 1 Seesaw mechanismsare not phenomenologically testable because the new Physics engendered by them willmanifest at 10 14 GeV scale which is completely out of the range of the current acceleratorexperiment.

So recently a new kind of seesaw was proposed ie. Inverse Seesaw Mechanism (ISS)where small neutrino masses arise as a result of new Physics at TeV scale which may beprobed at LHC experiment. The implementation of ISS mechanism requires the addition of three right handed neutrinos N R and the three extra SM gauge singlet neutral fermions S tothe three active neutrinos ϑ L.

After SSB the overall neutrino mass terms turn out to be

Lmass = 12

ν L N C R S C

0 m D 0mT D 0 M RS

0 M T

RS µ

νC L N R

S

(1.18)

Where µ is the mass of the neutrino singlet, also neutrino singlet has no Yukawa couplingto left handed neutrino but couple to N R.

A diagonalisation of the above 9 ×9 matrix leads to the e ff ective light neutrino massmatrix i.e.

mν = mT D( M T RS )−

1 µ( M RS )−1mT D (1.19)

Or

mν

0.1 eV

= m D

100 GeV

2

µ

1 keV M RS

10 T eV

−2(1.20)

Thus we see that Standard neutrinos with mass at sub eV scale are obtained for m D atelectroweak scale and M RS at TeV scale. The core of the ISS is that the smallness of theneutrino masses are guaranteed by assuming that µ scale is small and in order to bring theRH neutrinos at TeV scale, it has to be at KeV scale. ISS is also called double seesawbecause as seen from the above equation m D is doubly suppressed by M RS .

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1.6. Sterile Neutrino and Dark M atter

Mass eigen states that are dominantly linear combinations of LH neutrinos are calledActive Neutrinos and those are dominated by right handed neutrino components are called

STERILE neutrinos. Sterile neutrinos are hypothetical particles that interact only via gravityand do not interact via any of the of the Standard Model. The term sterile neutrino is used todistinguish them from the known active neutrinos in the Standard Model, which are chargedunder the weak interaction.This term usually refers to neutrinos with right-handed chiralitywhich may be added to the Standard Model. The existence of right-handed neutrinos istheoretically well-motivated, as all other known fermions have been observed with left andright chirality, and they can explain the observed active neutrino masses in a natural way.The mass of the right-handed neutrinos themselves is unknown and could have any valuebetween 10 15 GeV and less than one eV. The number of sterile neutrino types is unknown.The search for sterile neutrinos is an active area of particle physics. If they exist and theirmass is smaller than the energies of particles in the experiment, they can be produced in thelaboratory, either by mixing between active and sterile neutrinos or in high energy particlecollisions. If they are heavier, the only directly observable consequence of their existencewould be the observed active neutrino masses. They may, however, be responsible fora number of unexplained phenomena in physical cosmology and astrophysics, includingdark matter, baryogenesis or dark radiation. Dark matter can be divided into cold, warmand hot categories. If the dark matter is composed of abundant light particles whichremain relativistic until shortly before recombination, then it may be termed "hot". Thebest candidate for hot dark matter is a neutrino.But HDM cannot explain how individualgalaxies formed from the big bang, so active neutrinos are generally not well motivatedDark Matter candidates. Whereas Sterile Neutrinos with masses around the KeV can beviable warm dark matter candidates(WDM).They can potentially solve, even if providingonly a fraction of the total dark matter relic density.In addition a sterile neutrino at this scalecould in general decay into an ordinary neutrino and a photon which could be detected incosmic rays. This last possibility has recently triggered an interest in view of the indication( yet to be conrmed) of an unidentied photon line in galaxy cluster at an energy 3.5 KeV.Sterile Neutrino can mix with active neutrinos and in that case, oscillations of the activeneutrinos into the sterile neutrinos in the early universe can populate the number densityof sterile neutrino and by this mechanism, it is possible to explain observed relic densityof DM. But the same mechanism would make sterile neutrinos decay into photon and aneutrino. Such a monochromatic photon line can potentially be observed. Favoured mass

range of sterile neutrino is 1- 50 KeV, thus the photon line is predicted to fall into x-raydomain. Thus a sterile neutrino with a mass of 7 KeV could be a viable DM candidate forexplaining the recent detection of a 3.5 KeV x-ray emission line of the galaxy cluster.

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Chapter 2

Methodology

The “ (2,3) ISS “ Framework

The (2,3) ISS Realisation corresponds to an extension of the SM by two Right Handedneutrinos and three sterile states and this realization can provide a WDM candidate ( for amass of the sterile state in the KeV range). The lagrangian representing the neutrino mass is

Lm = dν̄ L ν R + m ν̄C Rν R + nν̄C RS + µS̄ S

Here the neutrino mass matrix M has the form:

M =0 d 0

d T m n

0 nT

µ

... (2.1)

The dirac mass matrix d arises from the Yukawa couplings to the SM Higgs boson

Y α i ¯lα L ˜ H ν i R + h.c (2.2)

where,(lα L) = ν

α Le

α L (2.3)

Which gives after electroweak symmetry breakingd α i = ν√ 2 Y α i

The mass m and µ represent Majorana mass terms for , respectively , right handed andsterile fermions.

Finally the matrix n represents lepton number conserving interactions between righthanded and new sterile fermions. The physical neutrino states are obtained upon diagonal-ization of the mass matrix M and feature the following mass pattern:

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mν ≈ O( µ) k 2

1 + k 2, k ≈

O(d )O(n)

(2.4)

• Pairs of heavy Dirac heavy neutrinos with masses O(n) + O (d).

• Light sterile states with masses O( µ)In order to be phenomenologically viable the matrix M associated to this (2,3) ISS

realization must exhibit upon diagonalization three light ≤ O (eV ) ,active eigen states withmass di ff erences in agreement with oscillation data and a mixing pattern compatible withthe experimental determination of the PMNS matrix.

As we will see, a good t is provided by the (2,3) ISS realization ( 2 right handed and3 additional sterile fermions); also an additional intermediate state with mass m4 = ms =O( µ) appears in the mass spectrum. However in order to comply with all constraints fromneutrino oscillations and laboratory experiments ,the coupling of this new state must be

highly suppressed, thus leading to a dominantly sterile state named as the Sterile Neutrinowith a mass ranging from eV to tens of KeV. As a consequence of its weak interactions thelifetime of the sterile neutrino largely exceeds the lifetime of the Universe and it thus play arelevant role in Dark Matter.

Since it is a (2,3) Realisation, we proceed to nd the texture of the individual matricesin the mass matrix M as follows:–

• d ( 3×2 matrix) =ν R1 Le ν R2 Leν R1 L µ ν R2 L µν R1 Lτ ν R2 Lτ

• µ ( 3×3 matrix ) = S 11 S 12 S 13S 21 S 22 S 23

S 31 S 32 S 33

• n ( 3×2 matrix) =ν R1S 1 ν R2S 1ν R1S 2 ν R2S 2ν R1S 3 ν R2S 3

Due to discrete avour symmetry like Z 2 , the coupling term of the Right Handed fermionsdoes not come in the nal expresssion and so –

The mass matrix M is now

0 d 0d T 0 n0 nT µ

This is a 8×8 leptonic mass matrix and considering the hierarchy µ d n, the block diagonalisation of this matrix provides the following e ff ective neutrino mass matrix for thestandard neutrinos.

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mν = dn−1 µ nT −1d T (2.5)

We carry out the Block Diagonalisation as follows

M =0 d 0

d T 0 n0 nT µ

= 0 R RT P (2.6)

where R = d 0

P= 0 nnT µ

If λ is the eff ective neutrino mass matrix ( lightest neutrino mass eigen value) representingthe the three active states, then

λ 2- Pλ – R RT = 0

Or λ = - RP−1

RT

mν = - RP−1 RT

= − d 0 1

−nn T µ −nT −n 0

d T

0

where

P−1 = 1

−nn T µ −nT −n 0

So the value of the e ff ective neutrino matrix comes out to bemν = − RP−1 RT

= − d 0 1−nn T µ

−nT

−n 0 d T

0

= − d 0 1−nn T µd T

−nd T = (

d µd T )nn T = dn−1 µ nT −

1d T

We can write it also as, mν = md M −1 µ M T −1

mT d

Therefore the resultant mass matrix mν is a 3×3 mixing matrix,like the PMNS matrixwhich is considered as leading order contribution to the neutrino mass.

The texture of the constituent matrices are taken such that md M −1 and M T −1

mT d are bothidentity matrix I..This can be framed in the following way—

md M −1 =1 0 00 1 00 0 1

= M T −1mT d

We take the µ matrix as a TBM ( Tribimaximal mixing) since it is compatible with allveried neutrino oscillation experiments until recently and may be used as a zeroth order

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approximation to more general forms of PMNS matrix. So the nal matrix mν being theproduct of two identity matrices and a TBM matrix; is also a TBM matrix.Therefore we get light neutrino mass matrix of TBM Type, namely

mν = t x y y y x + z y− z y y− z x + z

; where t is the Yukawa coupling.

Next we proceed with the calculations for studying the neutrino phenomenologyrelating to known mixing angles and squared mass di ff erences of the active neutrinos. If we can somehow restrict the eigenmass of the sterile neutrino µ to 0.1 KeV or less, thenit can be considered a potential dark matter candidate; as discussed by authors AsmaaAbada [5]. This is the prime objective of the proposed studies.The well known neutrinomixing matrix TBM predict the mixing angle angle θ 13 = 0.The non zero and relativelylarge mixing angle have already been reported by MINOS, Double Chooz„ Daya Bay, and

RENO collaborations. In order to accommodate non zero mixing angle θ 13 , we modify theTribimaximal mixing matrix (TBM) by introducing a simple perturbation matrix to perturbTBM matrix. Then we determine the neutrino mass spectrum in both normal and invertedhierarchy from the modied TBM matrix, keeping the sterile neutrino mass µ of the orderof 0.1 KeV or less; which can be achieved by varying the Yukawa coupling in order of 0.1. By considering the di ff erent Yukawa coupling of sterile neutrino, we have studied theneutrino phenomenology.

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Chapter 3

Results, Analysis and Conclusion

3.1. R esults and A nalysis

We are actually trying to establish a connection between neutrino phenomenology anddark matter. The µ matrix in the above light neutrino mass matrix mν representing sterilefermion will be a viable Dark Matter candidate if we can show that even after taking thesterile fermion into account we can reproduce the known neutrino parameters.

We start with the TBM form of the mass matrix..

mν = t x y y y x + z y− z y y− z x + z

; where t is the Yukawa coupling

The following rigorous calculations are done using the Mathematica Software.

• Finding the eigensystem of the above matrix—

t ( x− y) t ( x + 2 y) t ( x− y + 2 z){−2,1,1 } {1,1,1 } {0, −1, 1}

• The mass eigen values arem1= t(x-y)m2= t(x+ 2y)m3= t(x-y + 2z)

We take the Yukawa coupling value from 0.1 to 0.5 and run the following calculationsseparately for each case

• The Yukawa coupling actually represents the mass of the sterile fermion and so weare analyzing neutrino spectrum by varying the mass of µ.

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• Then we will observe if the parameters thus obtained falls in the 3 σ range of theneutrino oscillation parameters shown below:

Table 3.1: Neutrino Oscillation Data for Normal Hierarchy

Oscillation parameters Best t point(bfp) 3σ

∆ m221

[10−5eV 2]7.5 7.02-8.07

∆ m231

[10−3eV 2]2.457

2.317-2.607

sin 2θ 12 0.304 0.270-0.344

sin 2θ 13 0.0218 0.0186-

0.0250

sin 2θ 23 – 0.381-

0.643

Table 3.2: Neutrino Oscillation Data for Inverted Hierarchy

Oscillation parameters Best t point(bfp) 3σ

∆ m221

[10−5eV 2]7.5 7.02-8.07

∆ m231

[10−3eV 2]-2.449

-2.590,-2.307

sin 2θ 12 0.304 0.270

-0.34

sin 2θ 13 0.0219 0.0188-

0.0251

sin 2θ 23 – 0.388-

0.644

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For each set of t, the eigen values x, y,z are found to be only slightly di ff erent and so wediscuss the analysis taking –

Taking t = 0.3 (say for normal hierarchy)

∆ m221 = (m2)2

−(m1)2

= −0.09 ( x− y)2 + 0.09 ( x + 2 y)2

And ∆ m231 = ( m3)2

−(m2)2

= −0.09 ( x− y)2 + 0.09 ( x− y + 2 z)

2

Taking the known values of oscillation parameters ie

∆ m221 = 0.0000765,

∆ m231 = 0.0024

m1 = 0.00001

we get the numerical solution of the parameters x,y,z as

Table 3.3: Value of x,y,z for t = 0.3

x y z0.0097404 0.0097071 −0.08166630.0097404 0.0097071 0.0816329

−0.00969603 0.0097293 −0.0816663−0.00969603 0.0097293 0.0816329

We choose the values asx= 0.00974048y= 0.00970715z= -0.0816663

Similarly now for di ff erent values of the Yukawa coupling say 0.5, 0.4., 0.3 ..(say).theparameters x,y,z are found to be slightly di ff erent . These are tabulated below for NormalHierarchy

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Table 3.4: Value of x,y,z for normal hierarchy

Parameters t = 0.5 t = 0.4 t = 0.3x 0.00584429 0.00730536 0.00974048

y 0.00582429 0.00728036 0.00970715z -0.0489998 -0.0612497 -0.0816663

Similarly for the case of Inverted Hierarchy , we follow the same proceedings andfor di ff erent values of the Yukawa coupling t = 0.3,0.4,0.5, we tabulate the correspondingparameters x,y,z

Table 3.5: value of x,y,z for inverted hierarchy

Parameters t = 0.5 t = 0.4 t = 0.3x -0.0059943 -0.0344962 -0.0459949y -0.002338 -0.0022374 -0.0029833z 0.021555 -0.0459949 0.0215225

Now as we were considering the t = 0.3 case in NH, the eigen system of the TBM mass

matrix 0.3 x y y y x + z y− z y y− z x + z

comes out

as −0.0489897 0.00874643 9.9 ×10−6{1.13 ×10−16 , −0.70710, 0.7071 } {0.57735, 0.57736, 0.57735 } {0.8164, −0.4082, −0.4082 }

The mass matrix is still the TBM form with the third mixing angleθ 13 = 1.13 ×10−16 ≈ 0 ie

0.8164 0.57735 1.13 ×10−16−0.4082 0.57735 −0.70710−0.4082 0.57735 0.70710

However with the discovery of the non zero third mixing angle , it is necessary toinclude the type II perturbation and we vary the type ii seesaw strength from 10−6 to .01 toproduce non zero θ 13 .

The perturbation matrix takes the structure

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m II ν =0 −w w−w w 0w 0 −wAfter adding it, the new matrix becomes

mν = m II ν + m I ν

mν =tx ty −w ty + w

ty −w t ( x + z) + w t ( y− z)ty + w t ( y− z) t ( x + z) −w

(3.1)

After getting the complete mass matrices for each of the case t = 0.3,0.4, 0.5. . . ,wediagonalize them. Now the elements of the diagonalised matrix are associated with theparameters of the model. By varying the Type II perturbation strength w from 10−6 to0.01 in stepsize of 10−6 , we compute all the oscillation parameters.The variation of TypeII strength w with the non vanishing θ 13 has been shown in the rst two gures . Theproduction of other oscillation parameters ie the three mixing angles and the two masssquared splitting as a function of non zero θ 13 has been shown in the subsequent gures forboth NH and IH case.

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Figure 3.1: Type II perturbation versus sin 2θ 13

...

Figure 3.2: Type II perturbation versus sin 2θ 13

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...

Figure 3.3: Variation of sin 2θ 23 with sin 2θ 13

...


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...


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Figure 3.7: Variation of ∆ m223 with sin2θ 13


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Table 3.6: Summary of Results obtained from graphs

Model ∆ m221

∆ m231 or

∆ m223

θ 12 θ 13 θ 23

TBM(NH)

TBM( IH) ×

• We have calculated the oscillation parameters and from the graphs the followingobservations are being found—• The non zero value of θ 13 has been found to be consistent with the variation of Type

II seesaw strength.

• All the oscillation parameters are being generated in the correct 3 σ by the workingmatrix for any value of Yukawa coupling ranging from 0.1 to 0.5 in the NH case.

• However the matrix is unable to generate ∆ m221 for the IH case; other parameters aregenerated in the 3 σ range for the IH case.

• The Yukawa coupling has got a key role in generating the oscillation parameters aswell.3.2. C onclusion

Thus we have studied the prospect of producing non zero θ 13 by introducing a perturbationto the light neutrino mass matrix using Type II seesaw. We have also determined thestrength of the Type II seesaw term which is responsible for generating non zero θ 13 inthe correct 3 σ range. This model may have high relevance for future study so far as dark matter phenomenology and sterile neutrino is concerned.

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Chapter 4

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[15] M. Hirsch, S. Morisi , E. Peinado and J. W. F. Valle, Phys. Rev. D82 , (2010).

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