Project Report May 2016 Tezpur University

Contents
1

Chapter 1: Introduction to Neutrino Physics

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Origin and History . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Current Stautus of Neutrino Physics . . . . . . . . . . . . . . . . . .
1.4 Neutrino Masses and Mixing . . . . . . . . . . . . . . . . . . . . . .
1.4.1 Neutrino Oscillation . . . . . . . . . . . . . . . . . . . . . . .
1.4.2 Neutrinos Within Standard Model . . . . . . . . . . . . . . .
1.4.3 Neutrinos Beyond Standard Model: "See-saw Mechanism"
1.4.4 Theory of Type I See-saw Mechanism . . . . . . . . . . . . .

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2
2
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3
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4
5
6
6

Chapter 2: Baryogenesis
2.1 Baryon Asymmetry of The Universe .
2.2 Ingredient and Mechanism . . . . . .
2.2.1 Shakharavs conditions . . . .
2.2.2 Baryogenesis via Leptogenesis

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Chapter 3: Mathodology

11

Chapter 4: Numerical Analysis

4.1 Calculation For Normal Hierachy . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Analysis of Baryon Asymmetry Varying Parameters , m1 , and . . . . . .

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Project Report May 2016 Tezpur University

1.
1.1.

Chapter 1: Introduction to Neutrino Physics

Introduction

f we look deep into the universe, we see stars and galaxies of all shapes and sizes. What we
do not see, however, is that the universe is filled with particles called neutrinos. These particlewhich have no charge and have little mass were created less than one second after the Big
Bang, and large numbers of those primordial low-energy neutrinos remain in the universe today
because they interact very weakly with matter. Indeed every cubic centimetre of space contains
about 300 of these uncharged relics.
Trillions of neutrinos pass through our bodies every second almost all of these are produced
in fusion reactions in the Suns core. However, neutrino production is not just confined to our
galaxy. When massive stars die, most of their energy is released as neutrinos, in violent supernova
explosions.
Neutrinos are the least understood of the fundamental particles. They travel closed to speed of
light. They can able to pass through ordinary matter almost undisturbed and are thus extremely
difficult to detect. They obey Fermi statistics i.e. they are fermion of spin 1/2. Neutrinos are
created as a result of certain type of radioactive decay, or nuclear reactions in sun, in nuclear
reactors or when cosmic rays hit molecules. They interact with other particles only through weak
and gravitational interactions. They have mass but their masses are so tiny in reality that cant be
measured accurately. The neutrinos are leptons and corresponding to three generations of leptons
there are three types of neutrinos.
There are three flavours of neutrinos-

1. Electron neutrino (electron neutrinos are emitted in Beta-decay)

2. Muon Neutrino
3. Tau Neutrino
There are also three antineutrinos of the same flavours which are thought to be identical to the
neutrinos.

1.2.

Origin and History

Neutrinos have been shrouded in mystery ever since they were first suggested by Wolfgang Pauli
in 1930.At the time nuclear Physicists were puzzled because nuclear beta decay appeared to break
the law of energy conservation. Pauli also struggled with this mystery. His prediction was based
on the fact that in Beta decays, the energy and momentum did not appear to be conserved and
this missing energy might be carried off by neutral particle which was yet to be discovered.
Two years later, James Chadwick discovered what we now call neutron, but it was clear that this
particle was too heavy to be the neutron that Pauli had predicted. However, Paulis particle
played particle played a crucial role in the first theory of nuclear beta decay formulated by Enrico
Fermi in 1933 and which later became known as the weak force. Since Chad-wick had Taken the
name neutron for something else, Fermi had to invent a new name. Being Italian, neutrino
was the obvious choice: a little neutral one.
In 1956- Clyde Cowan, Frederick Reines, F.B. Harison, H.W Kruse, A.D. McGuire first detected
the neutrino and they published the article Detection of the free Neutrino: A confirmation for
which they got Nobel Prize in 1995.
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Project Report May 2016 Tezpur University

1n 1962-Leon Lederman, Melvin Schwartz and Jack Steinberger detecting interaction of muon
neutrino, for which they earned Nobel Prize in 1988.
In 1978- third type of lepton, the Tau particle was discovered at the Stanford Linear Accelerator.The
existence of the third type of neutrino with the tau particle was hence assumed but they were
detected in 2000 at Fermi Lab. By DONUT Collaboration.
In 1987- the beginning of the neutrino astronomy with the detaction of solar neutrinos and the
detection of neutrino of the SN 1987A.
In 1998- several experiments began to show that neutrino have mass and can change flavours
which resolved the solar neutrino problem i.e. electron neutrino produced in the sun had partly
changed into other flavours which the experiments could not detect.
In 2002-Raymond Davis Jr. and Masatoshi Koshiba were jointly awarded Nobel prize in physics
for their work on Solar neutrinos. Davis experiment confirmed that the sun produces neutrinos,
but only about one-third of the number of the neutrinos predicted by theory could be detected.

1.3.

Current Stautus of Neutrino Physics

As noted above the experimental establishment of neutrino oscillation has been made in a number
of experiments with various neutrino sources during the final decade of the last century. A
wealth of information on many aspects of neutrinos and neutrino oscillations can be found in the
Neutrino Oscillation Industry website. A felicitous testimony of the importance of the study of
the neutrino is the fact that there have been to date three Nobel prizes to its credit, once every
seven years since 1988: 1988, 1995, 2002. Recently the Nobel Prize in 2015 was awarded to Takaaki
Kajita and Aurther B. McDonald for the discovery of Neutrino Oscillation which have proved that
neutrino has mass.
We can summarize the recent stages of developmentof neutrino oscillation physics as follows:
The first convincing neutrino oscillation signal is provided by the atmospherical neutrino
experiment at Super-K, following several earlier indications of the oscillation. Now neutrino
oscillation has been clearly demonstratedin many experiments at many different laboratories,
with both cosmic and terrestrial neutrino sources from the sun, atmosphere, reactors, and
accelerators. Several detector types have beenused.
The experimental data available to date, which are limited to the accuracy of the dominant
mixing effect, can be understood in terms of theoretical frameworks of two-level vacuum
oscillations and adiabatic conversion in matter.
The large mixing angles (at least 2 of the 3) and small masses of the neutrino system are
in stark contrast to the mixing and mass patterns of the quarks, posting an interesting
challengeto a unified theoretical understanding.
The supernova events of SN1987A confirmed the existence of intergalactic neutrinos and
opened up a new area of astrophysics study, giving birth the so-called neutrino astronomy.
The advancement of our knowledge of neutrinos has greatly expanded our tools of study
for astrophysics. Neutrino detectors, located deep underground and in ocean, are used as
neutrino telescopes to probe regions of stars and the cosmos that are not accessible to the
electromagnetic radiation. Several experiments of such kinds are in progress.

1.4.

Neutrino Masses and Mixing

The mass of neutrino is zero in the Standard Model due to single helicity and lepton number
conservation. The evidence that neutrinos have the mass relies on two facts that 2nd and 3rd
generations of electrons (muon and tau) are unstable and short lived, it is near certainty that sun
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Project Report May 2016 Tezpur University

produces only one kind of neutrino i.e. the electron neutrino. If neutrinos were all massless then
it would be impossible for a given type of neutrino to become a different type of neutrino, the
interesting neutrino oscillation phenomena.
1.4.1

Neutrino Oscillation

Neutrino oscillation is a quantum mechanical phenomena that was predicted by Burno Pontecorvo
in 1968. Neutrino oscillation arises from a mixture between the flavour and mass eigenstates of
neutrinos. The three neutrino states that interact with the charged leptons in weak interactions are
each a different superposition of the three neutrino states of definite mass.Neutrinos are created
in weak decays and reactions in their flavour eigenstates. As neutrinos propagates through space,
the quantum mechanical phases of the three mass states advance at slightly different rates due
to the slight differences in the neutrino masses. This result in a changing mixture of ass states
as thae neutrino travels, but a different mixture of mass states. Since the quantum mechanical
phase advances in a periodic fashion, after some distance the state will nearly return to the oriinal
mixture. The electron flavour content of the neutrino will then continue to oscillate as long as
the quantum mechanical states maintains coherence. It is because the mass differences between
the neutrinos are small that the coherence length for neutrino oscillation is so long, making this
microscopic quantum effect observable over macroscopic distances. This phenomenon solves the
long standing solar neutrino problem and providing the non zero mass of neutrino.

e
1
The flavor eigen states of neutrino is and the mass eigen states is 2 . The two

3
states are related by a 33 mixing matrix :

e
1
= (UPMNS) 2

(1)

And,

UPMNS

Ue1
= U1
U1

Ue2
U2
U2

Ue3
U3
U3

sin
cos



(2)

Taking two generations,




=

cos
sin

1
2


(3)

At t=0, we have an electron neutrino and muon neutrino which are both mixtures of 1 and 2 .
e (t = 0) = cos1 + sin 2

(4)

(t = 0) = cos2 sin 1

(5)

e (t) = cos 1 eiE1 t + sin2 eiE2 t

(6)

(t) = cos 2 eiE2 t sin1 eiE1 t

(7)

At t=t

Project Report May 2016 Tezpur University

Taking approximations
E1 =
E2 =

m21 + p21
= p + m21 /2p,

m22 + p22
= p + m22 /2p

(8)
(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:
e (t) = e (0) [cos2 1 eiE1 t + sin2 2 eiE2 t ] + (0) [eiE1 t + eiE2 t ]cossin

(10)

Therefore,the probability of having e oscillation in time t is

P(e ) = [ sin2 sin (

E1 E2 2
t)]
2h

(11)

P( ) = 1 - P( e )

And,
Where,
E2

E1

m22 m21 3
m2 m21 4
c 2
c
2p
2E

(12)

Thus neutrino oscillation implies that there must be neutrino masses because the probability of
oscillation depends on the difference of their squared masses.
1.4.2

Neutrinos Within Standard Model

In standard model, neutrinos are known to be massless. Rather going to the quantitative treatment
we can explain why neutrinos are massless in standard model qualitatively as follows.
No matter how empty the vacuum looks, it is packed with particles called Higgs bosons that
have zero spin (and are therefore neither left or right handed). According to Higgs mechanism
in standard model, particles in the vacuum acquire their masses as they collide with the Higgs
boson. Quantum field theory and Lorentz invariance show that when a particle is injected into the

I,
its handedness changes when it interacts with a Higgs boson (figure-1).
AIJvacuum
A
For, example photons () are massless because they do not interact with the Higgs boson. But
left-handed electron will become right-handed after the first collision, then left-handed electron
will become right-handed after the first collision, then left-handed following a second collision,
and so on. Simply since the electron cannot travel through the vacuum at the speed of light; it has
to become massive. Similarly muons collide with Higgs bosons more frequently than electrons,
making them 200 times heavier than the electron, while the top quark interacts with the Higgs
boson almost all the time.
This picture also explains why neutrinos are massless. If a left-handed neutrino tried to collide
with the Higgs boson, it would have to become right-handed. But experiments show that neutrinos
are always left handed. Since right handed neutrinos do not exist in Standard Model, the theory
predicts that neutrinos can never acquire mass. In this way right- handed neutrino go hand in
hand with the absence of right-handed neutrinos in Standard Model.
To explain the masses of neutrinos we have to go beyond the Standard Model. One such beyond
standard model, theoretical framework is See-saw mechanism.
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Project Report May 2016 Tezpur University

1.4.3

Neutrinos Beyond Standard Model: "See-saw Mechanism"

Basically there are two ways to extend the Standard Model in order to make neutrinos massive.One
approach involves new particles called Dirac neutrinos, while the other approach involves a
completely different type of particle called the Majorana neutrino. The Dirac neutrino is a simple
idea with a serious flaw. According to this approach, the reason that right-handed neutrinos
have escaped detection so far is that their interactions are at least 26 orders of magnitude weaker
than ordinary neutrinos.The idea of the Dirac neutrino works in the sense that we can generate
neutrino masses via the Higgs mechanism (figure 1). However, it also suggests that neutrinos
should have similar masses to the other particles in the Standard Model.To avoid this problem,we
have to make the strength of neutrino interactions with the Higgs boson at least 1012 times weaker
than that of the top quark.
The second way to extend the Standard Model involves particles that are called Majorana
neutrinos. Massive neutrinos sit naturally within this framework. Earlier we argued that the
absence of right-handed neutrinos means that neutrinos are massless. But if neutrinos and
antineutrinos are the same particle, then we do not neeed such kind of urgument anymore.
So how is neutrino mass generated? In this scheme, it is possible for right-handed neutrinos to
have a mass of their own without relying on the Higgs boson. Unlike other quarks and leptons,
the mass of the right-handed neutrino,M, is not tied to the mass scale of the Higgs boson. Rather,
it can be much heavier than other particles.
When a left-handed neutrino collides with the Higgs boson, it acquires a mass, m, which
is comparable to the mass of other quarks and leptons. At the same time it transforms into a
right-handed neutrino, which is much heavier than energy conservation would normally allow
.However, the Heisenberg uncertainty principle allows this state to exist for a short time interval
t, given by ..........................................., after which the particle transforms back into a left-handed
neutrino with mass m by colliding with the Higgs boson again. Put simply, we can think of the
neutrino as having an average mass of m2 / M over time.
This so-called seesaw mechanism can naturally give rise to light neutrinos with normal-strength
interactions. Normally we would worry that neutrinos with a mass, m, that is similar to the masses
of quarks and leptons would be too heavy.However, we can still obtain light neutrinos if M is
much larger than the typical masses of quarks and leptons. Right-handed neutrinos must therefore
be very heavy, as predicted by grandunified theories that aim to combine electromagnetism with
the strong and weak interactions.

1.4.4

Theory of Type I See-saw Mechanism

The minute we talk about masses of spin 1/2 particles, we need both spin up and down states
because we can stop any massive particle. When the particle is at a relativistic speed, a more
useful label is left or right-handed states.For massless particles, left- and right-handed states are
completely independent from each other and you do not need both of them; this is how neutrinos
are described in the Standard Model. Once they are massive, though, we need both, so that we
can write a mass term called Dirac mass term using both of them:
1
C
D
Lmass
= m D R L + h.c =
(m D R L + m D CR L ) + h.c
2

(13)

But then the mass term is exactly the same as the other quarks and leptons, and why are neutrinos
so much lighter?
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Project Report May 2016 Tezpur University

The first step is to rewrite the mass term Eq. (5) in a matrix form





1
0
mD
L
+h.c.
Lmass =
L R
R
mD
0
2

(14)

Here, we have to put L and R (CP conjugate of #R ) together so that both of them are lefthanded and are allowed to be in the same multiplet. The problem is that we expect the Dirac mass
mD be of the same order of magnitudes as other quarks and lepton masses in the same generations
which would be way too large. The point is that the right-handed neutrino is completely neutral
under the standard-model
gauge groups and is not tied to the electroweak symmetry breaking (v = 246 GeV) to acquire a
mass. Therefore, it can have a mass much larger than the electroweak scale without violating
gauge invariance, and themass term is





1
L
0
mD
Lmass =
+H.c
(15)
L R
mD M
R
2
Because one of the mass eigenvalues is clearly dominated by MmD , while the determinant
m2

is m2D , the other eigenvalue mustbe suppressed, M D mD This way, physics at high-energy
scale M suppresses the neutrino mass in a natural way. In order to obtain the mass scale
2
for the atmospheric neutrino oscillation(m2atm ) 0.05 eVand taking the third generation mass
m2

mD mt 170 GeV we find M = mD# 0.61015 GeV

It is almost the grand-unification scale 21016 where all gauge coupling constants appear to
unify in the minimal supersymmetric standard model.

Project Report May 2016 Tezpur University

2.

2.1.

Chapter 2:

Baryogenesis

Baryon Asymmetry of The Universe

bservations indicate that the number of baryons (protons and neutrons) in the Universe
is unequal to the number of antibaryons (antiprotons and antineutrons). To the best of our
understanding, all the structures that we see in the Universe stars, galaxies, and clusters
consist of matter (baryons and electrons) and there is no antimatter (antibaryons and positrons)
in appreciable quantities. Since various considerations suggest that the Universe has started from
a state with equal numbers of baryons and antibaryons, the observed baryon asymmetry must
have been generated dynamically, a scenario that is known by the name of baryogenesis.

One may wonder why we think that the baryon asymmetry has been dynamically generated,
rather than being an initial condition. There are at least two reasons for that. First, if a baryon
asymmetry had been an initial condition, it would have been a highly fine-tuned one. For every
6,000,000 antiquarks, there should have been 6,000,001 quarks. Such a fine-tuned condition seems
very implausible. Second, and perhaps more important, we have excellent reasons, based on
observed features of the cosmic microwave background radiation, to think that inflation took
place during the history of the Universe. Any primordial baryon asymmetry would have been
exponentially diluted away by the required amount of inflation.
The baryon asymmetry of the Universe poses a puzzle in particle physics. The Standard Model
(SM) of particle interactions contains all the ingredients that are necessary to dynamically generate
such an asymmetry in an initially baryon-symmetric Universe. Yet, it fails to explain an asymmetry
as large as the one observed. New physics is called for. The new physics must first, distinguish
matter from antimatter in a more pronounced way than do the weak interactions of the SM.
Second, it should depart from thermal equilibrium during the history of the Universe.
The value of baryon asymmetry of the Universe is inferred from observations in two independent
ways.
The first way is via Big Bang Nucleosynthesis (BBN). One of the main successes of the
standard early-universe cosmology is the prediction of the abundances of the light elements,
D, He3 ,He4 andLi7 .Agreement between theory and observation is obtained for a certain
range of the parameter B , the ratio of baryon density and photon density,

BBBN = B = (2.6 6.21) 1010 ,

Where the photon number density of photons is .400cm3 . Since no significant amount
of antimatter is observed in the universe, the baryon density yields directly the cosmological

baryon asymmetry, = B B .
The second piece of evidence is from Cosmic Microwave Background Radiation (CMBR) .
On the experimental side, the precision of measurement of the baryon asymmetry has significantly improved with the observation of the acoustic peaks in the CMBR. The BOOMERanG
and DASI experiments have measured the baryon asymmetry as
B
= 61010

The most recent measurement of the WMAP Collaboration is consistent with this resut, with
an error of only 5%.
BCMB =

Project Report May 2016 Tezpur University

2.2.

Ingredient and Mechanism

2.2.1

Shakharavs conditions

The three ingredients required to dynamically generate a baryon asymmetry were given by
Sakharov:
(1) Baryon number violation: This condition is required in order to evolve from an initial state
with YB = 0 to a state with YB = 0.
(2) C and CP violation: If either C or CP were conserved, then processes involving baryons
would proceed at precisely the same rate as the C- or CP-conjugate processes involving
antibaryons, with the overall effects that no baryon asymmetry is generated.
(3) Out of equilibrium dynamics: In chemical equilibrium, there are no asymmetries in quantum
numbers that are not conserved (such as B, by the first condition.
These ingredients are all present in the Standard Model. However, no SM mechanism generating a large enough baryon asymmetry has been found. Therefore baryogenesis requires new
physics that extends the SM in at least two ways: It must introduce new sources of CP violation
and it must either provide a departure from thermal equilibrium in addition to the electroweak
phase transition (EWPT) or modify the EWPT itself. Some possible new physics mechanisms for
baryogenesis are the following:
GUT baryogenesis it generates the baryon asymmetry in the out-of-equilibrium decays of
heavy bosons in Grand Unified Theories (GUTs).
Leptogenesis from decay heavy right handed neutrinos by producing BL asymmetry above
TEW .
Electroweak baryogenesis by producing B=L at TEW , where the departure from thermal
equilibrium is provided by the electroweak phase transition
2.2.2

Baryogenesis via Leptogenesis

Leptogenesis is one of the most well motivated framework of producing baryon asymmetry of
the universe which creates an asymmetry in leptonic sector first and then converts it into baryon
assymmetry through B + L violating electroweak transitions.As pointed out first by Fukugita and
Yanagida , the out of equilibrium CP violating decay of heavy Majorana neutrinos provides a
natural way to create the required lepton asymmetry.This decay of right handed neutrinos obeys
the theee Sakharavs conditions because
Violation of B + L Guaranteed if neutrinos are Majorana particles. i.e. Lepton number is
violated(M)
C and CP violation. Guaranteed if the neutrino Yukawa couplings contain physical phases.
i.e. New sources of CP violation()
Departure from thermal equilibrium. Guaranteed, due to the expansion of the Universe.
Although the standard model satisfies the first two requirements and out of equilibrium
conditions in principle, can be achieved in an expanding Universe like ours, it turns out that the
amount of CP violation measured in the SM quark sector is too small to account for the entire
baryon asymmetry of the Universe.Since there can be more sources of CP violating phases in
the leptonic sector which are not yet from experimentally determined, leptogenesis provides
an indirect way of constrainingthese unknown phases from the requirement of producing the
observed baryon asymmetry. The salient feature of this mechanism is the way it relates two of the
most widely studied problems in particle physics: the origin of neutrino mass and the origin of
matter-antimatter asymmetry.
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Project Report May 2016 Tezpur University

Figure 1: The CP asymmetry in type-I seesaw leptogenesis results from the interference between tree and 1-loop wave
and vertex diagrams. For the 1-loop wave diagram, there is an additional contribution from L-conserving
diagram to the CP asymmetry which vanishes when summing over lepton avours.

Before going to complicated mathematical discussions of Leptogenesis, let us look at the basic
picture behind the theory of leptogenesis. Till now we have learnt that there could be RH neutrino
that is very heavy. A very heavy particle that could have been produced in early universe is the
connection to the baryogenesis via leptogenesis. Presumably there may be one RH neutrino for
each generation of neutrinos. One of them could be very long lived enough and once day produce
they hanging around the universe for a while, then eventually decay. But that will happen in
out of equilibrium condition and because of this right handed neutrinos turn out to be Majorana
fermions, which means that particles and antiparticles have no distinctions, when it decays it
will have 50:50 probability to go into ordinary lepton (R L + H) and ordinary antilepton
(R R + H ). That is given in the tree level lowest order diagram.
But if we look at the interference between the tree level lowest order diagram and higher order
diagram at one loop level we can find a CP violation. That is decay into ordinary lepton may be
supressed and decay into ordinary antilepton may be enhanced. So when these all decay we may
end up with the net ve lepton asymmetry.

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Project Report May 2016 Tezpur University

3.

Chapter 3: Mathodology

n a model with type I seesaw mechanism at work, the CP violating out of equilibrium decay of
the lightest right handed neutrino can give rise to the required lepton asymmetry. The neutrino
mass matrix in type I seesaw mechanism can be written as

1 T
M = m LR MRR
m LR

(16)

where m LR is the Dirac neutrino mass matrix and MRR is the right handed singlet neutrino mass
matrix. The Pontecorvo-Maki-Nakagawa-Sakata (PMNS) leptonic mixing matrix is related to the
diagonalizing matrices of neutrino and charged lepton mass matrices U , U l respectively, as
UPMNS = Ul U

(17)

In the diagonal charged lepton basis, UPMNS is same as the diagonalizing matrix U of the
neutrino mass matrix given by (. . . . . . . . . ..). The PMNS mixing matrix can be parametrized as

c12 c13
UPMNS s12 c23 c12 s23 s13 ei
s12 s23 c12 c23 s13 ei

s12 c13
c12 c23 s12 s23 s13 ei
c12 s23 s12 cc3 s13 ei

s13 ei
s23 c13 diag(1, ei , ei(+) )
c23 c13

(18)

Where,
cij =cosij , sij =sinij ,
is the Dirac CP phase. and, , are the Majorana phases.
In our work we are considering CP-violating out of equilibrium decay of heavy right handed
neutrinos into higgs and lepton within the framework type I seesaw mechanism.The lepton
asymmetry from the decay of right handed neutrino into leptons and Higgs scalar is given by-

NK =
i

(NK Li + H ) (NK Li + H)
( NK Li + H ) + (NK Li + H)

(19)

In a hierarchical pattern of three right handed neutrinos M2,3  M1, it is sufficient to consider
the lepton asymmetry produced by the decay of the lightest right handed neutrino N1. Now
the lepton asymmetry arising from the decay of N1 in the presence of type I seesaw only can be
written as-

1 =

1
1


2 m m
8v
LR LR 11





Im[(mLR )1 mLR m LR (m LR )j ]g x j +

j=2,3

1
1


8v2 mLR m LR 11

j=2,3

1j





Im[(mLR )1 mLR m LR (m LR )j ]g x j

(20)

j1

where v = 174 GeV is the vev of the Higgs bidoublets responsible for breaking the electroweak
symmetry,



1
1+x
g (x) = x 1 +
(1 + x)ln
(21)
1x
x
And, x j =

M2j
M12

. The second term in the expression for 1 above vanishes when

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Project Report May 2016 Tezpur University

summed over all the flavors = e, , . The sum over flavors is given by-

1 =

1
1


8v2 mLR m LR 11

2



Im[ mLR m LR ]g x j

j=2,3

ij

(22)

After determining the lepton asymmetry 1 , the corresponding baryon asymmetry can be obtained
by

YB = ck
g
Here, c is a measure of the fraction of lepton asymmetry being converted into baryon asymmetry
and is approximately equal to -0.55k is the dilution factor due to wash-out processes which erase
the produced asymmetry and can be parametrized as
"
#

4
k
, f or K 106
(23)
= 0.1Kexp
3 (0.1K )0.25

0.3
K (lnK )0.6

,
=
2 K2 + 9

f or 10 K 106
f or 0K10

(24)
(25)

Where K is given as


mLR m LR 11 M1
MPL
1
K=
=

2
H(T = M1 )
1.66 g M12
8v

(26)

For the calculation of baryon asymmetry, we first calculate the right handed singlet neutrino mass
matrix MRR as
UR MRR UR = diag(M1 , M2 , M3 )
(27)
In this diagonal M_RR basis, according to the type I seesaw formula , the Dirac neutrino mass
matrix also changes to
m LR = moLR UR
(28)
where, moLR is the Dirac neutrino mass matrix.

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4.

Chapter 4: Numerical Analysis

Using the parametric form of PMNS matrix shown in (. . . . . . ), the Majorana neutrino masscmatrix
M can be found as
Diag T
M = UPMNS M UPMNS
Where,

Diag

m1
= 0
0

0
m2
0

0
0
m3

where m1,m2 and m3 are the three neutrino mass eigenvalues. As mentioned earlier, here we
assume that the diagonalizing matrix of the neutrino mass matrix is M same as the PMNS mixing
matrix due to the chosen charged lepton mass matrix in the diagonal form.
For Normal Hierachy:
q
q
MDiag = diag(m1 , m21 + m221 , m21 + m231 )
For Inverted Hierachy:
MDiag = diag(

m23 + m223 m221 ,

m23 + m223 , m3 )

For illustrative purposes, we consider two different order of magnitude values for the lightest
neutrino mass m1 for NH and m3 for IH.In the first case we assume mlightest as large as possible so
that the sum of the absolute neutrino mass lie just below the cosmological upper bound and it
turns out to be 0.07 eV and 0.065 for NH and IH respectively.
Secondly we choose the lightest mass eigenvalue to be 106 for both NH and IH cases so that we
have a hierarchical pattern of neutrino masses.
The PMNS matrix is evaluated from the best fit values of the neutriono mixing angles given in the
Table
After using the best fit values of two mass squared differences and three mixing angles, the most
general neutrino mass matrix given by equation (. . . ) contain four parameters:
1. The lightest neutrino mass(m1 or m3 )
2. Dirac CP phase,
3. Majaran phase and
4. Majorana phase
Our objective is to calculate the variation of the Baryogenesis with respect to the variation of the
above four parameters, m1 or m3 , , and separately.

4.1.

Calculation For Normal Hierachy

0.823347

UPMNS = 0.385239 0.0886364ei

0.398092 0.0857747ei

ei
ei

0.54786ei


0.578954 0.0589792ei

0.598269 0.057075ei

i
0.148155e 2 +i(+)

0.710682ei(+)
i(+)
0.687737e

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Project Report May 2016 Tezpur University

Varying Dirac Phase Delta ()

= 0- 2
= 2 , = /2, m1 =0.001
In the calculation we have taken = 0- 2 (in Radian 0 - 6.28) with step size 0.001 for 6281
number of values of using Mathematica. Here we will show the calculation for only one particular value of 2 )

UPMNS

0. + 0.148155i
0.710682 + 0.i
0.687737 + 0.i

(29)

0.385239 0.0886364i 0.398092 0.0857747i

0.0589792 + 0.578954i
0.057075 0.598269i
0.710682 + 0.i
0.687737 + 0.i

(30)

0.823347 + 0.i
= 0.385239 0.0886364i
0.398092 0.0857747i

U T PMNS

0.823347 + 0.i
= 0. + 0.54786i
0. + 0.148155i

0. + 0.54786i
0.0589792 + 0.578954i
0.057075 0.598269i

The general form of MLR is given bym

=
0
0

MLR

0
n
0

0
0 mf
1

(31)

To calculate the baryon asymmetry in the appropriate flavor regime, we choose the diagonal
Dirac neutrino mass matrix in such a way that the lightest right handed singlet neutrino
mass lies in the same flavor regime.we choose mf = 82.43 GeV in the Dirac neutrino mass matrix
We also choose (m, n) = (1, 1) to keep the lightest right handed neutrino mass in one flavor regime.

MLR

0.22
= 0
0

1.81346 1010

0.
0.

0
0
0.22 0 82.43 109
0
1
0.
1.81346 1010
0.

0.

0.
10
8.243 10

(32)

(33)

Now,
MRR =MLR .UPMNS .M diag U T PMNS .MLR
2.1164 1023 + 0.i
= 1.1616 1023 2.31985 1022 i
5.45614 1023 1.02043 1023 i

0.456028 + 0.14776i
UR = 0.141544 0.0912156i
0.861307 + 0.i
14

1.1616 1023 2.31985 1022 i

5.45614 1023 1.02043 1023 i
3.57766 1022 + 2.50101 1022 i
1.72671 1023 3.73158 1021 i
23
21
1.72671 10 3.73158 10 i 7.90712 1023 5.16737 1023 i

0.869439 + 0.i
0.162591 0.13429i
0.419392 0.154004i

0.0705614 + 0.19253i

0.952037 + 0.i
0.226843 + 0.010992i

(34)

Project Report May 2016 Tezpur University

MLR
= M0 LR .UR
3.37995 1020 + 0.i

= 1.26309 1018 1.05478 1020 i

2.13084 1019 4.56842 1020 i

1.26309 1018 + 1.05478 1020 i 2.13084 1019 + 4.56842 1020 i

3.22023 1020 + 0.i
6.95725 1019 1.14652 1019 i
19
19
6.95725 10 + 1.14652 10 i
6.74739 1021 + 0.i

The eigenvalues of MRR are

{1.21483 1024 , 1.00774 1023 , 7.98593 1021 }

(35)

x2 = M2 2 / M1 2 = 159.237

(36)

x3 = M3 2 / M1 2 = 23140.9.

(37)

And,


1
1+x
g (x) = x 1 +
(1 + x)ln
1x
x



1 + x2
1
g ( x2 ) = x2 1 +
(1 + x2 )ln
= 0.119287
1 x2
x2



1
1 + x3
g ( x3 ) = x3 1 +
(1 + x3 )ln
= 0.00986079
1 x3
x3

(38)
(39)
(40)

Now we calculate the lepton asymmetry for Type I seesaw,

1 =

1
1


8v2 mLR m LR 11





Im[(mLR )1 mLR m LR (m LR )j ]g x j +
1j

j=2,3

1
1


8v2 mLR m LR 11





Im[(mLR )1 mLR m LR (m LR )j ]g x j

j=2,3

(41)

j1

1 = 3.11625 107

(42)

mLR m LR 11 M1
1
MPL
K=
=
= 19.4998

2
H(T = M1 )
1.66 g M12
8v

(43)

Where, Plank mass,MPL = 1.22 1019 Gev

Since, for 10 K 106
0.3
k =
= 0.00800575
=
K (lnK )0.6

(44)

Now we calculate the baryon asymmetry of the universe

YB = ck

= 1.24739 1011
g

where, g = 110, c=-0.55

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Project Report May 2016 Tezpur University

4.1.1

Analysis of Baryon Asymmetry Varying Parameters , m1 , and

A.For Normal Hierarchical(m3  m1 , m2 )

[1]Varying the Dirac phase,
Here, we have taken = 0- 2 (in Radian 0 - 6.28) with step size 0.001 and calculated the Baryon
Asymmetry YB for 6281 number of values of using Mathematica and plot results for Baryon
asymmetry YB vs Dirac phase, .

Figure 2: Variation of Baryon Asymmetry with respect to Dirac phase for Normal Hierarchy

[2]Varying the Lightest Neutrino Mass, m1

Here, we have taken the lower limit for the lightest neutrino mass to be 106 eV and upper limit
to be 0.07eV. Now taking the step size 0.0001 we have calculated the Baryon Asymmetry YB for
7000 number of values of m1 using Mathematica and plot the results for Baryon asymmetry YB vs
Lightest neutrino mass m1 .

Figure 3: Variation of Baryon Asymmetry with respect to lightest Neutrino Mass For Normal Hierarchy

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Project Report May 2016 Tezpur University

[3]Varying the Majorana Phase ,

Here, we have taken = 0- 2 (in Radian 0 - 6.28) with step size 0.001 and calculated the Baryon
Asymmetry YB for 6281 number of values of using Mathematica and plot results for Baryon
asymmetry YB vs Majoran Phase().

Figure 4: Variation of Baryon Asymmetry with respect to Majorana Phase For Normal Hierarchy

[4]Varying the Majorana Phase ,

Here, we have taken = 0- 2 (in Radian 0 - 6.28) with step size 0.001 and calculated the Baryon
Asymmetry YB for 6281 number of values of using Mathematica and plot results for Baryon
asymmetry YB vs Majoran Phase().

Figure 5: Variation of Baryon Asymmetry with respect to Majorana Phase For Normal Hierarchy

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Project Report May 2016 Tezpur University

B. For Inverted Hierarchical(m1  m2 , m3 )

[1]Varying the Dirac phase,
Here, we have taken = 0- 2 (in Radian 0 - 6.28) with step size 0.001 and calculated the Baryon
Asymmetry YB for 6281 number of values of using Mathematica and plot results for Baryon
asymmetry YB vs Dirac phase, .

Figure 6: Variation of Baryon Asymmetry with respect to Dirac phase for Inverted Hierarchy

[2]Varying the Lightest Neutrino Mass, m3

Here, we have taken the lower limit for the lightest neutrino mass to be 106 eV and upper limit to
be 0.065eV. Now taking the step size 0.0001 we have calculated the Baryon Asymmetry YB for 6500
number of values of m3 and plot the results for Baryon asymmetry YB vs Lightest neutrino mass
m3 .

Figure 7: Variation of Baryon Asymmetry with respect to lightest Neutrino Mass For Inverted Hierarchy

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Project Report May 2016 Tezpur University

[3]Varying the Majorana Phase ,

Here, we have taken = 0- 2 (in Radian 0 - 6.28) with step size 0.001 and calculated the Baryon
Asymmetry YB for 6281 number of values of using Mathematica and plot results for Baryon
asymmetry YB vs Majoran Phase().

Figure 8: Variation of Baryon Asymmetry with respect to Majorana Phase For Inverted Hierarchy

[4]Varying the Majorana Phase ,

Here, we have taken = 0- 2 (in Radian 0 - 6.28) with step size 0.001 and calculated the Baryon
Asymmetry YB for 6281 number of values of using Mathematica and plot results for Baryon
asymmetry YB vs Majoran Phase().

Figure 9: Variation of Baryon Asymmetry with respect to Majorana Phase For Inverted Hierarchy

19