inverse seesaw and standard model

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“NEUTRINO MASS (INVERSE SEESAW MECHANISM) and DARK

MATTER ”

A Minor Project Report submitted to the department of physics

at

The End of THIRD Semester

by

Mallika P. Shivam

PHY14002

Under the Supervision of

Dr. Mrinal Kumar Das

Associate Professor,

Department of Physics,

Tezpur University.



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ACKNOWLEDGEMENT

At the very onset, I would like to express my gratitude and thanks to

Dr. Mrinal Kumar Das, Associate Professor, Department of Physics, Tezpur

University for his supervision, encouragement and guidance throughout the

semester.

I would also like to thank my institution, my co-guide Ananya Mukherjee and

Happy Borgohain for their help in every step and last but not the least; I thank

my project partners, Pragyan Phukan and Papori Seal for their support and

enthusiasm.



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CERTIFICATE

This is to certify that Mallika P. Shivam , bearing Roll no: PHY14002 , a

student of Third semester of the master in science (M.Sc) programme in

Physics , Tezpur University has undertaken the project entitled ― Neutrino mass

( by Inverse Seesaw) and Dark Matter ‖ in the partial fulfilment of the

requirement for the degree of Master of Science in Physics.

Date: Dr. Mrinal Kr Das

Place: Project Supervisor



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ABSTRACT

Neutrinos are the only subatomic particle in the STANDARD MODEL which

fulfils many of the criteria to be a Dark Matter candidate. In the Standard

Model, neutrinos are massless due to the absence of right handed neutrinos, but

many later experiments gave the evidence of neutrino oscillations and proved

that neutrinos too have a tiny mass. Thus we need to go beyond Standard Model

to incorporate its mass.

In the present work, we have first introduced the STANDARD MODEL

through the gauge theories and gauge invariance and then we have proceeded to

Spontaneous Symmetry Breaking(SSB) and Higgs Mechanism by which gauge

bosons and fermions get mass. Then going beyond Standard Model, theneutrino mass is generated by a new mechanism called the Seesaw Mechanism,

which can explain tiny neutrino masses in sub eV scale. There are different

seesaw mechanisms i.e. Type I, Type II, Type III and Inverse Seesaw.

The present work will be focused on Inverse Seesaw mechanism which

incorporates new physics TeV scale and with this mechanism we will try to

establish a bridge between dark matter and neutrino oscillation.



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CONTENT

1. Introduction [ 6 ]

2. Dark Matter [ 6 ]

3. Neutrinos [ 6-7 ]

4. Standard Model

4. (a) Gauge theory — (Abelian and non Abelian) [7-9]

4. (b) SSB and Higgs Mechanism [10-11]

4. (c) Higgs Mechanism in Standard Model [11-14]

5. Beyond Standard Model

5. (a) Type 1 Seesaw Mechanism [14-15]

5. (b) Inverse Seesaw Mechanism [16]

6. Work for the next semester [17]

7. Conclusion [17]

7. References [18]



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1.INTRODUCTION

Particle physicists now believe that they can describe the behaviour of all

known subatomic particles within a single theoretical framework called the

Standard Model. This model incorporates the quarks and leptons as well as

their interactions through the strong, weak and electromagnetic forces. Gravity

remains outside the Standard Model. The basic forces are transmitted between

the quarks and leptons by a third family of particles called gauge bosons. There

are 6 leptons and 6 anti – leptons. There are 6 quark flavours but each quark and

anti-quark comes in three colours, so there are 36 quarks. There are 12

mediators (photon, W+, W

-, Z, gluons (8)). Thus the number of particles in

Standard Model (12 leptons +36 quarks +12 mediators +1 Higgs particle) comes

out to be 61.

2.DARK MATTER

The rotation speeds of outer stars in spiral galaxies are unexpectedly high whichsuggest that a spherical halo of invisible matter must surround each galaxy.

Similarly the motion of individual galaxies in clusters of them implies

gravitational fields about ten times more powerful than visible matter of galaxy

provides. This unknown form of matter accounts for 26.8% of the mass energy

content of the observable universe. Among all Standard Model particles,

neutrino is the only one to fulfil some of the criteria for a Dark Mattercandidate, so neutrinos may be a part of the answer, but only part.

3. NEUTRINOS

Neutrino in the Standard Model —

✓ A neutral lepton

✓ Massless Spin -1/2

✓ Only left handed neutrinos and right handed anti neutrinos exist.



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They are the least understood and most elusive elementary particle known to

exist. The Standard Model of particle physics can describe everything we know

about elementary particles. It says that neutrinos do not have mass because they

are all "left-handed" and cannot have a Dirac mass term. Later convincing

evidence was reported that neutrinos have oscillation among its flavours and if

neutrino oscillation exists, there must be a mass for it, as obvious from the

following equations —

Neutrino Oscillation arises from a mixture between the flavour and mass

eigenstates of neutrinos.

Two State mixing is

=

The two state probability oscillation formula –

Therefore | | must be non-zero if neutrino oscillation exists.

4. STANDARD MODEL (SM)

4.(a)GAUGE THEORIES

In particle physics, when we talk about conservation of electric charge, colour,

lepton number etc. there must be an internal symmetry according to Noether’s

theorem and this symmetry is nothing but Gauge symmetry or local Gauge

symmetry depending upon space and time. The present belief is that all particle

interaction may be dictated by the so called local Gauge symmetry. A Gauge

symmetry is a kind of phase transformation under which the lagrangian doesn't

change.



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Abelian Gauge Theory

The Lagrangian for a free electron field is [ ]

Clearly it has a global symmetry corresponding to the invariance under a phase

change.

Now, considering local symmetry Ψ(x) =

Now,

Which is not gauge invariant and hence we define a Gauge covariant derivative

Where,

The new Lagrangian is ( )

We add one kinetic energy term

Where Therefore the final lagrangian is

( )

The following features of the equation are

The photon is massless as the term is not Gauge invariant.

The Lagrangian does not have a gauge field self-coupling.



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NON ABELIAN GAUGE FIELD

If the fermion field is an isospin doublet then

Ψ=

Under SU(2) ,

U(

=

Here we define vector gauge field as

For an infinitesimal change

U(

And

The gauge field transform as

Here, ( )

And

The complete Gauge invariant Lagrangian is therefore

But we again got massless bosons because there is no mass term.



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4.(B) Spontaneous Symmetry Breaking and Higgs Mechanism

We saw in the previous section that the imposition of local symmetry implies

the existence of massless vector particles. If we want to avoid this feature, but

obtain massive particles while preserving gauge invariance, we have to

implement something called Spontaneous Symmetry Breaking (SSB).

The idea behind SSB is that the system obeys some symmetry but the ground

state doesn't. The non zero values of ground state energy breaks the symmetry.

This spoils the usual symmetry consequences of energy level degeneracies. But

according to Goldstone theorem this would imply the existence of a set of

massless scalar bosons. We will see here how the particles get mass.

This phenomenon is known as Higgs Mechanism.

Abelian Case

We consider the simple case of abelian U(1) Gauge theory

There will be two cases But since we want to generate the mass we are interested in

We shift the origin to

w if we expand the Lagrangian in terms of and ξ.

Then we can write

√ (x))

And.

……………….. + Other terms

Now we have a massive scalar field and more crucially a massive for what

we were searching for. But the Lagrangian has a kinetic energy term

but there



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is no mass term for it. But according to goldstone theorem it will contain

massless scalar bosons.

We can solve this problem by choosing two particular gauges and these are

√ and

So the Lagrangian becomes

=

So in the above expression the term

disappears. This is called Higgs

mechanism and thus we see

Massless vector boson + Goldstone boson = Massive Vector Boson

4.(C)Higgs Mechanism In Standard model (Generation of fermion and

Gauge boson masses)

The symmetry we use here is the SU(2)U(1) Gauge symmetry.

Spontaneous symmetry breaking makes SU(2)

U(1)

From SU(2), we get 3 gauge bosons and from U(1) we get one Gauge Boson,

Higgs mechanism gives mass to 3 of the 4 Gauge bosons.

The Higgs field is now assigned a SU(2) doublet

Φ=

Under SU(2)

U(1) local Gauge transformation

Where,

Y= weak hyper charge Now,

()



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Where the coupling constants of

Where,

= Generator of SU(2)

A simple and useful form of the Higgs field is Φ=

To generate masses we need to give a fluctuation to Φ=

We do in steps, first we don't take the fluctuation and generate the gauge boson

masses as follows

= (-ig- Y)

g = g

= ga

Y=

a

Therefore,

= -i

( )

Where, and



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We generated the masses of 3 bosons which are , Z.

field is orthogonal to Z

Where, ,

Thus SU(2) and U(1) mixes in a particular angle called Weinberg angle which

give rise to Z and field

=

By using Φ= in the second and third terms of

()

We get the mass of the Higgs boson as

Thus the three Goldstone boson are eaten by 3 gauge fields to become massive

which are , Z

And we still have one real scalar field left which is Higgs boson.

For Fermion masses we consider the interaction Lagrangian

= -(

Where is the Yukawa coupling and it relates how strongly the Higgs field

couples with leptonic and gauge field.

Φ= √

Φ = √



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Similarly,

=

√

= -[( √ ) + ( +√ ) ]

= -( - √ (

Thus electron acquire a mass m =

The second term is the electron Higgs vertex.

5.Beyond Standard model

5.(a)Type 1 seesaw

In the standard model, the matter fermions and the weak gauge bosons get their

masses from spontaneous breaking of weak gauge symmetry. Therefore all the

masses are limited by the symmetry breaking scale of 100 GeV. But the

neutrinos have no mass in the standard model because there is no right handed

neutrino. However from neutrino oscillation experiment we know that neutrino

has tiny non zero mass, more than billion time smaller than other fermion

masses.

So it raises the question why the observed neutrino mass is so small and a

simple explanation of this comes from the seesaw model. It goes a step beyond

standard model and assumes that besides the usual left handed neutrinos there

are also right handed neutrinos.

Therefore one can construct a Dirac mass term for neutrinos is

=

= ( ) + h.c



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Since neutrinos have non zero electric charge, Majorana mass terms are also

possible and the majorana mass is much larger than SM symmetry breaking

scale ie. M

Therefore

=

The left hand Majorana term is forbidden by SM Gauge symmetry ie.

and therefore the above mass matrix becomes

And after diagonalizing the matrix the following mass eigen states are obtained.

And

Or

So large Majorana masses of the right handed neutrino is responsible for

pushing down the left handed neutrino more than a billion times smaller than

other fermion masses. So this mechanism is known as SEESAW mechanism as

LH majorana masses are suppressed by the heavy scale .



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5.(b) Inverse Seesaw Model (ISS)

In spite of explaining the smallness of neutrino mass, such Type 1 Seesaw

mechanisms are not phenomenologically testable because the new Physics

engendered by them will manifest at 1014 GeV scale which is completely out of

the range of the current accelerator experiment.

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 be probed at LHC experiment. The implementation of

ISS mechanism requires the addition of three right handed neutrinos and the

three extra SM gauge singlet neutral fermions S to the three active neutrinos.

After SSB the overall neutrino mass terms turn out to be

=

Where µ is the mass of the neutrino singlet, also neutrino singlet has no Yukawa

coupling to left handed neutrino but couple to .

A diagonalisation of the above 9 matrix leads to the effective light neutrino

mass matrix i.e.

Or, =

Thus we see that Standard neutrinos with mass at sub ev scale are obtained for at electroweak scale and at Tev scale. The core of the ISS is that the

smallness of the neutrino masses are guaranteed by assuming that scale is

small and in order to bring the RH neutrinos at TeV scale, it has to be at KeV

scale. ISS is also called DOUBLE SEESAW because as seen from the above

equation is doubly suppressed by .



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6. Work for Next Semester

1. Study of Neutrino mass and Mixing in ISS.

2. Connection between Neutrino Mass and Dark Matter with flavour symmetry.

7.Conclusion

Thus I made a detailed study of the SM and then the Seesaw Mechanism and

found the essence of the controversial neutrino mass problem. There are several

dark matter candidates; one of them is the neutrino. If some discrete symmetry

forbids the Yukawa coupling relating to left handed and right handed neutrinos,

there could be a second Higgs doublet scalar which does not acquire any VEV

(Vacuum Expectation Value) or interact with the charged fermions and remain

inert. The lightest of this inert particle may be a dark matter candidate.....the

details of which will be studied and analysed by me in the next semester.



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\8.References

1.

Quarks and leptons by Halzen and Martin.

2.

Gauge Theory of elementary Particles by Chang & Li.

3.

Particle and astro-particle Physics by Utpal sarkar.

4.

Introduction to the standard model (PHYS4675) Lecture notes by

Lawrence Gibbons.

5. Elementary Gauge symmetry Moriyashu.

6.

Principle of relativistic and non relativistic quantum mechanics by K.D

Krori.

7.

Introduction to Particle Physics by Griffith.

8. Modern Elementary Particle Physics by Gordon Kane

9. Minimalistic dark matter extension of the Standard Model by Oliver

Fischer

10. Neutrino Mass Model by S F King

11.Trinification,the Hierarchy Problem and Inverse Seesaw Neutrino

Masses by Christophe Cauet and Heinrich Pas12.A Simple Realisation Of the Inverse Seesaw Mechanism by Dias and

Pires

13.Sterile Neutrino Dark Matter in Inverse Seesaw Realisations by Michele

Lucente

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