Summary of the PhD thesis: Construction and investigation of a neutrino mass model with A4 flavour symmetry by pertubation method - TRƯỜNG CÁN BỘ QUẢN LÝ GIÁO DỤC THÀNH PHỐ HỒ CHÍ MINH

Standard model have achieved great success in elementary particle physics, but in the model, neutrinos are considered as massless, this is a suggestion for physicists to extend the stand[r]

MINISTRY OF EDUCATION AND TRAINING

VIETNAM ACADEMY

OF SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY SCIENCE AND TECHNOLOGY

…… ….***…………

PHI QUANG VAN

CONSTRUCTION AND INVESTIGATION OF A

NEUTRINO MASS MODEL WITH A4 FLAVOUR SYMMETRY

BY PERTUBATION METHOD

                                                                                   Speciality: Theoretical and mathematical physics

Code: 62 44 01 03

SUMMARY OF THE PHD THESIS

This thesis was compled at the Graduate University Science and Technology, Viet Nam Academy of Science and Technology

Supervisors: Assoc Prof Dr Nguyen Anh Ky

Institute of Physics, Viet Nam Academy of S cience and Technology

Referee 1: Prof Dr Dang Van Soa

Referee 2: Assoc Prof Dr Nguyen Ai Viet Referee 3: Dr Tran Minh Hieu

This dissertation will be defended in front of the evaluating assembly at academy level, Place of defending: meeting room, Graduate

University Science and Technology, Viet Nam Academy of Science and Technology

This thesis can be studied at: - The Vietnam National Library

Introduction

Motivation of thesis topic

Neutrino masses and oscillations are always a challenge in elementary particle physics We have seen in the standard model (SM) that neutrinos not have mass, but the experiment has shown that neutrinos have mass The problem of neutrino masses and mixings is among the problems beyond the SM This problem is important for not only particle physics but also nuclear physics, astrophysics and cosmology, therefore, it has attracted much interest At present, there are many standard model extensions to studying neutrino masses and oscillations: the supersymmetry model, the grand unified theory, the left-right symmetry model, the 3-3-1 model, the mirror symmetry model, Zee model, Zee-Babu model, the flavour symmetry model, etc

One of the standard model extentions to explain neutrino mass is to add a flavor

symmetry to the SM symmetry, such as SU(3)C × SU(2)L ×U(1)Y ×GF, in which

GF is a flavour symmetry group, for example S3, S4, A4, A5, T7,∆(27)[1], A popular

flavour symmetry intensively investigated in the literature is that described by the

groupA4 (see, for instance, [2,3]) allowing obtaining a tribi-maximal (TBM) neutrino

mixing corresponding to the mixing angles θ12 ≈ 35.26◦ (sin2θ12 = 1/3), θ13 = 0◦ and

θ23 = 45◦ The recent experimental data that showing a non-zero mixing angle θ13

and a possible non-zero Dirac CP-violation (CPV) phase δCP, rejects, however, the

TBM scheme [4,5] There have been many attempts to explain these experimental

phenomena In particular, for this purpose, various models with a discrete flavour

symmetry [1], including anA4 flavor symmetry, have been suggested [1–3]

The objectives of the thesis

The thesis is devoted to constructing and evaluating versions of the SM extended with an A4 symmetry to explain some of the problems of neutrino physics Within these extended models, the results on neutrino masses and mixing, derived by the

perturbation method, are very closed to the global fit [4,5] In two models, a relation

between the Dirac CPV phase and the mixing angles is established In particular, the

models predict Dirac CPV phase δCP and effective mass of neutrinoless double beta

decay|hmeei|in good agreement with the current experimental data

Introduction

The main contents of the thesis

In general, the models, based onA4flavour symmetry, have extended lepton and scalar

sectors containing new fields in additions to the SM ones which now may have anA4

symmetry structure Therefore, base on anA4 flavour symmetry, these fields may also

transform underA4 At the beginning, the A4 based models were build to describe a

TBM neutrino mixing (see, for example, [2]) but later many attempts, such as those

in [1,3,6], to find a model fitting the non-TBM phenomenology, have been made On

these models, however, are often imposed some assumptions, for example, the vacuum expectation values (VEV’s) of some of the fields, especially those generating neutrino masses, have a particular alignment These assumptions may lead to a simpler diag-onalization of a mass matrix but restrict the generality of the model Since, according to the current experimental data, the discrepancy of a phenomenological model from a TBM model is quite small, we can think about a perturbation approach to building

a new, realistic, model [7,8]

The perturbative approach has been used by several authors (see for example, [9])

but their methods mostly are model-independent, that is, no model realizing the ex-perimentally established neutrino mixing has been shown On the other hand, most

of theA4-based models are analyzed in a non-perturbative way There are a few cases

such as [10] where the perturbative method is applied but their approach is different

from ours and their analysis, sometimes, is not precise (for example, the conditions

imposed in section IV of [10] are not always possible) Besides that, in many works

done so far, the neutrino mixing has been investigated with a less general vacuum structure of scalar fields

In this thesis we will introduce two versions ofA4flavor symmetric standard model,

which can generate a neutrino mixing, deviating from the TBM scheme slightly, as requested and explained above Since the deviation is small we can use a perturbation method in elaborating such a non-TBM neutrino mixing model The corresponding neutrino mass matrix can be developed perturbatively around a neutrino mass matrix diagonalizable by a TBM mixing matrix As a consequence, a relation between the

Dirac CPV phase δCP and the mixing angles θij, i, j = 1,2,3 (for a three-neutrino

mixing model) are established Based on the experimental data of the mixing angles,

this relation allows us to determine δCP numerically in both normal odering (NO)

and inverse ordering (IO) It is very important as the existence of a Dirac CPV phase

indicates a difference between the probabilities P(νl → νl0) and P(¯νl → ν¯l0), l 6= l0,

of the neutrino- and antineutrino transitions (oscillations) in vacuum νl → νl0 and

¯

νl → ν¯l0, respectively, thus, a CP violation in the neutrino subsector of the lepton

Introduction

knowing δCP we can determine the Jarlskog parameter (JCP) measuring a CP

viola-tion The determination ofδCP andJCP represents an application of the present model

and, in this way, verifies the latter (of course, it is not a complete verification)

Structure of thesis

Chapter presents the basis of the standard model and the problem of neutrino mass

Chapters and are designed for constructing and evaluating the two models A(1)4

andA(10)4 for neutrino masses and mixing Both models are constructed perturbatively

around a TBM model but objects of perturbation are different: vacuums in A14 and

Yukawa coupling coefficients inA104 In each model, physical quantities such as

neu-trino mass, mixing angles θij, δCP, JCP, and the relation between δCP with angle θij

are investigated and calculated Conclusions and discussion of the thesis’s results are presented in the final chapter

Chapter 1

Standard model and neutrino masses problem

1.1 Standard model

1.1.1 Local gauge invariance in Standard model

First, we can consider the free Lagrangian of the fieldψ(x)

L0 =ψ(x) iγλ∂λ−m

ψ(x), (1.1)

To the invariant theory with the SU(2) local gauge transformation ψ0(x) = U(x)ψ(x),

supposeψ(x)interacts with the vector field and has covariance derivative

Dλψ(x) =

∂λ +

2ig ~τ ~Aλ(x)

ψ(x), (1.2)

here, g is a dimensionless constant and Ai

λ(x) is the vector field Then the free

La-grangian becomes

LI =ψ(x) iγλDλ−m

ψ(x), (1.3)

and it will invariant to local gauge transformation

In the electroweak interaction model (GWS) with local gauge groupSU(2)L×U(1)Y,

the derivative ∂λψ(x) must be replaced by the covariant derivative Dλψ(x), where

Dλψ(x)is

Dλ(x) =

∂λ +ig

2~τ ~Aλ(x) +ig

01

2Y Bλ(x)

ψ(x), (1.4)

where Aλ(x) and Bλ(x) are the vector gauge fields of symmetry SU(2)L and U(1)Y, g

Standard model Standard model and neutrino masses problem

1.1.2 Spontaneous Symmetry Breaking Higgs Mechanism

After spontaneous symmetry breaking, the mass Lagrangian terms of W, Z and H

have the form

Lm =m2

WW

†

λW

λ+

2m

2

ZZλZλ−

1 2m

2

HH

2, (1.5)

here

m2W = 4g

2v2, m2

Z =

1 4(g

2+g02)v2, m2

H = 2λv2 = 2µ2 (1.6)

In summary, in the model after the spontaneous symmetry break, the vector bosons

W±, Z0 become a mass field, the fieldAλ has no mass

1.1.3 Yukawa interaction and fermion masses

Lagrangian of standard model

LSM =LF +LG+LS+LY, (1.7)

where LF is the kinetic Lagrangian of quark and lepton section, LG is the free

La-grangian of vector fields Bλ and Aiλ, LS is the Lagrangian of Higgs field and LY is

Lagrangian Yukawa interaction of quarks and leptons

FromLY can be obtained

LQ

mass=−U

0

m(U)U0 −D

0

m(D)D0 −L

0

m(lep)L0 (1.8)

We see that after spontaneous symmetry breaks, quarks and leptons become masses

1.1.4 The electroweak interaction current

From the Lagrangian interactive model can be written as interactive current

LI =

− g

2√2J CC

µ W

µ+h.c.

− g

2 cosθW

JµN CZµ−eJµEMAµ, (1.9)

where

JµN C = 2Jµ3−2 sin2θWJµEM, (1.10)

JµEM =

X

i=u,c,t

U

0 iγµU

0 i + −1 X

i=d,s,b

D

0 iγµD

0

i+ (−1)

X

l=e,µ,τ

lγµl (1.11)

Standard model have achieved great success in elementary particle physics, but in the model, neutrinos are considered as massless, this is a suggestion for physicists to extend the standard model to solve neutrino mass problems

Neutrino mass and osillation Standard model and neutrino masses problem

1.2 Neutrino mass and osillation

1.2.1 Dirac-Majorana mass term

We consider a neutrino mass term in the simplest case of two neutrino fields, Dirac and Majorana mass term in this case have the form

− Ldm =

2mLνLν c

L+mDνLνR+

1 2mRν

c

RνR+h.c (1.12)

We can rewrite the expression as a matrix

− Ldm =

2ηLMdm(ηL)

c+h.c., (1.13)

here

ηL=

νL

νRc

!

, andMdm =

mL mD

mD mR

!

(1.14)

The matrixMdmcan be diagonalized by the matrixU and obtained

M ≡ m1

0 m2 !

=UTMdmU, (1.15)

where m1,2 =|

1

2(mR+mL)±

q

(mR−mL)2+ 4m2D |, and U =

cosθ sinθ

−sinθ cosθ

!

, (1.16)

with tan 2θ= 2mD

mR−mL

, cos 2θ= p mR−mL (mR−mL)2+ 4m2D

(1.17)

From (1.13) and (1.15) we have

− Ldm=

2νmν =

X

i=1,2

miνiνi, (1.18)

here νM = U†nL + (U†nL)c =

ν1

ν2 !

, so νic = νi From here we have the mixing

expression

νL = cosθν1L+ sinθν2L, (1.19)

νRc =−sinθν1L+ cosθν2L (1.20)

We can see that the fields of the flavour neutrinos stateν are mixtures of the

Neutrino mass and osillation Standard model and neutrino masses problem

1.2.2 Seesaw mechanism

In the case of two neutrino fields, section 2.1, from (1.16) and conditionmD MR, mL=

0, we obtained the neutrino mass

m1 '

m2D MR

mD, m2 'MR mD (1.21)

From (1.16) we find θ 'mD/MR Thus, we obtain the mixing expression between

the flavour neutrino and the neutrino mass







νL =ν1L+mMD Rν2L

νRc =−mD

MRν1L+ν2L

(1.22)

The factormD/MR is characterized by the ratio of the electroweak scale and the scale

of the violation of thelepton number If we estimate mD ' mt ' 170GeV and m1 '

5.10−2eV, thenMR'm2D/m1 '1015GeV

From the above calculations, we can derive conditions for constructing the

mech-anism of neutrino mass generation, seesaw mechmech-anism, [14]: The left-handed

Majo-rana mass term equal to zero,mL = The Dirac mass termmD is generated by the

standard Higgs mechanism, i.e that mD is of the order of a mass of quark or lepton

The right-handed Majorana mass term breaks conservation of the lepton number, the lepton number is violated at a scale which is much larger than the electroweak scale, mR≡MR mD

1.2.3 Neutrino oscillations

From quantum field theory, states depend on time and satisfy Schrodinger’s equations

[14],

i∂|να(t)i

∂t =H|να(t)i, (1.23)

whereHis total Hamiltonian,α =e, µ, τ Here, we will consider state transformations

in a vacuum, in which caseHis a free Hamiltonian The equation (1.23) has a general

solution

|να(t)i=e−iHt|να(0)i, (1.24)

where,|να(0)iis the state at the initial timet=

From here, the neutrino and antineutrino left-handed states att ≥0are of the form

|να(t)i=e−iHt|ναi=

3 X

i=1

e−iEitU∗

αi|νii, |να(t)i=e−iHt|ναi=

3 X

i=1

e−iEitU

αi|νii (1.25)

With (1.25), we can obtain the amplitude of the transition να → να0 and να → να0 at

time t

Aνα→ν

α0(t) =

3 X

i=1

Uα0 ie

−iEitU∗

αi, Aνα→ν

α0(t) =

3 X

i=1

Uα∗0 ie

−iEitU

αi (1.26)

Neutrino mass and osillation Standard model and neutrino masses problem

In the quantum mechanics, probabilities of the transitions is equal to the squared

amplitude of the transition, thus probabilities of the transitionsνα →να0 andνα →να0

has the form

Pνα→να0(E, L) =δα0α+Bα0α+

1 2A

CP

α0α, Pνα→να0(E, L) =δα0α+Bα0α−

1 2A

CP

α0α, (1.27)

where

Bα0

α=−2

X

i>j

<Uα0

iU

∗

α0jU

∗

αiUαj

1−cos∆m

2 ji

2E L

!

, (1.28)

ACP α0α=

X

i>j

=Uα0iU∗

α0jU

∗

αiUαj

sin∆m

2 ji

2E L (1.29)

From theACP

α0α in the (1.29), we can calculate

ACPα0

α = 16Jsin

∆m212

2E Lsin

∆m223

2E Lsin

(∆m212+ ∆m223)

2E L (1.30)

where∆m2