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

Hanoi – 2017

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

- Library of the National Academy of Public Administration

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 do 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, T 7, ∆(27) [1], A popular flavour symmetry intensively investigated in the literature is that described by the group A4 (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 an A4 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

1

The main contents of the thesis

In general, the models, based on A4flavour symmetry, have extended lepton and scalar sectors containing new fields in additions to the SM ones which now may have an A4

symmetry structure Therefore, base on an A4 flavour symmetry, these fields may also transform under A4 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 the A4-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 of A4flavor 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 → νl 0) and P (¯νl → ¯νl 0), l 6= l0,

of the neutrino- and antineutrino transitions (oscillations) in vacuum νl → νl 0 and

¯l → ¯νl 0, respectively, thus, a CP violation in the neutrino subsector of the lepton sector We should note that for a three-neutrino mixing model, as considered in this paper, the mixing matrix in general has one Dirac- and two Majorana CPV phases

knowing δCP we can determine the Jarlskog parameter (JCP) measuring a CP viola-tion The determination of δCP and JCP 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 1 presents the basis of the standard model and the problem of neutrino mass Chapters 2 and 3 are designed for constructing and evaluating the two models A(1)4 and A(10)4 for neutrino masses and mixing Both models are constructed perturbatively around a TBM model but objects of perturbation are different: vacuums in A1

4 and Yukawa coupling coefficients in A10

4 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

3

Chapter 1

Standard model and neutrino

masses problem

First, we can consider the free Lagrangian of the field ψ(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), suppose ψ(x) interacts with the vector field and has covariance derivative

Dλψ(x) =



∂λ +1

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 group SU (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 + ig1

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 and g0 are the corresponding coupling constants

Standard model Standard model and neutrino masses problem

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

Lm = m2WWλ†Wλ+ 1

2m

2

ZZλZλ− 1

2m

2

HH2, (1.5) here

m2W = 1

4g

2v2, m2Z = 1

4(g

2+ g02)v2, m2H = 2λv2 = 2µ2 (1.6)

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

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

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 Ai

λ, 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

From the Lagrangian interactive model can be written as interactive current

LI =



− g

2√

2J

CC

µ Wµ+ h.c



− g

2 cos θWJ

N C

µ Zµ− eJEM

µ Aµ, (1.9)

where

JµN C = 2Jµ3− 2 sin2θWJµEM, (1.10)

JµEM = 2

3 X

i=u,c,t

U

0

iγµUi0 +



−1 3

 X

i=d,s,b

D

0

iγµD0i+ (−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

5

Neutrino mass and osillation Standard model and neutrino masses problem

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 = 1

2mLνLν

c

L+ mDνLνR+ 1

2mRν

c

RνR+ h.c (1.12)

We can rewrite the expression as a matrix

− Ldm = 1

2ηLMdm(ηL)

c+ h.c., (1.13) here

ηL= νL

νRc

! , and Mdm = mL mD

mD mR

! (1.14) The matrix Mdmcan be diagonalized by the matrix U and obtained

M ≡ m1 0

0 m2

!

= UTMdmU, (1.15) where

m1,2 =| 1

2(mR+ mL) ±

1 2

q (mR− mL)2+ 4m2

D |, and U = cos θ sin θ

− sin θ cos θ

! , (1.16)

with tan 2θ = 2mD

mR− mL

, cos 2θ = mR− mL

p(mR− mL)2+ 4m2

D

(1.17) From (1.13) and (1.15) we have

− Ldm= 1

2νmν =

1 2 X

i=1,2

miνiνi, (1.18)

here νM = U†nL + (U†nL)c = ν1

ν2

! , so νc

i = νi From here we have the mixing expression

νL = cos θν1L+ sin θν2L, (1.19)

νRc = − sin θν1L+ cos θν2L (1.20)

We can see that the fields of the flavour neutrinos state ν are mixtures of the

left-Neutrino mass and osillation Standard model and neutrino masses problem

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

0, we obtained the neutrino mass

m1 ' m

2 D

MR  mD, m2 ' MR  mD (1.21) From (1.16) we find θ ' mD/MR 1 Thus, we obtain the mixing expression between the flavour neutrino and the neutrino mass







νL = ν1L+mD

M Rν2L

νRc = −mD

M Rν1L+ ν2L

(1.22)

The factor mD/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 , then MR' m2

D/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 = 0 The Dirac mass term mD 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

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) whereH is total Hamiltonian, α = e, µ, τ Here, we will consider state transformations

in a vacuum, in which caseH is a free Hamiltonian The equation (1.23) has a general solution

|να(t)i = e−iHt|να(0)i, (1.24) where,|να(0)i is the state at the initial time t = 0

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

|να(t)i = e−iHt|ναi =

3

X

i=1

e−iEi tUαi∗|νii, |να(t)i = e−iHt|ναi =

3

X

i=1

e−iEi tUαi|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−iEi tUαi∗, Aνα→ν

α0(t) =

3

X

i=1

Uα∗0

ie−iEi tUαi (1.26) 7

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 transitions να → να0 and να → να0

has the form

Pνα→ν

α0(E, L) = δα0

α + Bα0

α +1

2 A CP

α0α , Pνα→ν

α0(E, L) = δα0

α + Bα0

α −1

2 A CP

where

Bα0

α = −2X

i>j

<Uα0

i Uα∗0 j Uαi∗U αj



1 − cos∆m

2 ji

!

A CP

α0α = 4X

i>j

=Uα0 i Uα∗0

j Uαi∗U αj

 sin∆m

2 ji

From theACP

α0α in the (1.29), we can calculate

ACPα0

α = 16J sin∆m

2 12

2E L sin

∆m223 2E L sin

(∆m212+ ∆m223) 2E L. (1.30) where ∆m2

ij = m2

j−m2

i and J = −c12c23c2

13s12s23s13sin δ is called the Jarlskog parameter