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]
(1)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
(2)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
(3)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
(4)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
(5)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
(6)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
(7)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
(8)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
(9)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)
(10)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