NEUTRINO MASSES AND MIXING INNGO THI THU DINH University of Natural Science, 334 Nguyen Trai, Thanh Xuan, Hanoi PHUNG VAN DONG Institute of Physics, VAST, 10 Dao Tan, Ba Dinh, Hanoi Abst
Trang 1NEUTRINO MASSES AND MIXING IN
NGO THI THU DINH University of Natural Science, 334 Nguyen Trai, Thanh Xuan, Hanoi
PHUNG VAN DONG Institute of Physics, VAST, 10 Dao Tan, Ba Dinh, Hanoi
Abstract Although the Standard Model is very successful, it also leaves many questions unan-swered, one of which is massless of neutrinos In this talk we introduce A 4 - flavour symmetry into the Standard Model with appropriate extension of scalar representations As a result, the neutrinos gain naturely small masses in agreement with experiment The neutrino mixing matrix
in terms of tribimaximal form is obtained.
I INTRODUCTION The neutrino experiments imply: masses of neutrinos are small, and tribimaximal mixing neutrinos as proposed by Harrison-Perkins-Scott is given by:
UHPS=
2
√ 6 1
√
−√1 6 1
√ 3 1
√ 2
−√1 6 1
√
3 −√1 2
The theories of neutrinos have recently been in trying to explain this form [1, 2, 3,
4, 5, 6, 7, 8, 9, 10, 11]
II THE MODEL II.1 Flavour symmetry A4
The finite group of the even permutation of four objects,A4, has 12 elements and
4 equivalence classes, with the number of elements 1, 4, 4, 3, respectively This means that there are 4 irreducible representations, with dimensions ni, such that Σin2i = 12 There
is only one solution: n1 = n2 = n3 = 1 and n4 = 3, and the character table of the 4 representations is shown in Table below:
Trang 2Table 1 Character table of A 4
Class χ1 χ1 0 χ1 00 χ3 n
C1 1 1 1 3 1
C2 1 ω ω2 0 4
C3 1 ω2 ω 0 4
C4 1 1 1 −1 3 The complex number ω is the cube root of unity, i.e., e2πi/3 Hence 1 + ω + ω2 = 0 Calling the 4 irreducible representations, and 3 respectively, we have the decomposition:
3 ⊗ 3 = 1(11 + 22 + 33) ⊕ 10(11 + ω222 + ω33)
⊕100(11 + ω22 + ω233) ⊕ 3(23, 31, 12) ⊕ 3(32, 13, 21) (2) The non-Abelian finite group A4 is also the symmetry group of the regular tetrahedron II.2 Lepton mass
Under A4, the fermions and scalars of this model transform as follows:
ψL=
νL
eL
∼ (2, −1, 3), (3)
l1R ∼ (1, −2, 1), l2R∼ (1, −2, 10), l3R ∼ (1, −2, 100), (4)
φ =
φ+
φ0
∼ (2, 1, 3) (5) The Yukawa interactions are:
LlepY = h1(ψLφ)1l1R+ h2(ψLφ)10l2R+ h3(ψLφ)100l3R+ H.c (6) The vacuum expectation value (VEV) of φ is (v1, v2, v3) under A4 It is now assumed that so that A4 is broken down to Z3 The mass Lagrangian for the changed leptons is:
Llepmass = h1v(l1L+ l2L+ l3L)l1R
+h2v(l1L+ ωl2L+ ω2l3L)l2R +h3v(l1L+ ω2l2L+ ωl3L)l3R
then the mass matrix is then diagonalized:
UL−1MlepUR=
√ 3h1v 0 0
0 √3h2v 0
0 0 √3h3v
=
me 0 0
0 mµ 0
0 0 mτ
, (8) where
UR= 1, UL= √1
3
1 1 1
1 ω ω2
1 ω2 ω
≡ UlL (9)
Trang 3II.3 Neutrino sector
To obtain arbitrary Majorana neutrino masses, four Higgs doublets are used:
σ =
σ011 σ12+
σ+12 σ22++
∼ (3∗, 2, 1), (10)
s =
s011 s+12
s+12 s++22
∼ (3∗, 2, 3) (11) The Yukawa interactions are:
LνY = x( ¯ψLcψL)1σ + y( ¯ψLcψL)3s + H.c (12) The VEV of σ is:
hσi =
u 0
0 0
The VEV of s is put as:
s = (hs1i, hs2i, hs3i) (14) with
hs2i = hs3i = 0, hs1i =
t 0
0 0
so that it is broken down to Z2 in neutrino sector
The mass Lagrangian for the neutrinos is:
LνY = x(ν1Lc ν1L+ ν2Lc ν2L+ ν3Lc ν3L)u + yν2Lc ν3Lt + H.c (16) The mass matrices are then obtained by
Mν =
xu 0 0
0 xu 12yt
0 12yt xu
= UνL
xu + 12yt 0 0
0 xu 0
0 0 xu − 12yt
UνLT , (17) where
ULν =
0 1 0
1
√
2 0 √1
2 1
√
2 0 −√1
2
The mismatch between Uν and UlL yields the tribimaximal mixing pattern as pro-posed by Harrison- Perkins- Scott:
UHP S= UlL†Uν =
p2/3 1/√3 0
−1/√6 1/√3 −1/√2
−1/√6 1/√3 1/√2
This is a main result of the paper
Trang 4II.4 Scalar Potential
We can separate the general scalar potential into
V = Vφ+ Vσs, (20) where
Vφ = µ2φ(φ+φ)1+ λφ1[(φ+φ)1(φ+φ)1] + λφ2[(φ+φ)1 0(φ+φ)1 00]
+λφ3[(φ+φ)3s(φ+φ)3a] + λφ4[(φ+φ)3s(φ+φ)3s+ H.c.], (21)
Vσs= V (σ) + V (s) + V (σ, s) + V (φ, σ) + V (φ, s) + V (φ, σ, s) + V , (22) with
V (σ) = µ2σT r(σ+σ) + λσT r(σ+σ)2+ λ0σ[T r(σ+σ)]2, (23)
V (s) = T rV (φ → s) + λ01sT r(s+s)1T r(s+s)1+ λ02sT r(s+s)10T r(s+s)100
+λ03sT r(s+s)3sT r(s+s)3a+ λ04s[T r(s+s)3sT r(s+s)3s+ H.c.], (24)
V (σ, s) = λσs1 T r[(σ+s)3(s+σ)3] + λ01σsT r(σ+s)3T r(s+σ)3
+λσs2 T r[(σ+σ)1(s+s)1] + λ02σsT r(σ+σ)1T r(s+s)1
+[λσs3 T r[(s+s)3s(s+σ)3] + λ03σsT r(s+s)3sT r(s+σ)3 +λσs4 T r[(s+s)3a(s+σ)3] + λ04σsT r(s+s)3aT r(s+σ)3+ H.c]
+[λσs5 T r[(s+σ)3(s+σ)3] + λ05σsT r(s+σ)3T r(s+σ)3+ H.c], (25)
V (φ, σ) = λφσ1 T r[(σ+φ)3(φ+σ)3] + λ01φσT r(σ+φ)3T r(φ+σ)3
+λφσ2 T r[(σ+σ)1(φ+φ)1] + λ02φσT r(σ+σ)1T r(φ+φ)1, (26)
V (φ, s) = λφs11T r[(φ+s)1(s+φ)1] + λ011φsT r(φ+s)1T r(s+φ)1
+λφs12T r[(φ+s)10(s+φ)100] + λ012φsT r(φ+s)10T r(s+φ)100
+λφs13T r[(φ+s)3s(s+φ)3a] + λ013φsT r(φ+s)3sT r(s+φ)3a +[λφs14T r[(φ+s)3s(s+φ)3s] + λ014φsT r(φ+s)3sT r(s+φ)3s+ H.c]
+λφs21T r[(φ+φ)1(s+s)1] + λ021φsT r(φ+φ)1T r(s+s)1
+λφs22T r[(φ+φ)10(s+s)100] + λ022φsT r(φ+φ)10T r(s+s)100
+λφs23T r[(φ+φ)3s(s+s)3a] + λ023φsT r(φ+φ)3sT r(s+s)3a +[λφs24T r[(φ+φ)3s(s+s)3s] + λ021φsT r(φ+φ)3sT r(s+s)3s+ H.c.], (27)
V (φ, σ, s) = µT r(φ+σ+sφ) + H.c., (28)
Trang 5V = µ0φTσφ + µ1φTs1φ (29) III CONCLUSION
We have shown the neutrinos gain naturely small masses in agreement with experiment The neutrino mixing matrix in terms of tribimaximal form is obtained Based
on the flavour symmetry A4, we can understand neutrino experiments [1, 2, 3, 4, 5, 6, 7,
8, 9, 10, 11]
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Received 30-09-2010