Adjoint SUSY $SU(5)$ Grand Unified Model with $S_4$ Flavor Symmetry

0.73 MB·

by  Gui-Jun Ding

SUSY Adjoint SU(5) Grand Unified Model with S
4
Flavor Symmetry
Gui-Jun Ding 1
1 Department of Modern Physics,
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University of Science and Technology of China, Hefei, Anhui 230026, China
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n
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J Abstract

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We construct a supersymmetric (SUSY) SU(5) model with the flavor symmetry
]
h
S Z Z . Three generations of adjoint matter fields are introduced to generate
p 4× 3× 4
- the neutrino masses via the combined type I and type III see-saw mechanism. The
p
e first two generations of the the 10 dimensional representation in SU(5) are assigned
h to be a doublet of S , the second family 10 is chose as the first component of the
[ 4
doublet, and the first family as the second component. Tri-bimaximal mixing in the

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neutrino sector is predicted exactly at leading order, charged lepton mixing leads
v
0 to small deviation from the tri-bimaximal mixing pattern. Subleading contributions
0
introduce corrections of order λ2 to all three lepton mixing angles. The model also
8 c
4 reproducesarealisticpatternofquarkandchargedleptonmassesandquarkmixings.
.
6 The phenomenological implications of the model are analyzed in datail.
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:
v
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1e-mail address: [email protected]
1 Introduction
So far there is convincing evidence that the so-called solar and atmospheric anomaly
can be well explained through the neutrino oscillation. The mass square differences ∆m2 ,
sol
∆m2 and the mixing angles have been measured with good accuracy [1–3]. Global fit to
atm
the current neutrino oscillation data demonstrates that the observed lepton mixing matrix
is remarkably compatible with the tri-bimaximal (TB) mixing pattern [4], which suggests
the following values of the mixing angles
1 1
sin2θ = , sin2θ = , sinθ = 0 (1)
12 23 13
3 2
This simple structure of the mixing matrix suggests that there may be some symmetry
underlying the lepton sector. Recent study [5–19](also in the context of grand unified
theories [20,21]) showed that the discrete group A is particularly suitable to produce the
4
TB mixing at leading order (LO), if it is properly managed to be broken differently in the
neutrino and the charged lepton sector. In the A based models, it seems very difficult and
4
unnatural to generate the correct mass hierarchies and mixing for quarks. An interesting
solution is to enlarge the symmetry group A , two non-Abelian discrete group T [22] and
4 ′
S [23–30]havebeeninvestigated. Wenotethatbothgroupshaveadoubletrepresentation,
4
which can be utilized to give the 2+1 representation assignments for the quarks. In the
context of U(2) flavor group, this assignment has been known to give realistic quark
mixing matrix and mass hierarchy [31]. The irreducible representations of T are those of
′
A plus three two dimensional representations 2, 2 and 2 with the multiplication rules
4 ′ ′′
2 2 = 2 2 = 3 1, 2 2 = 2 2 = 3 1 and 2 2 = 2 2 = 3 1 ,
′ ′′ ′ ′′ ′′ ′ ′′ ′ ′ ′′
⊗ ⊗ ⊕ ⊗ ⊗ ⊕ ⊗ ⊗ ⊕
these ingredients allow us to reproduce the successful U(2) predictions in the quark sector.
By working only with the triplet and singlet representations, T is indistinguishable from
′
A , thus we can replicate with T the successful construction realized within A in the
4 ′ 4
lepton sector. S is claimed to be the minimal group which can predict the TB mixing
4
in a natural way, namely without ad hoc assumptions, from the group theory point of
view [32]. Actually the exact TB mixing can be realized in the S flavor model [23,24].
4
Moreover, the group S as a flavor symmetry, as is shown for example in Ref. [27–30], can
4
also give a successful description of the quark and lepton masses and mixing angles within
the framework of SU(5) or SO(10) grand unified theory (GUT). For a review of discrete
flavor symmetry models, please see the Refs. [34,35].
The SU(5) GUT is the simplest grand unified theory [33], in this case each generation
of the standard model fields resides in 5 and 10 dimensional representations. To be spe-
cific, one family of right-handed down quarks and left-handed leptons are unified in a 5
and the rest fields of the family are in a 10. It is well-known that neutrino masses are zero
at renormalizable level in the minimal SU(5). In GUT neutrino masses come naturally
through the see-saw mechanism, where integrating out large masses leads to the appear-
ance of small masses. However, this requires some extra matter fields or Higgs to be added
below theGUTscale. Apopularchoice is toaddatleast two right-handedneutrinos which
are SU(5) singlets, neutrino masses are generated through type I see-saw mechanism. The
second choice is to introduce a 15-plet of Higgs, this is the SU(5) implementation of the
so-called type II see-saw mechanism. The third choice is to generate neutrino masses
1
through the combination of type I see-saw with type III see-saw mechanism by adding at
least one matter multiplet in the adjoint 24 representation [36–38].
In this work, we shall construct a supersymmetric SU(5) model, and the flavor symme-
try group is S Z Z , where the auxiliary symmetry Z Z plays an important role in
4 3 4 3 4
× × ×
eliminating unwanted couplings, ensuring the needed vacuum alignment and reproducing
the observed charged fermion mass hierarchies. We will introduce three generations of
adjoint matter fields to generate the neutrino masses. We remark that some variants of
SU(5) S flavor models have been proposed in Refs. [28,29], where three right handed
4
×
neutrinos are introduced and the neutrino masses are generated via the type I see-saw
mechanism.
The paper is organized as follows. In section 2 we discuss the structure of the model,
andtheleadingorder(LO)resultsforfermionmassesandmixingsarepresented. Insection
3 we justify the choice of the vacuum configuration assumed in the previous section, by
minimizing the scalar potential of the theory in the supersymmetric limit. In section 4
the subleading corrections to the vacuum alignment and the LO results of fermion masses
and flavor mixings are discussed. In section 5 we study the phenomenological predictions
of the model in detail. Finally, section 6 is devoted to our conclusion.
2 The structure of the model
Inthefollowingwepresent ourSU(5)modelintheframeworkofsupersymmetry, which
simplifies the minimization of the scalar potential greatly. The S group acts as a flavor
4
symmetry of our model, the group S has already been studied in literature [39,40], but
4
with different aims and different results. S isthe permutation group of four objects, it has
4
five irreducible representations 11, 12, 2, 31 and 32, the group theory of S4 is presented in
Appendix A. In addition to 5 matter fields denoted by F and the tenplet 10 dimensional
matter fields denoted by T , we introduce the chiral superfields A in the adjoint 24
1,2,3
representation [37]. In the Higgs sector we introduce H , H and H in order to break
24 5 5
the gauge symmetry SU(5) into the standard model symmetry and subsequently into the
residual SU(3) U(1) . Moreover, H and H are introduced to avoid the wrong
c × em 45 45
predictions MT = M , where M and M represent the mass matrix of down type quark
d ℓ d ℓ
andchargedleptonrespectively. Asusual, theflavonfieldsareintroducedtospontaneously
breaktheS flavorsymmetryproperly. Thetransformationrulesofthematterfields, Higgs
4
fields and the flavon fields under SU(5), S , Z and Z are summarized in Table 1. The
4 3 4
first and the second generation of the 10 dimensional representations are assigned to be
a doublet of S , and the third generation of 10 to 1 of S . This assignment is indicated
4 1 4
by the heaviness of the top quark. There is the freedom of choosing the first family or the
second family as the first component of the S doublet. In this work, the second family
4
10 is taken to be the first component of the doublet, and the first family as the second
component. If we assign the first family 10 as the first component of the doublet and
the second family as the second component as usual, the down quark and strange quark
2
masses would be of the same order1 without fine tuning unless some special mechanisms
are introduced such as Ref. [28]. Three generation of 5 fields F are assigned to 31 of S4,
and three generations of adjoint matter fields A are also assigned to be 31 of S4. Fermion
masses and mixings arise from the spontaneous breaking of the flavor symmetry by means
of the flavon fields. In thefollowing, we shall discuss the LO predictions forfermion masses
and flavor mixings. For the time being we assume that the scalar components of the flavon
fields acquire vacuum expectation values (VEV) according to the following scheme
v
χ
v
χ = v , ϕ = ϕ , ζ = 0
χ
h i   h i v h i
v (cid:18) ϕ (cid:19)
χ
 
0
0
φ = v , η =
φ
h i   h i v
0 (cid:18) η (cid:19)
 
v
∆
∆ = 0 , ξ = v (2)
ξ
h i   h i
0
 
We will prove it to be a natural solution of the minimization of the scalar potential in
section 3. Furthermorewe take the VEVs (scaled by thecutoff Λ) v /Λ, v /Λ, v /Λ, v /Λ,
χ ϕ φ η
v /Λ and v /Λ to be of the same order of magnitude about (λ2) with λ 0.22 being
∆ ξ O c c ≃
the Cabibbo angle, and we will parameterize the ratio VEV/Λ by the parameter ε. This
order of magnitude is indicated by the observed ratios of up quarks and down quarks and
charged lepton masses, by the scale of the light neutrino masses and is also compatible
with the current bounds on the deviations from TB mixing for the leptons. We note that
the assumed size of the VEVs can be partially explained by the minimization of the scalar
potential, as it will be clearer in the following.
2.1 Neutrino sector
The LO superpotential which contributes to the neutrino masses is given by
w = y (FA) H +λ (AA) χ+λ (AA) ϕ (3)
ν ν 11 5 1 31 2 2
1IfwetakeT1 andT2 asthefirstandthesecondcomponentsoftheS4 doublet,i.e.,(T1,T2)T 2,then
∼
(TF)31 ∼ (T1F2+T2F3,T1F3+T2F1,T1F1+T2F2)T and (TF)32 ∼ (T1F2−T2F3,T1F3−T2F1,T1F1−
T2F2)T. WeseethatT1F1 andT2F2,whicharerelatedtothedownandstrangequarkmassesrespectively,
appear simultaneouslyasthe thirdcomponentofboththe combinations(TF)31 and(TF)32. The opera-
torsTFH andTFH combiningwiththeflavonfieldsorthecompositionofflavons,whichtransformas
5 45
31 or32,contributetothefirsttwofamiliesdownquarkandchargedleptonmassesaftertheS4 andGUT
symmetry breaking. As a result, the down and strange quark messes would be of the same order except
for the case that the (TF)31 and (TF)32 relevant contributions to down quark or strange quark cancel
with each other. We notice that the same view has been put forward by Altarelli et al. [34]. Whereas
if we choose (T2,T1)T ∼ 2 as we proposed, then (TF)31 ∼ (T2F2 +T1F3,T2F3 +T1F1,T2F1 +T1F2)T
and(TF)32 ∼(T2F2−T1F3,T2F3−T1F1,T2F1−T1F2)T, the degeneracybetweenfirsttwofamilies down
quark (charged lepton) masses is dissolved.
3
Fields T (T ,T )T F A H H H H H χ ϕ ζ φ η ∆ ξ
3 2 1 5 45 5 45 24
SU(5) 10 10 5 24 5 45 5 45 24 1 1 1 1 1 1 1
S4 11 2 31 31 11 11 11 11 11 31 2 12 31 2 31 11
Z 1 ω 1 1 1 1 ω 1 1 1 1 1 ω2 ω2 ω ω
3
Z 1 i i i 1 -1 1 -1 1 -1 -1 1 i i i i
4
− −
U(1) 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
R
Table 1: Fields and their transformation properties under the symmetry groups SU(5), S4, Z3 and Z4,
where ω =ei2π/3 =( 1+i√3)/2.
−
TheadjointmatterfieldAdecomposesunderthestandardmodelasA = (ρ ,ρ ,ρ ,ρ ,ρ ) =
8 3 (3,2) (¯3,2) 0
(8,1,0) (1,3,0) (3,2, 5/6) (¯3,2,5/6) (1,1,0),obviouslytherearebothSU(2)triplet
⊕ ⊕ − ⊕ ⊕
ρ and singlet ρ with hypercharge Y = 0 in the model. The neutrino masses are generated
3 0
through the type I (mediated by the SU(2) singlet ρ of A) and type III (mediated by the
0
SU(2) triplet ρ of A) see-saw mechanism. The Dirac mass matrices is obtained from the
3
first term in Eq.(3),
1 0 0 1 0 0
1 √15
MD = y v 0 0 1 , MD = y v 0 0 1 (4)
ρ3 2 ν 5  ρ0 10 ν 5 
0 1 0 0 1 0
   
where MD and MD are the Dirac mass matrices associated with ρ and ρ respectively.
ρ3 ρ0 3 0
The last two terms in Eq.(3) lead to the Majorana mass matrices of ρ and ρ
3 0
2λ v λ v +λ v λ v +λ v
1 χ 1 χ 2 ϕ 1 χ 2 ϕ
− −
MM = λ v +λ v 2λ v +λ v λ v
ρ3  − 1 χ 2 ϕ 1 χ 2 ϕ − 1 χ 
λ v +λ v λ v 2λ v +λ v
1 χ 2 ϕ 1 χ 1 χ 2 ϕ
− −
 
MM = MM (5)
ρ0 ρ3
It is notable that the Majorana mass matrices of ρ and ρ are exactly the same. As a
3 0
result, the mass spectrums of ρ and ρ are degenerate. This degeneracy is violated at
3 0
NLO by the Higgs H . We note the VEVs χ and ϕ are invariant under the action of
24
h i h i
S elements TSTS2, TST and S2, consequently the S flavor symmetry is broken down
4 4
to the Klein four subgroup in the neutrino sector. The light neutrino mass matrix is the
sum of type I and type III see-saw contributions
M = (MD)T(MM) 1MD (MD)T(MM) 1MD
ν − ρ3 ρ3 − ρ3 − ρ0 ρ0 − ρ0
a b a+b a+b
− − − −
5b(3a b) 5b(3a b) 5b(3a b)
= a+−b 3a2 4−ab+b2 3a2+2−ab b2 y2v2 (6)
 5b−(3a b) −5b(9−a2 b2) −5b(9a2 b−2)  ν 5
a+−b 3a2+2−ab b2 3a2 4−ab+b2
 5b−(3a b) −5b(9a2 b−2) −5b(9−a2 b2) 
− − −
 
where
a λ v , b λ v (7)
1 χ 2 ϕ
≡ ≡
The above light neutrino mass matrix M is 2 3 invariant and it satisfies the magic
ν
↔
symmetry (M ) +(M ) = (M ) +(M ) . Therefore it is exactly diagonalized by the
ν 11 ν 13 ν 22 ν 32
4
TB mixing matrix
UTM U = diag(m ,m ,m ) (8)
ν ν ν 1 2 3
where m are the light neutrino masses, in unit of 2y2v2 they are
1,2,3 5 ν 5
1
m =
1
3a b
| − |
1
m =
2
2 b
| |
1
m = (9)
3
3a+b
| |
The neutrino mass spectrum can be normal hierarchy (NH) or inverted hierarchy (IH).
The unitary matrix U is given by
ν
U = U diag(e iα1/2,e iα2/2,e iα3/2) (10)
ν TB − − −
U is the well-known TB mixing matrix
TB
2 1 0
3 √3
UTB =  q1 1 1  (11)
−√6 √3 √2
1 1 1
 
 −√6 √3 −√2 
 
The phases α , α and α are
1 2 3
α = arg( y2v2/(3a b))
1 − ν 5 −
α = arg( y2v2/b)
2 − ν 5
α = arg( y2v2/(3a+b)) (12)
3 − ν 5
According to Eq.(9), the light neutrino mass spectrum is directly related to the heavy
neutrino (ρ or ρ ) masses, it is determined by only two complex parameters a and b. In
3 0
the following, we shall follow the method of Ref. [17] to analyze the light neutrino mass
spectrum in detail. For the sake of convenience, we define
a
= reiθ (13)
b
Note that the parameter r is real and positive and the phase θ is between 0 and 2π. We
can express r and the phase θ in term of light neutrino masses as follows
1 2m2 2m2
r = 2 + 2 1 (14)
3s m2 m2 −
1 3
m22 m22
cosθ = m23 − m21 (15)
2m22 + 2m22 1
m21 m23 −
q
Experimentally, only the mass square differences have been measured. For normal (in-
verted) hierarchy they are
∆m2 = m2 m2 = (7.67+0.22) 10 5eV2
sol 2 − 1 −0.21 × −
∆m2 = m2 m2(m2) = (2.46(2.45) 0.15) 10 3eV2 (16)
atm | 3 − 1 2 | ± × −
5
0.0 3.0
-0.5 Normalhierachy 2.5 Invertedhierachy
-1.0 2.0
ΘCos -1.5 ΘCos 1.5
-2.0 1.0
-2.5 0.5
-3.0 0.0
0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.02 0.04 0.06 0.08 0.10
m1HeVL m3HeVL
(a) (b)
Figure 1: cosθ as a function of the lightest neutrino mass m for both normal hierarchy and inverted
l
hierarchy spectrum, where we take ∆m2 = 7.67 10−5eV2 and ∆m2 = 2.46(2.45) 10−3eV2 for
sol × atm ×
normal hierarchy (inverted hierarchy).
By expressing the neutrino masses in terms of the lightest neutrino mass m (m = m for
l l 1
NH and m for IH), ∆m2 and ∆m2 , cosθ becomes a function of the lightest neutrino
3 sol atm
mass m . We display cosθ versus m in Fig.1 for both normal hierarchy and inverted hi-
l l
erarchy spectrum. Imposing the condition cosθ 1, we obtain the following constraints
| | ≤
on the lightest neutrino mass
m 0.011eV, for normal hierarchy
1
≥
m 0.028eV, for inverted hierarchy (17)
3
≥
The results presented so far are of course approximate since the model gets corrections
when higher dimensional operators are included in the Lagrangian. If the mixing of the
left handed charged lepton is neglected, we can straightforwardly derive the neutrinoless
double decay parameters m
ee
| |
m m2 m2
m = 1 2 2 +2 2 (18)
| ee| 3 s − m2 m2
1 3
2.2 Up quarks sector
The LO superpotential invariant under symmetry group SU(5) S Z Z , which
4 3 4
× × ×
gives rise to the masses of the up type quarks after S and SU(5) symmetry breaking, is
4
given by
4
y y y
ci (1) ut1 ut2
w = y T T H + TT H + TT (φχ) H + TT (ηϕ) H
u t 3 3 5 Λ2 Oi 5 Λ2 3 2 5 Λ2 3 2 5
i=1
X
y y
ut3 ct
+ TT ηζH + TT ηH (19)
Λ2 3 5 Λ 3 45
where
(1) = (φφ) ,(φφ) ,(ηη) ,(ηη)
O { 11 2 11 2}
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With the vacuum alignment in Eq.(2), it is immediate to derive the mass matrix as follows
0 0 4(y vφvχ +y vηvϕ)v
ut1 Λ2 ut2 Λ2 5
Mu =  0 8(yc2Λvφ22 +yc4Λvη22)v5 8yctvΛηv45 +4yut1vφΛv2χv5 
4(y vφvχ +y vηvϕ)v 8y vηv +4y vφvχv 8y v
 ut1 Λ2 ut2 Λ2 5 − ct Λ 45 ut1 Λ2 5 t 5 
 (20)
We note that the Higgs field H (H ) induces a symmetric (antisymmetric) contribution
5 45
T2
to M . Since the product decompositions (TT) T T + T T and (TT) 1
u 11 ∼ 1 2 2 1 2 ∼ T2
(cid:18) 2 (cid:19)
are both symmetric under the exchange of the two tenplets, the combinations of T T H
3 3 45
and TTH with arbitrary number of flavon fields don’t contribute to up quarks masses.
45
Therefore the corresponding terms are omitted from the beginning. The mass matrix M
u
is diagonalized by bi-unitary transformation
VRu†MuVLu = diag(mu,mc,mt) (21)
The up type quark masses are given by
2(y v2 +y v2)(y v v /Λ2 +y v v /Λ2)2
m c2 φ c4 η ut1 φ χ ut2 η ϕ v3
u ≃ y (y v2 +y v2)v2 +y2v2v2 5
t c2 φ c4 η 5 ct η 45
(cid:12) (cid:12)
(cid:12) v2 v2 y2 v2 v2 (cid:12)
m (cid:12)8(y φ +y η )v +8 ct η 45 (cid:12)
c ≃ c2Λ2 c4Λ2 5 y Λ2 v
t 5
(cid:12) (cid:12)
m (cid:12)8y v (cid:12) (22)
t (cid:12) t 5 (cid:12)
≃
(cid:12) (cid:12)
The mixing matrix Vu is(cid:12) (cid:12)
L (cid:12) (cid:12)
1 su su
12 13
Vu su 1 su (23)
L ≃  − 12∗ 23 
su +su su su 1
− 13∗ 12∗ 23∗ − 23∗
 
where
1 y (y v v +y v v )v v v
su = ct ut1 φ χ ut2 η ϕ 5 45 η ∗
12 −2 y (y v2 +y v2)v2 +y2v2v2 Λ
t c2 φ c4 η 5 ct η 45
h i
y v v y v v
su = ct 45 η + ut1 φ χ ∗
23 − y v Λ 2y Λ2
t 5 t
(cid:16) (cid:17)
1 y v v y v v
su = ut1 φ χ + ut2 η ϕ ∗ (24)
13 2 y Λ2 y Λ2
t t
(cid:16) (cid:17)
Wenotethatthereisamixing oforderλ2 between thefirst andthesecondfamily, although
c
the (12) and (21) elements of M vanish at LO. The top quark mass is generated at tree
u
level, and the mass hierarchies among the up quarks are reproduced naturally given the
VEVs v , v , v and v of order λ2Λ.
χ ϕ φ η c
2.3 Down type quarks and charged leptons sector
The superpotential generating the masses of down quarks and charged lepton is
9
y y y y
b s1 s2 di (2)
w = T FφH + (TF) (∆∆) H + (TF) ∆ξH + T F H
d Λ 3 5 Λ2 31 31 45 Λ2 31 45 Λ3 3 Oi 5
i=1
X
7
6 7
x z
di (3) di (4)
+ T F H + TF H +... (25)
Λ3 3 Oi 45 Λ3 Oi 5
i=1 i=1
X X
where dots stand for higher dimensional operators.
(2) = χ2φ,χ2η,ϕχφ,ϕχη,ϕ2φ,χφζ,χηζ,ϕφζ,φζ2
O { }
(3) = φ3,φ2η,φη2,∆3,∆2ξ,∆ξ2
O { }
(4) = φ2χ,φ2ϕ,φ2ζ,ηφχ,ηφϕ,ηφζ,η2χ (26)
O { }
For the last three terms in Eq.(25), one operator frequently induces several different con-
tractions, we shouldtakeinto account allpossible independent contractionsforeachopera-
tor. Note that the auxiliary symmetry Z Z imposes different powers of the flavon fields
3 4
×
for the bottom (tau), strange (muon) and down quark (electron) mass relevant terms. In
the expansion in powers of 1/Λ, the bottom (tau) mass is generated at order 1/Λ, the
strange (muon) and down quark (electron) masses are generated at order 1/Λ2 and 1/Λ3
respectively. As a result, if we only consider the LO operators suppressed by 1/Λ and
1/Λ2, the down quark and electron would be massless.
Recalling the vacuum configuration in Eq.(2), we can write down the mass matrix for
down quarks and charged leptons as follows
yd ε3v yd ε3v yd ε3v +2yd′ε3v
11 5 12 5 13 5 13 45
M = yd ε3v 2yd ε2v +yd′ε3v yd ε3v (27)
d  21 5 22 45 22 5 23 5 
2yd ε2v +yd′ε3v yd ε3v yd εv
22 45 31 5 32 5 33 5
 yd ε3v yd ε3v 6yd ε2v +yd′ε3v
11 5 21 5 − 22 45 31 5
M = yd ε3v 6yd ε2v +yd′ε3v yd ε3v (28)
ℓ  12 5 − 22 45 22 5 32 5 
yd ε3v 6yd′ε3v yd ε3v yd εv
13 5 − 13 45 23 5 33 5
 
where the coefficients yd (i,j = 1,2,3), yd′, yd′ and yd′ are linear combinations of the
ij 13 22 31
leading order coefficients, yd coincides with the LO parameter y up to corrections of
33 b
order ε2 which origins from the operators T F (3)H . The factor of 3 difference in the
3 O 5
(13),(22)and(31)elementsbetweenM andM istheso-calledGeorgi-Jarlskogfactor[41],
d ℓ
which is induced by the Higgs H . Similar to the up type quarks, the mass matrices M
45 d
and M can be diagonalized by the following transformations
ℓ
VRd†MdVLd = diag(md,ms,mb), VRℓ†MℓVLℓ = diag(me,mµ,mτ) (29)
The mass eigenvalues are given by
m yd ε3v yd yd ε4v2/(2yd v ) 2yd yd ε4v /yd 4yd′yd ε4v2 /(yd v )
d ≃ | 11 5 − 12 21 5 22 45 − 13 22 45 33 − 13 22 45 33 5 |
m 2yd ε2v +yd′ε3v
s ≃ | 22 45 22 5|
m yd εv (30)
b ≃ | 33 5|
and
m yd ε3v +yd yd ε4v2/(6yd v )+6yd yd ε4v /yd 36yd′yd ε4v2 /(yd v )
e ≃ | 11 5 12 21 5 22 45 13 22 45 33 − 13 22 45 33 5 |
m 6yd ε2v +yd′ε3v
µ ≃ |− 22 45 22 5|
m yd εv (31)
τ ≃ | 33 5|
8
The diagonalization matrices Vd and Vℓ are
L L
′
1 ( y2d1 v5 ε) (2y2d2v45ε+ y3d1ε2)
2y2d2v45 ∗ y3d3 v5 y3d3 ∗
Vd  y2d1 v5 ε 1 (y3d2ε2)  (32)
L ≃ −2y2d2 v45 y3d3 ∗
 2y2d2v45ε y3d1′ε2 y3d2ε2 y2d1∗y2d2 ε 2 1 
 − y3d3 v5 − y3d3 −y3d3 − y2d2∗y3d3| | 
 
′
1 ( y1d2 v5 ε) (y1d3ε2 6y1d3v45ε2)
− 6y2d2v45 ∗ y3d3 − y3d3 v5 ∗
Vℓ  y1d2 v5 ε 1 (y2d3ε2)  (33)
L ≃ 6y2d2v45 y3d3 ∗
′
 y1d3ε2 +6y1d3v45ε2 y2d3ε2 1 
 −y3d3 y3d3 v5 −y3d3 
 
As is shown in Eq.(30) and Eq.(31), obviously we have
m m , m 3m (34)
τ b µ s
≃ ≃
The well-known bottom-tau unification and the Georgi-Jarlskog relation [41] between the
down type quark and the charged lepton masses for the second generation are produced in
the present model. We note that generally the vanishing of the (11) elements of both the
down quark and charged lepton mass matrices is required, to obtain the Georgi-Jarlskog
relation for both the first and second generations simultaneously. The mass difference of
electron and down quark is induced by the Higgs field H , acceptable values of the masses
45
for electron and down quark can be accomodated due to the Georgi-Jarlskog factor. It
is well-known that the CKM mixing between the first and the second family is exactly
described by the Cabibbo angle. In order to satisfy this phenomenological constraint, for
the parameters yd and yd of order (1) we could choose
21 22 O
v λ v (35)
45 ∼ c 5
Furthermore we can see from Eq.(32) and Eq.(33) that the mixing angle between the first
and the second family charged leptons approximately is λ /3. The resulting quark mixing
c
matrix is given by
VCKM = VLu†VLd (36)
We can straightforwardly read the CKM matrix elements as follows
V V V 1
ud cs tb
≃ ≃ ≃
yd v 1 y (y v v +y v v )v v v
V V 21 5 ε+ ct ut1 φ χ ut2 η ϕ 5 45 η
u∗s ≃ − cd ≃ 2yd v 2y (y v2 +y v2)v2 +y2v2v2 Λ
22 45 t c2 φ c4 η 5 ct η 45
yd v yd′ y v v y v v 1 y2(y v v +y v v )v2 v2
V = 2 22 45ε+ 31ε2 ut1 φ χ ut2 η ϕ + ct ut1 φ χ ut2 η ϕ 45 η
u∗b yd v yd − 2y Λ2 − 2y Λ2 2y2(y v2 +y v2)v2 +y y2v2v2 Λ2
33 5 33 t t t c2 φ c4 η 5 t ct η 45
y v v
ct 45 η
V V
c∗b ≃ − ts ≃ y v Λ
t 5
yd v yd′ y v v y v v y yd v v v
V = 2 22 45ε 31ε2 + ut1 φ χ + ut2 η ϕ + ct 21 45 5 ηε (37)
td − yd v − yd 2y Λ2 2y Λ2 2y yd v v Λ
33 5 33 t t t 22 5 45
We note that the CKM elements V , V , V and V are dominantly determined by the
us cd ub td
mixing in the down type quark sector, V and V origin from the left handed up quarks
cb ts
9