MINISTRY OF
EDUCATION AND TRAINING
VIET NAM ACADEMY OF SCIENCE
AND TECHNOLOGY INSTITUTE OF PHYSICS
LE THO HUE LEPTON FLAVOR VIOLATING DECAYS IN SUPERSYMMETRIC
3-3-1 MODELS

Major subject: Theoretical and mathematical Physics
Code: 62 44 01 01

This dissertation will be defended in front of the evaluating assembly at academy
level
Place of defending: meeting room, floor , Institute of Physics, Viet Nam
Academy of Science and Technology.
No 10, Dao Tan, Ba Dinh, Ha Noi
Time: at…… … day… month……. year……
This thesis can be studied at the Vietnam National Library or

at Library of the National Academy of Public Administration
Introduction
Urgency of the topic
Particle physics are now strictly related with high energy col-
liders. All models are waiting for experimental significance of
New Physics to compare with all predictions as well as limiting
parameter space of each model. Specially, in the last 2012 and
early 2013 the Large Hadron Collider (LHC), placed in CERN-
Switzerland with two independent detectors CMS and ATLAS,
has discovered a new scalar particle inheriting many properties
of Higgs (Higgs-like) predicted by the Standard Model (SM) with
mass around 125-126 GeV. This is the last type of particle con-

meetric 3-3-1 (SUSY331) models, it is also necessary to predict
and investigate the parameter space in order to compare with
other models and experimental bounds. This is the main rea-
son why we concentrate on the study of the cLFV decays in the
SUSY331 models and publish obtained results in this disserta-
tion. We will consider two models supersymmetric economical
3-3-1 (SUSYE331) and supersymmetric reduced minimal 3-3-1
(SUSYRM331) models.
Research purpose
• We construct the SUSYRM331.
• We study the lepton flavor violation in the SUSYE331 model
through decays of Higgs, tau and Z boson.
Research object
• LFV verteices in the SUSYE331 and SUSYRM331.
2
• cLFV decays of H→ μτ , τ → μγ, τ → 3μ and Z → μτ in
the SUSYE331.
Research content
• The supersymmetric reduced minimal 3-3-1 model.
• Properties of LFV vertices in SUSY331 models
• The possibility of detection decay H→ μτ in recent colliders
• Discussing on regions of parameter space of SUSYE331 sat-
isfying the experimental bounds.
Research method
• Quantum field theoretical method.
• Numerical investigation by Mathematica 7.0.
Dissertation Structure
The dissertation contains the introduction, four chapters present-
ing main content and the conclusion listing all of new results
of our works. In addition, the dissertation contains three more

a right-handed neutrino to the bottom component of each lepton
triplet. In the quark sector, new quarks appearing as third com-
ponents of quark (anti) triplets are called exotic quarks, with the
lepton number L =2. One needs three Higgs (anti)triplets to gen-
erate masses of particles. Because neutrinos and anti-neutrinos
are in the same triplets then the lepton numbers are not con-
served. This is the common property of the 3-3-1 models. But
these models relate with a new conserved number called extended
lepton number, denoted L. It can be computed from the original
L number by a formula
L =
2
√
3
λ
8
+ LI. (1.1)
4
More detail, the table 1.1 lists particular values of L and baryon
number B = BI of multiplets contained in the considered model.
In addition, table 1.2 presents numerical values of components of
all multiplets.
Bảng 1.1: Values of B and L for 3-3-1 models with right-handed
neutrinos.
Multiplet χη ρQ
3L
Q
αL
u
aR

−
2
3
−
2
3
2
3
00−22
1
3
1
Bảng 1.2: Non-rezo values of L for fields in the 3-3-1 models with
right-handed neutrinos.
Field N
L
l
L
l
R
ρ
+
3
η
0
3
χ
0
1
χ

any new leptons. But the Higgs spectrum in this model is rather
complicated because of the appearance of a Higgs sextet. Table
1.3 lists values of B và L for multiplets in the model. Note that
Bảng 1.3: Values of B and L in the M331.
Đa tuyến χρ ηSQ
3L
Q
αL
u
aR
d
aR
T
R
D
αR
f
aL
Tích B 00 00
1
3
1
3
1
3
1
3
1
3
1

ple as that of the E331, even the VEVs are fewer. The super-
symmetric version, called the supersymmetric reduced minimal
(SUSYRM331) model, was also presented in 2013 by us and it is
reviewed in the chapter 2 of this dissertation.
6
Chương 2
Suppersymmetric 3-3-1
models
In this chapter, we concentrate on two supersymmetric models
constructed from two 3-3-1 models with the simplest Higgs spec-
trums, namely the E331 and RM331 models. The general basis
of the supersymmetric theory is not summarized here.
2.1 Supersymmetric economical 3-3-1 model
The SUSYE331 was introduced in 2007 as the supersymmetric
version of the E331. Similar to the case of MSSM, this model con-
tains the double number of Higgs multiplets as those in the non-
supersymmetric model. In previous works studying the SUSYE331,
the LFV vertices in the soft terms are not considered. In this dis-
sertation we assume that the LFV sources only appear in the soft
term of the lagrangian, namely
−L
˜μ˜τ
=(˜μ
∗
L
, ˜τ
∗
L
)



˜m
2
μ
R
˜m
2
R
μτ
˜m
∗2
R
μτ
˜m
2
τ
R


˜μ
c
L
˜τ
c
L

. (2.1)
7
Sleptons (˜μ
L

R
3

. The respective
masses are (˜m
2
L
2
, ˜m
2
L
3
) and (˜m
2
R
2
, ˜m
2
R
3
). The mass eigenstates
are mixing of flavor eigenstates ˜μ and ˜τ. The mixing is quan-
titatively determined through new parameters s
L
and s
R
which
satisfy,
s
L

. (2.2)
Eingenstates between two bases relate to each other by ˜μ
L
=
c
L
˜
l
L
2
−s
L
˜
l
L
3
, ˜τ
L
= s
L
˜
l
L
2
+ c
L
˜
l
L
3

l
R
2
+ c
R

l
R
3
. The lepton numbers are
conserved when s
L
= s
R
=0. Similar to the case of sneutrino
sector, the mixing between flavor eigenstates is parameterized by
s
ν
L
and s
ν
R
. Four parameters s
L
,s
R
,s
ν
L
and s

L
Lb
.
Also, this term relates with the measures of the neutrino oscil-
lation experiments so we can estimate some constrains to these
parameters. This problems are being studied and we will publish
in the near future.
In summary, in this chapter we concentrate on two results
1. We parameterize the LFV mixing in the soft term of the
SUSYE331 when assuming the existence of this mixing in
slepton sector.
2. We constructed the SUSYRM331 and discuss on the LFV
sources of the model.
9
Chương 3
H→ μτ decays in the
SUSYE331
3.1 Effective operators and branching ra-
tios
The low energy effective operators in the general case are used
from previous works. The branching ratios of neutral Higgses are
determined as
BR(Φ
0
→ τ
+
μ
−
)=BR(Φ
0


,
Δ
ρ
L
and Δ
ρ
R
are one loop effective LFV coefficients, Φ
0
denote
the mass eigenstates of Higgses in the SUSYE331, Φ
0
= ϕ
S
a36
or φ
S
a36
. Feynmann digrams presenting contributions to Δ
ρ
L
and
Δ
ρ
R
are shown in Fig. 3.1. The formulas are as follows,
Δ
ρ
L

Lc
, Δ
ρ
Ld
, Δ
ρ
Le
, Δ
ρ
Lf
và Δ
ρ
Lk
receive one loop
contributions from diagrams in Fig. 3.1. They are computed in
10
μ
τ
c
˜ρ
0
˜ρ
0
λ
B
ρ
0∗
˜
l
L

1
˜
W
−
˜
W
+
ρ
0∗
˜ν
Lα
(c)
μ
τ
c
˜ρ
−
2
˜ρ
+
2
˜
Y
−
˜
Y
+
ρ
0∗
˜ν

0∗
˜ν
R
α
˜ν
L
β
(f)
τ
μ
c
˜ρ
0
˜ρ
0
λ
B
ρ
0∗
˜
l
R
α
(i)
μ
τ
c
λ
B
ρ

Δ
ρ
R
[(i), (l)].
the dissertation. They depend on only the function I
3
(x, y, z)
where the precise formula is,
I
3
(x, y, z)=
xy ln(x/y)+yz ln(y/z)+zxln(z/x)
(x − y)(y − z)(z −x)
. (3.3)
3.2 Numerical investigation
We just investigate the case of maximal mixing. The LFV de-
cays Br(H → μτ) are large when tan γ is large enough so we
choose tan γ =50. Other parameters are assumed correspond-
ing to the values shown in Figs. refFDeltaRhoR1, 3.3, 3.4, 3.5
and 3.6. These Figs. show that |Δ
ρ
R
|
2
obtains the maximal value
∼ 10
−3
when |μ
ρ
|/ ˜m

| is the
dominant contribution when |μ
ρ
|/ ˜m
L
is small. |Δ
ρ
R
| gives large
contributions when |μ
ρ
|/ ˜m
L
are very large. To compare the cor-
0 5 10 15 20 25 30
10
8
10
7
10
6
10
5
10
4
0.001
Μ
Ρ
m


ρ
|/ ˜m
R
. The respective parameters: 1) Blue–m

=
˜m
R
=˜m
L
; 2) green–3m

=˜m
R
=˜m
L
; 3) yellow- m

=˜m
R
= 3 ˜m
L
; 4) red–m

=
˜m
R
=˜m
L
/3. Two black lines correspond to two values 10

0.01
Μ
Ρ
m

L
50L
2
0 2 4 6 8 10
10
7
10
6
10
5
10
4
0.001
0.01
Μ
Ρ
m

L
50L
2
Hình 3.3: |Δ
ρ
L
|

R
/3. The black lines correspond to values 10
−3
of |50Δ
ρ
L
|
2
.
relation between Δ
L
and Δ
R
, we consider two Figs. 3.4 and 3.5.
In the left panel of the Fig. 3.4, it can be see that if |μ
ρ
|/ ˜m
L
≤ 8
then
|Δ
ρ
R
|
2
|Δ
ρ
L
|
2

50R
2
50L
2
Hình 3.4: Plots of |Δ
ρ
R
|
2
/Δ
ρ
L
|
2
as function of |μ
ρ
|/ ˜m
L
. Numerical values of pa-
rameters: 1) blue –m

=˜m
R
=˜m
L
; 2) green–3m

=˜m
R
=˜m

ρ
R
|
2
|Δ
ρ
L
|
2
.
In the right panel, there is a region satisfying 10 ≤|μ
ρ
|/ ˜m
L
≤ 30
which Δ
R
are much larger than Δ
R
. This is the region where dia-
grams cancel to each other and there appears minima which each
minimum divides the plot into two regions. Therefore, both Δ
L
and Δ
R
are very suppressed. With |μ
ρ
|/ ˜m
L
≥ 30, the dominant

R
|
2
|Δ
ρ
L
|
2
is large when two following con-
ditions are satisfied: 1) μ
ρ
is very large ;2) if ˜m
R
> ˜m
L
then the
ratio ˜m
R
/ ˜m
L
increases according to the increasing values of μ
ρ
,
if ˜m
R
< ˜m
L
then
|Δ
ρ

1
2
3
4
5
6
7
Μ
Ρ
mSUSY
m

Rm

L
0.0005
0.0005
0.0005
0.001
0.00
0
.001
0.01 0.1
0.1
0 2 4 6 8 10
0.0
0.5
1.0
1.5
2.0

R
=˜m
ν
R
,
m

= m
λ
=˜m
L
=˜m
ν
L
= m
SUSY
. The red region correspond to values of
|Δ
ρ
R
|
2
|Δ
ρ
L
|
2
≥
0.5.
Bảng 3.1: Higgs-fermion-fermion vertices in the SUSYE331 comparing with those

Higgs with mass 125 GeV discoverd recently by LHC can decay to
dominant decays of fermion-antifermion b
¯
b and τ ¯τ. For example,
the branching ratio of light Higgs ϕ
Sa36
is Br(ϕ
Sa36
→ τ ¯τ )  8%.
This leads to Br(ϕ
Sa36
→ μτ )  8 ×10
−3
%. This is a significance
which can be detected in recent colliders. For heavy Higgses, the
main decays are decays to other heavy boson such as W
+
W
−
,
ZZ, so the LFV decays are very suppressed.
Numerical investigation in Fig. 3.6 shows the region where
the branching ratio BR(H → μτ)/BR(H → ττ) cực đại cỡ 10
−3
when 0.1 ≤|μ
ρ
|/M
SUSY
≤ 6 and 0.1 ≤|˜m
g

0.5
1.0
1.5
2.0
2.5
3.0
Μ
Ρ
M
SUSY
m

g
M
SUSY
Hình 3.6: Contour plots of BR(H → μτ )/BR(H → ττ) as function of ˜m
g
and
|μ
ρ
|/m
SUSY
. Other parameters are fixed: m

= m
λ
=˜m
g
and ˜m
R

−8
, (4.1)
BR(τ
−
→ μ
−
μ
+
μ
−
) < 2.1 × 10
−8
, (4.2)
BR(Z → μ
+
τ
−
) < 1.2 × 10
−5
. (4.3)
These three processes are studied at the same time because the
experimental bounds are very clear and the analytic formulas
calculating them closely relate to one another. We will investigate
these three decays in the framework of the SUSYE331.
16
0.5
2.5
5
10
15

0.5
1
2.5
5
0.25
0.5
1
2.5
5
100 200 300 400 500
100
150
200
250
m
Λ
GeV
mL

3
GeV
D
L
Γ b
10
9
GeV
2

Hình 4.1: Contour plots of D

˜ν
L
= θ
˜ν
R
= π/4 and μ
ρ
= 140 GeV
(1TeV) for the left (right) panel. The black and dashed curves correspond to values
of m
B
= 300 GeV and m
B
= −300 GeV.
4.1 Effective operators and branching ra-
tios
In the SUSYE331, the branching ratios of τ → μγ, Z → μτ,
Z

→ τμ and τ → 3μ in the limit of effective low energy theory
have the same formulas as those of MSSM.
Contributions from Hμτ effective vertices to the branching
ratio of τ → μμμ decays in our investigation is very suppressed.
Hence, they are ignored in this our calculation.
4.2 Numerical calculation and discussion
To guarantee the vacuum stability of the model constructed in
previous works, we add the B/μ-term types in the soft term of
the original model. As the result, the charged Higgses naturelly
satisfy the lower bounds of experiments. Investigation of neutral
Higgs sector shows that t

erate values.
Fig. 4.2 draws the regions of mass parameters containing
heavy sleptons, order of TeV. The results show that the satisfying
regions are expanded in the regions of heavier sleptons masses.
Fig. 4.3 presents left-right mixing A
τ
as function of μ
ρ
in the
20
30
40
50
80
100
20
30
40
50
80
100
100 200 300 400 500
100
150
200
250
300
m
Λ
GeV

mL

3
GeV
D
L
Γ b
10
9
GeV
2

Hình 4.2: Contour plots of D
γ(b)
L
with tan γ = 3.0, m
˜
L
2
= m
˜ν
L
2
= m
˜ν
R
2
and
m
˜

L
, A
τ
must be large to satisfy experiments as well as
cancel tachyon sleptons. In general, the existence of three LFV
sources will exclude the mass parameter space of light sleptons.
If we consider the case of cLFV effects caused by only right-
handed charged slepton sector, s
R
= c
R
=1/
√
2, then only D
γ
R
=
0. Numerical investigation in Fig. 4.4 indicates that the regions
of parameter space containing light sleptons allow D
γ
R
to easily
18
2.5
0
2.5
1000 500 0 500 1000
200
400
600

L
= θ
˜ν
R
=0, A
L
τμ
=0. Other parameters are chosen as
(m
B
,m
λ
,m
˜
L
3
,m
˜
R
)[GeV]: (200, 300, 300, 200)–black, (100, 400, 100, 200)–dashed,
(100, 500, 300, 100)–dotted. In particular, the center curves correspond to D
γ
L
=0
while two other curves limit |D
γ
L
|≤2.5 × 10
−9
[GeV

→ μ
−
μ
+
μ
−
), we define two con-
tribution coefficients f
A
Z
and f
D
γ
. The precise formulas of these
coefficients are shown in the dissertation. We denote A
Z
are dom-
inant contribution if 1.05 ≥ f
A
Z
≥ 0.95. The regions of parameter
19
0.25
1
2
0
.25
1
300 200 100 0 100 200 300
100

4.5
300 200 100 0 100 200 300
100
150
200
250
300
m
B
GeV
m R

3
GeV
D
R
Γ b
10
9
GeV
2

Hình 4.4: Contour plots of D
γ(a)
R
(left) and D
γ(b)
R
(right) as functions of two
parameters m

). If f
D
γ
≤ 0.05, the main con-
tributions to the Br(τ → 3μ) is F
μ
L(R)
L(R)
and the respective regions
are called F −domination. In contrast, if 1.05 ≥ f
D
γ
≥ 0.95 then
the regions of parameter space are D-domination.
Maximal mixing in left-handed sector of (˜μ, ˜τ)
This case corresponds to s
L
= c
L
=
1
√
2
và s
R
= s
˜ν
L
= s
˜ν

has both upper
and lower bounds because of the condition BR(τ → μγ) < 4.4 ×
10
−8
. There do not exist any A
Z
-domination regions.
Now we come to two decays Z → μτ and τ → μμμ with
investigation shown in Fig. 4.6. The mass parameter of gaugino
m
λ
is chosen according to the limit bound of experiment. Other
parameters are chosen in order of O(100) GeV. From Fig. 4.6 the
20
0.05
0.05
0.3
0.3
0.9
5
1.05
1
1
4.4
0.9
0
.91
100 150 200 250 300 350 400
100
200

1.05
1.2
1.2
1
2
4.4
0.0
5
0.1
0.7
0.75
200 400 600 800 1000
200
400
600
800
1000
1200
Μ
Ρ
GeV
mL

3
GeV
f
A
z
, f
D

,m
˜
L
2
m
˜
L
R
)
are (100, 300, 1000, 100)[GeV] (left ) and (100, 500, 1000, 100) [GeV] (right).
maximal value of BR(Z → μτ ) is about 5.10
−10
and it is very
small comparing with the recent experimental bounds. While the
values of BR(τ → 3μ) can approach the limit detection of collid-
ers.
Fig. 4.7 shows the numerical resulus of two branching ratios
as functions of m
B
−μ
ρ
with A
τ
=0. It is shown that the upper
experimental bound of BR(τ → μγ) directly affects to the values
of other cLFV decays. Specifically, BR(τ → 3μ) < 0.5×10
−9
and
BR(Z → μτ) < 10
−10

numerical investigation shows that f
A
Z
as well as BR Z → μτ
are still very small.
21
100 200 300 400 500 600 700 800
1.0  10
10
5.0  10
10
2.0  10
10
3.0  10
10
1.5  10
10
m
B
BRZΜΤ
100 200 300 400 500 600 700 800
1 10
11
5 10
11
1 10
10
5 10
10
1 10

3
,m
˜
R
) [GeV] is
chosen for three cases : (300, 150, 1000, 100, 100)-black,(400, 200, 1000, 100, 100)-
green, (500, 150, 1000, 100, 100)-blue.
Results and discussion
The main result of our work are summarized as follows
1. We constructed the 3-3-1 supersymmetric model with the
simplest particle content, called the SUSYRM331.
2. We established and parameterize analytic formulas present-
ing cLFV vertices in the SUSYE331. This kind of vertices
in the SUSYRM331 are also discussed.
3. In the frame work of the SUSYE331, we constructed ana-
lytic formulas of quantities needed for calculating branching
ratios of the cLFV decays at one loop level, namely effec-
tive operators, Lagrangian and branching ratios of cLFV
decays.
4. We numerically investigated four particular LFV decays:
H → μτ, τ → μγ, Z → μτ and τ → 3μ. The results are
used to compare with bounds of experiments and find the
allowed regions of the parameter space in the SUSYE331.
We also predict that the branching ratios of H
0
→ μτ ,
τ → μγ and τ → 3μ are in the detectable limit of recent
colliders. In contrast the BRZ → μτ are very suppressed
22
0.5

Μ

10
9
, ΤΜ Γ 10
8

Hình 4.7: Contour plots of branching ratios of τ
−
→ μ
−
μ
+
μ
−
(dotted), Z →
μτ ( dashed) with τ → μγ (black) with A
τ
=0and (m
λ
,m
˜
L
2
,m
˜
L
3
,m
˜